Schrödinger system; Sign-changing solution; Shrinking region; Attraction; Concentration.
35B44; 35B40; 35B20; 35J50

1 Introduction↩︎

In this paper we consider the following competitive Schrödinger system \[\label{eq1461} \begin{cases} -\Delta u_i + u_i = \mu_i Q_\varepsilon(x-y_i) |u_i|^{2p-2}u_i + \sum_{j\neq i} \lambda_{ij} |u_j|^{p} |u_i|^{p-2}u_i, \\[6pt] u_i \in H^1(\mathbb{R}^N),\;i=1,\dots,m. \end{cases}\tag{1}\] where \(N\ge1\), \(m\ge2\). Nonlinear Schrödinger systems with competing interactions have been extensively studied in recent years, motivated both by physical models such as Bose–Einstein condensates and nonlinear optics, and by the rich mathematical structure of coupled elliptic equations. Among the various phenomena that may occur in this context, a particularly interesting situation arises when the nonlinear potentials are localized in small regions that shrink to isolated points as a small parameter \(\varepsilon>0\) tends to zero. In this case, solutions may concentrate around finitely many prescribed points and exhibit a delicate interaction between the different components.

We now state the structural assumptions that will be used throughout the paper. They are standard in this context and reflect the subcritical regime and the competitive nature of the system.

• \(1<p<\frac{2^*}{2}\) if \(N\ge3\), and \(p>1\) if \(N=1,2\), where \(2^*=\frac{2N}{N-2}\) if \(N\ge3\).

• \(\mu_i>0\), \(i=1,\dots,m\), \(\lambda_{ij}=\lambda_{ji}<0\), for all \(i\neq j\).

• Let \(Q_\varepsilon(x-y_i):=Q(\varepsilon x-y_i)\), where \(Q\in C(\mathbb{R}^N)\) is nonnegative, \(\operatorname{supp}(Q)\) is bounded, and there exist exactly pairwise distinct points \(y_1,\dots,y_m\in\mathbb{R}^N\) such that \[Q(-y_i)=\|Q\|_\infty=\max\limits_{x\in\mathbb{R}^N}Q(x)~\text{for all }i=1,\dots,m.\]

In this setting, the solutions \(u_i\) may be interpreted as standing wave profiles of different species which are attracted to regions where \(Q_\varepsilon\) is positive and repelled from their complement. The assumption \(\lambda_{ij}<0\) means that distinct components repel each other, which in turn favors spatial segregation, so the system 1 is competitive.

In the scalar case, a fundamental contribution is due to Ackermann and Szulkin [1], who showed that when the positive region of the nonlinear coefficient collapses to isolated points, every nontrivial solution concentrates at one of these cores, and ground states select a single core without splitting their mass. In this framework, the sign structure of the nonlinearity is the sole driver of localization, without any periodicity or symmetry assumption on the linear part. Since then, their approach has been extended in various directions, including problems on the whole space and coupled systems; see, e.g., [2]–[4] and the references therein. We point out that these works focus primarily on positive solutions or the least energy solutions, for which the Mountain Pass Theorem ([5]) and the maximum principles are available. Very recently, in [3], Clapp, Saldaña and Szulkin studied the system 1 with a single or multiple shrinking domains, the existence of nonnegative least energy solution and concentration behavior were obtained as \(\varepsilon\to0\). Furthermore, the authors characterized the limit profile and described how the components either decouple or remain coupled depending on the geometry of the attraction centers. Related results for scalar equations, including concentration of semiclassical states and the description of their limit profiles, were obtained in earlier works such as [1], [6], and see [7] and the references therein for phase separation phenomena of competitive systems. We refer [8] to the existence of concentrating positive solutions via a Lyapunov Schmidt reduction strategy.

However, much less is known about sign-changing solutions for systems of the form 1 . Even for the scalar equation, constructing nodal solutions requires a refined variational approach based on nodal Nehari sets and careful control of the positive and negative parts of the solutions. As far as we known, there are only two papers [2], [8] concerned with this topic. In [2], Clapp, Hernández-Santamaría and Saldaña obtained the existence and concentration of nodal solutions via the nodal Nehari manifold method, and characterized the symmetries and the polynomial decay of the least-energy nodal limiting profiles. In [8], Clapp, Pistoia and Saldaña established the existence of concentrating nodal solutions via a Lyapunov Schmidt reduction method. But for systems, the situation is substantially more delicate, since one has to keep track simultaneously of the sign structure of each component and of the competitive couplings between different components. It seems that there is no work concerned with the existence and concentration of sign-changing solutions of the system 1 .

First, we state the meaning of sign-changing solutions of 1 .

The purpose of this paper is to develop a variational framework to seek for infinitely many sign-changing solutions of 1 , and to analyze their concentration behavior as \(\varepsilon\to 0\). The main technical difficulties arise from the sign-changing nature of the solutions. First, the natural Nehari manifold associated with nonnegative solutions does not capture directly sign changing solutions, so for elements in the Nehari manifold, one has to impose constraints separately on the positive and negative parts of each component. This leads to a nonlinear and nonconvex constraint set on which the direct minimization is not available. Second, the competitive couplings \(\lambda_{ij}<0\) mix the components in a nontrivial way, and one must show that on the nodal constraint the energy still controls the \(H^{1}\)-norm uniformly, so that Palais-Smale sequences are bounded. Third, compactness issues are more severe in the nodal setting, one has to prevent vanishing of some sign components and to rule out loss of mass at infinity while keeping track of the nodal structure. So it is interesting to seek for sign-changing solutions of 1 since the above difficulties prevent one to use techniques based on the maximum principles, the comparison arguments, or the monotonicity methods. These obstacles make it necessary to develop a variational strategy that is tailored to the nodal structure of the problem and that remains effective in the presence of competitive couplings.

We are now in a position to state the main results. Our first theorem shows that, for small \(\varepsilon>0\), system 1 admits a sequence of sign-changing solutions with unbounded energies.

To treat sign-changing solutions of competitive Schrödinger systems with shrinking nonlinear potentials, we develop a variational framework tailored to the nodal setting of 1 , inspired by the ideas in [1], [3] but adapted to the present system. We work in the product space \(H:=(H^1(\mathbb{R}^N))^m\) endowed with the natural norm, and we consider the energy functional associated with 1 \(J_\varepsilon:H\to\mathbb{R}\). Our first step is to introduce a fully sign-changing Nehari set \(\mathcal{N}^{\mathrm{sc}}_\varepsilon\) (see §2), by imposing Nehari type constraints separately on the positive and negative parts of each component. On a suitable open subset \(\mathcal{A}\) of \(H\), rather than on a linear subspace, we define a nodal projection \[m_\varepsilon:\mathcal{A}\longrightarrow \mathcal{N}^{\mathrm{sc}}_\varepsilon,\] and we use it to construct an even reduced functional \(\Psi_\varepsilon:=J_\varepsilon\circ m_\varepsilon\) on \(\mathcal{A}\). The symmetry of \(\Psi_\varepsilon\) allows us to apply Krasnosel’skii genus theory and to implement a symmetric minimax scheme, which provide infinitely many critical values and yield a sequence of nodal solutions with unbounded energies. This feature is essential for applying genus theory in a genuinely nodal setting and allows us to recover a symmetric minimax structure despite the strong nonlinearity of the constraints. The proof of Theorem 1 will be carried out in Section 4.

The second result in this paper concerns the asymptotic behavior of the above sign-changing solutions as \(\varepsilon\to 0^+\). After a suitable rescaling, the potentials \(Q_\varepsilon\) converge to the constants \(Q(-y_i)\), and one is naturally led to the limiting autonomous system \[\label{eq1462} \begin{cases} -\Delta U_i + U_i = \mu_i Q(-y_i)\,|U_i|^{2p-2}U_i + \sum_{j\neq i}\lambda_{ij}|U_j|^p|U_i|^{p-2}U_i,\\[6pt] U_i\in H^1(\mathbb{R}^N),\;i=1,\dots,m. \end{cases}\tag{2}\] The energy functional associated with 2 is denoted by \(J_0(\mathbf{u}): H\to\mathbb{R}\).

The proof of Theorem 2 is based on a concentration–compactness argument for suitable translations of sign-changing solutions \(\{\mathbf{u}_n\}\), which yields a nontrivial limiting profile solving 2 and the strong convergence in (ii) via a Brézis–Lieb type energy decomposition. The decay property in (iii) follows from localized estimates with cut–off functions, interpolation, and Agmon-type exponential decay bounds; the case \(q=\infty\) is obtained by a translation argument combined with local elliptic regularity and the compact support of \(Q\). It is carried out in Section 5.

The paper is organized as follows. In Section 2 we introduce the variational setting, the fully nodal Nehari set and the reduced functional, and state our assumptions on the nonlinear potentials. In Section 3 we establish a compactness result for the reduced functional \(\Psi_{\varepsilon}\), which will be crucial in the construction of sign-changing solutions. Section 4 is devoted to a genus-based minimax scheme on \(\mathcal{N}_{\varepsilon}^{\mathrm{sc}}\), and hence obtain infinitely many sign-changing critical points. In Section 5 we prove the concentration result Theorem 2, including the description of the limiting profiles for the semiclassical states.

Notation.

• Let \(w^+(x):=\max\{w(x),0\}\) and \(w^-(x):=\min\{w(x),0\}\) be the positive and negative parts of \(w\), respectively.

• For \(r>0\) and \(x\in\mathbb{R}^N\), \(B_r(x)\) is the open ball centered at \(x\) with radius \(r\).

• Boldface \(\mathbf{u}\) denote a \(\mathbb{R}^m\) vector-value function, \(u_i\) denotes the \(i\)-th component function of \(\mathbf{u}\).

• "\(\to\)" denotes the strong convergence and "\(\rightharpoonup\)" denotes the weak convergence.

• For \(k\in\mathbb{Z}^N\), we denote by \(k*\mathbf{u}\) the integer translation of \(\mathbf{u}\): \[(k*\mathbf{u})(x):=\mathbf{u}(x-k),~ x\in\mathbb{R}^N .\]

• For a function space \(E\), \(E^m\) denotes the product space \(\underbrace{E\times\cdots\times E}_{m}\).

2 Variational setting and the fully sign-changing Nehari constrain↩︎

In this section, we introduce the variational framework and the fully sign-changing Nehari set which will be used in the construction of sign-changing solutions. For \(\varepsilon>0\), we define the family of shrinking potentials \[Q_\varepsilon(x-y_i):=Q(\varepsilon x-y_i), ~ i=1,\dots,m.\] Under \((A_3)\), the supports of \(Q_\varepsilon(\cdot-y_i)\) concentrate and shrink to the points \(y_i\) as \(\varepsilon\to 0^+\) for \(i=1,2,\cdots,m\), creating \(m\) distinct regions of attraction. Let \(H^1(\mathbb{R}^N)\) be the usual Sobolev space with the norm \[\|v\|_{H^1}^2 :=\int_{\mathbb{R}^N}(|\nabla v|^2+|v|^2)dx,~v\in H^1(\mathbb{R}^N).\] As we mentioned in Section 1, we will work in the product Hilbert space \(H:=H^1(\mathbb{R}^N)^m\) equipped with the norm \[\|\mathbf{u}\|^2 := \sum_{i=1}^m\int_{\mathbb{R}^N}\bigl(|\nabla u_i|^2+|u_i|^2\bigr)\,dx~\text{for}~\mathbf{u}=(u_1,\dots,u_m)\in H.\] By the Sobolev embedding theorem, \((A_1)\) implies that the embedding \[H^1(\mathbb{R}^N) \hookrightarrow L^{2p}(\mathbb{R}^N)\] is continuous and locally compact. The energy functional associated with 1 is \[\label{eq2461} \begin{align} J_\varepsilon(\mathbf{u}) &= \frac{1}{2}\|\mathbf{u}\|^2 - \frac{1}{2p}\sum_{i=1}^m \mu_i \int_{\mathbb{R}^N} Q_\varepsilon(x-y_i)\,|u_i|^{2p}\,dx \\ &\phantom{=}\; - \frac{1}{2p}\sum_{i\neq j}\lambda_{ij} \int_{\mathbb{R}^N}|u_i|^p |u_j|^p\,dx,~\mathbf{u}=(u_1,\dots,u_m)\in H. \end{align}\tag{3}\] Assumptions \((A_1)\)–\((A_3)\) and the embedding above imply that \(J_\varepsilon\) is well defined and of class \(C^1(H,\mathbb{R})\). Moreover, \(J_\varepsilon\) is even, i.e. \(J_\varepsilon(-\mathbf{u})=J_\varepsilon(\mathbf{u})\) for all \(\mathbf{u}\in H\).

A standard computation shows that \[\label{eq2462} \begin{align} \langle J_\varepsilon^\prime(\mathbf{u}),\boldsymbol{\varphi}\rangle &= \sum_{i=1}^m \int_{\mathbb{R}^N} \bigl(\nabla u_i\cdot\nabla\varphi_i + u_i\varphi_i\bigr)\,dx - \sum_{i=1}^m \mu_i\int_{\mathbb{R}^N} Q_\varepsilon(x-y_i)|u_i|^{2p-2}u_i\varphi_i\,dx \\ &\phantom{=}\; - \sum_{i\neq j}\lambda_{ij}\int_{\mathbb{R}^N} |u_j|^{p}|u_i|^{p-2}u_i\,\varphi_i\,dx, \end{align}\tag{4}\] for all \(\mathbf{u},\boldsymbol{\varphi}\in H\). Hence \(\mathbf{u}\in H\) is a critical point of \(J_\varepsilon\) if and only if it is a weak solution of 1 . In particular, sign-changing solutions of 1 correspond to sign-changing critical points of \(J_\varepsilon\).

Recall the standard Nehari manifold is defined by \[\mathcal{N}_\varepsilon :=\{\mathbf{u}\in H\setminus\{\mathbf{0}\}:\langle J_\varepsilon^\prime(\mathbf{u}), \mathbf{u}\rangle=0\}.\] It is obvious that \(\mathcal{N}_\varepsilon\) contains all nontrivial solutions of 1 , and it is usually used for seeking for positive or nonnegative solutions (see e.g. [3] and [5]). But this manifold does not work well for sign-changing solutions since the constrained minimizer on it not necessary to be sign-changing. To capture sign-changing solutions, we introduce a sign-changing analogue that imposes orthogonality conditions separately on the positive and negative parts of each component. For \(\mathbf{u}=(u_1,\ldots,u_m)\in H\), we denote by \(u_i^\pm\) the positive and negative parts of the \(i\)-th component \(u_i\). We start with an energy splitting formula for sign-decomposed functions.

To simplify the notations, we identify \(u_i^\pm\) with the element \((\underbrace{0,\ldots,0,u_i^\pm,0,\ldots,0}_{m})\in H\), and write \[\langle J^\prime_\varepsilon(\mathbf{u}),u_i^\pm\rangle := \langle J^\prime_\varepsilon(\mathbf{u}),(0,\ldots,0,u_i^\pm,0,\ldots,0)\rangle.\]

It is obvious that \(\mathcal{N}^{\mathrm{sc}}_\varepsilon\) contains all sign-changing solutions \(\mathbf{u}=(u_1,\cdots,u_m)\) with each component \(u_i\) is sign-changing. We will show below that \(\mathcal{N}_\varepsilon^{\mathrm{sc}}\) is nonempty (see Lemma 4).

By 4 , \(\langle J_\varepsilon^\prime(\mathbf{u}),u_i^\pm \rangle=0\) can be written as \[\label{eq2463} \|u_i^\pm\|_{H^1}^2 = \mu_i\int_{\mathbb{R}^N}Q_\varepsilon(x-y_i)|u_i^\pm|^{2p}\,dx + \sum_{j\neq i} \lambda_{ij} \int_{\mathbb{R}^N}|u_j|^p|u_i^\pm|^p\,dx,\tag{5}\] for each \(i=1,\dots,m\). Next we establish a coercivity estimate on \(\mathcal{N}^{\mathrm{sc}}_\varepsilon\).

.  For \(\mathbf{u}\in\mathcal{N}^{\mathrm{sc}}_\varepsilon\), by 5 , for all \(i\) and \(\sigma\in\{+,-\}\), \[\|u_i^\sigma\|_{H^1}^2 = \mu_i\int_{\mathbb{R}^N}Q_\varepsilon(x-y_i)|u_i^\sigma|^{2p}\,dx + \sum_{j\neq i}\lambda_{ij}\int_{\mathbb{R}^N}|u_j|^p|u_i^\sigma|^p\,dx.\] Summing over \(i\) and both signs and recalling that \(u_i=u_i^+ + u_i^-\) yields \[\|\mathbf{u}\|^2 = \sum_{i=1}^m \mu_i\int_{\mathbb{R}^N}Q_\varepsilon(x-y_i)|u_i|^{2p}\,dx + \sum_{i\neq j}\lambda_{ij}\int_{\mathbb{R}^N}|u_i|^p|u_j|^p\,dx.\] Substituting this identity into 3 , we obtain \[\begin{align} J_\varepsilon(\mathbf{u}) &= \Bigl(\frac{1}{2}-\frac{1}{2p}\Bigr)\|\mathbf{u}\|^2 - \frac{1}{2p}\sum_{i\neq j}\lambda_{ij}\int_{\mathbb{R}^N}|u_i|^p|u_j|^p\,dx \\ &\ge \Bigl(\frac{1}{2}-\frac{1}{2p}\Bigr)\|\mathbf{u}\|^2, \end{align}\] because \(\lambda_{ij}<0\). The conclusion follows with \(\alpha=\frac{1}{2}-\frac{1}{2p}>0\).

In particular, the functional \(J_\varepsilon\) is coercive on \(\mathcal{N}_\varepsilon^{\mathrm{sc}}\) and controls the \(H\)-norm from above. Since \(\lambda_{ij}<0\), the right-hand side in 5 is strictly smaller than in the decoupled case, which is favorable for coercivity along \(\mathcal{N}^{\mathrm{sc}}_\varepsilon\).

Fix \(\mathbf{u}\in H\) such that \(u_i^\pm\neq 0\) for all \(i=1,\dots,m\). For each fixed \(i\) we consider the two parameter fiber map \[\Phi_i(t^+,t^-) := J_\varepsilon\bigl(u_1,\dots,t^+u_i^+ + t^- u_i^-,\dots,u_m\bigr),~(t^+,t^-)\in(0,\infty)^2.\] Since \(u_i^+u_i^-=0\) a.e. in \(\mathbb{R}^N\), the decomposition holds \[\label{eq2464} \Phi_i(t^+,t^-)=g_i^+(t^+)+g_i^-(t^-) +J_\varepsilon(u_1,\dots,u_{i-1},0,u_{i+1},\dots,u_m),\tag{6}\] where, for \(\sigma\in\{+,-\}\), \[\label{eq2465} g_i^\sigma(t) = \frac{1}{2} a_i^\sigma t^2 - \frac{\mu_i}{2p} b_i^\sigma t^{2p} - \frac{1}{2p}\sum_{j\neq i}\lambda_{ij} c_{ij}^\sigma t^p,~t>0,\tag{7}\] with coefficients \[a_i^\sigma:=\|u_i^\sigma\|_{H^1}^2,~ b_i^\sigma:=\int_{\mathbb{R}^N}Q_\varepsilon(x-y_i)|u_i^\sigma|^{2p}\,dx, ~ c_{ij}^\sigma:=\int_{\mathbb{R}^N}|u_j|^p|u_i^\sigma|^p\,dx.\] Obviously, \(a_i^\sigma>0\), \(b_i^\sigma\ge0\) and \(c_{ij}^\sigma\ge0\).

.  By 6 –7 , the stationarity conditions \(\partial_{t^\pm}\Phi_i(t^+,t^-)=0\) decouple into the two one-dimensional equations \((g_i^\sigma)'(t^\sigma)=0\), \(\sigma\in\{+,-\}\). A direct computation yields \[\label{eq2467} (g_i^\sigma)^\prime(t) = a_i^\sigma t - \mu_i b_i^\sigma t^{2p-1} - \frac{1}{2}\sum_{j\neq i}\lambda_{ij} c_{ij}^\sigma t^{p-1},~ t>0.\tag{8}\] Since \(t>0\), dividing 8 by \(t^{p-1}\) gives exactly ?? . Thus critical points of \(\Phi_i\) correspond to positive zeros of \(f_i^\sigma\).

Fix \(i\) and \(\sigma\) and consider \(f:=f_i^\sigma\) on \((0,\infty)\). Since \(p>1\), we have \(t^p \to 0\) as \(t\to0^+\). Using \(a_i^\sigma>0\), \(\lambda_{ij}<0\) and \(c_{ij}^\sigma\ge0\), we obtain \[\liminf_{t\to 0} f(t) \;\ge\; -\frac{1}{2} \sum_{j\ne i}\lambda_{ij} c_{ij}^\sigma \;>\; 0.\]

In particular, \(f(t)>0\) for all \(t>0\) sufficiently small. Hence the continuous function \(f\) has at least one positive zero since \(\lim\limits_{t\to\infty}f_i^\sigma(t)=-\infty\). Direct calculations give \[f^\prime(t) = (2-p)a_i^\sigma t^{1-p} - \mu_i p b_i^\sigma t^{p-1}\] and \[f^{\prime\prime}(t) = (2-p)(1-p)a_i^\sigma t^{-p} - \mu_i p(p-1)b_i^\sigma t^{p-2}.\] Since \(p>1\) and \(a_i^\sigma>0\), \(b_i^\sigma\ge0\), we have \(f''(t)<0\) for all \(t>0\), so \(f\) is strictly concave on \((0,\infty)\) and has exactly one positive zero point \(t_i^\sigma\). The pair \((t_i^+,t_i^-)\) is therefore uniquely determined and yields the desired element in \(\mathcal{N}^{\mathrm{sc}}_{\varepsilon,i}\).

We now show that all components of an element of \(\mathcal{N}^{\mathrm{sc}}_\varepsilon\) cannot be arbitrarily small, in the \(H^{1}\)-norm sense.

.  For \(\mathbf{u}\in\mathcal{N}^{\mathrm{sc}}_\varepsilon\), one has \(u_i^\pm\not\equiv0\) for each \(i\). Suppose by contradiction that there exists \(\{\mathbf{u}^{(n)}\}\subset\mathcal{N}^{\mathrm{sc}}_\varepsilon\) and indices \(i_n\) and signs \(\sigma_n\in\{+,-\}\) such that \(\|u^{(n)\,\sigma_n}_{i_n}\|_{H^1}\to0\). It follows from 5 and the Sobolev inequality that \[\begin{align} \|u^{(n)\,\sigma_n}_{i_n}\|_{H^1}^2 &=\mu_{i_n}\int_{\mathbb{R}^N}Q_\varepsilon|u^{(n)\,\sigma_n}_{i_n}|^{2p}\,dx + \sum_{j\neq i_n}\lambda_{i_n j} \int_{\mathbb{R}^N}|u^{(n)}_j|^p|u^{(n)\,\sigma_n}_{i_n}|^p\,dx\\ &\leq \mu_{i_n}\int_{\mathbb{R}^N}Q_\varepsilon|u^{(n)\,\sigma_n}_{i_n}|^{2p}\,dx\\ & \le C\|u^{(n)\,\sigma_n}_{i_n}\|_{H^1}^{2p} \end{align}\] since \(\lambda_{ij}<0\), which yields a uniform lower bound \(\delta>0\) for all \(\|u_i^\pm\|_{H^1}\).

To obtain sign-changing solutions with all components are sign-changing, we introduce the following subset of \(H\) \[\mathcal{A} :=\bigl\{\mathbf{u}\in H:\;u_i^\pm\neq 0,~\forall~i\in\{1,\dots,m\}\bigr\}.\]

We claim that \(\mathcal{A}\) is open in \(H\). Indeed, for \(w\in H^1(\mathbb{R}^N)\), the truncation mappings \(w\mapsto w^\pm\) are Lipschitz on \(H^1(\mathbb{R}^N)\), namely \[\|w^\pm-z^\pm\|_{H^1}\le \|w-z\|_{H^1},~\forall\, w,z\in H^1(\mathbb{R}^N).\] Hence, for each fixed \(i\) and \(\sigma\in\{+,-\}\), the mapping \(\mathbf{u}\mapsto u_i^\sigma\) is continuous from \(H\) to \(H^1(\mathbb{R}^N)\). Since \(H^1(\mathbb{R}^N)\setminus\{0\}\) is open, it follows that \[\mathcal{A}=\bigcap_{i=1}^m\bigcap_{\sigma\in\{+,-\}} \{\mathbf{u}\in H|u_i^\sigma\in H^1(\mathbb{R}^N)\setminus\{0\}\}\] is open in \(H\), since the right hand set is the intersection of finite open sets. Or equivalently, one can check \(\mathcal{A}\) is open in a direct way. Indeed, for \(\mathbf{u}\in \mathbf{A}\), define \[r = \min\limits_{1\le i\le m,\sigma\in\{+,-\}}\|u_i^\sigma\|_{H^1}>0,\] then for \(\mathbf{v}\in B_{r/2}(\mathbf{u})\), there holds \[\|v_i^\sigma\|_{H^1} \geq \|u_i^\sigma\|_{H^1} - \|v_i^\sigma-u_i^\sigma\|_{H^1} \geq \|u_i^\sigma\|_{H^1} - \|v_i - u_{i}\|_{H^1} > \delta - \frac{\delta}{2} = \frac{\delta}{2} > 0,\] which implies \(v_i^\sigma\neq 0\) and hence \(\mathbf{v}\in \mathcal{A}\). Therefore \(\mathcal{A}\) is open. Consequently, \(\mathcal{A}\) is a smooth Hilbert manifold modeled on \(H\) and the tangent space \(T_u\mathcal{A}=H\). Moreover, \[\label{eq2468} \partial\mathcal{A}=\{\mathbf{u}\in H:~\text{there exists}~i\in\{1,\cdots,m\}~\text{and}~\sigma\in\{+,-\}~\text{such that}~u_i^\sigma=0\}.\tag{9}\] Here we point out that \(\mathcal{A}\) is not convex.

We now introduce the nodal projection and the reduced functional. The uniqueness in Lemma 4 allows us to define \((t_i^+(\mathbf{u}),t_i^-(\mathbf{u}))\) for \(\mathbf{u}\in\mathcal{A}\).

.  Replacing \(\mathbf{u}\) by \(-\mathbf{u}\) swaps \(u_i^+\) and \(u_i^-\) but leaves ?? invariant, so \(t_i^\pm(-\mathbf{u})=t_i^\mp(\mathbf{u})\), and hence \(m_\varepsilon\) is odd.

By the definition of \(m_\varepsilon\), \(m_\varepsilon(\mathbf{u})\) is a linear combination of \(t_i^+(\mathbf{u}) u_i^+\) and \(t_i^-(\mathbf{u}) u_i^-\) for each \(i\). Since \(u_i^\sigma\) is Lipschitz continuous on \(H^1(\mathbb{R}^N)\), and hence we only need to show \(t_i^\sigma(\mathbf{u}) \in C^1(\mathcal{A}, (0, \infty))\).

For fixed \(i\) and \(\sigma \in \{+, -\}\), define \[F_i^\sigma(\mathbf{u}, t) = a_i^\sigma(\mathbf{u}) t^{2-p} - \mu_i b_i^\sigma(\mathbf{u}) t^p - \frac{1}{2} \sum_{j \neq i} \lambda_{ij} c_{ij}^\sigma(\mathbf{u}),~(\mathbf{u}, t) \in \mathcal{A} \times (0, \infty).\] By Lemma 4, \(F_i^\sigma(\mathbf{u}, t_i^\sigma(u)) = 0\) for all \(\mathbf{u} \in \mathcal{A}\). Obviously, \(F_i^\sigma \in C^1(\mathcal{A} \times (0, \infty), \mathbb{R})\) and \[\frac{\partial F_i^\sigma}{\partial t}(\mathbf{u}, t)|_{(\mathbf{u},t_i^\sigma(\mathbf{u}))} = (2 - p) a_i^\sigma(\mathbf{u}) (t_i^\sigma(\mathbf{u}))^{1-p} - \mu_i p b_i^\sigma(\mathbf{u}) (t_i^\sigma(\mathbf{u}))^{p-1}<0\] since \(a_i^\sigma(\mathbf{u}) > 0\), \(b_i^\sigma(\mathbf{u}) \geq 0\) and \(p > 1\). By the Implicit Functiona Theorem, \(t_i^\sigma(\mathbf{u})\) is \(C^1\) in \(\mathbf{u} \in \mathcal{A}\). Since this holds for all \(i\) and \(\sigma\), \(m_\varepsilon(\mathbf{u})\) is \(C^1\) on \(\mathcal{A}\).

Next, we show that \(m_\varepsilon: \mathcal{A} \to \mathcal{N}_\varepsilon^{sc}\) is bijective. Using Lemma 4 again, \(m_\varepsilon\) is one to one. For any \(\mathbf{v} \in \mathcal{N}_\varepsilon^{sc}\), \(v_i^+ \neq 0\) and \(v_i^- \neq 0\), so \(\mathbf{v} \in \mathcal{A}\) and \(\langle J_\varepsilon^\prime(\mathbf{v}),v_i^+\rangle= 0\) and \(\langle J_\varepsilon^\prime(\mathbf{v}),v_i^-\rangle = 0\), and the scaling factors for \(\mathbf{v}\) are \(t_i^+(\mathbf{v}) = 1\) and \(t_i^-(\mathbf{v}) = 1\). Thus, \(m_\varepsilon(\mathbf{v}) = 1 \cdot v_i^+ + 1 \cdot v_i^- = \mathbf{v}\). Therefore, \(m_\varepsilon\) is surjective.

The above fats implies that the inverse map \(m_\varepsilon^{-1}: \mathcal{N}_\varepsilon^{sc} \to \mathcal{A}\) is well defined. At last we show that \(m_\varepsilon^{-1}\) is \(C^1\).

Indeed, recall that \(\mathcal{A}\) is an open subset of \(H\), hence a Hilbert manifold modeled on \(H\) and the tangent space \(T_u\mathcal{A}=H\). By the Regular Value Theorem, the constraints \(G_i^\sigma(w) = \langle J_\varepsilon'(w), w_i^\sigma \rangle = 0\) (for \(i = 1, \dots, m\), \(\sigma \in \{+, -\}\)) are regular, so \(\mathcal{N}_\varepsilon^{sc}\) is a \(C^1\)-submanifold of \(H\). Let \(T_{m_\varepsilon(u)}\mathcal{N}_\varepsilon^{sc}\) be the tangent space to \(\mathcal{N}_\varepsilon^{sc}\) at \(m_\varepsilon(u)\), which has codimension \(2m\) since there are \(2\) constraints per component. The derivative \(m_\varepsilon^\prime(u)\) encodes the variation of scaling factors \(t_i^\sigma(u)\) and the original components \(u_i^\sigma\). By the non-degeneracy of \(\frac{\partial F_i^\sigma}{\partial t}\) and bijectivity of \(m_\varepsilon\), the Fréchet derivative \(m_\varepsilon^\prime(u): T_u\mathcal{A} \to T_{m_\varepsilon(u)}\mathcal{N}_\varepsilon^{sc}\) is both injective and surjective, hence a linear isomorphism.

By the Inverse Function Theorem, \(m_\varepsilon\) is locally \(C^1\)-invertible at every \(u \in \mathcal{A}\). Since \(m_\varepsilon\) is globally bijective, the inverse map \(m_\varepsilon^{-1}\) is globally \(C^1\)-smooth on \(\mathcal{N}_\varepsilon^{sc}\).

.  (a) Obviously, \(\Psi_\varepsilon\) is even since \(m_\varepsilon\) is odd. By Lemma 6, \(m_\varepsilon\) is \(C^1\), and hence \(\Psi_\varepsilon(u)=J_\varepsilon(m_\varepsilon(u))\) is \(C^1\).

(b) Let \(\mathbf{u}\in\mathcal{A}\) be a critical point of \(\Psi_{\varepsilon}\) and set \(\mathbf{v}:=m_{\varepsilon}(\mathbf{u})\in\mathcal{N}_{\varepsilon}^{\mathrm{sc}}\). By the chain rule we have \[\langle\Psi_{\varepsilon}^\prime(\mathbf{u}),\boldsymbol{\varphi}\rangle =\langle J_{\varepsilon}^\prime(\mathbf{v}), {m_{\varepsilon}}^\prime(\mathbf{u})\boldsymbol{\varphi}\rangle,~\boldsymbol{\varphi}\in T_{\mathbf{u}}\mathcal{A}.\] Since \(\Psi_{\varepsilon}^{\prime}(\mathbf{u})=0\) and \(m_{\varepsilon}\) is a \(C^1\)-diffeomorphism between \(\mathcal{A}\) and \(\mathcal{N}_{\varepsilon}^{\mathrm{sc}}\), the range of \({m_{\varepsilon}}^\prime(\mathbf{u})\) coincides with the tangent space \(T_{\mathbf{v}}\mathcal{N}_{\varepsilon}^{\mathrm{sc}}\). Hence \[\langle J_{\varepsilon}^\prime(\mathbf{v}),\mathbf{w}\rangle=0,~\mathbf{w}\in T_{\mathbf{v}}\mathcal{N}_{\varepsilon}^{\mathrm{sc}},\] that is, \(\mathbf{v}\) is a constrained critical point of \(J_{\varepsilon}\) on \(\mathcal{N}_{\varepsilon}^{\mathrm{sc}}\).

Next we show that \(\mathbf{v}\) is in fact an unconstrained critical point of \(J_{\varepsilon}\). Recall that the sign-changing Nehari manifold can be written as \[\mathcal{N}_{\varepsilon}^{\mathrm{sc}} = \Bigl\{ \mathbf{w}\in H : G_i^\sigma(\mathbf{w})=0, \forall i\in\{1,\dots,m\},~\sigma\in\{+,-\} \Bigr\},\] where \[G_i^\sigma(\mathbf{w}) := \bigl\langle J_{\varepsilon}^\prime(\mathbf{w}), w_i^\sigma\bigr\rangle, ~i=1,\dots,m,\;\sigma\in\{+,-\}.\]

By the Lagrange multiplier rule, there exist real numbers \(\lambda_i^\sigma\) such that \[\label{eq24610} J_{\varepsilon}'(\mathbf{v}) = \sum_{i=1}^m\sum_{\sigma\in\{+,-\}} \lambda_i^\sigma\, G_i^\sigma{}'(\mathbf{v})~\text{in}~H^\prime.\tag{10}\]

For each \(j=1,\dots,m\) and \(\tau\in\{+,-\}\), we evaluate 10 in the direction \(v_j^\tau\). Since \(\mathbf{v}\in\mathcal{N}_{\varepsilon}^{\mathrm{sc}}\), we have \(\langle J_{\varepsilon}'(\mathbf{v}), v_j^\tau\rangle = 0\), and thus \[0 = \bigl\langle J_{\varepsilon}'(\mathbf{v}), v_j^\tau\bigr\rangle = \sum_{i=1}^m\sum_{\sigma\in\{+,-\}} \lambda_i^\sigma\, \langle (G_i^\sigma)^\prime(\mathbf{v}),v_j^\tau\rangle.\]

This yields a homogeneous linear system for the coefficients \(\lambda_i^\sigma\) with coefficient matrix \[A_{(j,\tau),(i,\sigma)} := \langle (G_i^\sigma)^\prime(\mathbf{v}),v_j^\tau\rangle,~i,j=1,\dots,m,\;\sigma,\tau\in\{+,-\}.\] On the other hand, by the definition of the fiber maps \(g_i^\sigma\) in Lemma 4 and a straightforward computation using 1 and 3 , the matrix \(A\) can be identified (up to positive factors) with the Hessian of the function \[(t_i^\sigma)_{i,\sigma}\longmapsto J_{\varepsilon}\Bigl( \sum_{i=1}^m \sum_{\sigma\in\{+,-\}} t_i^\sigma v_i^\sigma \Bigr)\] at the point corresponding to \(\mathbf{v}\). By Lemma 4, this Hessian is negative definite, since \(\mathbf{v}\) is the unique maximizer of \(J_{\varepsilon}\) along its fiber. In particular, the matrix \(A\) is invertible. Hence the only solution of the above linear system is \(\lambda_i^\sigma=0\) for all \(i\) and \(\sigma\), and from 10 we conclude that \[J_{\varepsilon}'(\mathbf{v})=0~\text{in}~H^\prime.\] Thus \(\mathbf{v}=m_{\varepsilon}(\mathbf{u})\) is an critical point of \(J_{\varepsilon}\), and by construction, it is sign-changing. This proves part (b).

(c) This is a direct consequence of \((b)\), see also [10].

(d) Let \(\mathbf{v}:=m_{\varepsilon}(\mathbf{u})\in\mathcal{N}_{\varepsilon}^{\mathrm{sc}}\). By 5 and summing over \(i=1,\dots,m\) and over \(\sigma\in\{+,-\}\) we obtain \[\label{eq24611} \|\mathbf{v}\|^{2} = \sum_{i=1}^{m}\mu_{i} \int_{\mathbb{R}^{N}}Q_{\varepsilon}(x-y_{i})\,|v_{i}|^{2p}\,dx + \sum_{i\neq j}\lambda_{ij} \int_{\mathbb{R}^{N}}|v_{i}|^{p}|v_{j}|^{p}\,dx .\tag{11}\] Substituting 11 into 3 yields \[\begin{align} J_{\varepsilon}(\mathbf{v}) &= \frac{1}{2}\|\mathbf{v}\|^{2} - \frac{1}{2p}\sum_{i=1}^{m}\mu_{i} \int_{\mathbb{R}^{N}}Q_{\varepsilon}(x-y_{i})\,|v_{i}|^{2p}\,dx - \frac{1}{2p}\sum_{i\neq j}\lambda_{ij} \int_{\mathbb{R}^{N}}|v_{i}|^{p}|v_{j}|^p\,dx \\[2mm] &= \frac{1}{2}\|\mathbf{v}\|^{2} - \frac{1}{2p}\Biggl( \|\mathbf{v}\|^{2} - \sum_{i\neq j}\lambda_{ij} \int_{\mathbb{R}^{N}}|v_{i}|^{p}|v_{j}|^p\,dx \Biggr) - \frac{1}{2p}\sum_{i\neq j}\lambda_{ij} \int_{\mathbb{R}^{N}}|v_{i}|^{p}|v_{j}|^p\,dx \\[2mm] &= \Bigl(\frac{1}{2}-\frac{1}{2p}\Bigr)\|\mathbf{v}\|^{2}. \end{align}\]

Therefore, taking \(\alpha=\frac{1}{2}-\frac{1}{2p}>0\) we obtain \[\Psi_{\varepsilon}(\mathbf{u}) = J_{\varepsilon}(\mathbf{v}) = \alpha\,\|\mathbf{v}\|^{2},\] which clearly implies the desired coercivity on \(\mathcal{N}_{\varepsilon}^{\mathrm{sc}}\).

3 The \((PS)_c\) condition of \(\Psi_\varepsilon\)↩︎

In order to apply the minimax scheme to construct sign-changing solutions, we must establish the compactness property for the reduced functional \(\Psi_\varepsilon\). The main difficulties arise from the lack of compactness of the Sobolev embedding and the presence of sign components. To obtain the compactness, we combine concentration compactness arguments with the fully sign-changing Nehari structure introduced in Section 2. As we will see below, the compactness for \((PS)\) sequences relies on the interplay between the potential \(Q\) and the competitive couplings \(\lambda_{ij}<0\).

Recall a sequence \(\{\mathbf{u}_n\}\subset\mathcal{A}\) is called a \((PS)_c\) sequence for \(\Psi_\varepsilon\) if \[\Psi_\varepsilon(\mathbf{u}_n)\to c~\text{and}~ \Psi_\varepsilon^\prime(\mathbf{u}_n)\to 0~\text{in}~H^\prime.\] The functional \(\Psi_\varepsilon\) satisfies the \((PS)_c\) condition on \(\mathcal{A}\) if every \((PS)_c\)- sequence has a convergent subsequence in \(\mathcal{A}\) and its limit also contained in \(\mathcal{A}\).

.  Let \(\{\mathbf{u}_n\}\subset \mathcal{A}\) be a \((PS)_c\) sequence for \(\Psi_\varepsilon\), i.e. \[\Psi_\varepsilon(\mathbf{u}_n)\to c~\text{and}~(\Psi_\varepsilon)'(\mathbf{u}_n)\to 0 \;\text{in } H^\prime.\] Set \(\mathbf{v}_n:=m_\varepsilon(\mathbf{u}_n)\in \mathcal{N}^{\mathrm{sc}}_\varepsilon\). By Lemma 7(c), \[J_\varepsilon(\mathbf{v}_n)=\Psi_\varepsilon(\mathbf{u}_n)\to c~\text{and}~ J_\varepsilon'(\mathbf{v}_n)\to 0 \;\text{in } H^\prime.\] In another word, \(\{\mathbf{v}_n\}\) is a \((PS)_c\) sequence for \(J_\varepsilon\) in \(H\).

Our main strategy is to prove that \(\{\mathbf{v}_n\}\) has a strongly convergent subsequence in \(H\) and the limit lies in \(\mathcal{N}^{\mathrm{sc}}_\varepsilon\subset \mathcal{A}\). Jointly with Lemma 7, \(\Psi_\varepsilon\) satisfies the \((PS)_c\) condition.

We first show that \(\{\mathbf{v}_n\}\) is bounded in \(H\). In fact, using \(J_\varepsilon(\mathbf{v}_n)\to c\) and \(J_\varepsilon^\prime(\mathbf{v}_n)\to0\), we obtain \[\frac{p-1}{2p}\,\|\mathbf{v}_n\|^2 = J_\varepsilon(\mathbf{v}_n) - \frac{1}{2p} \langle J_\varepsilon^\prime(\mathbf{v}_n),\mathbf{v}_n\rangle \le c + o(1) + o(1)\,\|\mathbf{v}_n\|,\] which implies that \(\{\mathbf{v}_n\}\) is bounded in \(H\). Up to a subsequence, \(\mathbf{v}_n\rightharpoonup \mathbf{v}\) in \(H\) and \(\mathbf{v}_n\to \mathbf{v}\) in \(L^q_{\rm loc}(\mathbb{R}^N)^m\) for all \(q\in[1,2^*)\). Since \(J_\varepsilon'(\mathbf{v}_n) \to 0\) and \(\mathbf{v}_n \rightharpoonup \mathbf{v}\), using the fact that \(\mathbf{v}_n \to \mathbf{v}\) in \(L^{2p}_{\text{loc}}(\mathbb{R}^N)^m\) and \((A_3)\), one can check \(\mathbf{v}\) is a critical point of \(J_\varepsilon\) in a standard way (see [5]).

Set \(\mathbf{w}_n:=\mathbf{v}_n-\mathbf{v}\). Obviously, \(\mathbf{w}_n\) is bounded in \(H\). Next we will show that, up to a subsequence, \(\mathbf{w}_n\to \mathbf{0}\) in \(H\). Assume by contradiction that \(\|\mathbf{w}_n\|\nrightarrow 0\).

If \(\{\mathbf{w}_n\}\) is vanishing, then for every \(R>0\), \[\sup_{y\in\mathbb{R}^N} \int_{B_R(y)} |\mathbf{w}_n(x)|^2\,dx \to 0 ~ \text{as } n\to\infty.\] By the vanishing Lemma (see [11] or [5]), this would imply \(\mathbf{w}_n\to0\) in \(L^{2p}(\mathbb{R}^N)^m\).

Because \(J_\varepsilon'(\mathbf{v}_n) \to 0\) and \(J_\varepsilon'(\mathbf{v}) = 0\), we have \[\label{eq3461} \begin{align} o(1)&=\langle J_\varepsilon'(\mathbf{v}_n) - J_\varepsilon'(\mathbf{v}),\, \mathbf{w}_n\rangle\\ &= \|\mathbf{w}_n\|^2- \sum_{i=1}^m \mu_i \int_{\mathbb{R}^N} Q_\varepsilon(x-y_i)\bigl(|v_{n,i}|^{2p-2}v_{n,i} - |v_i|^{2p-2}v_i\bigr)w_{n,i}\,dx \\ &~ - \sum_{i\neq j} \lambda_{ij} \int_{\mathbb{R}^N} \bigl(|v_{n,j}|^p|v_{n,i}|^{p-2}v_{n,i} - |v_j|^p|v_i|^{p-2}v_i\bigr) w_{n,i}\,dx\\ &= \|\mathbf{w}_n\|^2+o(1), \end{align}\tag{12}\] where we use the fact \(|v_{n,i}|^{2p-2}v_{n,i} \to |v_i|^{2p-2}v_i\) in \(L^{\frac{2p}{2p-1}}(\mathbb{R}^N)\), the Hölder inequality and \(Q_\varepsilon \in L^\infty\). This yields that \(\|\mathbf{w}_n\| \to 0\), contradicts \(\|\mathbf{w}_n\| \nrightarrow 0\). Therefore, by [11], \(\{\mathbf{w}_n\}\) is non-vanishing, i.e., there exist constants \(R>0\), \(\delta>0\) and a sequence \(\{z_n\}\subset\mathbb{R}^N\) such that \[\label{eq3462} \int_{B_R(z_n)} |\mathbf{w}_n|^2\,dx \ge \delta~\text{for all}~n.\tag{13}\] Observe that \(|z_n|\to\infty\). Otherwise, up to a subsequence, \(z_n\to z\in\mathbb{R}^N\). Then for \(n\) large, \(B_R(z_n)\subset B_{R+1}(z)\), and since \(\mathbf{w}_n\to \mathbf{0}\) in \(L^2(B_{R+1}(z))^m\), hence \[\int_{B_R(z_n)} |\mathbf{w}_n(x)|^2\,dx \le \int_{B_{R+1}(z)} |\mathbf{w}_n(x)|^2\,dx \to 0,\] contradicting 13 . Thus, up to a subsequence, \(|z_n|\to\infty\).

Define two translated sequences \[\widetilde{\mathbf{v}}_n(x):=\mathbf{v}_n(x+z_n)~\text{and}~ \widetilde{\mathbf{w}}_n(x):=\mathbf{w}_n(x+z_n),~x\in\mathbb{R}^N.\] Note that \(\widetilde{\mathbf{v}}_n=\widetilde{\mathbf{w}}_n+\mathbf{v}(\,\cdot+z_n)\), combined with the facts that \(\mathbf{v}\in L^2(\mathbb{R}^N)^m\) and \(|z_n|\to\infty\), one has \[\|\mathbf{v}(\,\cdot+z_n)\|_{L^2(B_R(0))}^2 =\int_{B_R(0)}|\mathbf{v}(x+z_n)|^2\,dx =\int_{B_R(z_n)}|\mathbf{v}(x)|^2dx\to 0~(n\to\infty).\] Thus for all sufficiently large \(n\), \[\|\mathbf{v}(\,\cdot+z_n)\|_{L^2(B_R(0))}<\frac{\sqrt{\delta}}{2},\] which implies that for large \(n\) there holds \[\sqrt{\delta}\le\|\widetilde{\mathbf{w}}_n\|_{L^2(B_R(0))} \le\|\widetilde{\mathbf{v}}_n\|_{L^2(B_R(0))}+\|\mathbf{v}(\,\cdot+z_n)\|_{L^2(B_R(0))} <\|\widetilde{\mathbf{v}}_n\|_{L^2(B_R(0))}+\frac{\sqrt{\delta}}{2}.\] Consequently, \[\|\widetilde{\mathbf{v}}_n\|_{L^2(B_R(0))}>\frac{\sqrt{\delta}}{2}~\text{for large}~n.\] Obviously, \(\{\widetilde{\mathbf{v}}_n\}\) is bounded in \(H\). Going to a subsequence if necessary, we assume \[\widetilde{\mathbf{v}}_n\rightharpoonup\widetilde{\mathbf{v}}~\text{in}~H~\text{and}~ \widetilde{\mathbf{v}}_n\to\widetilde{\mathbf{v}}~\text{strongly in }L^2(B_R(0))^m.\] Thus \[\|\widetilde{\mathbf{v}}\|_{L^2(B_R(0))} =\lim_{n\to\infty}\|\widetilde{\mathbf{v}}_n\|_{L^2(B_R(0))} \ge\frac{\sqrt{\delta}}{2}>0,\] which implies \(\widetilde{\mathbf{v}}\neq\mathbf{0}\).

By \((A_3)\), there exists \(R_0>0\) with \(\operatorname{supp}(Q)\subset B_{R_0}(0)\). Hence, for each \(i=1,\dots,m\), \[Q_\varepsilon(x-y_i)=Q(\varepsilon x-y_i)=0 ~\text{whenever}~ |x|>\frac{R_0+\max_{1\le i\le m}|y_i|}{\varepsilon}=:R_\varepsilon.\] In particular, \[\label{eq3463} \operatorname{supp}\bigl(Q_\varepsilon(\cdot-y_i)\bigr)\subset B_{R_\varepsilon}(0),~ i=1,\dots,m.\tag{14}\]

Fix \(\varphi\in C_c^\infty(\mathbb{R}^N)\) with \(\operatorname{supp}(\varphi)\subset B_{R_1}(0)\) and test \(J_\varepsilon'(\mathbf{v}_n)\to0\) with \[\boldsymbol{\phi}_n:=\bigl(\varphi(\cdot-z_n) w_1,\dots,\varphi(\cdot-z_n) w_m\bigr)\in H .\] Therefore, by 4 , we have \[\begin{align} \label{eq3464} \langle J_\varepsilon^\prime(\mathbf{v}_n),\boldsymbol{\phi}_n\rangle &=\sum_{i=1}^m\int_{\mathbb{R}^N}\Bigl(\nabla v_{n,i}\cdot\nabla\bigl(\varphi(\cdot-z_n)w_i\bigr) +v_{n,i}\,\varphi(\cdot-z_n)w_i\Bigr)\,dx \nonumber\\ &~-\sum_{i=1}^m\mu_i\int_{\mathbb{R}^N}Q_\varepsilon(x-y_i)\,|v_{n,i}|^{2p-2}v_{n,i}\, \varphi(x-z_n)w_i(x)\,dx \nonumber\\ &~-\sum_{i\neq j}\lambda_{ij}\int_{\mathbb{R}^N}|v_{n,j}|^{p}|v_{n,i}|^{p-2}v_{n,i}\, \varphi(x-z_n)w_i(x)\,dx \;=\;o(1). \end{align}\tag{15}\]

Since \(|z_n|\to\infty\), for sufficiently large \(n\) we have \(|z_n|>R_1+R_\varepsilon\). Then for any \(x\in B_{R_1}(z_n)\), \[|x|\ge |z_n|-|x-z_n|> (R_1+R_\varepsilon)-R_1=R_\varepsilon .\] Hence, by 14 we obtain \(Q_\varepsilon(x-y_i)=0\) for all \(i=1,\dots,m\) and all such \(x\). Therefore, \[Q_\varepsilon(\cdot-y_i)\,\varphi(\cdot-z_n)\equiv0 \qquad \text{for all }i=1,\dots,m\] for \(n\) sufficiently large. Consequently, for \(n\) large 15 reduces to \[\begin{align} \label{eq3465} &\sum_{i=1}^m\int_{\mathbb{R}^N}\Bigl(\nabla v_{n,i}\cdot\nabla\bigl(\varphi(\cdot-z_n)w_i\bigr) +v_{n,i}\,\varphi(\cdot-z_n)w_i\Bigr)\,dx \nonumber\\ &~-\sum_{i\neq j}\lambda_{ij}\int_{\mathbb{R}^N}|v_{n,j}|^{p}|v_{n,i}|^{p-2}v_{n,i}\, \varphi(x-z_n)w_i(x)\,dx=o(1). \end{align}\tag{16}\] Changing variables \(x\mapsto x+z_n\) in 16 , we obtain \[\label{eq:shifted-test} \sum_{i=1}^m\int_{\mathbb{R}^N}\Bigl(\nabla \tilde{v}_{n,i}\cdot\nabla(\varphi w_i) +\tilde{v}_{n,i}\,\varphi w_i\Bigr)\,dx \nonumber\\ -\sum_{i\neq j}\lambda_{ij}\int_{\mathbb{R}^N}|\tilde{v}_{n,j}|^{p}|\tilde{v}_{n,i}|^{p-2}\tilde{v}_{n,i}\, \varphi w_i\,dx=o(1),\tag{17}\] which implies that \(\widetilde{\mathbf{v}}\) is a weak solution of the limit system by letting \(n\to \infty\) \[\label{eq3466} -\Delta \widetilde{v}_i+\widetilde{v}_i=\sum_{j\neq i}\lambda_{ij}|\widetilde{v}_j|^{p}|\widetilde{v}_i|^{p-2}\widetilde{v}_i ~\text{in}~\mathbb{R}^N,~i=1,\dots,m.\tag{18}\] Testing 18 by \(\widetilde{v}_i\) and summing over \(i\), we obtain \[\sum_{i=1}^m\int_{\mathbb{R}^N}\bigl(|\nabla \widetilde{v}_i|^2+|\widetilde{v}_i|^2\bigr)\,dx = \sum_{i\neq j}\lambda_{ij}\int_{\mathbb{R}^N}|\widetilde{v}_i|^{p}|\widetilde{v}_j|^{p}\,dx \le 0,\] because \(\lambda_{ij}<0\) by \((A_2)\). Therefore \(\widetilde{\mathbf{v}}=\mathbf{0}\), and this is absurd. This contradiction implies that \(\mathbf{v}_n\to\mathbf{v}\) in \(H\).

Since \(\mathbf{v}_n\in\mathcal{N}^{\mathrm{sc}}_\varepsilon\), Lemma 5 gives a uniform lower bound \(\|v_{n,i}^\pm\|_{H^1}\ge\delta>0\) for each \(i\). This yields that \(\|v_i^\pm\|_{H^1}\ge\delta>0\), hence \(v_i^\pm\neq 0\) for all \(i\). Moreover, \(J_\varepsilon'(\mathbf{v})=0\). Testing this with \(v_i^\pm\) yields \(\langle J_\varepsilon^\prime(\mathbf{v}),v_i^\pm\rangle=0\), so \(\mathbf{v}\in\mathcal{N}^{\mathrm{sc}}_\varepsilon\subset\mathcal{A}\).

Finally, by Lemma 6, \(m_\varepsilon:\mathcal{A}\to\mathcal{N}^{\mathrm{sc}}_\varepsilon\) is a \(C^1\)-diffeomorphism, we have \(\mathbf{u}_n=m_\varepsilon^{-1}(\mathbf{v}_n)\to m_\varepsilon^{-1}(\mathbf{v})\) strongly in \(H\), and \(m_\varepsilon^{-1}(\mathbf{v})\in\mathcal{A}\). Therefore \(\{\mathbf{u}_n\}\) has a convergent subsequence in \(\mathcal{A}\), proving that \(\Psi_\varepsilon\) satisfies the \((PS)_c\) condition.

4 Existence of infinitely many sign-changing solutions↩︎

In this section we prove Theorem 1. By Lemma 7(a), the reduced functional \(\Psi_\varepsilon\) is an even \(C^1\) functional on the symmetric set \(\mathcal{A}\subset H\). Our argument follows a symmetric minimax scheme for the reduced nodal functional \(\Psi_{\varepsilon}\) based on the Krasnosel’skii genus. The crucial compactness is provided by Lemma 8, which ensures that \(\Psi_{\varepsilon}\) satisfies the \((PS)_C\) condition at every level \(c\) for each fixed \(\varepsilon>0\). With this at hand, the genus minimax levels yield infinitely many critical points of \(\Psi_{\varepsilon}\), and via the nodal projection \(m_{\varepsilon}\), we obtain infinitely many nodal solutions of 1 .

First, we recall the standard Krasnosel’skii genus of symmetrical sets.

Proof of Theorem 1. Fix \(k\in\mathbb{N}^{*}\), we construct a symmetric finite dimensional subspace. Choose functions \[\boldsymbol{\varphi}_{1},\dots,\boldsymbol{\varphi}_{k}\in C_{c}^{\infty}(\mathbb{R}^{N})^m\] with pairwise disjoint supports and such that each component \(\varphi_{j,i}\) changes sign in \(\boldsymbol{\varphi}_{j}\)’s support for \(i=1,\cdots,m\), so that \(\varphi_{j,i}^{+}\) and \(\varphi_{j,i}^{-}\) are both nontrivial for \(j=1,\cdots,k\) and \(i=1,\cdots,m\). Let \[E_{k}:=\operatorname{span}\{\boldsymbol{\varphi}_{1},\dots,\boldsymbol{\varphi}_{k}\}\] and consider the sphere \[S_{k}:=\{\,\mathbf{u}\in E_{k}:\|\mathbf{u}\|=1\,\}.\] Since \(E_{k}\) is finite dimensional, \(S_{k}\) is compact and symmetric. For each \(\sigma=(\sigma_1,\dots,\sigma_k)\in\{\pm1\}^{k}\), set \[\mathbf{u}_\sigma:=\sum_{j=1}^{k}\sigma_j\,\boldsymbol{\varphi}_j\in H.\] \(\mathbf{u}_\sigma\neq 0\) for every \(\sigma\in\{\pm1\}^{k}\). Moreover, the mapping \[\boldsymbol{\sigma}\longmapsto \frac{\mathbf{u}_{\boldsymbol{\sigma}}}{\|\mathbf{u}_{\boldsymbol{\sigma}}\|}\] induces an odd homeomorphism between the discrete set \(\{\pm1\}^{k}\) and a subset of \(S_{k}\) and hence \(\gamma(S_{k})\ge k\).

By construction every nonzero \(\mathbf{u}\in E_{k}\) has all components sign‑changing on their supports, hence \(S_{k}\subset\mathcal{A}\).

Set \[B_{k}:=m_{\varepsilon}(S_{k})\subset\mathcal{N}_{\varepsilon}^{\mathrm{sc}} .\] Because \(m_{\varepsilon}\) is odd and continuous, the monotonicity of the genus under odd maps yields \[\gamma(B_{k})\;\ge\;\gamma(S_{k})\;\ge\;k .\] Denote \[\label{eq4461} \Sigma_{k}=\{A\subset\mathcal{A}: A~\text{is compact and symmetric with}~\gamma(A)\ge k\},\tag{19}\] and define the minimax level \[\label{eq4462} c_{\varepsilon,k}:=\inf_{A\in\Sigma_{k}}\;\sup_{\mathbf{u}\in A}\Psi_{\varepsilon}(\mathbf{u}).\tag{20}\] Since \(B_{k}\in\Sigma_{k}\) and \(\Psi_{\varepsilon}\) is continuous and bounded from below on \(B_{k}\), so each \(c_{\varepsilon,k}\) is finite.

Next, we claim that \(c_{\varepsilon,k}\) is a critical value of \(\Psi_{\varepsilon}\).

In fact, by Lemma 8, \(\Psi_{\varepsilon}\) satisfies the \((PS)_c\) condition for every \(c\in\mathbb{R}\) on the open set \(\mathcal{A}\subset H\). Hence the symmetric minimax theorem in terms of the Krasnosel’skii genus (e.g. [12]) with modifications can be applied. For completeness we give a deformation argument on the open constraint set \(\mathcal{A}\).

Assume by contradiction that \(c_{\varepsilon,k}\) is not a critical value of \(\Psi_{\varepsilon}\). Then there exists \(\delta>0\) such that \(\Psi_{\varepsilon}\) has no critical values in \([c_{\varepsilon,k}-2\delta,\,c_{\varepsilon,k}+2\delta]\). Moreover, there exists \(\sigma>0\) such that \[\label{eq4463} \bigl\|\Psi_{\varepsilon}^\prime(\mathbf{u})\bigr\|_{H^\prime}\ge \sigma ,~\forall\mathbf{u}\in\Psi_\varepsilon^{-1}([c_{\varepsilon,k}-2\delta,\,c_{\varepsilon,k}+2\delta]) =\{\mathbf{u}\in \mathcal{A}:~\Psi_{\varepsilon}(\mathbf{u})\in I\}.\tag{21}\] Indeed, if 21 is false, we can find a \((PS)_c\)-sequence \((\mathbf{u}_n)\subset\mathcal{A}\) of \(\Psi_\varepsilon\) for some \(c\in I\). By Lemma 8, up to a subsequence, \(\mathbf{u}_n\to \mathbf{u}\) in \(H\) and \(\mathbf{u}\in\mathcal{A}\). Therefore \(\Psi_{\varepsilon}'(\mathbf{u})=0\) and \(\Psi_{\varepsilon}(\mathbf{u})=c\in I\), which contradicts the choice of \(I\). Thus 21 holds.

Next we will apply a variant of the quantitative deformation lemma [5] on \(\mathcal{A}\). Define the set \(\mathcal{S} := \Psi_\varepsilon^{-1}([c_{\varepsilon,k}-2\delta,\,c_{\varepsilon,k}+2\delta])\). In a standard way ([5]), one can construct a locally Lipschitz continuous pseudo-gradient vector field \(V: \mathcal{A} \to H\) satisfying \[\label{eq4464} \langle \Psi_{\varepsilon}(\mathbf{u}),V(\mathbf{u})\rangle \ge \frac{1}{2} \|\Psi_\varepsilon^\prime(\mathbf{u})\|_{H^\prime}^2,~ \|V(\mathbf{u})\| \le 2 \|\Psi_{\varepsilon}^\prime(\mathbf{u})\|_{H^\prime},~\mathbf{u}\in \mathcal{S},\tag{22}\] \(V(-\mathbf{u}) = -V(\mathbf{u})\) for \(\mathbf{u}\in \mathcal{A}\), and \(V(\mathbf{u})=0\) for \(\mathbf{u}\in\mathcal{A}\setminus \mathcal{S}\).

Choose an even Lipschitz cut-off function \(\chi: \mathcal{A} \to [0,1]\) such that \[\chi(\mathbf{u}) = \begin{cases} 1, & \mathbf{u}\in\Psi_\varepsilon^{-1}([c_{\varepsilon,k} - \delta, c_{\varepsilon,k} + \delta]), \\ 0, & \mathbf{u}\in \mathcal{A}\setminus \mathcal{S}. \end{cases}\]

Consider the initial value problem \[\label{eq4465} \begin{cases} \displaystyle \frac{d}{dt} \eta(t, \mathbf{u}) = -\chi(\eta(t, \mathbf{u}))V(\eta(t, \mathbf{u})), \\[6pt] \eta(0, \mathbf{u}) = \mathbf{u}. \end{cases}\tag{23}\] Since the right-hand side of 23 is locally Lipschitz on \(\mathcal{A}\), \(\chi V\) vanishes near \(\partial\mathcal{A}\), and Remark 4 implies that \(\Psi_\varepsilon\to+\infty\) near \(\partial\mathcal{A}\), the solution \(\eta(t,\mathbf{u})\) exists for all \(t\ge 0\) and stays in \(\mathcal{A}\). From the symmetry \(V(-\mathbf{u}) = -V(\mathbf{u})\) and \(\chi(-\mathbf{u}) = \chi(\mathbf{u})\), one has \(\eta(t, -\mathbf{u}) = -\eta(t, \mathbf{u})\).

Along the flow lines, by 22 and 23 , we have \[\begin{align} \frac{d}{dt} \Psi_{\varepsilon}(\eta(t, \mathbf{u})) &= \langle \Psi_{\varepsilon}^\prime (\eta(t, \mathbf{u})),\frac{d}{dt} \eta(t, \mathbf{u})\rangle \\ &= -\chi(\eta) \langle\Psi_{\varepsilon}^\prime (\eta(t, \mathbf{u})),V(\eta(t, \mathbf{u}))\rangle\\ &\le -\frac{1}{2} \chi(\eta) \|\Psi_{\varepsilon}^\prime (\eta(t, \mathbf{u}))\|_{H^\prime}^2\\ &\le -\frac{\sigma^2}{2} \chi(\eta). \end{align}\] In particular, if \(\Psi_{\varepsilon}(\mathbf{u}) \in [c_{\varepsilon,k} - \delta, c_{\varepsilon,k} + \delta]\), then \(\chi(\mathbf{u}) = 1\) and thus \[\Psi_{\varepsilon}(\eta(t, \mathbf{u})) \le \Psi_{\varepsilon}(\mathbf{u}) - \frac{\sigma^2}{2} t.\] Set \(T := \frac{4\delta}{\sigma^2}\). Then for any \(\mathbf{u} \in \Psi_{\varepsilon}^{-1}([c_{\varepsilon,k} - \delta, c_{\varepsilon,k} + \delta])\), we have \[\Psi_{\varepsilon}(\eta(T, \mathbf{u})) \le c_{\varepsilon,k} + \delta - \frac{\sigma^2}{2} \cdot T = c_{\varepsilon,k} - \delta.\] Define \(\varphi: \mathcal{A} \to \mathcal{A}\) by \[\varphi(\mathbf{u}) := \eta(T, \mathbf{u}).\] Then \(\varphi\) satisfies \[\begin{align} &\text{(a)}~\varphi(\mathbf{u})=\mathbf{u}~\text{if}~t=0~\text{or}~ \mathbf{u}\notin \Psi_{\varepsilon}^{-1}\bigl([c_{\varepsilon,k}-2\delta,c_{\varepsilon,k}+2\delta]\bigr),\\ &\text{(b)}~\varphi(\{\mathbf{u}\in \mathcal{A}:\Psi_{\varepsilon}(\mathbf{u})\le c_{\varepsilon,k}+\delta\}) \subset \{\mathbf{u}\in \mathcal{A}:\Psi_{\varepsilon}(\mathbf{u})\le c_{\varepsilon,k}-\delta\},\\ &\text{(c)}~\varphi~\text{is continuous and odd}. \end{align}\]

By the definition of \(c_{\varepsilon,k}\), choose \(A\in\Sigma_k\) such that \(\sup_{\mathbf{u}\in A}\Psi_{\varepsilon}(\mathbf{u})\le c_{\varepsilon,k}+\delta\). Then by (b), \[\sup_{\mathbf{u}\in \varphi(A)}\Psi_{\varepsilon}(\mathbf{u})\le c_{\varepsilon,k}-\delta.\] By (c), \(\gamma(\varphi(A))\ge \gamma(A)\ge k\), so \(\varphi(A)\in\Sigma_k\). Therefore, \[c_{\varepsilon,k}\le \sup_{\mathbf{u}\in \varphi(A)}\Psi_{\varepsilon}(\mathbf{u})\le c_{\varepsilon,k}-\delta,\] a contradiction. Hence \(c_{\varepsilon,k}\) is a critical value of \(\Psi_{\varepsilon}\).

Finally, we show that \(c_{\varepsilon,k}\to+\infty\) as \(k\to\infty\). This is a variant of a standard argument for unboundedness of genus minimax levels (see, e.g., [13]). Assume by contradiction that \(\{c_{\varepsilon,k}\}\) is bounded above and set \[\bar c_\varepsilon:=\sup_{k\ge1}c_{\varepsilon,k}<+\infty .\] Since \(\Sigma_{k+1}\subset\Sigma_k\), the sequence \(\{c_{\varepsilon,k}\}\) is nondecreasing and hence \(c_{\varepsilon,k}\uparrow\bar{c}_\varepsilon\) as \(k\to\infty\). If \(c_{\varepsilon,k}=\bar{c}_\varepsilon\) for all \(k\) large enough, then \(\gamma(\mathcal{K}_{\bar{c}_\varepsilon})=\infty\) (see [13]), where \[\mathcal{K}_{\bar{c}_\varepsilon}=\{\mathbf{u}\in \mathcal{A}:~\Psi_\varepsilon^\prime(\mathbf{u})=0 ~\text{and}~\Psi_\varepsilon(\mathbf{u})=\bar{c}_\varepsilon\}.\] But by Lemma 8, \(\mathcal{K}_{\bar{c}_\varepsilon}\) is compact and so \(\gamma(\mathcal{K}_{\bar{c}_\varepsilon})<\infty\). This contradiction implies that \(c_{\varepsilon,k}<\bar{c}_\varepsilon\) for all \(k\).

Fix \(\delta\in(0,1)\) and choose \(k_0\ge1\) such that \(c_{\varepsilon,k_0}>\bar{c}_\varepsilon-\delta\). Consider the symmetric critical set \[\mathcal{K}:=\Bigl\{\mathbf{u}\in\mathcal{A}:\;\Psi_{\varepsilon}'(\mathbf{u})=\mathbf{0},\; \Psi_{\varepsilon}(\mathbf{u})\in[\bar{c}_\varepsilon-\delta,\bar{c}_\varepsilon]\Bigr\}.\] Using Lemma 8 again, \(\mathcal{K}\) is compact in \(H\). Moreover, \(\mathcal{K}\) is symmetric since \(\Psi_{\varepsilon}\) is even. Hence there exists an open symmetric neighborhood \(\mathcal{O}\subset\mathcal{A}\) of \(\mathcal{K}\) such that \(q:=\gamma(\mathcal{O})<\infty\) (see [13]). Then, arguing as in 21 , there exists \(\sigma_\delta>0\) such that \[\|\Psi_\varepsilon^\prime(u)\|_{H^\prime}\ge \sigma_\delta ~\text{for all}~\mathbf{u}\in \Psi_\varepsilon^{-1}([\bar{c}_\varepsilon-\delta,\bar{c}_\varepsilon+\delta])\setminus\mathcal{O}.\] Therefore one can repeat the deformation construction above on the open set \(\mathcal{A}\) (based on 21 –23 ) with the cut-off supported in \(\Psi_{\varepsilon}^{\,\bar{c}_\varepsilon+\delta}\setminus\mathcal{O}\) and obtain an odd continuous map \(\varphi:\mathcal{A}\to\mathcal{A}\) such that \[\label{eq4466} \varphi\bigl(\Psi_{\varepsilon}^{\,\bar{c}_\varepsilon+\delta}\setminus\mathcal{O}\bigr)\subset \Psi_{\varepsilon}^{\,\bar{c}_\varepsilon-\delta}.\tag{24}\] Next, by the definition of \(c_{\varepsilon,k_0+q}\), choose \(A\in\Sigma_{k_0+q}\) satisfying \[\sup_{\mathbf{u}\in A}\Psi_{\varepsilon}(\mathbf{u})< c_{\varepsilon,k_0+q}+\delta< \bar{c}_\varepsilon+\delta,\] so \(A\subset \Psi_{\varepsilon}^{\,\bar{c}_\varepsilon+\delta}\). Set \(A_1:=A\setminus\mathcal{O}\). Then \(A_1\) is compact and symmetric. By the subadditivity of the genus and the choice \(q=\gamma(\mathcal{O})\), we have \[k_0+q\le \gamma(A)\le \gamma(A\cap\mathcal{O})+\gamma(A_1)\le q+\gamma(A_1),\] which implies \(\gamma(A_1)\ge k_0\), i.e.\(A_1\in\Sigma_{k_0}\). Since \(\varphi\) is odd and continuous, the genus monotonicity yields \(\gamma(\varphi(A_1))\ge\gamma(A_1)\ge k_0\), and thus \(\varphi(A_1)\in\Sigma_{k_0}\). On the other hand, by 24 , we have \(\varphi(A_1)\subset\Psi_{\varepsilon}^{\,\bar{c}_\varepsilon-\delta}\), which implies \[c_{\varepsilon,k_0}\le \sup_{\mathbf{u}\in\varphi(A_1)}\Psi_{\varepsilon}(\mathbf{u})\le \bar{c}_\varepsilon-\delta,\] contradicting \(c_{\varepsilon,k_0}>\bar{c}_\varepsilon-\delta\). This contradiction shows that \(\{c_{\varepsilon,k}\}\) is unbounded, and therefore \(c_{\varepsilon,k}\to+\infty\) as \(k\to\infty\).

Consequently, for each \(k\in\mathbb{N}^{*}\) there exists \(\mathbf{v}_\varepsilon^{(k)}\in\mathcal{A}\) such that \[\Psi_{\varepsilon}\bigl(\mathbf{v}_\varepsilon^{(k)}\bigr)=c_{\varepsilon,k} ~\text{and}~ \Psi_{\varepsilon}^\prime\bigl(\mathbf{v}_\varepsilon^{(k)}\bigr)=0 .\]

Set \[\mathbf{u}_\varepsilon^{(k)}:=m_{\varepsilon}\bigl(\mathbf{v}_\varepsilon^{(k)}\bigr)\in\mathcal{N}_{\varepsilon}^{\mathrm{sc}} .\] By Lemma 7(b), \(\mathbf{u}_\varepsilon^{(k)}\) is a critical point of \(J_{\varepsilon}\) in \(H\), hence a weak solution of system 1 . Moreover, \[J_\varepsilon(\mathbf{u}_\varepsilon^{(k)}) =J_\varepsilon\!\bigl(m_\varepsilon(\mathbf{u}^{(k)})\bigr) =\Psi_\varepsilon(\mathbf{u}^{(k)})=c_{\varepsilon,k}.\]

Since \(c_{\varepsilon,k}\to+\infty\) as \(k\to\infty\), after relabeling, we may assume the energies are strictly increasing: \[0<J_\varepsilon\bigl(\mathbf{u}_\varepsilon^{(1)}\bigr) <J_\varepsilon\bigl(\mathbf{u}_\varepsilon^{(2)}\bigr) <\cdots<J_\varepsilon\bigl(\mathbf{u}_\varepsilon^{(k)}\bigr) <\cdots, ~ \lim_{k\to\infty}J_\varepsilon\bigl(\mathbf{u}_\varepsilon^{(k)}\bigr)=+\infty.\] This completes the proof of Theorem 1.

5 Concentration of sign-changing solutions as \(\varepsilon\to 0^+\)↩︎

In this section we prove Theorem 2, which describes the concentration behavior of sign-changing solutions obtained in Theorem 1 and the structure of their limiting profiles as \(\varepsilon \to 0\).

First, we recall the limiting autonomous system 2 \[\notag \begin{cases} -\Delta U_i + U_i = \mu_i Q(-y_i)\,|U_i|^{2p-2}U_i + \displaystyle\sum_{j\neq i}\lambda_{ij}\,|U_j|^p\,|U_i|^{p-2}U_i, \\[2mm] U_i \in H^1(\mathbb{R}^N), ~ i=1,\dots,m. \end{cases}\] The corresponding energy functional associated with 2 is \[J_0(\mathbf{U}) = \frac{1}{2}\sum_{i=1}^m\int_{\mathbb{R}^N}\bigl(|\nabla U_i|^2+|U_i|^2\bigr)\,dx -\frac{1}{2p}\sum_{i=1}^m \mu_i Q(-y_i)\int_{\mathbb{R}^N}|U_i|^{2p}\,dx -\frac{1}{2p}\sum_{i\neq j}\lambda_{ij}\int_{\mathbb{R}^N}|U_i|^p|U_j|^p\,dx,\] and the corresponding Nehari manifold is \[\mathcal{N}_0 := \bigl\{\mathbf{U}\in H\setminus\{\mathbf{0}\}:\;\langle J_0^\prime(\mathbf{U}),\mathbf{U}\rangle=0\bigr\}.\] Denote \(\alpha:=\inf\limits_{\mathbf{U}\in\mathcal{N}_0}J_0(\mathbf{U})\), then \(\alpha>0\) (see [3]). All definitions and properties introduced in Section 2 for \(J_\varepsilon\) and \(\Psi_\varepsilon\) can be repeated for \(\varepsilon=0\). In particular, the fully sign-changing Nehari set is denoted by \(\mathcal{N}^{\mathrm{sc}}_0\), the reduced functional \(\Psi_0:\mathcal{A}\to \mathbb{R}\) is defined by \[\Psi_0(\mathbf{u}) = J_0\bigl(m_0(\mathbf{u})\bigr),~ \mathbf{u}\in\mathcal{A},\] where \(m_0:\mathcal{A}\to \mathcal{N}^{\mathrm{sc}}_0\) is the corresponding nodal projection.

However, the functional \(\Psi_0\) does not satisfy the \((PS)_c\) condition since it is invariant under translations. Therefore, we establish below the \((PS)_c\) property modulo \(\mathbb{Z}^N\)-translations, which is sufficient for the genus minimax argument.

For every \(a\in\mathbb{R}\), set \[\mathcal{K}^a:=\{\mathbf{u}\in \mathcal{A}:~\Psi_0^\prime(\mathbf{u})=0~\text{and}~\Psi_0(\mathbf{u})\le a\}.\]

.  Let \(\{\mathbf{u}_n\}\subset \mathcal{K}^a\) and set \(\mathbf{v}_n:=m_0(\mathbf{u}_n)\in \mathcal{N}_{0}^{\mathrm{sc}}\), then \(\Psi_0(\mathbf{u})=J_0\bigl(m_0(\mathbf{u})\bigr)\) for \(\mathbf{u}\in\mathcal{A}\), we obtain \[J_0(\mathbf{v}_n)=\Psi_0(\mathbf{u}_n)\le a.\] Moreover, since \(\Psi_0^\prime(\mathbf{u}_n)=0\), we have \(J_0^\prime(\mathbf{v}_n)=0\) and \[\label{eq5461} J_0(\mathbf{v}_n)=\Bigl(\frac{1}{2}-\frac{1}{2p}\Bigr)\|\mathbf{v}_n\|^2.\tag{25}\] Therefore, \(\{\mathbf{v}_n\}\) is a bounded sequence of critical points of \(J_0\). Going to a subsequence if necessary, we may assume \(J_0(\mathbf{v}_n)\to c\) for some \(c\le a\), and hence \(\{\mathbf{v}_n\}\) is a bounded \((PS)_c\) sequence for \(J_0\). Therefore, by the global compactness principle for translation invariant problems (see [14] or [5]), there exist an integer \(\ell\ge 1\), sequences \(\{z_n^{(j)}\}\subset\mathbb{R}^N\) for \(j=1,\dots,\ell\) satisfying \[|z_n^{(i)}-z_n^{(j)}|\to\infty~\text{as }n\to\infty \;\text{ whenever }i\neq j,\] and nontrivial critical points \(\mathbf{V}^{(1)},\dots,\mathbf{V}^{(\ell)}\in H\) of \(J_0\) such that \[\label{eq5462} \mathbf{v}_n=\sum_{j=1}^{\ell}\mathbf{V}^{(j)}(\cdot-z_n^{(j)})+\mathbf{r}_n, ~\mathbf{r}_n\to\mathbf{0}~\text{in}~H,\tag{26}\] and the energy splitting holds: \[\label{eq5463} J_0(\mathbf{v}_n)=\sum_{j=1}^{\ell}J_0(\mathbf{V}^{(j)})+o(1)~\text{as}~n\to\infty.\tag{27}\] In particular, for each \(j=1,\dots,\ell\), we have \(J_0^\prime(\mathbf{V}^{(j)})=\mathbf{0}\) in \(H^\prime\) and \(\mathbf{V}^{(j)}\neq\mathbf{0}\). Furthermore, since \(\mathbf{V}^{(j)}\neq \mathbf{0}\) is a critical point of \(J_0\), we have \(J_0(\mathbf{V}^{(j)})\ge \alpha\) for each \(j\). Hence, by 27 and \(J_0(\mathbf{v}_n)\le a\), we obtain \[\sum_{j=1}^{\ell}J_0(\mathbf{V}^{(j)})=\lim_{n\to\infty}J_0(\mathbf{v}_n)\le a,\] and therefore necessarily \(\ell<\infty\).

For each \(j\in\{1,\dots,\ell\}\), choose \(k_n^{(j)}\in\mathbb{Z}^N\) such that \[|z_n^{(j)}-k_n^{(j)}|\le \frac{\sqrt N}{2}~\text{for all}~n,\] and write \[z_n^{(j)}=k_n^{(j)}+\xi_n^{(j)}~\text{with}~\xi_n^{(j)}\in\Bigl[-\frac{1}{2},\frac{1}{2}\Bigr]^N.\] Passing to a subsequence if necessary, we may assume that \[\xi_n^{(j)}\to\xi^{(j)}\in\Bigl[-\frac{1}{2},\frac{1}{2}\Bigr]^N ~\text{for each}~j=1,\dots,\ell.\]

Define \(\tilde{\mathbf{r}}_n^{(j)}:=k_n^{(j)}*\mathbf{r}_n\), then \(\tilde{\mathbf{r}}_n^{(j)}\to\mathbf{0}\) in \(H\). Moreover, for each fixed \(j\) we consider the translated sequence \[\mathbf{v}_n^{(j)}:=k_n^{(j)}*\mathbf{v}_n=\mathbf{v}_n(\cdot-k_n^{(j)}).\] Using 26 and \(z_n^{(j)}-k_n^{(j)}=\xi_n^{(j)}\), we obtain \[\label{eq5464} \mathbf{v}_n^{(j)}=\mathbf{V}^{(j)}(\cdot+\xi_n^{(j)}) +\sum_{\substack{i=1\\ i\neq j}}^{\ell}\mathbf{V}^{(i)}\bigl(\cdot-(z_n^{(i)}-k_n^{(j)})\bigr) +\tilde{\mathbf{r}}_n^{(j)} .\tag{28}\]

We claim that the other profiles vanish locally after this discretization. Indeed, since \(|z_n^{(i)}-z_n^{(j)}|\to\infty\) for \(i\neq j\) and \(z_n^{(j)}-k_n^{(j)}=\xi_n^{(j)}\) is bounded, we have \[|z_n^{(i)}-k_n^{(j)}| =\bigl|z_n^{(i)}-z_n^{(j)}+\xi_n^{(j)}\bigr|\to\infty ~\text{for every}~i\neq j.\] Consequently, for every \(R>0\) and every \(2\le q<2^*\), \[\bigl\|\mathbf{V}^{(i)}(\cdot-(z_n^{(i)}-k_n^{(j)}))\bigr\|_{L^q(B_R)^m} =\|\mathbf{V}^{(i)}\|_{L^q(B_R(z_n^{(i)}-k_n^{(j)}))^m}\to 0, ~ i\neq j,\] because \(B_R(z_n^{(i)}-k_n^{(j)})\) drifts to infinity and \(\mathbf{V}^{(i)}\in L^q(\mathbb{R}^N)^m\).

On the other hand, since \(\xi_n^{(j)}\to\xi^{(j)}\), \[\mathbf{V}^{(j)}(\cdot+\xi_n^{(j)})\to \mathbf{V}^{(j)}(\cdot+\xi^{(j)}) ~\text{in}~H.\] Combining these facts with 28 and \(\tilde{\mathbf{r}}_n^{(j)}\to 0\) in \(H\), we conclude that for every \(R>0\) and every \(2\le q<2^*\), \[\mathbf{v}_n^{(j)} \to \mathbf{V}^{(j)}(\cdot+\xi^{(j)}) ~\text{in}~L^q(B_R)^m ~\text{and a.e.\;in }B_R.\] In particular, \(\mathbf{v}_n^{(j)}\rightharpoonup \mathbf{V}^{(j)}(\cdot+\xi^{(j)})\) in \(H\). It is obvious that each \(\mathbf{V}^{(j)}(\cdot+\xi^{(j)})\) is again a nontrivial critical point of \(J_0\), since \(J_0^\prime(\mathbf{V}^(j))=\mathbf{0}\) and \(J_0\) is translation invariant.

Fix \(j=1\) and set \(k_n:=k_n^{(1)}\in\mathbb{Z}^N\). Then, by the decomposition 26 and the separation property \(|z_n^{(i)}-z_n^{(1)}|\to\infty\) for \(i\neq 1\), we have \[k_n*\mathbf{v}_n=\mathbf{V}^{(1)}(\cdot+\xi_n^{(1)})+\tilde{\mathbf{r}}_n^{(1)}+o(1)~\text{in}~H,\] where \(\tilde{\mathbf{r}}_n^{(1)}\to\mathbf{0}\) in \(H\). Hence \[k_n*\mathbf{v}_n\to \mathbf{V}^{(1)}(\cdot+\xi^{(1)})~\text{in}~H.\]

We first verify the equivariance of \(m_0\), i.e., for every \(k\in\mathbb{Z}^N\) and every \(\mathbf{u}\in\mathcal{A}\), \[\label{eq5465} m_0(k*\mathbf{u})=k*m_0(\mathbf{u}).\tag{29}\] Indeed, note that \((k*u_i)^\pm = k*(u_i^\pm)\) for each \(i\). Write \[m_0(\mathbf{u}) =\sum_{i=1}^m\Bigl(t_i^+(\mathbf{u})\,u_i^+ + t_i^-(\mathbf{u})\,u_i^-\Bigr),\] where \((t_i^+(\mathbf{u}),t_i^-(\mathbf{u}))\) is the unique scaling pair sending \(\mathbf{u}\) onto \(\mathcal{N}_0^{\mathrm{sc}}\). Since \(J_0\) is translation invariant and the defining equations for \((t_i^+,t_i^-)\) involve only integrals of \(u_i^\pm\) and \(|u_i^\pm u_j^\pm|\) (which are invariant under translations), the same pair \((t_i^+(\mathbf{u}),t_i^-(\mathbf{u}))\) also sends \(k*\mathbf{u}\) onto \(\mathcal{N}_0^{\mathrm{sc}}\). By uniqueness of the scalings, \(t_i^\pm(k*\mathbf{u})=t_i^\pm(\mathbf{u})\) for all \(i\), and therefore \[m_0(k*\mathbf{u}) =\sum_{i=1}^m\Bigl(t_i^+(\mathbf{u})\,(k*u_i^+) + t_i^-(\mathbf{u})\,(k*u_i^-)\Bigr) =k*\sum_{i=1}^m\Bigl(t_i^+(\mathbf{u})\,u_i^+ + t_i^-(\mathbf{u})\,u_i^-\Bigr) =k*m_0(\mathbf{u}),\] which proves 29 .

We now deduce the compactness of critical orbits for \(\Psi_0\). Let \(\{\mathbf{u}_n\}\subset \mathcal{K}^a\) be arbitrary and set \[\mathbf{v}_n:=m_0(\mathbf{u}_n)\in\mathcal{N}^{\mathrm{sc}}_0.\] Then \(J_0(\mathbf{v}_n)=\Psi_0(\mathbf{u}_n)\le a\), and moreover, \(\Psi_0^\prime(\mathbf{u}_n)=\mathbf{0}\) implies \(J_0^\prime(\mathbf{v}_n)=\mathbf{0}\). Hence \(\{\mathbf{v}_n\}\) is a bounded sequence of critical points of \(J_0\) in \(H\). By the above arguments, there exists a sequence \(\{k_n\}\subset\mathbb{Z}^N\) and a nontrivial critical point \(\mathbf{V}\in H\) of \(J_0\) such that \[k_n*\mathbf{v}_n \to \mathbf{V}~\text{in}~H.\] Using the equivariance 29 , we have \[m_0(k_n*\mathbf{u}_n)=k_n*m_0(\mathbf{u}_n)=k_n*\mathbf{v}_n \to \mathbf{V}~\text{in}~H.\] Finally, since \(m_0:\mathcal{A}\to\mathcal{N}^{\mathrm{sc}}_0\) is a homeomorphism, we obtain \[k_n*\mathbf{u}_n = m_0^{-1}(k_n*\mathbf{v}_n)\to m_0^{-1}(\mathbf{V})~\text{in}~H,\] which implies that the \(\mathbb{Z}^N\)–orbit of \(\{\mathbf{u}_n\}\) is precompact in \(H\), and hence \(\mathcal{K}^a/\mathbb{Z}^N\) is compact in the quotient topology induced by \(H\). This completes the proof of Lemma 9.

Therefore, as in section 4, we can construct a sequence of critical values \(\{c_{0,k}\}\) of \(\Psi_0\).

.  For \(k\in\mathbb{N}^{*}\), define \[c_{0,k} :=\inf_{A\in\Sigma_{k}}\;\sup_{\mathbf{u}\in A}\Psi_{0}(\mathbf{u}),\] where \(\Sigma_{k}\) is given in 19 . Then \[0<c_{0,1}\le c_{0,2}\le \cdots \le c_{0,k}\le\cdots,\] since \[\label{eq5466} \Psi_0(\mathbf{u}) =J_0\bigl(m_0(\mathbf{u})\bigr) = \Bigl(\frac{1}{2}-\frac{1}{2p}\Bigr)\,\bigl\|m_0(\mathbf{u})\bigr\|^2>0, ~\forall\,\mathbf{u}\in\mathcal{A},\tag{30}\] and \(\Sigma_{k+1}\subset\Sigma_{k}\).

Next, similar to [14] and [5], we show that \(c_{0,k}\) is a critical value of \(\Psi_0\) via an equivariant deformation. Fix \(k\in\mathbb{N}^{*}\), assume by contradiction that \(c_{0,k}\) is not a critical value of \(\Psi_0\). Then there exists \(\delta>0\) such that \(\Psi_0\) has no critical values in \([c_{0,k}-\delta,\;c_{0,k}+\delta]\). Set \[\mathcal{S}=\Psi_0^{-1}([c_{0,k}-2\delta,c_{0,k}+2\delta]).\] Then \(\mathcal{S}\) contains no critical points of \(\Psi_0\). We claim that there exists \(\sigma_\rho>0\) such that \[\label{eq5467} \|\Psi_0'(\mathbf{u})\|\ge \sigma_\rho~\text{for all}~\mathbf{u}\in \mathcal{S}.\tag{31}\] Indeed, argue by contradiction. Then there exists a sequence \(\{\mathbf{u}_n\}\subset S\) such that \[\|\Psi_0'(\mathbf{u}_n)\|\to0,~ c_{0,k}-2\delta\le \Psi_0(\mathbf{u}_n)\le c_{0,k}+2\delta~\text{for all}~n.\] Set \(\mathbf{v}_n:=m_0(\mathbf{u}_n)\in\mathcal{N}_0^{\mathrm{sc}}\). Then \[J_0(\mathbf{v}_n)=\Psi_0(\mathbf{u}_n)\in[c_{0,k}-2\delta,\;c_{0,k}+2\delta],\] so \(\{J_0(\mathbf{v}_n)\}\) is bounded and \(J_0^\prime(\mathbf{v}_n)\to 0\). Hence \(\{\mathbf{v}_n\}\subset \mathcal{N}_0^{\mathrm{sc}}\) is a \((PS)_c\) sequence of \(J_0\) for some \(c\in[c_{0,k}-2\delta,\;c_{0,k}+2\delta]\). Similar to 25 , \(\{\mathbf{v}_n\}\) is bounded. Using a standard translation argument ([5]), we find a sequence \(\{k_n\}\subset\mathbb{Z}^N\) and a critical point \(\mathbf{V}\) of \(J_0\) such that \[k_n*\mathbf{v}_n\to\mathbf{V}~\text{in}~H.\] Jointly with 29 , we have \[m_0(k_n*\mathbf{u}_n)=k_n*m_0(\mathbf{u}_n)=k_n*\mathbf{v}_n\to\mathbf{V} ~\text{in}~H.\] Since \(m_0:\mathcal{A}\to\mathcal{N}_0^{\mathrm{sc}}\) is a homeomorphism, it follows that \[k_n*\mathbf{u}_n\to m_0^{-1}(\mathbf{V})~\text{in }H.\] It follows from the continuities of \(\Psi_0\) and \(\Psi_0^\prime\) that \[\Psi_0\bigl(m_0^{-1}(\mathbf{V})\bigr)=\lim_{n\to\infty}\Psi_0(k_n*\mathbf{u}_n) =\lim_{n\to\infty}\Psi_0(\mathbf{u}_n)\in[c_{0,k}-2\delta,\;c_{0,k}+2\delta],\] and \(\Psi_0^\prime\bigl(m_0^{-1}(\mathbf{V})\bigr)=\mathbf{0}\), contradicts the fact that \(\mathcal{S}\) contains no critical points of \(\Psi_0\). This proves 31 .

In a standard way ([14],[5]), one can construct a locally Lipschitz continuous, even and \(\mathbb{Z}^N\)-equivariant pseudo-gradient vector field \(V: \mathcal{A} \to H\) satisfying \[\langle \Psi_0(\mathbf{u}),V(\mathbf{v})\rangle \ge \frac{1}{2} \|\Psi_0^\prime(\mathbf{u})\|_{H^\prime}^2,~ \|V(\mathbf{u})\| \le 2 \|\Psi_0^\prime(\mathbf{u})\|_{H^\prime},~\mathbf{u}\in \mathcal{S},\] \(V(-\mathbf{u}) = -V(\mathbf{u})\) for \(\mathbf{u}\in \mathcal{A}\), \(V(\mathbf{u})=0\) for \(\mathbf{u}\in\mathcal{A}\setminus \mathcal{S}\). Repeat the arguments in Section 4, one can construct a deformation \(\varphi:\mathcal{A}\to \mathcal{A}\) satisfies \[\begin{align} &\text{(a)}~\varphi(\mathbf{u})=\mathbf{u}~\text{if}~ \mathbf{u}\notin \Psi_{0}^{-1}\bigl([c_{0,k}-2\delta,\;c_{0,k}+2\delta]\bigr),\\ &\text{(b)}~\varphi\bigl(\{\mathbf{u}\in \mathcal{A}:\Psi_{0}(\mathbf{u})\le c_{0,k}+\delta\}\bigr) \subset \{\mathbf{u}\in \mathcal{A}:\Psi_{0}(\mathbf{u})\le c_{0,k}-\delta\},\\ &\text{(c)}~\varphi~\text{is}~\mathbb{Z}^N\text{-equivariant, continuous and odd}. \end{align}\]

By the definition of \(c_{0,k}\), choose \(A\in\Sigma_k\) such that \(\sup_{\mathbf{u}\in A}\Psi_{0}(\mathbf{u})\le c_{0,k}+\delta\). Then by (b), \[\sup_{\mathbf{u}\in \varphi(A)}\Psi_{0}(\mathbf{u})\le c_{0,k}-\delta.\] By (c), the set \(\varphi(A)\) is compact and symmetric, and by the monotonicity of the genus under odd maps we have \(\gamma(\varphi(A))\ge \gamma(A)\ge k\), so \(\varphi(A)\in\Sigma_k\). Therefore, \[c_{0,k}\le \sup_{\mathbf{u}\in \varphi(A)}\Psi_{0}(\mathbf{u})\le c_{0,k}-\delta,\] a contradiction. Hence \(c_{0,k}\) must be a critical value of \(\Psi_0\).

Thus, for every \(k\in\mathbb{N}^{*}\), there exists \(\mathbf{u}^{(k)}_{0}\in\mathcal{A}\) such that \[\Psi_0^\prime(\mathbf{u}^{(k)}_{0})=\mathbf{0} ~\text{and}~ \Psi_0(\mathbf{u}^{(k)}_{0})=c_{0,k}.\]

Set \[\mathbf{U}^{(k)}:=m_0\bigl(\mathbf{u}^{(k)}_{0}\bigr)\in \mathcal{N}^{\mathrm{sc}}_0.\] Observe that critical points of \(\Psi_0\) correspond to critical points of \(J_0\) on \(\mathcal{N}^{\mathrm{sc}}_0\), hence \[J_0^\prime\bigl(\mathbf{U}^{(k)}\bigr)=\mathbf{0}~\text{in}~H^\prime.\] Therefore, \(\mathbf{U}^{(k)}\) is a weak solution of the limit system 2 . Moreover, because \(\mathbf{u}^{(k)}_{0}\in\mathcal{A}\), we have \((u^{(k)}_{0,i})^\pm\neq 0\) for \(i=1,\dots,m\). Since \(m_0\) is defined by independent positive scalings on \(u_i^{+}\) and \(u_i^{-}\), the same property holds for \(\mathbf{U}^{(k)}\), and hence \(\mathbf{U}^{(k)}\) is fully sign-changing and satisfies \[J_0\bigl(\mathbf{U}^{(k)}\bigr) =\Psi_0\bigl(\mathbf{u}^{(k)}_{0}\bigr) =c_{0,k}.\]

Finally, we show that \(\Psi_0\) has a sequence of pairwise distinct critical values. For the case that \(c_{0,k}\to +\infty\) as \(k\to\infty\), after relabeling, we may assume the energies are strictly increasing and the conclusion holds.

So we only consider the case that \(\sup_{k\ge1}c_{0,k}\le C\) for some \(C>0\). Then for every \(k\) we can choose \(A_k\in\Sigma_k\) such that \[\sup_{\mathbf{u}\in A_k}\Psi_0(\mathbf{u})\le C+1.\] Hence \(A_k\subset \Psi_0^{\,C+1}:=\{\mathbf{u}\in\mathcal{A}:\;\Psi_0(\mathbf{u})\le C+1\}\) and \[\gamma(\Psi_0^{\,C+1})\ge \gamma(A_k)\ge k~\text{for all}~k,\] which implies \(\gamma(\Psi_0^{\,C+1})=+\infty\). Let \[\mathcal{K}^{C+1} :=\Bigl\{\mathbf{u}\in\mathcal{A}:\;\Psi_0'(\mathbf{u})=0,\;\Psi_0(\mathbf{u})\le C+1\Bigr\}.\] By Lemma 9, the set of critical orbits \(\mathcal{K}^{C+1}/\mathbb{Z}^N\) is compact. Fix \(\rho>0\) and set \[\mathcal{U}_\rho:=U_\rho(\mathcal{K}^{C+1}) =\bigl\{\mathbf{u}\in\mathcal{A}:\;\operatorname{dist}_H(\mathbf{u},\mathcal{K}^{C+1})<\rho\bigr\},~ \mathcal{U}_{2\rho}:=U_{2\rho}(\mathcal{K}^{C+1}).\] Since \(\Psi_0\) is even and \(\mathbb{Z}^N\)–invariant on \(\mathcal{A}\) and the action of \(\mathbb{Z}^N\) on \(H\) is isometric, we may choose \(\mathcal{U}_\rho\) and \(\mathcal{U}_{2\rho}\) symmetric and \(\mathbb{Z}^N\)–invariant. Define \[\Sigma:=\Psi_0^{\,C+1}\setminus \mathcal{U}_{2\rho}.\] By definition, \(\Sigma\) contains no critical points of \(\Psi_0\). Similar to 31 , there exists \(\sigma_\rho>0\) such that \[\label{eq5468} \|\Psi_0^\prime(\mathbf{u})\|\ge \sigma_\rho~\text{for all}~\mathbf{u}\in\Sigma.\tag{32}\] Since \(\Psi_0\) is even and \(\mathbb{Z}^N\)–invariant on \(\mathcal{A}\), we can choose a locally Lipschitz pseudo-gradient vector field \(V:\mathcal{A}\setminus U\to H\) which is odd and \(\mathbb{Z}^N\)–equivariant and satisfies \[\langle \Psi_0^\prime(\mathbf{u}),V(\mathbf{u})\rangle \ge \frac{1}{2}\|\Psi_0^\prime(\mathbf{u})\|^2,~ \|V(\mathbf{u})\|\le 2\|\Psi_0'(\mathbf{u})\| ~\text{for all}~\mathbf{u}\in\mathcal{A}\setminus \mathcal{U}_\rho.\] Let \(\chi:\mathcal{A}\to[0,1]\) be an even, \(\mathbb{Z}^N\)–invariant Lipschitz cut-off function such that \[\chi\equiv 0~\text{on}~\mathcal{U}_{\rho}, ~\chi\equiv 1\;\text{on}~\Sigma.\] Consider the initial value problem \[\begin{cases} \displaystyle \frac{d}{dt} \eta(t, \mathbf{u}) = -\chi(\eta(t, \mathbf{u}))V(\eta(t, \mathbf{u})), \\[6pt] \eta(0, \mathbf{u}) = \mathbf{u}_0\in\Psi_0^{\,C+1}. \end{cases}\] The vector field \(\mathbf{u}\mapsto -\chi(\mathbf{u})V(\mathbf{u})\) is locally Lipschitz on \(\mathcal{A}\), odd and \(\mathbb{Z}^N\)–equivariant, and it vanishes on \(\mathcal{U}_{\rho}\). Hence the corresponding flow \(\varphi(t,\mathbf{u}_0)\) is globally defined, odd and \(\mathbb{Z}^N\)–equivariant, and satisfies \(\varphi(t,\mathbf{u}_0)=\mathbf{u}_0\) for all \(\mathbf{u}_0\in \mathcal{U}_{\rho}\) and \(t\ge0\). Moreover, for \(\mathbf{u}(t):=\varphi(t,\mathbf{u}_0)\) we have \[\frac{d}{dt}\Psi_0(\mathbf{u}(t)) =\langle \Psi_0'(\mathbf{u}(t)),\dot{\mathbf{u}}(t)\rangle =-\chi(\mathbf{u}(t))\langle \Psi_0'(\mathbf{u}(t)),V(\mathbf{u}(t))\rangle \le -\frac{1}{2}\,\chi(\mathbf{u}(t))\|\Psi_0'(\mathbf{u}(t))\|^2\le 0.\] In particular, if \(\mathbf{u}(t)\in\Sigma\), then \(\chi(\mathbf{u}(t))=1\) and by 32 , \[\frac{d}{dt}\Psi_0(\mathbf{u}(t))\le -\frac{1}{2}\sigma_\rho^2,\] and hence \[0\le \Psi_0(\mathbf{u}(t))\le C+1-\frac{1}{2}\sigma_\rho^2 t,\] which implies that \(t\le T_\rho:=\frac{2(C+1)}{\sigma_\rho^2}\). Since \(\Psi_0\ge0\) on \(\mathcal{A}\) and \(\Psi_0(\mathbf{u}_0)\le C+1\), this implies that the trajectory cannot remain in \(\Sigma\) for all \(t\ge0\). More precisely, it must enter \(\mathcal{U}_{2\rho}\) no later than \(T_\rho\). Define the deformation \(\eta_\rho:\Psi_0^{\,C+1}\to\Psi_0^{\,C+1}\) by \[\eta_\rho(\mathbf{u}):=\varphi(T_\rho,\mathbf{u}).\] Then \(\eta_\rho\) is continuous, odd and \(\mathbb{Z}^N\)–equivariant, \(\eta_\rho(\mathbf{u})=\mathbf{u}\) for all \(\mathbf{u}\in \mathcal{U}_\rho\), and \(\Psi_0(\eta_\rho(\mathbf{u}))\le \Psi_0(\mathbf{u})\) for all \(\mathbf{u}\in\Psi_0^{\,C+1}\). Moreover, by the definition of \(T_\rho\), we have \[\label{eq5469} \eta_\rho\bigl(\Psi_0^{\,C+1}\bigr)\subset \mathcal{U}_{2\rho}.\tag{33}\] By monotonicity of the Krasnosel’skii genus under odd continuous maps, 33 yields \[\gamma\bigl(\Psi_0^{\,C+1}\bigr)\le \gamma\bigl(\mathcal{U}_{2\rho}\bigr).\]

It remains to show that \(\gamma(\mathcal{U}_{2\rho})<\infty\) for \(\rho>0\) chosen sufficiently small. First, for \(\rho>0\) small we have \(\mathcal{U}_{2\rho}\subset\mathcal{A}\) and \(0\notin \mathcal{U}_{2\rho}\). Moreover, since \(\mathcal{K}^{C+1}/\mathbb{Z}^N\) is compact, the same holds for \(\mathcal{U}_{2\rho}/\mathbb{Z}^N\). Observe that the map \(\mathbf{u}\mapsto \|\mathbf{u}\|_H\) is continuous and \(\mathbb{Z}^N\)-invariant, hence it induces a continuous function on the compact set \(\mathcal{U}_{2\rho}/\mathbb{Z}^N\) and attains its positive because \(0\notin \mathcal{U}_{2\rho}\). Thus there exists \(\delta_0>0\) such that \[\|\mathbf{u}\|_H \ge \delta_0~\text{for all}~\mathbf{u}\in \mathcal{U}_{2\rho}.\] By the compactness of \(\mathcal{U}_{2\rho}/\mathbb{Z}^N\), we may choose \(\ell\in\mathbb{N}\), elements \(\mathbf{u}^1,\dots,\mathbf{u}^\ell\in\mathcal{U}_{2\rho}\) and a radius \(r\in(0,\delta_0/2)\) such that \[\label{eq54610} \mathcal{U}_{2\rho}\subset \bigcup_{j=1}^\ell\;\bigcup_{k\in\mathbb{Z}^N} B_r(k*\mathbf{u}^j).\tag{34}\] For each \(j=1,\dots,\ell\), set \(E_j:=\mathrm{span}\{\mathbf{u}^j\}\) and let \(P_{E_j}:H\to E_j\) be the orthogonal projection. If \(\mathbf{v}\in B_r(\mathbf{u}^j)\), then \[\|P_{E_j}\mathbf{v}\|_{H}\ge \|\mathbf{u}^j\|_{H}-\|\mathbf{v}-\mathbf{u}^j\|_{H} \ge \delta_0-r>0,\] hence \(P_{E_j}(\mathbf{v})\neq 0\). Moreover, since \(\|\mathbf{u}^j\|_H\ge \delta_0\) and \(r<\delta_0/2\), we have \[\langle \mathbf{v},\mathbf{u}^j\rangle_H =\|\mathbf{u}^j\|_H^2+\langle \mathbf{v}-\mathbf{u}^j,\mathbf{u}^j\rangle_H \ge \|\mathbf{u}^j\|_H^2-\|\mathbf{v}-\mathbf{u}^j\|_H\,\|\mathbf{u}^j\|_H > \|\mathbf{u}^j\|_H(\|\mathbf{u}^j\|_H-r)>0,\] and therefore \[P_{E_j}(\mathbf{v})=\frac{\langle \mathbf{v},\mathbf{u}^j\rangle_H}{\|\mathbf{u}^j\|_H^2}\,\mathbf{u}^j ~\text{is a positive multiple of }\mathbf{u}^j .\]

To construct a global odd continuous map, consider the \(\mathbb{Z}^N\)-invariant open cover \(\{B_r(k*\mathbf{u}^j)\}_{1\le j\le \ell,\;k\in\mathbb{Z}^N}\) of \(\mathcal{U}_{2\rho}\). Since \(H\) is a metric space, it is paracompact; thus there exists a locally finite \(\mathbb{Z}^N\)-invariant partition of unity \(\{\vartheta_{j,k}\}_{j,k}\) subordinate to this cover. Replacing \(\vartheta_{j,k}\) by \(\frac{1}{2}(\vartheta_{j,k}(\cdot)+\vartheta_{j,k}(-\cdot))\), we may assume that each \(\vartheta_{j,k}\) is even.

Let \(E:=E_1\oplus\cdots\oplus E_\ell\) and define \(F:\mathcal{U}_{2\rho}\to E\) by \[F(\mathbf{u}) :=\Bigl(\sum_{k\in\mathbb{Z}^N}\vartheta_{1,k}(\mathbf{u})\,P_{E_1}(k^{-1}*\mathbf{u}),\;\dots,\; \sum_{k\in\mathbb{Z}^N}\vartheta_{\ell,k}(\mathbf{u})\,P_{E_\ell}(k^{-1}*\mathbf{u})\Bigr).\] The sums are finite for each fixed \(\mathbf{u}\) by local finiteness, hence \(F\) is well-defined and continuous. Moreover, \(F\) is odd because each \(\vartheta_{j,k}\) is even and each \(P_{E_j}\) is linear.

Finally, we show that \(F(\mathbf{u})\neq 0\) for all \(\mathbf{u}\in\mathcal{U}_{2\rho}\). Fix \(\mathbf{u}\in\mathcal{U}_{2\rho}\), by 34 , there exist \(j\in\{1,\dots,\ell\}\) and \(k_0\in\mathbb{Z}^N\) such that \(\mathbf{u}\in B_r(k_0*\mathbf{u}^j)\), i.e. \(k_0^{-1}*\mathbf{u}\in B_r(\mathbf{u}^j)\). Since \(\{\vartheta_{j,k}\}_k\) is a partition of unity subordinate to \(\{B_r(k*\mathbf{u}^j)\}_k\), we have \(\vartheta_{j,k_0}(\mathbf{u})>0\), and for every \(k\) with \(\vartheta_{j,k}(\mathbf{u})>0\) we also have \(k^{-1}*\mathbf{u}\in B_r(\mathbf{u}^j)\). Hence each vector \(P_{E_j}(k^{-1}*\mathbf{u})\) appearing in the \(j\)-th component is a positive multiple of \(\mathbf{u}^j\), and at least one of them is nonzero. Since all coefficients \(\vartheta_{j,k}(\mathbf{u})\) are nonnegative, the \(j\)-th component of \(F(\mathbf{u})\) is a positive multiple of \(\mathbf{u}^j\). Therefore \(F(\mathbf{u})\neq 0\).

Therefore, \(F:\mathcal{U}_{2\rho}\to E\setminus\{0\}\) is odd and continuous, and hence \[\gamma(\mathcal{U}_{2\rho})\le \gamma(E\setminus\{0\})=\dim E<\infty.\] Consequently, \(\gamma(\Psi_0^{\,C+1})<\infty\), which contradicts \(\gamma(\Psi_0^{\,C+1})=+\infty\). This proves that \(c_{0,k}\to +\infty\) as \(k\to\infty\).

Combining the above, we obtain a sequence of fully sign-changing solutions \(\{\mathbf{U}^{(k)}\}\subset H\) of 2 such that \[J_0\bigl(\mathbf{U}^{(k)}\bigr)=c_{0,k} ~\text{and}~ c_{0,k}\to+\infty \;\text{as }k\to\infty.\] Since \((c_{0,k})\) is nondecreasing and unbounded, we can extract a strictly increasing subsequence. Thus, relabeling if necessary, we may assume \[0<c_{0,1}<c_{0,2}<\cdots<c_{0,k}<\cdots.\] This completes the proof.

From now on, we fix \(k\in\mathbb{N}^*\). Let \(\{\varepsilon_n\}\subset(0,+\infty)\) be a sequence with \(\varepsilon_n\downarrow0\), \(\{\mathbf{u}^{(k)}_{\varepsilon_n}\bigr\}\) be the sequence of sign-changing solutions given by Theorem 1 with \(\Psi_{\varepsilon_n}(\mathbf{u}^{(k)}_{\varepsilon_n})=c_{\varepsilon_n,k}\). With the critical values \(\{c_{0,k}\}\) and the corresponding sign-changing critical points \(\{\mathbf{U}^{(k)}\}\) at hand, we now prove Theorem 2 by a concentration compactness analysis for the sequence \(\{\mathbf{u}^{(k)}_{\varepsilon_n}\}\). To simplify the notations, we omit superscript \((k)\).

.  Fix \(i\in\{1,\dots,m\}\). To apply the concentration compactness argument, we first establish the boundedness of \(\{\mathbf{u}_{\varepsilon_n}\}\) in \(H\). We claim that \[\label{eq54611} \lim_{n\to\infty} c_{\varepsilon_n,k}=c_{0,k}.\tag{35}\]

In fact, by the definition of \(c_{0,k}\), for any \(\delta>0\), there exists a compact and symmetric set \(A_\delta\subset\mathcal{A}\) such that \(\gamma(A_\delta)\ge k\) and \[\label{eq54612} \sup_{\mathbf{u}\in A_\delta}\Psi_0(\mathbf{u})\le c_{0,k}+\delta.\tag{36}\] Moreover, since \(\sup_{A_\delta}\Psi_0<\infty\), the coercivity identity 30 implies that \(A_\delta\) is bounded in \(H\). By a standard implicit function argument, the nodal projection \(m_\varepsilon\) depends continuously on \((\varepsilon,\mathbf{u})\). In particular, \[\label{eq54613} m_{\varepsilon_n}(\mathbf{u})\to m_0(\mathbf{u})~\text{in}~H, ~\forall\,\mathbf{u}\in A_\delta.\tag{37}\] Furthermore, for every \(R>0\) and \(i\in\{1,\dots,m\}\), \[Q_{\varepsilon_n}(x-y_i)=Q(\varepsilon_n x-y_i)\to Q(-y_i) ~\text{uniformly for}~x\in B_R \text{ as }n\to\infty.\] Combining 37 with the Sobolev embedding \(H\hookrightarrow L^{2p}(\mathbb{R}^N)^m\) and the uniform convergence of \(Q(\varepsilon_n x-y_i)\) on bounded sets, we deduce \[\label{eq54614} \sup_{\mathbf{u}\in A_\delta}\Bigl|\Psi_{\varepsilon_n}(\mathbf{u})-\Psi_0(\mathbf{u})\Bigr| =\sup_{\mathbf{u}\in A_\delta}\Bigl|J_{\varepsilon_n}\bigl(m_{\varepsilon_n}(\mathbf{u})\bigr) -J_0\bigl(m_0(\mathbf{u})\bigr)\Bigr|\to0~\text{as}~n\to\infty.\tag{38}\] By 36 –38 , we obtain for \(n\) large \[c_{\varepsilon_n,k}\le \sup_{\mathbf{u}\in A_\delta}\Psi_{\varepsilon_n}(\mathbf{u}) \le \sup_{\mathbf{u}\in A_\delta}\Psi_0(\mathbf{u})+\delta \le c_{0,k}+2\delta.\] Letting \(n\to\infty\) and then \(\delta\to 0^+\) gives \[\label{eq54615} \limsup_{n\to\infty}c_{\varepsilon_n,k}\le c_{0,k}.\tag{39}\]

On the other hand, for any \(n\), choose \(A_n\subset\mathcal{A}\) such that \[\sup_{\mathbf{u}\in A_n}\Psi_{\varepsilon_n}(\mathbf{u})\le c_{\varepsilon_n,k}+\frac{1}{n}.\] Then \(A_n\subset\{\mathbf{u}\in\mathcal{A}:\;\Psi_{\varepsilon_n}(\mathbf{u})\le c_{\varepsilon_n,k}+1\}\). By 39 , \(\{c_{\varepsilon_n,k}\}\) is bounded above, and hence there exists \(M_k>0\) such that, up to discarding finitely many indices, \[A_n\subset \{\mathbf{u}\in\mathcal{A}:\;\Psi_{\varepsilon_n}(\mathbf{u})\le M_k\}~\text{for all large}~n.\] Lemma 7(d) yields that \((A_n)\) is bounded in \(H\) uniformly for \(n\). Therefore, arguing exactly as in 38 , we obtain \[\sup_{\mathbf{u}\in A_n}\bigl|\Psi_{\varepsilon_n}(\mathbf{u})-\Psi_0(\mathbf{u})\bigr|\to 0 ~\text{as}~n\to\infty.\] Consequently, \[c_{0,k} =\inf_{A\in\Sigma_k}\sup_{\mathbf{u}\in A}\Psi_0(\mathbf{u}) \le\sup_{\mathbf{u}\in A_n}\Psi_0(\mathbf{u}) \le \sup_{\mathbf{u}\in A_n}\Psi_{\varepsilon_n}(\mathbf{u})+o_n(1) \le c_{\varepsilon_n,k}+\frac{1}{n}+o_n(1).\] Letting \(n\to\infty\) yields \[\label{eq54616} \liminf_{n\to\infty}c_{\varepsilon_n,k}\ge c_{0,k}.\tag{40}\] Combining 39 with 40 , we obtain 35 . Combined with Lemma 3, \(\{\mathbf{u}_{\varepsilon_n}\}\) is bounded in \(H\).

Assume by contradiction that the conclusion fails. Then for every \(R>0\), \[\sup_{y\in\mathbb{R}^N} \int_{B_R(y)} |u_{i,\varepsilon_n}|^{2p} dx \to 0~\text{as}~n\to\infty.\] By the vanishing lemma (see [11], [5]) we obtain \[u_{i,\varepsilon_n} \to 0~\text{in}~L^{2p}(\mathbb{R}^N)~\text{as}~n\to\infty.\]

Since \(\mathbf{u}_{\varepsilon_n}\) is a critical point of \(J_{\varepsilon_n}\), \(\langle J_{\varepsilon_n}^\prime(\mathbf{u}_{\varepsilon_n}),(0,\dots,u_{i,\varepsilon_n},\dots,0)\rangle=0\), i.e., \[\|u_{i,\varepsilon_n}\|_{H^1}^2 = \mu_i\int_{\mathbb{R}^N}Q_\varepsilon\,|u_{i,\varepsilon}|^{2p}\,dx + \sum_{j\neq i}\lambda_{ij}\int_{\mathbb{R}^N}|u_{i,\varepsilon}|^{p}|u_{j,\varepsilon}|^{p}\,dx .\] Note that the coupling term is nonpositive because of \(\lambda_{ij}<0\), and \(0\le Q_{\varepsilon_n}(x)\le\|Q\|_\infty\) a.e. in \(\mathbb{R}^N\), so we have \[\label{eq54617} \| u_{i,\varepsilon_n} \|_{H^1}^2 \le \mu_i \| Q\|_\infty \| u_{i,\varepsilon_n} \|_{L^{2p}}^{2p} \to 0~\text{as}~n\to\infty,\tag{41}\] which contradicts Lemma 5.

Therefore, there exist a subsequence \(\{\varepsilon_n\}\), constants \(R_i,\eta_i>0\) and points \(\{x_{i,\varepsilon_n}\}\) such that \[\int_{B_{R_i}(x_{i,\varepsilon_n})} |u_{i,\varepsilon_n}|^{2p} dx \ge \eta_i.\] Taking \(R=\max\limits_{1\le i\le m} R_i\), \(\eta=\min\limits_{1\le i\le m}\eta_i\) and passing to a further common subsequence yields the conclusion.

We finally show that the concentration centers converge to the origin.

.  By Lemma 11, there exist \(R>0\), \(\eta>0\) and \(\varepsilon_0>0\) such that for every \(\varepsilon\in(0,\varepsilon_0)\) one can find points \(x_{i,\varepsilon}\in\mathbb{R}^N\) satisfying \[\label{eq54618} \int_{B_R(x_{i,\varepsilon})}|u_{i,\varepsilon}|^{2p}\,dx\ge \eta, ~ i=1,\dots,m.\tag{42}\] Fix \(i\in\{1,\dots,m\}\) and let \(\varepsilon_n\downarrow0\). Set \(x_{i,n}:=x_{i,\varepsilon_n}\) and define \[\tilde{u}_{\ell,n}(x):=u_{\ell,\varepsilon_n}(x+x_{i,n}),~ \tilde{\mathbf{u}}_n:=(\tilde{u}_{1,n},\dots,\tilde{u}_{m,n}) .\] Then 42 yields \[\int_{B_R(0)}|\tilde{u}_{i,n}|^{2p}\,dx =\int_{B_R(x_{i,n})}|u_{i,\varepsilon_n}|^{2p}\,dx\ge \eta,\] jointly with \(\{\tilde{\mathbf{u}}_n\}\) is bounded in \(H\), up to a subsequence, \[\tilde{\mathbf{u}}_n\rightharpoonup \mathbf{U} \;\text{in }H, ~ \tilde{\mathbf{u}}_n\to \mathbf{U} \;\text{in }L^{2p}_{\rm loc}(\mathbb{R}^N)^m,\] for some \(\mathbf{U}=(U_1,\dots,U_m)\in H\) with \(\mathbf{U}\neq \mathbf{0}\).

We next verify that \(\mathbf{U} \in \mathcal{A}\), i.e., \(U_i^\pm \neq 0\) for each \(i = 1, \dots, m\).

Recall that \(\mathbf{u}_{\epsilon_n} \in \mathcal{N}_{\varepsilon_n}^{\mathrm{sc}} \subset \mathcal{A}\) for each \(n\). By Lemma 5, there exists \(\delta > 0\) independent of \(n\) such that \[\|u_{i,\varepsilon_n}^\pm\|_{H^1} \geq \delta, ~ i = 1, \dots, m,\] which implies \[\label{eq54619} \|\tilde{u}_{i,n}^\pm\|_{H^1} = \|u_{i,\varepsilon_n}^\pm\|_{H^1} \geq \delta.\tag{43}\]

Suppose to the contrary that there exist some \(j \in \{1, \dots, m\}\) and sign \(\sigma \in \{+, -\}\) such that \(U_j^\sigma = 0\) in \(H^1(\mathbb{R}^N)\), then \(\tilde{u}_{j,n}^\sigma \rightharpoonup 0\) in \(H^1(\mathbb{R}^N)\) and \(\tilde{u}_{j,n}^\sigma \to 0\) in \(L_{\mathrm{loc}}^{2p}(\mathbb{R}^N)\).

By 5 for \(\mathbf{u}_{\varepsilon_n}\) and make a change of variables we obtain \[\label{eq54620} \begin{align} \|\tilde{u}_{j,n}^\sigma\|_{H^1}^2 &= \mu_j \int_{\mathbb{R}^N} Q_{\varepsilon_n}(x + x_{i,n} - y_j) |\tilde{u}_{j,n}^\sigma|^{2p} \, dx + \sum_{k \neq j} \lambda_{jk} \int_{\mathbb{R}^N} |\tilde{u}_{k,n}|^p |\tilde{u}_{j,n}^\sigma|^p \, dx\\ &\le \mu_j \int_{\mathbb{R}^N} Q_{\varepsilon_n}(x + x_{i,n} - y_j) |\tilde{u}_{j,n}^\sigma|^{2p} \, dx, \end{align}\tag{44}\] since \(\lambda_{jk} < 0\).

For any \(M>0\), it follows from \(\tilde{u}_{j,n}^\sigma \to 0\) in \(L^{2p}(B_M(0))\) and \(|Q({\varepsilon_n}(x + x_{i,n} - y_j)| \leq \|Q\|_\infty\) that \[\label{eq54621} \int_{B_M(0)} Q_{\varepsilon_n}(x + x_{i,n} - y_j) |\tilde{u}_{j,n}^\sigma|^{2p} \, dx \to 0.\tag{45}\] On the other hand, by the standard Agmon-type estimates (see [15]), there exist \(C,\alpha>0\) independent of \(n\), such that \[|\tilde{u}_{j,n}(x)|\leq C e^{-\alpha|x|} \quad \text{for all } x\in\mathbb{R}^N.\] Consequently, \[\int_{|x|>M} Q_{\epsilon_n}(x+x_{i,n}-y_j)|\tilde{u}_{j,n}^\sigma|^{2p}dx \leq \|Q\|_\infty \int_{|x|>M} |\tilde{u}_{j,n}^\sigma|^{2p}dx \leq \|Q\|_\infty\, C^{2p} \int_{|x|>M} e^{-2p\alpha|x|}dx.\] For any \(\eta>0\), choose \(M=M(\eta)>0\) such that \[\int_{|x|>M} e^{-2p\alpha|x|}dx < \eta,\] jointly with 44 and 45 , implies that \(\|\tilde{u}_{j,n}^\sigma\|_{H^1}\to0\). This contradicts 43 . Therefore, \(U_j^\sigma \neq 0\) for every \(j\) and \(\sigma\), i.e., \(\mathbf{U} \in \mathcal{A}\).

We claim that the sequence \(\{\varepsilon_n x_{i,n}\}\) is bounded. Assume by contradiction that, up to a subsequence, \(|\varepsilon_n x_{i,n}|\to\infty\). Fix \(\varphi\in C_c^\infty(\mathbb{R}^N)\), and set \(K:=\operatorname{supp}\varphi\). Since \(\mathbf{u}_{\varepsilon_n}\) is a weak solution of the system, changing variables gives \[\begin{align} \int_{\mathbb{R}^N}\bigl(\nabla \tilde{u}_{\ell,n}\cdot\nabla\varphi+\tilde{u}_{\ell,n}\varphi\bigr)\,dx &= \mu_\ell\int_{\mathbb{R}^N} Q\bigl(\varepsilon_n(x+x_{i,n})-y_\ell\bigr)\, |\tilde{u}_{\ell,n}|^{2p-2}\tilde{u}_{\ell,n}\,\varphi\,dx \\ &~ +\sum_{j\neq \ell}\lambda_{\ell j}\int_{\mathbb{R}^N} |\tilde{u}_{j,n}|^{p}|\tilde{u}_{\ell,n}|^{p-2}\tilde{u}_{\ell,n}\,\varphi\,dx . \end{align}\] By \((A_3)\), \(\operatorname{supp}(Q)\subset B_{R_Q}(0)\) for some \(R_Q>0\). Since \(K\) is bounded and \(|\varepsilon_n x_{i,n}|\to\infty\), for all large \(n\) we have \[\bigl|\varepsilon_n(x+x_{i,n})-y_\ell\bigr| \ge |\varepsilon_n x_{i,n}|-|y_\ell|-\varepsilon_n|x|>R_Q ~\text{for all }x\in K,\] hence \(Q(\varepsilon_n(x+x_{i,n})-y_\ell)\equiv0\) on \(K\) for all large \(n\). Letting \(n\to\infty\) and using \(\tilde{\mathbf{u}}_n\to\mathbf{U}\) in \(L^{2p}_{\rm loc}(\mathbb{R}^N)^m\), we obtain \[\int_{\mathbb{R}^N}\bigl(\nabla U_\ell\cdot\nabla\varphi+U_\ell\varphi\bigr)\,dx = \sum_{j\neq \ell}\lambda_{\ell j}\int_{\mathbb{R}^N}|U_j|^{p}|U_\ell|^{p-2}U_\ell\,\varphi\,dx, ~\forall\,\varphi\in C_c^\infty(\mathbb{R}^N).\] Choose \(\chi_R\in C_c^\infty(\mathbb{R}^N)\) such that \(0\le \chi_R\le 1\), \(\chi_R\equiv 1\) on \(B_R(0)\), \(\operatorname{supp}(\chi_R)\subset B_{2R}(0)\) and \(|\nabla\chi_R|\le C/R\). Testing the above identity with \(\varphi=\chi_R^2U_\ell\) and using \(\lambda_{\ell j}<0\), we obtain \[\int_{\mathbb{R}^N}\Bigl(\nabla U_\ell\cdot\nabla(\chi_R^2U_\ell)+\chi_R^2|U_\ell|^2\Bigr)\,dx =\sum_{j\neq \ell}\lambda_{\ell j}\int_{\mathbb{R}^N}\chi_R^2|U_j|^p|U_\ell|^p\,dx\le 0.\] Moreover, by the product rule, \[\int_{\mathbb{R}^N}\nabla U_\ell\cdot\nabla(\chi_R^2U_\ell)\,dx =\int_{\mathbb{R}^N}\Bigl(|\nabla(\chi_RU_\ell)|^2-|U_\ell|^2|\nabla\chi_R|^2\Bigr)\,dx.\] Consequently, \[\int_{\mathbb{R}^N}\Bigl(|\nabla(\chi_RU_\ell)|^2+\chi_R^2|U_\ell|^2\Bigr)\,dx \le \int_{\mathbb{R}^N}|U_\ell|^2|\nabla\chi_R|^2\,dx.\] Since \(|\nabla\chi_R|\le C/R\) and \(\operatorname{supp}(\nabla\chi_R)\subset B_{2R}(0)\setminus B_R(0)\), we have \[\int_{\mathbb{R}^N}|U_\ell|^2|\nabla\chi_R|^2\,dx \le \frac{C}{R^2}\int_{B_{2R}(0)\setminus B_R(0)}|U_\ell|^2\,dx\to 0\] as \(R\to\infty\), which implies \[\int_{\mathbb{R}^N}\bigl(|\nabla U_\ell|^2+|U_\ell|^2\bigr)\,dx=0,\] i.e., \(U_\ell= 0\), which contradicts \(U_\ell\neq 0\). Consequently, passing to a subsequence, there exists \(\xi_i\in\mathbb{R}^N\) such that \[\label{eq54622} \varepsilon_n x_{i,n}\to \xi_i~\text{as}~n\to\infty.\tag{46}\]

Next, we show that \(\xi_i = 0\) by contradiction. Suppose that \(\xi_i \neq 0\).

We first claim that \[\label{eq54623} Q(\xi_i - y_\ell)<Q(-y_\ell)~\text{for some}~\ell\in\{1,\cdots,m\}.\tag{47}\] Suppose to the contrary that \(Q(\xi_i-y_\ell)=Q(-y_\ell)=\|Q\|_\infty\) for all \(\ell\). Then, by \((A_3)\), for each \(\ell\), there exists\(j_\ell\in{1,\dots,m}\) such that \(\xi_i-y_\ell=-y_{j_\ell}\). If \(j_\ell=\ell\) for some \(\ell\), then \(\xi_i=0\), contradicting \(\xi_i\neq0\). Hence \(j_\ell\neq\ell\) for all \(\ell\). Define a map \(\sigma:\{1,\dots,m\}\to\{1,\dots,m\}\) by \(\sigma(\ell)=j_\ell\). Since the points \(y_1,\dots,y_m\) are pairwise distinct, \(\sigma\) is a permutation of \(\{1,\dots,m\}\) without fixed points. Moreover, we have \[y_{\sigma(\ell)}=y_\ell-\xi_i,~\text{for each}~\ell.\] Iterating this relation gives \(y_{\sigma^k(\ell)}=y_\ell-k\xi_i\) for any positive integer \(k\). Because the set \(\{y_1,\dots,y_m\}\) is finite, there exist \(k_1\neq k_2\) such that \(y_{\sigma^{k_1}(\ell)}=y_{\sigma^{k_2}(\ell)}\), which implies \((k_1-k_2)\xi_i=0\) and hence \(\xi_i=0\), a contradiction. Therefore, 47 must hold.

Observe that \(\mathbf{U}\) is a weak solution of \[\label{eq54624} -\Delta U_\ell + U_\ell = \mu_\ell Q(\xi_i - y_\ell) |U_\ell|^{2p-2}U_\ell + \sum_{j\neq\ell}\lambda_{\ell j}|U_j|^p |U_\ell|^{p-2}U_\ell, ~ \ell=1,\dots,m .\tag{48}\] Denote by \(J_\xi\) the energy functional associated with 48 and recall \(J_0\) be the original limiting functional with coefficients \(Q(-y_\ell)\) \[-\Delta U_\ell + U_\ell = \mu_\ell Q(- y_\ell) |U_\ell|^{2p-2}U_\ell + \sum_{j\neq\ell}\lambda_{\ell j}|U_j|^p |U_\ell|^{p-2}U_\ell, ~ \ell=1,\dots,m .\]

From Lemma 11 we have \(c_{\varepsilon_n,k} \to c_{0,k}\). The concentration-compactness principle together with the Brézis–Lieb lemma yield \[\label{eq54625} c_{0,k} = \lim_{n\to\infty} J_{\varepsilon_n}(\mathbf{u}_{\varepsilon_n}) \ge J_\xi(\mathbf{U}).\tag{49}\]

By 47 and \(U_\ell \neq 0\), we obtain \[\label{eq54626} \begin{align} J_\xi(\mathbf{U}) - J_0(\mathbf{U}) &= \frac{1}{2p}\sum_{\ell=1}^m \mu_\ell\bigl[Q(-y_\ell) - Q(\xi_i - y_\ell)\bigr] \int_{\mathbb{R}^N} |U_\ell|^{2p}\\ &\ge \frac{1}{2p}\mu_\ell\bigl[Q(-y_\ell) - Q(\xi_i-y_\ell)\bigr] \int |U_\ell|^{2p} > 0. \end{align}\tag{50}\]

Let \(t_0>0\) be the unique number such that \(t_0\mathbf{U} \in \mathcal{N}_0^{\mathrm{sc}}\). Then \[J_0(t_0\mathbf{U}) = \max_{t>0} J_0(t\mathbf{U}) < J_\xi(\mathbf{U}).\]

For \(k=1\), the critical value \(c_{0,1}\) coincides with the minimum of \(J_0\) on \(\mathcal{N}_0^{\mathrm{sc}}\). Indeed, by Theorem 1, \(c_{0,1}\) is the first critical value of \(\Psi_0\), and since \(\Psi_0(\mathbf{u}) = J_0(m_0(\mathbf{u}))\) and \(m_0\) is a bijection between \(\mathcal{A}\) and \(\mathcal{N}_0^{\mathrm{sc}}\), we have \(c_{0,1} = \inf_{\mathcal{N}_0^{\mathrm{sc}}} J_0\). Consequently, \[c_{0,1} \le J_0(t_0\mathbf{U}).\] Combining this with 49 and 50 we obtain \[c_{0,1} \le J_0(t_0\mathbf{U}) < J_\xi(\mathbf{U}) \le c_{0,1},\] a contradiction. Hence \(\xi_i = 0\) for \(k=1\).

Assume now that \(k\ge2\) and still \(\xi_i\neq0\). We claim that for every \(\delta>0\) there exists a symmetric compact set \(A_k\subset\mathcal{A}\) with \(\gamma(A_k)\ge k\) such that \[\sup_{\mathbf{w}\in A_k}\Psi_0(\mathbf{w})\le J_0(t_0\mathbf{U})+\delta.\] Once this is proved, the definition of \(c_{0,k}\) yields \(c_{0,k}\le \sup_{A_k}\Psi_0\le J_0(t_0\mathbf{U})+\delta\), and letting \(\delta\to0\) we get \(c_{0,k}\le J_0(t_0\mathbf{U})\), which contradicts \(J_0(t_0\mathbf{U})<J_\xi(\mathbf{U})\le c_{0,k}\).

We begin by constructing a compactly supported, fully sign-changing test function whose reduced energy is arbitrarily close to \(J_0(t_0\mathbf{U})\). Fix a cut-off function \(\chi\in C_c^\infty(\mathbb{R}^N)\) such that \(0\le\chi\le1\), \(\chi\equiv1\) on \(B_1(0)\) and \(\chi\equiv0\) on \(\mathbb{R}^N\setminus B_2(0)\), and set \(\chi_R(x):=\chi(x/R)\) for \(R>0\). Since \(t_0\mathbf{U}\in\mathcal{N}^{\mathrm{sc}}_0\subset\mathcal{A}\) is fully sign-changing, for each \(i\) one has \(U_i^\pm\not\equiv0\), hence there exists \(R_0>0\) such that \[\|U_i^\pm\|_{H^1(B_{R_0}(0))}>0~\text{for all }i=1,\dots,m.\] Choosing \(R\ge R_0\) and defining \[\widetilde{\boldsymbol{\psi}}_R:=\chi_R\, t_0\mathbf{U}\in H,\] we have \((\widetilde{\psi}_{R,i})^\pm\not\equiv0\) for all \(i\) (because \(\chi_R\equiv1\) on \(B_{R_0}(0)\)), hence \(\widetilde{\boldsymbol{\psi}}_R\in\mathcal{A}\), and moreover \(\widetilde{\boldsymbol{\psi}}_R\to t_0\mathbf{U}\) in \(H\) as \(R\to\infty\). By Lemma 7(a) and Lemma 6, \(\Psi_0=J_0\circ m_0\) is continuous on \(\mathcal{A}\), so \[\Psi_0(\widetilde{\boldsymbol{\psi}}_R)\longrightarrow \Psi_0(t_0\mathbf{U})=J_0(t_0\mathbf{U}) ~\text{as }R\to\infty.\] Therefore, for the given \(\delta>0\) we can fix \(R\ge R_0\) such that \[\label{eq54627} \Psi_0(\widetilde{\boldsymbol{\psi}}_R)\le J_0(t_0\mathbf{U})+\frac{\delta}{4}.\tag{51}\] In particular, \(\widetilde{\boldsymbol{\psi}}_R\) has compact support contained in \(B_{2R}(0)\).

Next, we place \(k-1\) far-away translates of \(\widetilde{\boldsymbol{\psi}}_R\) so that all supports are mutually disjoint and lie outside a large ball on which \(\mathbf{U}\) has small tail. Fix \(\eta>0\) (to be chosen depending on \(\delta\)). Since \(\mathbf{U}\in H\) and \(\mathbf{U}\in L^{2p}(\mathbb{R}^N)^m\), there exists \(R_1>2R\) such that \[\label{eq54628} \sum_{\ell=1}^m\int_{\mathbb{R}^N\setminus B_{R_1}(0)} \Bigl(|\nabla U_\ell|^2+|U_\ell|^2+|U_\ell|^{2p}\Bigr)\,dx<\eta.\tag{52}\] Choose points \(z_1,\dots,z_{k-1}\in\mathbb{R}^N\) satisfying \[|z_j|\ge R_1+2R,~ |z_j-z_{j'}|>4R~ (j\neq j').\] Define \[\boldsymbol{\psi}_j(x):=\widetilde{\boldsymbol{\psi}}_R(x-z_j),~ j=1,\dots,k-1.\] Then \(\operatorname{supp}(\boldsymbol{\psi}_j)\subset \mathbb{R}^N\setminus B_{R_1}(0)\) for each \(j\), and the supports \(\operatorname{supp}(\boldsymbol{\psi}_1),\dots,\operatorname{supp}(\boldsymbol{\psi}_{k-1})\) are pairwise disjoint; moreover, each \(\boldsymbol{\psi}_j\in\mathcal{A}\) and \(\Psi_0(\boldsymbol{\psi}_j)=\Psi_0(\widetilde{\boldsymbol{\psi}}_R)\) by translation invariance of \(J_0\) (hence of \(\Psi_0\)).

We now set \[E_k:=\mathrm{span}\bigl\{t_0\mathbf{U},\boldsymbol{\psi}_1,\dots,\boldsymbol{\psi}_{k-1}\bigr\}, ~ A_k:=\{\mathbf{w}\in E_k:\;\|\mathbf{w}\|=1\}.\] Then \(A_k\) is compact and symmetric, and since \(E_k\) is \(k\)-dimensional we have \(\gamma(A_k)=k\). We also have \(A_k\subset\mathcal{A}\): indeed, if \(\mathbf{w}=a\,t_0\mathbf{U}+\sum_{j=1}^{k-1}b_j\boldsymbol{\psi}_j\in A_k\), then on \(B_{R_0}(0)\) all \(\boldsymbol{\psi}_j\) vanish (because \(|z_j|\ge R_1+2R\ge R_0+2R\)), hence \(\mathbf{w}=a\,t_0\mathbf{U}\) on \(B_{R_0}(0)\). If \(a\neq0\) then \((w_i)^\pm\not\equiv0\) for each \(i\) since \(U_i^\pm\not\equiv0\) on \(B_{R_0}(0)\); if \(a=0\), then \(\mathbf{w}=\sum b_j\boldsymbol{\psi}_j\neq0\), and because the supports of the \(\boldsymbol{\psi}_j\) are disjoint and each \(\boldsymbol{\psi}_j\) is fully sign-changing, it follows that \((w_i)^\pm\not\equiv0\) for every \(i\) as well. Thus \(\mathbf{w}\in\mathcal{A}\) for all \(\mathbf{w}\in A_k\).

It remains to estimate \(\Psi_0\) on \(A_k\). Write \(\mathbf{w}=a\,t_0\mathbf{U}+\sum_{j=1}^{k-1}b_j\boldsymbol{\psi}_j\in A_k\). Because the supports of \(\boldsymbol{\psi}_j\) are pairwise disjoint and contained in \(\mathbb{R}^N\setminus B_{R_1}(0)\), all interaction integrals among different \(\boldsymbol{\psi}_j\) vanish. Moreover, by 52 and \(\operatorname{supp}(\boldsymbol{\psi}_j)\subset\mathbb{R}^N\setminus B_{R_1}(0)\), all mixed integrals involving \(\mathbf{U}\) and \(\boldsymbol{\psi}_j\) (quadratic terms and \(L^{2p}\)-terms) can be made arbitrarily small by choosing \(\eta\) sufficiently small. Since \(A_k\) is compact in the finite-dimensional space \(E_k\), the coefficients \((a,b_1,\dots,b_{k-1})\) stay in a bounded set, hence the above smallness is uniform on \(A_k\). Using the continuity of \(m_0\) on \(\mathcal{A}\) (Lemma 6) and the continuity of \(J_0\) on \(H\), we can thus fix \(\eta=\eta(\delta)>0\) and then choose \(R_1\) and the points \(z_j\) as above so that \[\Psi_0(\mathbf{w})=J_0\bigl(m_0(\mathbf{w})\bigr) \le \max\Bigl\{J_0(t_0\mathbf{U}),\,\Psi_0(\widetilde{\boldsymbol{\psi}}_R)\Bigr\}+\frac{\delta}{4} ~\text{for all }\mathbf{w}\in A_k.\] Combining this with 51 yields \[\sup_{\mathbf{w}\in A_k}\Psi_0(\mathbf{w})\le J_0(t_0\mathbf{U})+\delta,\] which proves the claim and hence gives the desired contradiction.

Therefore \(\xi_i=0\) also for \(k\ge2\). In all cases we conclude that \(\xi_i=0\), i.e. \(\varepsilon_n x_{i,n}\to0\) as \(n\to\infty\).

Proof of Theorem 2. Fix \(k\in\mathbb{N}^*\) and take a sequence \(\{\varepsilon_n\}\subset(0,+\infty)\) with \(\varepsilon_n\to 0\). Let \[\mathbf{u}_n:=\mathbf{u}_{\varepsilon_n}^{(k)}=(u_{1,n},\dots,u_{m,n})\in H\] be the \(k\)-th sign-changing solution given by Theorem 1, so that \[J_{\varepsilon_n}(\mathbf{u}_n)=c_{\varepsilon_n,k}.\] For simplicity, we omit the superscript \((k)\) in Steps 1–2 and keep the notation \(u_{i,n}:=u_{i,\varepsilon_n}\) for the components of \(\mathbf{u}_n\).

Proof of (i). Since \(\mathbf{u}_n\) is a critical point of \(J_{\varepsilon_n}\) at level \(c_{\varepsilon_n,k}\) and \(\mathbf{u}_n\in\mathcal{N}^{\mathrm{sc}}_{\varepsilon_n}\), the Nehari identities on \(\mathcal{N}^{\mathrm{sc}}_{\varepsilon_n}\) and the Sobolev inequality imply that \(\{\mathbf{u}_n\}\) is bounded in \(H\).

We now fix a concentration sequence. By Lemma 11 (applied, for instance, to the first component), there exist \(R>0\), \(\eta>0\) and a sequence \(\{x_n\}\subset\mathbb{R}^N\) such that \[\int_{B_R(x_n)} |u_{1,n}|^{2p}\,dx \ge \eta ~ \text{for all }n .\] Moreover, by Lemma 12, we have \[\label{eq54629} \varepsilon_n x_n \to 0 ~\text{as }n\to\infty .\tag{53}\] To match the notation in Theorem 2, we set \[x^{(k)}_{\varepsilon_n,1}=\cdots=x^{(k)}_{\varepsilon_n,m}:=x_n ~ \text{for all }n.\] Then (i) follows from 53 .

Proof of (ii). Define, for \(i=1,\dots,m\), \[\tilde{u}_{i,n}(x):=u_{i,n}(x+x_n),~ \tilde{\mathbf{u}}_n:=(\tilde{u}_{1,n},\dots,\tilde{u}_{m,n}).\] Since translations preserve the \(H\)–norm, \(\{\tilde{\mathbf{u}}_n\}\) is bounded in \(H\). Hence, up to a subsequence, there exists \(\mathbf{U}=(U_1,\dots,U_m)\in H\) such that \[\tilde{\mathbf{u}}_n \rightharpoonup \mathbf{U} ~\text{in }H, ~ \tilde{\mathbf{u}}_n \to \mathbf{U} ~\text{in }L^{2p}_{\mathrm{loc}}(\mathbb{R}^N)^m.\] In particular, by the choice of \(\{x_n\}\) we have \(\int_{B_R(0)}|\tilde{u}_{1,n}|^{2p}\,dx\ge\eta\) for all \(n\), so \(U_1\not\equiv 0\) and \(\mathbf{U}\not\equiv \mathbf{0}\).

We now identify the equation solved by \(\mathbf{U}\). Fix \(\ell\in\{1,\dots,m\}\) and \(\varphi\in C_c^\infty(\mathbb{R}^N)\). Testing the \(\ell\)-th Euler–Lagrange equation for \(\mathbf{u}_n\) with \(\varphi(\cdot-x_n)\) and changing variables, we obtain \[\begin{align} \int_{\mathbb{R}^N}\bigl(\nabla \tilde{u}_{\ell,n}\cdot\nabla \varphi+\tilde{u}_{\ell,n}\varphi\bigr)\,dx &= \mu_\ell\int_{\mathbb{R}^N} Q\bigl(\varepsilon_n(x+x_n)-y_\ell\bigr) |\tilde{u}_{\ell,n}|^{2p-2}\tilde{u}_{\ell,n}\varphi\,dx \\ &~ + \sum_{j\neq \ell}\lambda_{\ell j}\int_{\mathbb{R}^N} |\tilde{u}_{j,n}|^p\,|\tilde{u}_{\ell,n}|^{p-2}\tilde{u}_{\ell,n}\varphi\,dx . \end{align}\] Choose \(R_\varphi>0\) such that \(\operatorname{supp}(\varphi)\subset B_{R_\varphi}(0)\). Since \(|\varepsilon_n x|\le \varepsilon_n R_\varphi\to0\) for all \(x\in B_{R_\varphi}(0)\) and \(\varepsilon_n x_n\to0\) by 53 , we have \[\varepsilon_n(x+x_n)-y_\ell \to -y_\ell ~\text{uniformly for }x\in B_{R_\varphi}(0),\] and hence, by continuity of \(Q\), \[Q\bigl(\varepsilon_n(x+x_n)-y_\ell\bigr)\to Q(-y_\ell)~\text{uniformly on }B_{R_\varphi}(0).\] Together with \(\tilde{u}_{i,n}\to U_i\) in \(L^{2p}(B_{R_\varphi}(0))\) for all \(i\), we may pass to the limit and conclude that \(\mathbf{U}\) is a weak solution of the limiting system 2 . In particular, \(\mathbf{U}\) is a critical point of \(J_0\).

We next show that the convergence is in fact strong in \(H^1(\mathbb{R}^N)\), namely \[\label{eq54630} u_{i,n}(\cdot+x_n)\to U_i ~\text{in}~H^1(\mathbb{R}^N)\;\text{as }n\to\infty,~ i=1,\dots,m.\tag{54}\] Indeed, by 35 we have \(c_{\varepsilon_n,k}\to c_{0,k}\) as \(n\to\infty\). Moreover, applying the concentration–compactness principle to the bounded sequence \(\{\tilde{\mathbf{u}}_n\}\) and using the Brézis–Lieb lemma for the quadratic term, the \(2p\)–powers and the coupling \(p\)–powers, we obtain the energy decomposition along the extracted profile: \[\lim_{n\to\infty}J_{\varepsilon_n}(\mathbf{u}_n) =\lim_{n\to\infty}J_{0}(\tilde{\mathbf{u}}_n) =J_0(\mathbf{U})+\lim_{n\to\infty}J_0(\tilde{\mathbf{u}}_n-\mathbf{U}),\] and the corresponding splitting for the Nehari-type identities. Since \(\mathbf{U}\) is a critical point of \(J_0\), the above identities force \(\lim_{n\to\infty}J_0(\tilde{\mathbf{u}}_n-\mathbf{U})=0\), and hence \(\tilde{\mathbf{u}}_n\to \mathbf{U}\) in \(H\). This yields 54 .

As a consequence, the truncation maps \(u\mapsto u^\pm\) being continuous on \(H^1(\mathbb{R}^N)\) give \(u_{i,n}^\pm(\cdot+x_n)\to U_i^\pm\) in \(H^1(\mathbb{R}^N)\) for every \(i\). By Lemma 5, we have \(u_{i,n}^\pm\not\equiv0\) for all \(i\) and all \(n\), hence \(U_i^\pm\not\equiv0\) for all \(i\). Therefore, \(\mathbf{U}\) is fully sign-changing, and (ii) follows.

Finally, combining \(c_{\varepsilon_n,k}\to c_{0,k}\) with the above strong convergence and the Brézis–Lieb lemma, we obtain \[c_{0,k}=\lim_{n\to\infty}J_{\varepsilon_n}(\mathbf{u}_n)=J_0(\mathbf{U}).\]

Proof of (iii). Fix \(i\in\{1,\dots,m\}\) and keep \(x_n\) as in Step 1. By 53 and 54 , we have \[\varepsilon_n x_n\to 0 ~\text{and}~ u_{i,n}(\cdot+x_n)\to U_i~\text{in}~H^1(\mathbb{R}^N).\] Define \[v_n(x):=u_{i,n}(x+x_n).\] Then \(v_n\to U_i\) in \(H^1(\mathbb{R}^N)\) in particular, \[\label{eq54631} v_n\to U_i ~\text{strongly in }L^2(\mathbb{R}^N)\;\text{and in }L^{2p}(\mathbb{R}^N).\tag{55}\]

Fix \(q\in[1,\infty)\) and \(R>0\), and set \[\Omega_{n,R}:=\mathbb{R}^N\setminus B_{R/\varepsilon_n}(0), ~ d_n:=|x_n|.\] Since \(\varepsilon_n d_n=|\varepsilon_n x_n|\to0\), we have \(d_n=o(1/\varepsilon_n)\). For any \(x\in\Omega_{n,R}\) one has \(|x|\ge R/\varepsilon_n\), hence \[|x-x_n|\ge |x|-|x_n|\ge \frac{R}{\varepsilon_n}-d_n,\] and therefore \[\Omega_{n,R}-x_n \subset \mathbb{R}^N\setminus B_{R/\varepsilon_n-d_n}(0).\] Since \(R/\varepsilon_n-d_n\to+\infty\), for every \(R_0>0\) there exists \(n_0\) such that \(R/\varepsilon_n-d_n\ge R_0\) for all \(n\ge n_0\), and consequently \[\label{eq54632} \Omega_{n,R}-x_n\subset \mathbb{R}^N\setminus B_{R_0}(0)~ \text{for all }n\ge n_0.\tag{56}\]

We claim that \(\int_{\Omega_{n,R}}|u_{i,n}|^q\,dx\to0\) for every fixed \(R>0\).

If \(q\in[2,2p]\), choose \(\theta\in[0,1]\) such that \(\frac{1}{q}=\frac{\theta}{2}+\frac{1-\theta}{2p}\). Then, by interpolation and 56 , for all \(n\ge n_0\), \[\|u_{i,n}\|_{L^q(\Omega_{n,R})} =\|v_n\|_{L^q(\Omega_{n,R}-x_n)} \le \|v_n\|_{L^2(\mathbb{R}^N\setminus B_{R_0})}^{\theta} \|v_n\|_{L^{2p}(\mathbb{R}^N\setminus B_{R_0})}^{1-\theta}.\] Taking \(\limsup_{n\to\infty}\) and using 55 , we obtain \[\limsup_{n\to\infty}\|u_{i,n}\|_{L^q(\Omega_{n,R})} \le \|U_i\|_{L^2(\mathbb{R}^N\setminus B_{R_0})}^{\theta} \|U_i\|_{L^{2p}(\mathbb{R}^N\setminus B_{R_0})}^{1-\theta}.\] Letting \(R_0\to\infty\) yields \[\limsup_{n\to\infty}\int_{\Omega_{n,R}}|u_{i,n}(x)|^q\,dx=0 ~\text{for every fixed }R>0\text{ and every }q\in[2,2p].\]

If \(q>2p\), since \(\{\mathbf{u}_n\}\) is bounded in \(H\) and the system is subcritical, standard elliptic \(L^\infty\) estimates yield that there exists \(C>0\) independent of \(n\) such that \(\|u_{i,n}\|_{L^\infty(\mathbb{R}^N)}\le C\) for all \(n\). Hence, \[\int_{\Omega_{n,R}}|u_{i,n}|^q\,dx \le \|u_{i,n}\|_{L^\infty(\mathbb{R}^N)}^{\,q-2p}\int_{\Omega_{n,R}}|u_{i,n}|^{2p}\,dx \le C^{q-2p}\int_{\Omega_{n,R}}|u_{i,n}|^{2p}\,dx\to0.\] If \(q\in[1,2]\), fix \(\sigma\in(0,1)\) and set \(v_n(x):=u_{i,n}(x+x_n)\). Since \(\{u_n\}\) is bounded in \(H\) and the right-hand side has subcritical growth, standard Agmon-type weighted estimates for \(-\Delta w+w=g\) applied componentwise yield a constant \(C_\sigma>0\), independent of \(n\), such that \[\label{eq54633} \int_{\mathbb{R}^N}e^{2\sigma|x|}\bigl(|\nabla v_n|^2+|v_n|^2\bigr)\,dx\le C_\sigma .\tag{57}\] Let \(R_0>0\) and \(E_{R_0}:=\mathbb{R}^N\setminus B_{R_0}(0)\). For \(q\in[1,2)\), writing \(|v_n|^q=(e^{\sigma|x|}|v_n|)^q e^{-\sigma q|x|}\) and applying Hölder’s inequality, we obtain \[\int_{E_{R_0}}|v_n|^q\,dx \le \Bigl(\int_{\mathbb{R}^N}e^{2\sigma|x|}|v_n|^2\,dx\Bigr)^{q/2} \Bigl(\int_{E_{R_0}}e^{-\frac{2\sigma q}{2-q}|x|}\,dx\Bigr)^{(2-q)/2}.\] By 57 the first factor is uniformly bounded in \(n\), while the second factor tends to \(0\) as \(R_0\to\infty\) since the exponential weight is integrable on \(\mathbb{R}^N\). Hence, \[\lim_{R_0\to\infty}\;\limsup_{n\to\infty}\int_{\mathbb{R}^N\setminus B_{R_0}(0)}|v_n(x)|^q\,dx=0, \qquad q\in[1,2).\] For \(q=2\), we simply write \[\int_{\mathbb{R}^N\setminus B_{R_0}(0)}|v_n(x)|^2\,dx \le e^{-2\sigma R_0}\int_{\mathbb{R}^N}e^{2\sigma|x|}|v_n(x)|^2\,dx \le e^{-2\sigma R_0} C_\sigma,\] and hence the same conclusion holds for \(q=2\). Consequently, \[\lim_{R_0\to+\infty}\;\limsup_{n\to\infty}\int_{\mathbb{R}^N\setminus B_{R_0}(0)}|v_n(x)|^q\,dx=0 ~\text{for every }q\in[1,2].\]

Finally, by 56 , \(\Omega_{n,R}-x_n\subset\mathbb{R}^N\setminus B_{R/\varepsilon_n-d_n}(0)\) with \(R/\varepsilon_n-d_n\to+\infty\), we deduce \[\limsup_{n\to\infty}\int_{\Omega_{n,R}}|u_{i,n}(x)|^q\,dx = \limsup_{n\to\infty}\int_{\Omega_{n,R}-x_n}|v_n(x)|^q\,dx \le \limsup_{n\to\infty}\int_{\mathbb{R}^N\setminus B_{R/\varepsilon_n-d_n}(0)}|v_n(x)|^q\,dx\to0,\] for every fixed \(R>0\) and every \(q\in[1,2]\). Together with the already proved cases \(q\in[2,2p]\) and \(q>2p\), this yields (iii) for all \(q\in[1,\infty)\).

Altogether, for every \(q\in[1,\infty)\) and every fixed \(R>0\), \[\limsup_{n\to\infty}\int_{\mathbb{R}^N\setminus B_{R/\varepsilon_n}(0)}|u_{i,n}(x)|^q\,dx=0,\] and therefore \[\lim_{R\to\infty}\;\limsup_{n\to\infty} \int_{\mathbb{R}^N\setminus B_{R/\varepsilon_n}(0)}|u_{i,n}(x)|^q\,dx=0, ~\forall\,q\in[1,\infty).\]

It remains to prove the case \(q=\infty\). Assume by contradiction that there exist \(\delta_0>0\), \(R_n\to\infty\) and \(z_n\in\mathbb{R}^N\setminus B_{R_n/\varepsilon_n}(0)\) such that \[|u_{i,n}(z_n)|\ge \delta_0 .\] Define \[\hat{\mathbf{u}}_n(x):=\mathbf{u}_n(x+z_n)=(\hat{u}_{1,n}(x),\dots,\hat{u}_{m,n}(x)).\] Then \(\{\hat{\mathbf{u}}_n\}\) is bounded in \(H\) and \(|\hat{u}_{i,n}(0)|\ge\delta_0\) for all \(n\).

Fix \(R>0\) and consider the system satisfied by \(\hat{\mathbf{u}}_n\) on \(B_R(0)\) \[-\Delta \hat{u}_{\ell,n}+\hat{u}_{\ell,n} =\mu_\ell\,Q(\varepsilon_n(x+z_n)-y_\ell)\,|\hat{u}_{\ell,n}|^{2p-2}\hat{u}_{\ell,n} +\sum_{j\neq \ell}\lambda_{\ell j}|\hat{u}_{j,n}|^{p}|\hat{u}_{\ell,n}|^{p-2}\hat{u}_{\ell,n},~\ell=1,\dots,m.\] Since \(Q\in L^\infty(\mathbb{R}^N)\) and \(\{\hat{\mathbf{u}}_n\}\) is bounded in \(H\), we have \(\{\hat{u}_{\ell,n}\}\) bounded in \(L^{2^*}(B_R)\) for every \(\ell\). Hence the right-hand side is bounded in \(L^{t}(B_R)\) for some \(t>1\) independent of \(n\), and therefore, by standard local elliptic \(W^{2,t}\)-estimates ([16]) we obtain that \(\{\hat{u}_{\ell,n}\}\) is bounded in \(W^{2,t}(B_{R/2})\) for each \(\ell\). Choosing \(t>N\), the Sobolev–Morrey embedding ([16]) yields that, up to a subsequence, \[\hat{u}_{\ell,n}\to \hat{W}_\ell ~\text{in }C^{1,\alpha}(B_{R/2})\;\text{for some }\alpha\in(0,1), ~ \ell=1,\dots,m,\] and in particular \[\hat{u}_{i,n}(0)\to \hat{W}_i(0)~\text{as }n\to\infty.\] Since \(|\hat{u}_{i,n}(0)|\ge\delta_0\) for all \(n\), it follows that \(\hat{W}_i(0)\neq0\).

Moreover, by boundedness in \(H\), up to a subsequence, we may assume that \[\hat{\mathbf{u}}_n\rightharpoonup \hat{\mathbf{W}}\;\text{in }H, ~ \hat{\mathbf{u}}_n\to \hat{\mathbf{W}}\;\text{in }L^r_{\mathrm{loc}}(\mathbb{R}^N)^m \;\text{for every }r\in[2,2^*),\] where \(\hat{\mathbf{W}}=(\hat{W}_1,\dots,\hat{W}_m)\).

Since \(|z_n|\ge R_n/\varepsilon_n\), we have \(|\varepsilon_n z_n|\ge R_n\to\infty\). Let \(K\subset\mathbb{R}^N\) be bounded. Then \(\varepsilon_n x\to0\) uniformly on \(K\), and thus for each \(\ell\), \[|\varepsilon_n(x+z_n)-y_\ell| \ge |\varepsilon_n z_n|-|y_\ell|-\varepsilon_n|x| \to\infty ~\text{uniformly for }x\in K.\] By \((A_3)\), there exists \(R_Q>0\) such that \(\operatorname{supp}(Q)\subset B_{R_Q}(0)\). Hence, for all large \(n\) and all \(x\in K\), \(|\varepsilon_n(x+z_n)-y_\ell|>R_Q\), which implies \[Q(\varepsilon_n(x+z_n)-y_\ell)=0~\text{on }K \;\text{for all large }n \text{ and all }\ell.\] Passing to the limit in the weak formulation, using the local strong convergence above, we infer that \(\hat{\mathbf{W}}\) solves \[-\Delta \hat{W}_\ell+\hat{W}_\ell=\sum_{j\neq \ell}\lambda_{\ell j}|\hat{W}_j|^p|\hat{W}_\ell|^{p-2}\hat{W}_\ell, ~ \ell=1,\dots,m.\]

Testing each equation with \(\hat{W}_\ell\) and summing over \(\ell\) yields \[\sum_{\ell=1}^m\int_{\mathbb{R}^N}\bigl(|\nabla \hat{W}_\ell|^2+|\hat{W}_\ell|^2\bigr)\,dx =\sum_{\ell=1}^m\sum_{j\neq\ell}\lambda_{\ell j}\int_{\mathbb{R}^N}|\hat{W}_\ell|^p|\hat{W}_j|^p\,dx \le 0,\] because \(\lambda_{\ell j}<0\) by \((A_2)\). Hence \(\hat{\mathbf{W}}=\mathbf{0}\), which contradicts \(\hat{W}_i(0)\neq0\). Therefore, \[\lim_{R\to\infty}\;\limsup_{n\to\infty} \|u_{i,n}\|_{L^\infty(\mathbb{R}^N\setminus B_{R/\varepsilon_n}(0))}=0,\] which implies (iii) holds.

Proof of (iv). In Steps 1–2 we fixed \(k\) and omitted the superscript \((k)\). In this step \(k\) varies, so we restore the superscript to avoid ambiguity. Since \(\{c_{0,k}\}_{k\ge1}\) is nondecreasing and unbounded, we can choose a strictly increasing subsequence \(\{c_{0,k_j}\}_{j\ge1}\) such that \[0<c_{0,k_1}<c_{0,k_2}<\cdots<c_{0,k_j}<\cdots\to+\infty.\] For simplicity of notation, we relabel this subsequence and still denote it by \(\{c_{0,k}\}_{k\ge1}\).

Fix \(k\ge1\). Choose an arbitrary sequence \(\{\varepsilon_n^{(k)}\}\subset(0,+\infty)\) with \(\varepsilon_n^{(k)}\downarrow0\), and let \(\mathbf{u}_{\varepsilon_n^{(k)}}^{(k)}\) be the \(k\)-th sign-changing solution given by Theorem 1, namely \[J_{\varepsilon_n^{(k)}}\bigl(\mathbf{u}_{\varepsilon_n^{(k)}}^{(k)}\bigr)=c_{\varepsilon_n^{(k)},k}.\] Applying Steps 1–2 to the sequence \(\{\mathbf{u}_{\varepsilon_n^{(k)}}^{(k)}\}_n\), we obtain a limiting profile \(\mathbf{U}^{(k)}\in H\) such that \(\mathbf{U}^{(k)}\) is a fully sign-changing weak solution of 2 and \[J_0(\mathbf{U}^{(k)}) =\lim_{n\to\infty}J_{\varepsilon_n^{(k)}}\bigl(\mathbf{u}_{\varepsilon_n^{(k)}}^{(k)}\bigr) =\lim_{n\to\infty}c_{\varepsilon_n^{(k)},k} =c_{0,k},\] where the last equality follows from 35 . In particular, if \(k\neq \ell\), then \[J_0(\mathbf{U}^{(k)})=c_{0,k}\neq c_{0,\ell}=J_0(\mathbf{U}^{(\ell)}),\] and hence \(\mathbf{U}^{(k)}\neq \mathbf{U}^{(\ell)}\). Consequently, the family \(\{\mathbf{U}^{(k)}\}_{k\ge1}\) is pairwise distinct and satisfies \[J_0(\mathbf{U}^{(1)})<J_0(\mathbf{U}^{(2)})<\cdots\to+\infty.\] This proves (iv).

Conflict Of Interest Statement. The authors declare that there are no conflict of interests, we do not have any possible conflicts of interest.

Data Availability Statement. Our manuscript has non associated data.

References↩︎

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1. Corresponding authors. E-mail addresses: fukunzhao@163.com (F. Zhao).↩︎

2. This work is supported by NSFC 12261107, China.↩︎