1 Introduction↩︎

This paper deals with the following superlinear Brezis-Nirenberg problem driven by the mixed local-nonlocal operator: \[_{2^*}}\label{main95PDE} \begin{cases} -\Delta u+(-\Delta)^su=\lambda |u|^{p-2}u+|u|^{2^\ast-2}u \; &\text{in }\Omega,\\ u=0&\text{in }\mathbb{R}^N\setminus \Omega, \end{cases}\tag{1}\] where \(\Omega\subset \mathbb{R}^N\) is a bounded open set with \(\mathcal{C}^{1,\alpha}\) boundary, \(N \ge 3\), \(s\in(0,1)\), \(\lambda> 0\) is a parameter, \(2\leq p<2^*\) where \(2^* = \frac{2N}{N-2}\) is the critical Sobolev exponent, and fractional Laplacian is defined for smooth enough functions as \[(-\Delta)^su(x)=c_{N,s}\,\text{P.V.}\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,{\rm d}x\,{\rm d}y,\] where \(c_{N,s}\) is a normalization constant.

1.1 Main results↩︎

We first consider the space \[X_0(\Omega):=\left\{u\in H^1(\mathbb{R}^N):u|_{\Omega}\in H_0^1(\Omega),\, u=0\text{ a.e. in }\mathbb{R}^N\setminus\Omega\right\}.\] It is easy to see that \(X_0(\Omega)\) is a Hilbert space with the norm \[\rho(u):=\left(\|\nabla u\|_2^2+[u]_s^2\right)^{\frac{1}{2}},\] which is associated with the inner product \[\langle u,v\rangle=\int_{\Omega}\nabla u\cdot\nabla v\,{\rm d}x+\iint_{\mathbb{R}^{2N}}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2s}}\,{\rm d}x\,{\rm d}y.\] Here \([\cdot]_s\), called the Gagliardo seminorm, is defined as \[[u]_s^2=\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,{\rm d}x\,{\rm d}y.\] Note that that the norm \(\rho\) is equivalent to the gradient norm \(\norm{\nabla\cdot}_{L^2(\Omega)}\) and as a consequence, \(X_0(\Omega)\) is continuously embedded into \(L^r(\Omega)\) for every \(r\in[1,2^*]\) and the embedding is compact for \(r<2^*\). A function \(u\in X_0(\Omega)\) is a said to be a weak solution to \[-\Delta u+(-\Delta)^su=f(u)\text{ in }\Omega, \quad u=0\text{ in }\mathbb{R}^N \setminus \Omega,\] if for every \(v\in X_0(\Omega)\) it holds \[\label{weak} \int_{\Omega}\nabla u\cdot\nabla v\,{\rm d}x+\iint_{\mathbb{R}^N}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2s}}\,{\rm d}x\,{\rm d}y=\int_{\Omega} f(u)v\,{\rm d}x.\tag{2}\] We now present our main theorem, which asserts that every bounded sequence of weak solutions associated with a family of superlinear perturbation of subcritical/critical problems converges strongly in \(X_0(\Omega)\) to a weak solution of the critical problem 1 . This is known as ‘compactness’ in the literature.

Regarding the compactness result, we make a few remarks. Set \[p_0:=\begin{cases} 2+\frac{4s}{N-2},& \text{if }N>6-4s;\\ 2^*-1, &\text{if }N\leq6-4s. \end{cases}\]

Due to the presence of the critical nonlinearity, the energy functional associated to 1 fails to satisfy the Palais-Smale condition for every \(c\in\mathbb{R}\). Consequently, the minimax theorem cannot be applied directly to obtain infinitely many solutions. As an application of Theorem 1, we also obtain the existence of infinitely many sign-changing solutions as stated in the following theorem:

1.2 Literature review↩︎

Consider the purely local problem \[\label{loc95BrNi2} -\Delta u=\lambda|u|^{p-2}u+|u|^{2^*-2}u \; \text{ in }\Omega,\quad u=0\;\text{ on }\partial\Omega.\tag{6}\] The case \(p=2\) is corresponds to the classical Brezis-Nirenberg problem, while \(p>2\) gives a superlinear perturbation.

In the case \(p=2\), Brezis and Nirenberg, in their seminal work [2], showed that

1. 6 has no nontrivial solution on any star-shaped domains for \(\lambda\leq0\);

2. There exists \(\lambda^*\in[0,\mu_1)\) such that 6 has a positive solution for all \(\lambda\in [\lambda^*,\mu_1)\), where \(\mu_1\) is the first Dirichlet eigenvalue of \((-\Delta,\Omega)\). If \(N\geq4\), \(\lambda^*=0\) and if \(N=3\) and \(\Omega\) is a ball then \(\lambda^*=\frac{\mu_1}{4}\);

3. 6 does not admit any positive solution for \(\lambda\geq\mu_1\).

Since then, various multiplicity results have been obtained by many authors. One of the important contributions in this direction was made by Devillanova and Solimini. In [3], they proved the following: let \(N\geq7\), \(\lambda>0\) and \(U\) be a bounded set in \(H_0^1(\Omega)\) whose elements are weak solutions to \[\label{loc95BrNi1} -\Delta u=\lambda u+|u|^{q-2}u \; \text{ in }\Omega,\quad u=0\;\text{ on }\partial\Omega,\tag{7}\] for \(q\) varying in \([2,2^*]\). Then \[\sup_{u\in U}\sup_{x\in\Omega}|u(x)|\leq C.\] This in turn implies a compactness result similar to Theorem 1. As an application of the compactness result, using standard minimization arguments, they proved the existence of infinitely many nontrivial solutions to 6 for any \(\lambda>0\) provided \(N\geq7\). This is an important contribution because the existence of infinitely many nontrivial solutions to 6 was known only when \(\Omega\) is a symmetric domain. For instance, in [4], Fortunato and Jannelli proved the existence of infinitely many non-trivial solutions of 6 for \(N\geq4\), \(\lambda>0\) and \(\Omega\) satisfying some symmetry conditions. However, the result of Devillanova and Solimini does not require any symmetry assumption on \(\Omega\).

Observe that, the restriction \(N\geq7\) in the compactness theorem of Devillanova and Solimini can not be removed for \(\lambda>0\) small. To see this, note that when \(\Omega\) is a ball, minimizing over the space of radial functions, compactness theorem implies the existence of infinitely many radial solutions to 6 for any \(\lambda>0\). For \(N\geq3\) and \(\lambda>0\), Srikanth [5] proved that 6 admits at most one positive solution when \(\Omega\) is a ball. Hence, 6 has infinitely many radial sign-changing solutions. However, for \(4\leq N \leq 6\) and \(\Omega\) is a ball, Atkinson, Brezis and Peletier in [6] showed that 6 does not admit any radial sign-changing solution if \(\lambda>0\) is small enough. Hence, \(N\geq 7\) is crucial in the theorem of Devillanova and Solimini.

Schechter and Zou in [7] proved an abstract critical point theorem and using it along with the compactness theorem of Devillanova and Solimini, they established the existence of infinitely many sign-changing solutions of 6 for all \(\lambda>0\) when \(N\geq7\).

Next, for the superlinear perturbation case (i.e. \(p>2\)), Brezis and Nirenberg in [2] also showed that

1. If \(N\geq4\) or \(N=3\) and \(p\in(4,6)\), 6 has a positive solution for every \(\lambda>0\).

2. If \(N=3\) and \(p\in(2,4]\) , 6 has a positive solution for \(\lambda\) large enough.

Servadei and Valdinoci in [8], [9] studied the following fractional analog of 6 , namely \[\label{nonloc95BrNi1} (-\Delta)^su=\lambda|u|^{p-2}u+|u|^{2_s^*-2}u\;\text{ in } \Omega,\quad u=0\; \text{ in }\mathbb{R}^N\setminus\Omega,\tag{8}\] where \(2_s^*=\frac{2N}{N-2s}\) is the nonlocal critical Sobolev exponent.
In the case \(p=2\), they proved that

1. for \(N\geq4s\) and for any \(\lambda\in(0,\lambda_{1,s})\), 8 admits a positive solution;

2. for \(2s< N<4s\) there exists \(\lambda_s>0\) such that for any \(\lambda>\lambda_s\), 8 has a nontrivial solution.

When \(p\in(2,2_s^*)\), Barrios, Colorado, Servadei and Soria in [10] showed that

1. for \(N>\frac{2s(p+2)}{p}\) and \(\lambda>0\) or \(N<\frac{2s(p+2)}{p}\) and \(\lambda>0\) sufficiently large, 8 admits a positive solution.

Following the arguments of [3] and using the Caffarelli-Silvestre extension method, Yan, Yang and Yu [11] demonstrated a compactness result, comparable to Theorem 1, for the spectral Laplace counterpart of 8 when \(N>6s\) . Building on the compactness result of [11], Li, Su and Tersian [12] leveraged those findings to prove the existence of infinitely many sign-changing solutions. However, to our knowledge, it remains an open question whether this compactness result holds for the case \(N \leq 6s\).

Finally, we mention some known existence results for the mixed local nonlocal Brezis-Nirenberg problems. Biagi, Dipierro, Valdinoci and Vecchi [13] recently proved that
when \(p=2\),

1. for \(\lambda\leq0\), 1 has no nontrivial solution;

2. there exists \(\lambda^*\in[\lambda_{1,s},\lambda_1)\) such that for any \(\lambda\in(\lambda^*,\lambda_1)\), 1 has a positive solution;

3. for \(0<\lambda\leq \lambda_{1,s}\), 1 has no positive solution belonging to a certain ball in \(L^{2^*}(\mathbb{R}^N)\);

4. for \(\lambda\geq\lambda_1\), 1 does not admit any positive solution;

and when \(p \in (2,2^*)\), set \[\label{Biagi95cond} \kappa_{s,N}:=\min\{2-2s,N-2\},\quad\beta_{p,N}:=N-\frac{p(N-2)}{2},\tag{9}\]

1. if \(\kappa_{s,N}>\beta_{p,N}\), then 1 admits a positive solution for every \(\lambda>0\);

2. if \(\kappa_{s,N}\leq\beta_{p,N}\), then 1 admits a positive solution for \(\lambda\) large enough.

Regarding the multiplicity of nontrivial solutions to the mixed local-nonlocal problems, da Silva, Fiscella and Viloria in [14] showed that

(1) for \(p=2\), 1 admits \(m\) pairs of nontrivial solutions when \(\lambda>\lambda_{k+1}-\mathcal{S}_N|\Omega|^{-\frac{2}{N}}\) and \(\lambda_{k}\leq \lambda<\lambda_{k+1}=\cdots=\lambda_{k+m}<\lambda_{k+m+1}\), where \(\mathcal{S}_N\) is the best constant in the Sobolev inequality 21 ;

(2) for \(p>2\), there exists \(\lambda_{**}>0\) such that for any \(\lambda\in(0,\lambda_{**})\), 1 admits at least \(\text{cat}_{\Omega}(\Omega)\) many nontrivial solutions. They employed Ljusternik-Schnirelmann category theory to establish this result.

1.3 Difficulties and Strategies↩︎

To prove Theorem 1, we follow the strategy by Devillanova and Solimini in [3]. The same strategy has been modified and applied to prove similar compactness theorems for other elliptic operators, such as the \(p\)-Laplace operator [15], spectral Laplace operator [11] and other quasilinear operators (see e.g. [16]). We briefly outline the main idea. Observe that \(\{u_n\}\) is a Palais-Smale sequence of \(I_\lambda\). Thus, using the Palais-Smale decomposition of the energy functional associated with 1 , (see Proposition 12, also see [17] for \(p=2\)) \(u_n\) can be decomposed as the sum of the weak limit of \(u_n\) and finitely many local Talenti bubbles. To establish strong convergence, it remains to prove that no such bubbles arise in the decomposition.

For the classical Brezis-Nirenberg problem 6 with \(p=2\), Devillanova and Solimini considered a safe region suitably away from any concentration points of the Talenti bubbles. In that safe region, they showed boundedness and some gradient estimates of \(u_n\). Using the local Pohozaev’s identity on a ball centred at a concentration point, with the boundary lying inside the safe region, they obtain control over the \(L^2\) mass of \(u_n\) in the whole ball via boundary integrals inside the safe region. Exploiting the estimates in the safe region yields an upper bound, while the Palais-Smale decomposition together with the asymptotic behaviour of the bubbles provides a corresponding lower bound. A comparison of these bounds ultimately shows that no bubbles can occur when \(N\geq7\).

Next, we discuss some key steps and difficulties for proving Theorem 1:

1. To get estimates on \(u_n\), Devillanova and Solimini used the following identity \[\label{form951} \frac{\,{\rm d}}{\,{\rm d}t}\left(\frac{1}{t^{N-1}}\int_{\partial B_t^N(x_0)}u(y)\,{\rm d}S\right)+\left(\frac{1}{t^{N-1}}\int_{B_t^N(x_0)}(-\Delta u(y))\,{\rm d}y\right)=0,\tag{10}\] where \(B_t^N(x_0)\subset\Omega\). Similarly, for 8 , Yan et al. employed a related identity based on Caffarelli-Silvestre’s extension of \(u\). Inspired by [15], we obtain the following expression through the application of the weak Harnack inequality [18] and the Wolff potential estimates [19]: \[\label{form952} \left(\frac{1}{r^N}\int_{B_r^N(x_0)}u^\gamma\,{\rm d}x\right)^{\frac{1}{\gamma}}\leq C+C\int_r^R\left(\frac{1}{\rho^{N-1}}\int_{B_\rho^N(x_0)}|(-\Delta u+(-\Delta)^su)|\,{\rm d}x\right)\,{\rm d}\rho,\tag{11}\] for any \(\gamma\in[1,\frac{N}{N-2})\).

2. After establishing \(L^\gamma\) estimate for some \(\gamma\), [11], [15] used the Moser iteration to get the local boundedness. However, due to the presence of a nonlocal term, Moser iteration for the mixed local-nonlocal problem introduces a tail term (see [18]). We observe that, due to the presence of the tail term, the method of Devillanova and Solimini can not be applied directly.

3. To the best of our knowledge, the local Pohozaev’s identity is not known for the fractional Laplacian operator and also for the mixed operator.

4. In view of the above two difficulties, following [11], we consider Caffarelli-Silvestre’s (CS) extension for 1 . Although a Moser iteration scheme can still be implemented in the CS extension set-up, the nonhomogeneous nature of the operator prevents a direct application of the techniques used in [11], [15] to obtain the desired \(L^\gamma\) estimates. We also mention that the use of CS extension has recently been applied in the context of mixed local-nonlocal operators, see [20].

5. In Theorem 1, we prove a local Pohozaev’s identity for the CS extension of 1 . In this case, the local Pohozaev’s identity yields (see Subsection 2.5): \[\lambda\int_{B^N_{\sigma_n^{-\frac{1}{2}}}(x_n)}|u_n|^p\,{\rm d}x\leq \text{Boundary integrals}+C\int_{B^{N+1}_{\sigma_n^{-\frac{1}{2}}}(x_n,0)\cap\{y>0\}}y^{1-2s}|\nabla U_n|^2\,{\rm d}x\,{\rm d}y,\] where \(U_n\) is the Caffarelli-Silvestre extension of \(u_n\), and \(\sigma_n, \,(x_n,0)\) are respectively the blow-up rate and the concentration point of the slowest concentrating bubble in the (PS) decomposition of \(\{U_n\}\) (see the Subsection 2.2). Using the estimates obtained on the safe region, we have \(\text{Boundary integrals}\leq C\sigma_n^{1-\frac{N}{2}}\). Also, using the bubble behaviour, we get \[\begin{align} & \lambda\int_{B^N_{\sigma_n^{-\frac{1}{2}}}(x_n)}|u_n|^p\,{\rm d}x\geq C\lambda\sigma_n^{\frac{(N-2)p}{2}-N}, \quad\text{ and } \\ &\int_{B^{N+1}_{\sigma_n^{-\frac{1}{2}}}(x_n,0)\cap\{y>0\}}y^{1-2s}|\nabla U_n|^2\,{\rm d}x\,{\rm d}y\sim \sigma_n^{2s-2}. \end{align}\] This implies \[\sigma_n^{\frac{N-2}{2}p-N}\leq C\sigma_n^{2s-2}+C\sigma_n^{1-\frac{N}{2}}\leq\begin{cases} C\sigma_n^{2s-2},&\text{ if }N>6-4s;\\ C\sigma_n^{1-\frac{N}{2}},&\text{ if }N\leq6-4s. \end{cases}\] We observe that the above inequality, with the fact that \(\{\sigma_n\}\) is unbounded, implies \[\begin{cases} \frac{N-2}{2}p-N\leq 2s-2\Longrightarrow p\leq 2+\frac{4s}{N-2},&\text{ if }N>6-4s;\\ \frac{N-2}{2}p-N\leq1-\frac{N}{2}\implies p\leq 2^*-1,&\text{ if }N\leq6-4s. \end{cases}\] Therefore, in the ranges mentioned in the hypothesis of Theorem 1, no bubble can appear in the decomposition. We emphasize that this is the only step where the restriction on \(p\) arises.

As mentioned before, a key step in proving the existence of infinitely many solutions to 1 is the analysis of the compactness established in Theorem 1. The same strategy has been implemented to obtain the existence of infinitely many solutions for different elliptic operators (see e.g. [3], [11], [15]). To prove the existence of infinitely many sign changing solutions, we apply the methods developed by Schechter and Zou [7].

1.4 Outline and Notation↩︎

The paper is organized as follows: Section 2 establishes a (PS) decomposition for the CS extension of 1 and provides several key estimates over the safe regions to rule out bubbles in the decomposition. This section concludes with the proof of the compactness theorem. In Section 3, we utilize these results to prove the existence of infinitely many sign-changing solutions to 1 . Appendix 4 contains some technical lemmas.

We use the following notation and convention:

(i) We denote \(\,{\rm d}\mu := \frac{\,{\rm d}x\,{\rm d}y}{|x-y|^{N+2s}}.\)

(ii) We denote \[\mathbb{A}(u,v) := \iint_{\mathbb{R}^{2N}}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2s}}\,{\rm d}x\,{\rm d}y.\]

(iii) \(B_r^N\) is a ball centered at \(x_0\in\mathbb{R}^N\) and radius \(r\) in \(\mathbb{R}^N\). Every point in \(\mathbb{R}^{N+1}\) is written as \((x,y)\), where \(x\in\mathbb{R}^N\) and \(y\in\mathbb{R}\). We further denote \[\mathbb{R}_+^{N+1}:=\mathbb{R}^{N+1}\cap\{y>0\},\, B_r^+(x_0,0):=B_r^{N+1}(x_0,0)\cap\{y>0\}.\]

(iv) For any open set \(U\subset\mathbb{R}^N\), we denote \[X_0(U):=\left\{u\in H^1(\mathbb{R}^N):u|_{U}\in H_0^1(U),\, u=0\text{ a.e. in }\mathbb{R}^N\setminus U\right\}.\]

(v) We define the space \(\mathcal{D}^{1,2}(\mathbb{R}_+^{N+1},y^{1-2s})\) as \[\begin{align} \mathcal{D}^{1,2}(\mathbb{R}_+^{N+1},y^{1-2s}) := \overline{\mathcal{C}_c^{\infty}(\mathbb{R}_+^{N+1})}^{\|\cdot\|_{\mathcal{D}^{1,2}(\mathbb{R}_+^{N+1},y^{1-2s})}}, \end{align}\] where \[\|\cdot\|_{\mathcal{D}^{1,2}(\mathbb{R}_+^{N+1},y^{1-2s})}:=\left(\int_{\mathbb{R}_+^{N+1}}y^{1-2s}\abs{\nabla \cdot}^2\,{\rm d}x\,{\rm d}y\right)^{\frac{1}{2}}.\]

(vi) For \(U\in \mathcal{D}^{1,2}(\mathbb{R}_+^{N+1},y^{1-2s})\), we denote the trace of \(U\) on the boundary \(\mathbb{R}^N\times\{y=0\}\) by \(\text{Tr}(U)\).

(vii) We denote \[\mathcal{X}^s(\mathbb{R}_+^{N+1}):=\left\{U\in \mathcal{D}^{1,2}(\mathbb{R}_+^{N+1},y^{1-2s}):\text{Tr}(U)\in \mathcal{D}^{1,2}(\mathbb{R}^N)\right\}.\]

(viii) We also denote \[\mathcal{X}_{\Omega}^s(\mathbb{R}_+^{N+1}):=\left\{U\in \mathcal{D}^{1,2}(\mathbb{R}_+^{N+1},y^{1-2s}):\text{Tr}(U)\in X_0(\Omega)\right\}.\]

(ix) The Tail term is denoted as \[\text{Tail}(u;x_0,R):=R^2\int_{\mathbb{R}^N\setminus B_R^N(x_0)}\frac{|u(x)|}{|x-x_0|^{N+2s}}\,{\rm d}x.\]

(x) The excess functional is defined as \[E(u;x_0,R):=\fint_{B_R^N(x_0)}|u-(u)_{B_R^N(x_0)}|\,{\rm d}x+\text{Tail}(u-(u)_{B_R^N(x_0)};x_0,R).\]

(xi) \(\norm{\cdot}_p := \norm{\cdot}_{L^p(\mathbb{R}^N)}\) for \(p \in (0, \infty)\).

(xii) \(\,{\rm d}\mathcal{H}^k\) denotes the \(k\)-dimensional Hausdorff measure.

(xiii) \(C\) denotes a generic positive constant.

2 Compactness↩︎

This section is devoted to proving Theorem 1. As discussed in the introduction, we begin with Caffarelli-Silvestre’s extension [21].

2.1 Caffarelli-Silvestre’s extension↩︎

Let \(u\in \dot{H}^s(\mathbb{R}^N)\) and \(E_s(u):=U\in \mathcal{D}^{1,2}(\mathbb{R}_+^{N+1},y^{1-2s})\) be the \(s\)-harmonic extension of \(u\) to the upper half space \(\mathbb{R}_+^{N+1}\), i.e. \(U\) weakly solves \[\begin{cases} \text{div}(y^{1-2s}\nabla U)=0\text{ in }\mathbb{R}_+^{N+1},\\ U(x,0)=u(x)\text{ in }\mathbb{R}^N. \end{cases}\] Then we have the following properties:

(a) \(U\) can be expressed explicitly by means of the Poisson kernel. For \(x\in\mathbb{R}^N,y>0\) , \[\label{Poisson95formula} U(x,y)=\int_{\mathbb{R}^N}P(x-\xi,y)u(\xi)\,{\rm d}\xi,\tag{12}\] where the Poisson kernel \(P(x,y)\) is defined as \[\label{Poisson95kernel} P(x,y)=C_{n,s}\frac{y^{2s}}{(|x|^2+y^2)^{\frac{N+2s}{2}}}, \quad x\in\mathbb{R}^N,\, y>0.\tag{13}\]

(b) The extension is related to the fractional Laplacian in the following way \[\label{DtoN} -C_{N,s}\lim_{y\to0^+}y^{1-2s}\frac{\partial U}{\partial y}(x,y)=(-\Delta)^su(x),\quad x\in\mathbb{R}^N.\tag{14}\]

(c) Due to the relation \[\label{isometry} [u]_s^2=C_{N,s}\int_{\mathbb{R}_+^{N+1}}y^{1-2s}|\nabla U|^2\,{\rm d}x\,{\rm d}y,\tag{15}\] the extension map \(E_s:\dot{H}^s(\mathbb{R}^N)\to \mathcal{D}^{1,2}(\mathbb{R}_+^{N+1},y^{1-2s})\) sending \(u\) to its \(s\)-extension \(U\) is an isometry.

(d) Trace inequality: \[\label{trace95ineq} \|u\|_{L^{2^*}(\mathbb{R}^N)}\leq C\|U\|_{\mathcal{D}^{1,2}(\mathbb{R}_+^{N+1},y^{1-2s})}.\tag{16}\]

(e) Let \(U\in \mathcal{D}^{1,2}(\mathbb{R}_+^{N+1},y^{1-2s})\) and \(B_r^+(x,0)\subset\mathbb{R}_+^{N+1}\). Then (see [22]) there exist \(C(r),\,\delta=\delta(N,s)>0\) such that for any \(1\leq t\leq\frac{N+1}{N}+\delta\), \[\label{Tan-Xiong} \|U\|_{L^{2t}(B_r^+(x,0),y^{1-2s})}\leq C(r)\|\nabla U\|_{L^2(B_r^+(x,0),y^{1-2s})}.\tag{17}\]

We define the Hilbert space \[\mathcal{X}_{\Omega}^s(\mathbb{R}_+^{N+1}):=\left\{U\in \mathcal{D}^{1,2}(\mathbb{R}_+^{N+1},y^{1-2s}):\text{Tr}(U)\in X_0(\Omega)\right\},\] equipped with the norm \[\tilde{\rho}(U):=\left(\int_{\Omega}|\nabla_x U(x,0)|^2\,{\rm d}x+C_{N,s} \int_{\mathbb{R}_+^{N+1}}y^{1-2s}|\nabla U|^2\,{\rm d}x\,{\rm d}y\right)^{\frac{1}{2}},\quad U\in \mathcal{X}_{\Omega}^s(\mathbb{R}_+^{N+1}).\] We also define the Hilbert space \(\mathcal{X}^s(\mathbb{R}_+^{N+1})\) as \[\mathcal{X}^s(\mathbb{R}_+^{N+1}):=\left\{U\in \mathcal{D}^{1,2}(\mathbb{R}_+^{N+1},y^{1-2s}):\text{Tr}(U)\in \mathcal{D}^{1,2}(\mathbb{R}^N)\right\},\] equipped with the norm \[\|U\|_{\mathcal{X}^s(\mathbb{R}_+^{N+1})}^2:=\|U\|_{\mathcal{D}^{1,2}(\mathbb{R}_+^{N+1},y^{1-2s})}^2+\|\text{Tr}(U)\|_{\mathcal{D}^{1,2}(\mathbb{R}^N)}^2.\] Using the isometry of the extension map, we have \[\rho(\text{Tr}(U))=\tilde{\rho}(U),\quad U\in \mathcal{X}_{\Omega}^s(\mathbb{R}_+^{N+1}).\] Let \(u_n\in X_0(\Omega)\) weakly solve 4 . Then its Caffarelli-Silvestre’s extension \(U_n\in \mathcal{X}_{\Omega}^s(\mathbb{R}_+^{N+1})\) satisfies \[\label{ext95sub95pro} \begin{cases} \text{div}(y^{1-2s}\nabla U_n)=0\text{ in }\mathbb{R}_+^{N+1},\\ U_n(x,0)=u_n(x)\text{ in }\mathbb{R}^N,\\ -\Delta u_n(x)- C_{N,s} \lim_{y\to0^+}y^{1-2s}\frac{\partial U_n}{\partial y}(x,y)=\lambda|u_n|^{p-2}u_n+|u_n|^{p_n-2}u_n. \end{cases}\tag{18}\] The weak formulation of 18 is \[\begin{align} \int_{\Omega}\nabla u_n(x)\cdot\nabla_x\varphi (x,0)\,{\rm d}x& + C_{N,s} \int_{\mathbb{R}_+^{N+1}}y^{1-2s}\nabla U_n(x,y)\cdot\nabla \varphi (x,y)\,{\rm d}x\,{\rm d}y\nonumber\\ &=\int_{\Omega}(\lambda|u_n|^{p-2}u_n+|u_n|^{p_n-2}u_n)\varphi (x,0)\,{\rm d}x,\; \forall \, \varphi \in \mathcal{X}_{\Omega}^s(\mathbb{R}_+^{N+1}). \end{align}\] We consider the energy functional \[\label{ext95func} \tilde{I}_\lambda(U)=\frac{1}{2}\tilde{\rho}(U)^2-\frac{\lambda}{p}\int_{\Omega}|U(x,0)|^p\,{\rm d}x-\frac{1}{2^*}\int_{\Omega}|U(x,0)|^{2^*}\,{\rm d}x,\quad U\in \mathcal{X}_{\Omega}^s(\mathbb{R}_+^{N+1}).\tag{19}\] Critical points of \(\tilde{I}_\lambda\) weakly solve the extended problem associated with 1 , i.e. \[\label{ext95main95pro} \begin{cases} \text{div}(y^{1-2s}\nabla U)=0\text{ in }\mathbb{R}_+^{N+1},\\ U(x,0)=u(x)\text{ in }\mathbb{R}^N,\\ -\Delta u(x)- C_{N,s} \lim_{y\to0^+}y^{1-2s}\frac{\partial U}{\partial y}(x,y)=\lambda|u|^{p-2}u+|u|^{2^*-2}u\text{ in }\Omega. \end{cases}\tag{20}\]

In view of the above discussion, the statement of Theorem 1 can be formulated as follows.

We claim that, up to a subsequence, \(\{U_n\}\) in Theorem 10 is a (PS) sequence for \(\tilde{I}_\lambda\). Using the uniform boundedness of \(\{U_n\}\subset \mathcal{X}_{\Omega}^s(\mathbb{R}_+^{N+1})\) and the trace inequality, we see that \(\{\tilde{I}_\lambda(U_n)\}_n\) is bounded. Observe that for any \(V\in \mathcal{X}_{\Omega}^s(\mathbb{R}_+^{N+1})\), \[\langle \tilde{I}_\lambda'(U_n),V\rangle=\int_{\Omega}(|u_n(x)|^{p_n-2}u_n(x)-|u_n(x)|^{2^*-2}u_n(x))V(x,0)\,{\rm d}x,\] and \[\||u_n(x)|^{p_n-2}u_n(x)-|u_n(x)|^{2^*-2}u_n(x)\|_{L^{\frac{2^*}{2^*-1}}(\Omega)}\leq C\||u_n|^{2^*-1}+1\|_{L^{\frac{2^*}{2^*-1}}(\Omega)}\leq C.\] Using \(\text{Tr}(\mathcal{X}_{\Omega}^s(\mathbb{R}_+^{N+1}))\underset{\text{continuous}}{\hookrightarrow} H_0^1(\Omega)\), \(u_n\rightharpoonup u\) in \(H_0^1(\Omega)\) and \(u_n\to u\) a.e. on \(\Omega\). Thus, \(|u_n(x)|^{p_n-2}u_n(x)-|u_n(x)|^{2^*-2}u_n(x)\to0\) a.e. on \(\Omega\) and up to a subsequence \(|u_n(x)|^{p_n-2}u_n(x)-|u_n(x)|^{2^*-2}u_n(x)\rightharpoonup0\) in \(L^{\frac{2^*}{2^*-1}}(\Omega)\). Hence, \(\langle \tilde{I}_\lambda'(U_n),V\rangle\to0\) as \(n\to\infty\) and \(\{U_n\}\) is a (PS) sequence of \(\tilde{I}_\lambda\).

It is worth mentioning that every (PS) sequence of \(\tilde{I}_\lambda\) may not converge strongly due to the noncompactness of the trace embedding \(\mathcal{X}_{\Omega}^s(\mathbb{R}_+^{N+1}) \hookrightarrow L^{2^*}(\Omega)\). Moreover, the weak limit of the (PS) sequence can be zero even if \(\eta>0\). This necessitates studying the profile decomposition of the (PS) sequences for the energy functional 19 associated with 20 . We define a \(C^1\) functional \(\tilde{I}\) on \(\mathcal{D}^{1,2}(\mathbb{R}^N)\) as \[\tilde{I}(u):=\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2\,{\rm d}x-\frac{1}{2^*}\int_{\mathbb{R}^N}|u|^{2^*}\,{\rm d}x, \; u \in \mathcal{D}^{1,2}(\mathbb{R}^N).\]

In the following proposition, we state the profile decomposition of the (PS) sequences for 19 (see also [17]). We need the following remark.

2.2 Safe regions↩︎

For every \(n\), we denote \(\sigma_n\) to be the minimum of \(\sigma_n^i\) i.e. \[\sigma_n:=\min_{1\leq i\leq k}\sigma_n^i\] and \((x_n,0)\) be the corresponding concentration point. For each \(1\leq i\leq k\), choose \(c_i\geq1\) with \(c_{i+1} = c_{i} +6\). Then for each \(n\), consider the half-annuli of the form \[\mathcal{sec:A}_i:=B_{(c_i+5)\sigma_n^{-\frac{1}{2}}}^+(x_n,0)\setminus \overline{B^+}_{c_i\sigma_n^{-\frac{1}{2}}}(x_n,0).\] Since excluding \((x_n,0)\), there are \(k-1\) concentration points and \(\overline{\mathcal{sec:A}_i} \cap \overline{\mathcal{sec:A}_j} = \{ \cdot \}\) for \(1\le i \neq j \le k\), for each \(n\) we can find an annulus whose flat boundary \(\partial\mathcal{A}_i\cap\{y=0\}\) does not contain any concentration points. Since there are only \(k\) many \(c_i\)’s, one \(c_i\) will be chosen infinitely many times. Further, using Proposition 12, \(\tilde{I}(\Psi^i) \ge \frac{1}{N}S^{\frac{N}{2}}\) and \(\tilde{I}_{\lambda}(U_n) \le C\) for some \(C>0\), we get for large \(n\), \[\begin{align} C + |\tilde{I}_{\lambda}(U_{\infty})| \ge \frac{k}{N}S^{\frac{N}{2}}, \end{align}\] which infers that \(k\) is bounded by a fixed constant. Hence passing through a subsequence, we find \(\overline{C}\geq1\), independent of \(n\), such that for any \(n\), the flat boundary of the annulus \[\mathcal{A}_n^1:=\left(B^+_{(\overline{C}+5)\sigma_n^{-\frac{1}{2}}}(x_n,0)\setminus \overline{B^+}_{\overline{C}\sigma_n^{-\frac{1}{2}}}(x_n,0)\right)\cap(\Omega\times(0,\infty)),\] does not contain any concentration points \((x_n^i,0)\). We further define the following thinner subsets of \(\mathcal{A}_n^1\): \[\begin{align} \mathcal{A}_n^2:=\left(B^+_{(\overline{C}+4)\sigma_n^{-\frac{1}{2}}}(x_n,0)\setminus \overline{B^+}_{(\overline{C}+1)\sigma_n^{-\frac{1}{2}}}(x_n,0)\right)\cap(\Omega\times(0,\infty)),\\ \mathcal{A}_n^3:=\left(B^+_{(\overline{C}+3)\sigma_n^{-\frac{1}{2}}}(x_n,0)\setminus \overline{B^+}_{(\overline{C}+2)\sigma_n^{-\frac{1}{2}}}(x_n,0)\right)\cap(\Omega\times(0,\infty)). \end{align}\]

2.3 Estimates under auxiliary norm↩︎

We recall an elementary inequality: Let \(q \in [2,2^*]\). Then there exists \(A(\lambda)>1\) independent of \(q\) such that \[\begin{align} \label{ineq-1} \left| |t|^{q-2}t+\lambda|t|^{p-2}t \right| \leq A|t|^{2^*-1}+A, \text{ for all } t\in\mathbb{R}. \end{align}\tag{23}\] We consider the following superset of \(\Omega\):

1. Let \(\tilde{\Omega} \supsetneq \Omega\) be such that \(\text{dist}(\Omega,\partial\tilde{\Omega})>0\).

Let \(u_n\) be solutions of 4 uniformly bounded in \(X_0(\Omega)\). For each \(n \in \mathbb{N}\), let \(v_n\in X_0(\tilde{\Omega})\) weakly solve the following problem (whose existence and uniqueness are guaranteed by the Riesz representation theorem): \[\label{PDE1} -\Delta u+(-\Delta)^su=A|u_n|^{2^*-1}+A \text{ in } \tilde{\Omega}, \; u=0\text{ in } \mathbb{R}^N\setminus \tilde{\Omega}.\tag{24}\] Taking \(v_n^-\in X_0(\tilde{\Omega})\) as a test function in the weak formulation, \[-\rho(v_n^-)^2-\iint_{\mathbb{R}^{2N}}[v_n^+(x)v_n^-(y)+v_n^+(y)v_n^-(x)]\,{\rm d}\mu=\int_{\Omega}(A|u_n|^{2^*-1}+A)v_n^-\,{\rm d}x.\] Hence, \(v_n^-\equiv0\) and \(v_n\) is a non-negative solution. Applying the strong maximum principle [26], \(v_n\) is a positive solution. Now, for any \(0\leq \varphi \in \mathcal{C}_c^{\infty}(\Omega)\), \[\begin{align} \int_{\Omega}\nabla v_n\cdot\nabla\varphi \,{\rm d}x+\mathbb{A}(v_n,\varphi )&=\int_{\tilde{\Omega}}\nabla v_n\cdot\nabla\varphi \,{\rm d}x+\mathbb{A}(v_n,\varphi )=\int_{\Omega}(A|u_n|^{2^*-1}+A)\varphi \,{\rm d}x\\& \geq \int_{\Omega}(|u_n|^{p_n-2}u_n+\lambda|u_n|^{p-2}u_n)\varphi \,{\rm d}x=\int_{\Omega}\nabla u_n\cdot\nabla\varphi \,{\rm d}x+\mathbb{A}(u_n,\varphi ). \end{align}\] Similarly, we see that \[\int_{\Omega}\nabla v_n\cdot\nabla\varphi \,{\rm d}x+\mathbb{A}(v_n,\varphi )\geq \int_{\Omega}\nabla (-u_n)\cdot\nabla\varphi \,{\rm d}x+\mathbb{A}(-u_n,\varphi ).\] Since \(|u_n|\leq v_n\) a.e. in \(\mathbb{R}^N\setminus\Omega\), using the weak comparison principle [27], \[\begin{align} \label{est-1} |u_n|\leq v_n \; \text{ a.e. in }\Omega. \end{align}\tag{25}\] In view of 25 , it is sufficient to estimate \(v_n\).

Notice that \[\begin{align} \label{ineq-2} \|u\|_{p_1,p_2,\sigma}\leq \min\left\{\|u\|_{p_1},\|u\|_{p_2}\sigma^{\frac{N}{p_2}-\frac{N}{2^*}}\right\}. \end{align}\tag{27}\]

2.4 Integral estimates over the safe regions↩︎

In this subsection, we aim to derive several integral estimates of \(u_n\) and \(U_n\) over the safe regions \(\mathcal{A}_n^i\). We recall a few facts from [19]. Suppose \(u\) is a weak solution to \[-\Delta u+(-\Delta)^su=f \text{ in }\Omega,\, u=0\text{ in }\mathbb{R}^N\setminus \Omega,\] where \(f\in L^1(\Omega)\). Let \(B_{\tilde{r}}^N(x_0)\subset\Omega\). Then for any \(0<\rho<\tilde{r}\), using [19], one has \[\label{byso1} \int_\rho^{\tilde{r}} E(u;x_0,t)\frac{\,{\rm d}t}{t}\leq cE(u;x_0,\tilde{r})+c\int_{\rho}^{\tilde{r}}\left(\frac{1}{t^{N-1}}\int_{B_t^N}|f|\,{\rm d}x\right)\,{\rm d}t,\tag{28}\] where \(c=c(N,s,\text{diam}(\Omega))\). Now for any \(0<\tilde{\rho}\leq \frac{\rho}{2}<\frac{\tilde{r}}{8}\) and \(\theta\in(\frac{1}{4},\frac{1}{2}]\), by [19] and 28 , we get \[\begin{align} |(u)_{B_\rho^N(x_0)}-(u)_{B_{\tilde{\rho}}^N(x_0)}|&\leq \int_{\tilde{\rho}}^{\frac{\rho}{\theta}}E(u;x_0,t)\frac{\,{\rm d}t}{t}\leq \int_{\tilde{\rho}}^{\tilde{r}}E(u;x_0,t)\frac{\,{\rm d}t}{t}\nonumber\\ &\leq cE(u;x_0,\tilde{r})+c\int_{\tilde{\rho}}^{\tilde{r}}\left(\frac{1}{t^{N-1}}\int_{B_t^N}|f|\,{\rm d}x\right)\,{\rm d}t.\nonumber \end{align}\] Since, the above inequality is true for any \(\rho<\frac{\tilde{r}}{4}\), taking \(\rho\uparrow\frac{\tilde{r}}{4}\) we have \[\label{byso2} |(u)_{B_\frac{\tilde{r}}{4}^N(x_0)}-(u)_{B_{\tilde{\rho}}^N(x_0)}|\leq cE(u;x_0,\tilde{r})+c\int_{\tilde{\rho}}^{\tilde{r}}\left(\frac{1}{t^{N-1}}\int_{B_t^N}|f|\,{\rm d}x\right)\,{\rm d}t\tag{29}\]

We also observe that for any \(\hat{\rho}\in[\frac{\tilde{r}}{4},\tilde{r}]\), \[\begin{align} \label{byso3} |(u)_{B_{\hat{\rho}}^N(x_0)}-(u)_{B_{\tilde{r}}^N(x_0)}|&\leq\fint_{B_{\hat{\rho}}^N(x_0)}|u-(u)_{B_{\tilde{r}}^N(x_0)}|\,{\rm d}x\nonumber\\ &\leq 4^N\fint_{B_{\tilde{r}}^N(x_0)}|u-(u)_{B_{\tilde{r}}^N(x_0)}|\,{\rm d}x\leq 4^N E(u;x_0,\tilde{r}). \end{align}\tag{30}\] Recall that \(v_n\) weakly solves \[-\Delta u+(-\Delta)^su=A|u_n|^{2^*-1}+A\text{ in }\tilde{\Omega}, \; u=0\text{ in } \mathbb{R}^N\setminus \tilde{\Omega},\] where \(\tilde{\Omega}\) is given in [O951] and \(\{v_n\}\) is also uniformly bounded in \(X_0(\tilde{\Omega})\).

Next, we prove a similar estimate for the extension \(U_n\). For that, we impose the second condition on \(\tilde{\Omega}\):

1. Let \(\tilde{\Omega}\) be such that \(\text{dist}(\Omega,\partial\tilde{\Omega})\geq16\text{ diam}(\Omega)\).

In the next proposition, we get the estimate of \(u_n\) and \(U_n\) over the safe region \(\mathcal{A}_n^2\) for any \(q \ge 1\).

Finally, we obtain the following gradient estimate of \(u_n\) and \(U_n\).

2.5 Proof of compactness results↩︎

This subsection aims to prove Theorem 10.

Proof of Theorem 10: We start with the local Pohozaev’s identity. Consider a cut-off function \(\varphi _n\) with \[\begin{align} 0\leq \varphi _n\leq 1, \quad \varphi _n=1 \text{ in } B_{(\overline{C}+2)\sigma_n^{-\frac{1}{2}}}^{N+1}(x_n,0), \quadand\quad \text{supp}(\varphi _n)\subseteq B_{(\overline{C}+3)\sigma_n^{-\frac{1}{2}}}^{N+1}(x_n,0). \end{align}\] Set \(B_n^+:=B_{(\overline{C}+3)\sigma_n^{-\frac{1}{2}}}^+(x_n,0)\). Multiplying \(((X-X_0)\cdot\nabla U_n)\varphi _n\) with \(\text{div}(y^{1-2s}\nabla U_n)\) and integrating over \(B_n^+\), we have \[\begin{align} 0&=\int_{B_n^+}\text{div}(y^{1-2s}\nabla U_n)((X-X_0)\cdot\nabla U_n)\varphi _n\,{\rm d}x\,{\rm d}y\\ &=-\int_{B_n^+}y^{1-2s}\nabla U_n\cdot\nabla(((X-X_0)\cdot\nabla U_n)\varphi _n)\,{\rm d}x\,{\rm d}y\\&\quad+\int_{\partial B_n^+}((X-X_0)\cdot\nabla U_n)\varphi _n y^{1-2s}(\nabla U_n\cdot\nu_N)\,{\rm d}\mathcal{H}^N. \end{align}\] Expanding we see \[\begin{align} &-\int_{B_n^+}y^{1-2s}\nabla U_n\cdot\nabla(((X-X_0)\cdot\nabla U_n)\varphi _n)\,{\rm d}x\,{\rm d}y\\ &=-\int_{B_n^+}y^{1-2s}\nabla U_n\cdot\left[\nabla((X-X_0)\cdot\nabla U_n)\varphi _n+\nabla\varphi _n((X-X_0)\cdot\nabla U_n)\right]\,{\rm d}x\,{\rm d}y\\ &=-\int_{B_n^+}y^{1-2s}|\nabla U_n|^2\varphi _n\,{\rm d}x\,{\rm d}y-\int_{B_n^+}y^{1-2s}\nabla\left(\frac{1}{2}|\nabla U_n|^2\right)\cdot(X-X_0)\varphi _n\,{\rm d}x\,{\rm d}y\\ &\quad-\int_{B_n^+}y^{1-2s}((X-X_0)\cdot\nabla U_n)(\nabla U_n\cdot\nabla\varphi _n)\,{\rm d}x\,{\rm d}y\\ &=-\int_{B_n^+}y^{1-2s}|\nabla U_n|^2\varphi _n\,{\rm d}x\,{\rm d}y+\int_{B_n^+}\frac{1}{2}|\nabla U_n|^2\nabla\cdot((X-X_0)y^{1-2s}\varphi _n)\,{\rm d}x\,{\rm d}y\\ &\quad-\int_{\partial B_n^+}y^{1-2s}\frac{1}{2}|\nabla U_n|^2\varphi _n((X-X_0)\cdot\nu_N)\,{\rm d}\mathcal{H}^N-\int_{B_n^+}y^{1-2s}((X-X_0)\cdot\nabla U_n)(\nabla U_n\cdot\nabla\varphi _n)\,{\rm d}x\,{\rm d}y\\ &=\frac{N-2s}{2}\int_{B_n^+}y^{1-2s}|\nabla U_n|^2\varphi _n\,{\rm d}x\,{\rm d}y+\frac{1}{2}\int_{B_n^+}y^{1-2s}|\nabla U_n|^2(X-X_0)\cdot\nabla\varphi _n\,{\rm d}x\,{\rm d}y\\ &\quad-\int_{B_n^+}y^{1-2s}((X-X_0)\cdot\nabla U_n)(\nabla U_n\cdot\nabla\varphi _n)\,{\rm d}x\,{\rm d}y. \end{align}\] For the first term in the sixth line of the above identities, we use \(X-X_0\perp\nu_N\) on \(\partial B_n^+\cap\{y=0\}\) and \(\varphi _n=0\) on \(\partial B_n^+\cap\{y>0\}\). For brevity, we set \[B_n^N:=\partial B_n^+\cap\{y=0\}=B_{(\overline{C}+3)\sigma_n^{-\frac{1}{2}}}^N(x_n),\] \(f_n(t):=\lambda|t|^{p-2}t+|t|^{p_n-2}t\) and \(F_n(t):=\int_0^tf_n(s)\,{\rm d}s\). Then we have \[\begin{align} &\int_{\partial B_n^+}((X-X_0)\cdot\nabla U_n)\varphi _n y^{1-2s}(\nabla U_n\cdot\nu_N)\,{\rm d}\mathcal{H}^N\\ &=\left(\int_{\partial B_n^+\cap\{y>0\}}+\int_{\partial B_n^+\cap\{y=0\}}\right)((X-X_0)\cdot\nabla U_n)\varphi _n y^{1-2s}(\nabla U_n\cdot\nu_N)\,{\rm d}\mathcal{H}^N\\ &=\int_{\partial B_n^+\cap\{y=0\}}-y^{1-2s}\frac{\partial U_n}{\partial y}((X-X_0)\cdot\nabla U_n)\varphi _n \,{\rm d}\mathcal{H}^N\\ &=\int_{B_n^N\cap\Omega}(f_n(u_n)+\Delta u_n)((x-x_0)\cdot\nabla u_n)\varphi _n(x,0) \,{\rm d}x\\ &=-\int_{B_n^N\cap\Omega}F_n(u_n)\nabla\cdot((x-x_0)\varphi _n)\,{\rm d}x+\int_{\partial(B_n^N\cap\Omega)}F_n(u_n)\varphi _n(x-x_0)\cdot\nu_{N-1}\,{\rm d}\mathcal{H}^{N-1}\\ &\quad-\int_{B_n^N\cap\Omega}\nabla u_n\cdot\nabla(((x-x_0)\cdot\nabla u_n)\varphi _n)\,{\rm d}x+\int_{\partial(B_n^N\cap\Omega)}(\nabla u_n\cdot\nu_{N-1})((x-x_0)\cdot\nabla u_n)\varphi _n\,{\rm d}\mathcal{H}^{N-1}\\ &=-N\int_{B_n^N\cap\Omega}F_n(u_n)\varphi _n(x,0)\,{\rm d}x-\int_{B_n^N\cap\Omega}F_n(u_n)((x-x_0)\cdot\nabla_x\varphi _n(x,0))\,{\rm d}x\\ &\quad+\frac{N-2}{2}\int_{B_n^N\cap\Omega}|\nabla u_n|^2\varphi _n(x,0)\,{\rm d}x+\frac{1}{2}\int_{B_n^N\cap\Omega}|\nabla u_n|^2((x-x_0)\cdot\nabla_x\varphi _n(x,0))\,{\rm d}x\\ &-\int_{B_n^N\cap\Omega}(\nabla u_n\cdot\nabla_x\varphi _n(x,0))((x-x_0)\cdot\nabla u_n))\,{\rm d}x-\int_{ B_n^N\cap\partial\Omega}\frac{1}{2}|\nabla u_n|^2\varphi _n((x-x_0)\cdot\nu_{N-1})\,{\rm d}\mathcal{H}^{N-1}\\ &\quad +\int_{ B_n^N\cap\partial\Omega}(\nabla u_n\cdot\nu_{N-1})((x-x_0)\cdot\nabla u_n)\varphi _n\,{\rm d}\mathcal{H}^{N-1}. \end{align}\] Combining the above identities, we get \[\begin{align} 0&=\frac{N-2s}{2}\int_{B_n^+}y^{1-2s}|\nabla U_n|^2\varphi _n\,{\rm d}x\,{\rm d}y+\frac{1}{2}\int_{B_n^+}y^{1-2s}|\nabla U_n|^2(X-X_0)\cdot\nabla\varphi _n\,{\rm d}x\,{\rm d}y\nonumber\\ &\quad-\int_{B_n^+}y^{1-2s}((X-X_0)\cdot\nabla U_n)(\nabla U_n\cdot\nabla\varphi _n)\,{\rm d}x\,{\rm d}y\nonumber\\ &\quad-N\int_{B_n^N\cap\Omega}F_n(u_n)\varphi _n(x,0)\,{\rm d}x-\int_{B_n^N\cap\Omega}F_n(u_n)((x-x_0)\cdot\nabla_x\varphi _n(x,0))\,{\rm d}x\nonumber\\ &\quad+\frac{N-2}{2}\int_{B_n^N\cap\Omega}|\nabla u_n|^2\varphi _n(x,0)\,{\rm d}x+\frac{1}{2}\int_{B_n^N\cap\Omega}|\nabla u_n|^2((x-x_0)\cdot\nabla_x\varphi _n(x,0))\,{\rm d}x\nonumber\\ &-\int_{B_n^N\cap\Omega}(\nabla u_n\cdot\nabla_x\varphi _n(x,0))((x-x_0)\cdot\nabla u_n))\,{\rm d}x-\int_{ B_n^N\cap\partial\Omega}\frac{1}{2}|\nabla u_n|^2\varphi _n((x-x_0)\cdot\nu_{N-1})\,{\rm d}\mathcal{H}^{N-1}\nonumber\\ &\quad +\int_{ B_n^N\cap\partial\Omega}(\nabla u_n\cdot\nu_{N-1})((x-x_0)\cdot\nabla u_n)\varphi _n\,{\rm d}\mathcal{H}^{N-1}.\label{poh951461} \end{align}\tag{40}\] On the other hand multiplying \(U_n\varphi _n\) with \(\text{div}(y^{1-2s}\nabla U_n)\) and integrating by parts over \(B_n^+\) yields \[\begin{align} 0&=\int_{B_n^+}\text{div}(y^{1-2s}\nabla U_n)U_n\varphi _n\,{\rm d}x\,{\rm d}y\nonumber\\ &=-\int_{B_n^+}y^{1-2s}\nabla U_n\cdot\nabla(U_n\varphi _n)\,{\rm d}x\,{\rm d}y+\int_{\partial B_n^+}y^{1-2s}(\nabla U_n\cdot\nu_N)U_n\varphi _n\,{\rm d}\mathcal{H}^{N}\nonumber\\ &=-\int_{B_n^+}y^{1-2s}|\nabla U_n|^2\varphi _n\,{\rm d}x\,{\rm d}y-\int_{B_n^+}y^{1-2s}U_n(\nabla U_n\cdot\nabla\varphi _n)\,{\rm d}x\,{\rm d}y\\&\quad+\int_{\partial B_n^{+}\cap\{y=0\}}-y^{1-2s}\frac{\partial U_n}{\partial y}U_n\varphi _n\,{\rm d}\mathcal{H}^{N}\nonumber\\ &=-\int_{B_n^+}y^{1-2s}|\nabla U_n|^2\varphi _n\,{\rm d}x\,{\rm d}y-\int_{B_n^+}y^{1-2s}U_n(\nabla U_n\cdot\nabla\varphi _n)\,{\rm d}x\,{\rm d}y\\&\quad+\int_{B_n^{N}\cap\Omega}(f_n(u_n)+\Delta u_n)u_n\varphi _n(x,0)\,{\rm d}x\nonumber\\ &=-\int_{B_n^+}y^{1-2s}|\nabla U_n|^2\varphi _n\,{\rm d}x\,{\rm d}y-\int_{B_n^+}y^{1-2s}U_n(\nabla U_n\cdot\nabla\varphi _n)\,{\rm d}x\,{\rm d}y+\int_{ B_n^{N}\cap\Omega}f_n(u_n)u_n\varphi _n(x,0)\,{\rm d}x\nonumber\\ &\quad-\int_{B_n^N\cap\Omega}\nabla u_n\cdot\nabla(u_n\varphi _n(x,0))\,{\rm d}x+\int_{\partial(B_n^N\cap\Omega)}(\nabla u_n\cdot\nu_{N-1})u_n\varphi _n(x,0)\,{\rm d}\mathcal{H}^{N-1}\nonumber\\ &=-\int_{B_n^+}y^{1-2s}|\nabla U_n|^2\varphi _n\,{\rm d}x\,{\rm d}y-\int_{B_n^+}y^{1-2s}U_n(\nabla U_n\cdot\nabla\varphi _n)\,{\rm d}x\,{\rm d}y+\int_{ B_n^{N}}f_n(u_n)u_n\varphi _n(x,0)\,{\rm d}x\nonumber\\ &\quad-\int_{B_n^N\cap\Omega}|\nabla u_n|^2\varphi _n(x,0)\,{\rm d}x-\int_{B_n^N\cap\Omega}u_n\nabla u_n\cdot\nabla_x\varphi _n(x,0)\,{\rm d}x.\label{poh951462} \end{align}\tag{41}\] Adding 40 and \(\frac{N-2}{2}\)41 , we obtain \[\begin{align} &\int_{B_n^N\cap\Omega}\left(NF_n(u_n)-\frac{N-2}{2}u_nf_n(u_n)\right)\varphi _n(x,0)\,{\rm d}x-(1-s)\int_{B_n^+}y^{1-2s}|\nabla U_n|^2\varphi _n\,{\rm d}x\,{\rm d}y\\ &=\frac{1}{2}\int_{B_n^+}y^{1-2s}|\nabla U_n|^2(X-X_0)\cdot\nabla\varphi _n\,{\rm d}x\,{\rm d}y+\frac{1}{2}\int_{B_n^N\cap\Omega}|\nabla u_n|^2(x-x_0)\cdot\nabla_x\varphi _n(x,0)\,{\rm d}x\\ &\quad-\int_{B_n^+}y^{1-2s}((X-X_0)\cdot\nabla U_n)(\nabla U_n\cdot\nabla\varphi _n)\,{\rm d}x\,{\rm d}y-\int_{B_n^N\cap\Omega}(\nabla u_n\cdot\nabla_x\varphi _n(x,0))((x-x_0)\cdot\nabla u_n)\,{\rm d}x\\ &\quad-\int_{B_n^N\cap\Omega}F_n(u_n)((x-x_0)\cdot\nabla_x\varphi _n(x,0))\,{\rm d}x\\ &\quad-\int_{ B_n^N\cap\partial\Omega}\frac{1}{2}|\nabla u_n|^2\varphi _n((x-x_0)\cdot\nu_{N-1})\,{\rm d}\mathcal{H}^{N-1}+\int_{ B_n^N\cap\partial\Omega}(\nabla u_n\cdot\nu_{N-1})((x-x_0)\cdot\nabla u_n)\varphi _n\,{\rm d}\mathcal{H}^{N-1}.\\ &\quad-\frac{N}{2^*}\left(\int_{B_n^+}y^{1-2s}U_n(\nabla U_n\cdot\nabla\varphi _n)\,{\rm d}x\,{\rm d}y+\int_{B_n^N\cap\Omega}u_n(\nabla u_n\cdot\nabla_x\varphi _n(x,0))\,{\rm d}x\right). \end{align}\] For large \(n\), we choose \(X_0=(x_0,0)\in\Omega^c\) such that \(|x-x_0|\leq C\sigma_n^{-\frac{1}{2}}\) in \(B_n^N\) and \((x-x_0)\cdot\nu_{N-1}\leq0\) on \(B_n^N\cap\partial\Omega\). Combining this with the fact that \(u_n=0\) on \(\partial\Omega\) yields \[\begin{align} &-\int_{ B_n^N\cap\partial\Omega}\frac{1}{2}|\nabla u_n|^2\varphi _n((x-x_0)\cdot\nu_{N-1})\,{\rm d}\mathcal{H}^{N-1}+\int_{ B_n^N\cap\partial\Omega}(\nabla u_n\cdot\nu_{N-1})((x-x_0)\cdot\nabla u_n)\varphi _n\,{\rm d}\mathcal{H}^{N-1}\\ &=\frac{1}{2}\int_{ B_n^N\cap\partial\Omega}|\nabla u_n|^2\varphi _n((x-x_0)\cdot\nu_{N-1})\,{\rm d}\mathcal{H}^{N-1}\leq0. \end{align}\] Furthermore, using \((\frac{N}{p_n}-\frac{N}{2^*})\int_{B_n^N}|u_n|^{p_n}\,{\rm d}x\geq0\), \[\begin{align} &\left(\frac{N}{p}-\frac{N}{2^*}\right)\lambda\int_{B_n^N \cap \Omega}|u_n|^p\varphi _n(x,0)\,{\rm d}x\\ &\leq\frac{1}{2}\int_{B_n^+}y^{1-2s}|\nabla U_n|^2(X-X_0)\cdot\nabla\varphi _n\,{\rm d}x\,{\rm d}y+\frac{1}{2}\int_{B_n^N\cap\Omega}|\nabla u_n|^2(x-x_0)\cdot\nabla_x\varphi _n(x,0)\,{\rm d}x\\ &\quad-\int_{B_n^+}y^{1-2s}((X-X_0)\cdot\nabla U_n)(\nabla U_n\cdot\nabla\varphi _n)\,{\rm d}x\,{\rm d}y-\int_{B_n^N\cap\Omega}(\nabla u_n\cdot\nabla_x\varphi _n(x,0))((x-x_0)\cdot\nabla u_n)\,{\rm d}x\\ &\quad-\int_{B_n^N\cap\Omega}F_n(u_n)((x-x_0)\cdot\nabla_x\varphi _n(x,0))\,{\rm d}x+(1-s)\int_{B_n^+}y^{1-2s}|\nabla U_n|^2\varphi _n\,{\rm d}x\,{\rm d}y\\ &\quad-\frac{N}{2^*}\left(\int_{B_n^+}y^{1-2s}U_n(\nabla U_n\cdot\nabla\varphi _n)\,{\rm d}x\,{\rm d}y+\int_{B_n^N\cap\Omega}u_n(\nabla u_n\cdot\nabla_x\varphi _n(x,0))\,{\rm d}x\right). \end{align}\] We set \[\mathcal{B}_n:=\left(B^+_{(\overline{C}+3)\sigma_n^{-\frac{1}{2}}}(x_n,0)\setminus \overline{B^+}_{(\overline{C}+2)\sigma_n^{-\frac{1}{2}}}(x_n,0)\right)\cap\{y>0\}.\] Now we use that \(\text{supp}(\nabla\varphi _n)\cap\mathbb{R}_+^{N+1}\subset\mathcal{B}_n\), \(|\nabla\varphi _n|\leq C\sigma_n^{\frac{1}{2}}\). Since, \(|x-x_0|\leq C\sigma_n^{-\frac{1}{2}}\) in \(B_n^N\), it holds \(|X-X_0|\leq C\sigma_n^{-\frac{1}{2}}\) in \(B_n^+\). Hence, using Propositions 20 and 21, we obtain \[\begin{align} &\left(\frac{N}{p}-\frac{N}{2^*}\right)\lambda\int_{B_n^N \cap \Omega}|u_n|^p\varphi _n(x,0)\,{\rm d}x\\ &\leq C\int_{B_n^+}y^{1-2s}|\nabla U_n|^2\,{\rm d}x\,{\rm d}y+\int_{\partial\mathcal{A}_n^3\cap\{y=0\}}(|u_n|^p+|u_n|^{p_n}+|\nabla u_n|^2)\,{\rm d}x\\ &\quad+C\int_{\mathcal{B}_n}y^{1-2s}(\sigma_n|U_n|^2+|\nabla U_n|^2)\,{\rm d}x\,{\rm d}y+\int_{\partial\mathcal{A}_n^3\cap\{y=0\}}(\sigma_n|u_n|^2+|\nabla u_n|^2)\,{\rm d}x\\ &\leq C\int_{B_n^+}y^{1-2s}|\nabla U_n|^2\,{\rm d}x\,{\rm d}y+C\sigma_n^{1-\frac{N}{2}}+C\sigma_n^{-\frac{N}{2}}\\ &\leq C\int_{B_n^+}y^{1-2s}|\nabla U_n|^2\,{\rm d}x\,{\rm d}y+C\sigma_n^{1-\frac{N}{2}}. \end{align}\] For the first term in the third line of the above inequalities, we have used Hölder’s inequality and the inequality (see [22]) \[\int_{\mathcal{B}_n}y^{1-2s}|U|^{2}\,{\rm d}x\,{\rm d}y\leq C\sigma_n^{-1}\int_{\mathcal{B}_n}y^{1-2s}|\nabla U|^{2}\,{\rm d}x\,{\rm d}y.\] Using the (PS) decomposition, \(p>\frac{N}{N-2}\) and arguing as in [15], \[\begin{align} \lambda\int_{B_n^N}u_n^p\varphi _n(x,0)\,{\rm d}x\ge\lambda\int_{B_{\sigma_n^{-1}}(x_n)\cap\Omega}u_n^p\,{\rm d}x\geq C\sigma_n^{\frac{N-2}{2}p-N}. \end{align}\] Using Remark 14, we see that \[\begin{align} \int_{B_n^+}y^{1-2s}|\nabla U_n|^2\varphi _n\,{\rm d}x\,{\rm d}y&\leq \int_{B_n^+}y^{1-2s}|\nabla U_n|^2\,{\rm d}x\,{\rm d}y\\ &=\int_{B_n^+}y^{1-2s}|\nabla U_\infty|^2\,{\rm d}x\,{\rm d}y+O\left(\sum_{i=1}^k(\sigma_{n}^i)^{2s-2}\right)\\ &\leq C\sigma_n^{-\frac{N+2-2s}{2}}+C\sigma_n^{2s-2}\leq C\sigma_n^{2s-2}. \end{align}\] Case-1: Suppose \(N>6-4s\), then combining all the estimates we get \[\sigma_n^{\frac{N-2}{2}p-N}\leq C\sigma_n^{2s-2}+C\sigma_n^{1-\frac{N}{2}}\leq C\sigma_n^{2s-2}.\] This implies that \[p\leq\frac{2(N-2+2s)}{N-2},\] which is a contradiction to the given hypothesis on \(p\).
Case-2: Now suppose \(N\leq 6-4s\), then we get \[\sigma_n^{\frac{N-2}{2}p-N}\leq C\sigma_n^{2s-2}+C\sigma_n^{1-\frac{N}{2}}\leq C\sigma_n^{1-\frac{N}{2}}.\] This implies that \[p\leq \frac{N+2}{N-2},\] which contradicts the given hypothesis on \(p\).

Therefore, the (PS) decomposition (Proposition 12) holds without any \(\Psi_n^i\) and as a consequence, \(U_n\) strongly converges in \(\mathcal{X}_{\Omega}^s(\mathbb{R}_+^{N+1})\). This completes the proof. 0◻

3 Infinitely many sign-changing solutions↩︎

In this section, as an application of Theorem 1 and following the strategy developed in [7], we prove the existence of infinitely many sign-changing solutions to \[\begin{cases} -\Delta u+(-\Delta)^su=\lambda |u|^{p-2}u+|u|^{2^\ast-2}u&\text{in }\Omega,\\ u=0&\text{in }\mathbb{R}^N\setminus \Omega, \end{cases}\] where \(N,s,p\) be as in Theorem 5.

Let \(0<\lambda_1<\lambda_2\leq\lambda_3\leq\cdots\) be the eigenvalues of \((-\Delta+(-\Delta)^s,\Omega)\) and \(\varphi _k\) be the eigenfunction corresponding to \(\lambda_k\). The set \(\{\varphi _k\}\) forms an orthonormal basis of \(L^2(\Omega)\) and (after a rescaling) also an orthonormal basis of \(X_0(\Omega)\). Define \(E_k=\text{span}\{\varphi _1,\ldots,\varphi _k\}\). These properties follow from standard variational methods and the theory of compact self-adjoint operators. Thus \[X_0(\Omega)=\overline{\cup_{k=1}^\infty E_k}^{\rho(\cdot)},\quad \text{dim}(E_k)=k,\quad E_k\subset E_{k+1}.\] We choose a sequence \(p_n\in(p,2^*)\) such that \(p_n\to 2^*\). As \(p_n\in(p,2^*)\), there exists \(C_0>0\) independent of \(n\) such that \[\label{A950} \|\cdot\|_{p_n}\leq C_0\rho(\cdot) \text{ in }X_0(\Omega),\quad X_0(\Omega)\Subset L^{p_n}(\Omega).\tag{42}\] We consider the functional \[I_{n,\lambda}(u):=\frac{1}{2}\rho(u)^2-\frac{\lambda}{2}\|u\|_p^p-\frac{1}{p_n}\|u\|_{p_n}^{p_n}, \; \forall \, u \in X_0.\] Notice that for any \(u,v \in X_0(\Omega)\), \[\begin{align} I_{n,\lambda}'(u)(v)=\int_{\Omega}\nabla u\cdot\nabla v\,{\rm d}x+\mathbb{A}(u,v)-\lambda\int_{\Omega}|u|^{p-2}uv\,{\rm d}x-\int_{\Omega}|u|^{p_n-2}uv\,{\rm d}x. \end{align}\] By Riesz representation theorem, there exists \(L(u), G(u)\in X_0(\Omega)\) such that \[\langle L(u),v\rangle=\lambda\int_{\Omega}|u|^{p-2}uv\,{\rm d}x,\quad \langle G(u),v\rangle=\int_{\Omega}|u|^{p_n-2}uv\,{\rm d}x.\] Hence \(\nabla I_{n,\lambda}=u- K_{n,\lambda}(u)\) where \(K_{n,\lambda}(u)=L(u)+G(u)\). Observe that both \(L\), \(G\) are continuous and as a result \(K_{n,\lambda}:X_0(\Omega)\to X_0(\Omega)\) is continuous. Let \(u_n\in X_0(\Omega)\) be a critical point of \(I_{n,\lambda}\) i.e. \(u_n\) weakly solves \[-\Delta u+(-\Delta)^su=\lambda|u|^{p-2}u+|u|^{p_n-2}u \text{ in }\Omega,\quad u=0\text{ in }\mathbb{R}^N\setminus \Omega.\] By [28], \(u_n\in \mathcal{C}^{1,\alpha}(\overline{\Omega})\) for any \(\alpha\in(0,\min\{1,2-2s\})\). Now consider the map \(I_{n,\lambda}''(u_n):X_0(\Omega)\to X_0(\Omega)^*\), defined for any \(v,w\in X_0(\Omega)\) by \[I_{n,\lambda}''(u_n)[v,w]=\int_{\Omega}\nabla v\cdot\nabla w\,{\rm d}x+\mathbb{A}(v,w)-\lambda(p-1)\int_{\Omega}|u_n|^{p-2}vw\,{\rm d}x-(p_n-1)\int_{\Omega}|u_n|^{p_n-2}vw\,{\rm d}x.\] Observe that if \(v\in \text{ker}(I_{n,\lambda}''(u_n))\) then \(v\) weakly solves \[-\Delta v+(-\Delta)^sv=\lambda(p-1)|u_n|^{p-2}v+(p_n-1)|u_n|^{p_n-2}v \text{ in }\Omega,\quad v=0\text{ in }\mathbb{R}^N\setminus \Omega.\] For \(f\in L^2(\Omega)\), we define \(K_nf\in X_0(\Omega)\) to be the unique solution of the problem \[-\Delta u+(-\Delta)^su=\lambda(p-1)|u_n|^{p-2}f+(p_n-1)|u_n|^{p_n-2}f \text{ in }\Omega,\quad u=0\text{ in }\mathbb{R}^N\setminus \Omega.\] Now, \(K_n\) is well defined by the Riesz Representation theorem. Since \(u_n\in L^\infty(\Omega)\), \[\begin{align} \rho(K_nf)^2&\leq \left( \lambda(p-1)\|u_n\|_{\infty}^{p-2}+(p_n-1)\|u_n\|_{\infty}^{p_n-2}\right)\|f\|_2\|K_nf\|_2\\ &\leq\left(\lambda(p-1)\|u_n\|_{\infty}^{p-2}+(p_n-1)\|u_n\|_{\infty}^{p_n-2}\right)\|f\|_{2}\rho(K_nf). \end{align}\] Using the fact that \(X_0(\Omega)\Subset L^2(\Omega)\), each \(K_n\) is a bounded, linear, and compact operator. Thus, by Fredholm alternative, \(I_{n,\lambda}''(u_n)\) is a Fredholm operator.

We define \[\begin{align} C_{k}^{**}(n,\lambda):=\sup_{E_{k}}I_{n,\lambda}. \end{align}\] By Lemma 5, \(C_{k}^{**}(n,\lambda)\) is well defined and \(C_{k}^{**}(n,\lambda)<\infty\).

Proof of Theorem 5: Observe that \(I_{n,\lambda}\) satisfies the (PS)\(_c\) condition for every \(c\in\mathbb{R}\). Applying [7], \(I_{n,\lambda}\) has a nontrivial sign-changing critical point \(u^*_{n,k}\in \mathcal{S}^*\) at the level \(C^*(n,\lambda,k)\), where \[C^*(n,\lambda,k)\in[-\Lambda_0,C_{k+1}^{**}(n,\lambda)]\subset[-\Lambda_0,T_1\lambda_k^{\frac{p}{p-2}}].\] Furthermore the augmented Morse index (the number of non-positive eigenvalues of the linearized operator) \(m^*(u^*_{n,k})\geq k\) and for fixed \(\lambda,k\), \(\{u^*_{n,k}\}\) weakly solves 4 for \(p_n \in (p,2^*)\). For fixed \(\lambda,k\), using \(C^*(n,\lambda,k)\leq T_1\lambda_{k+1}^{\frac{p}{p-2}}\), we see that \[\begin{align} \left(\frac{1}{2}-\frac{1}{p}\right)\rho(u^*_{n,k})^2&\leq \left(\frac{1}{2}-\frac{1}{p}\right)\rho(u^*_{n,k})^2+\left(\frac{1}{p}-\frac{1}{p_n}\right)\|u^*_{n,k}\|_{p_n}^{p_n}\\&= \frac{1}{2}\rho(u^*_{n,k})^2-\frac{\lambda}{p}\|u^*_{n,k}\|_p^p-\frac{1}{p_n}\|u^*_{n,k}\|_{p_n}^{p_n}\leq T_1\lambda_{k+1}^{\frac{p}{p-2}}. \end{align}\] Hence the sequence \(\{u^*_{n,k}\}_n\) is uniformly bounded in \(X_0(\Omega)\). By Theorem 1, up to a subsequence, we get \(u^*_{n,k}\to u^*_k\) in \(X_0(\Omega)\) and \(u^*_k\) weakly solves 1 . Since no solution of 1 has negative energy, \[0\leq C^*(\lambda,k)=\lim_{n\to\infty} C^*(n,\lambda,k)\leq T_1\lambda_{k+1}^{\frac{p}{p-2}}.\] Now we show that \(u^*_k\) is also sign-changing. Since \(u^*_{n,k}\) weakly solves 4 , we have \[\begin{align} &\rho((u^*_{n,k})^{\pm})^2+2\iint_{\mathbb{R}^{2N}}(u^*_{n,k})^+(x)(u^*_{n,k})^-(y)\,{\rm d}\mu\\ &=\lambda\|(u^*_{n,k})^{\pm}\|_p^p+\|(u^*_{n,k})^{\pm}\|_{p_n}^{p_n}\leq \lambda C\rho((u^*_{n,k})^\pm)^p+C\rho((u^*_{n,k})^\pm)^{p_n}. \end{align}\] This implies \[1\leq\lambda C\rho((u^*_{n,k})^\pm)^{p-2}+C\rho((u^*_{n,k})^\pm)^{p_n-2}.\] Since \(2<p<p_n\), \(\rho((u^*_{n,k})^\pm)\geq C>0\), where \(C\) is independent of \(n\), we take the limit as \(n\to\infty\), to get \(\rho((u^*_k)^\pm)\geq C>0\). Hence \(u^*_k\) is sign-changing. To prove the infinitude of sign-changing solutions, we prove that \(\lim_{k\to\infty}C^*(\lambda,k)=\infty\). Suppose \(\lim_{k\to\infty}C^*(\lambda,k)=c'<\infty\). Then there exists \(n_k\in\mathbb{N}\) such that \(\lim_{k\to\infty}C^*(n_k,\lambda,k)=\lim_{k\to\infty}C^*(\lambda,k)=c'\). Since \(I_{n_k,\lambda}(u^*_{n_k,k})=C^*(n_k,\lambda,k)\) and \(I_{n_k,\lambda}'(u^*_{n_k,k})=0\), \(\{u^*_{n_k,k}\}\) is uniformly bounded in \(X_0(\Omega)\) and using Theorem 1, \(\{u^*_{n_k,k}\}\) converges strongly to \(u_0\) in \(X_0(\Omega)\) as \(k\to\infty\) for some \(u_0\in X_0(\Omega)\). Set \[\begin{align} V_k&:=\lambda(p-1)|u^*_{n_k,k}|^{p-2}-(p_{n_k}-1)|u^*_{n_k,k}|^{p_{n_k}-2}, \text{ and } \\ V_0&:=\lambda(p-1)|u_0|^{p-2}-(2^*-1)|u_0|^{2^*-2}. \end{align}\] Consider the linearized operators \[\begin{align} S_k\varphi :=\left(-\Delta+(-\Delta)^s-V_k\right)\varphi =0, \, \quad k\geq0. \end{align}\] By the Courant-Fischer min-max theorem, we write the eigenvalues as \[\lambda_m(S_k):=\min_{\substack{W_m\subset X_0(\Omega)\\\text{dim}(W_m)=m}}\,\max_{\substack{\varphi \in W_m\\\|\varphi \|_2=1}}\left(\rho(\varphi )^2-\int_{\Omega}V_k\varphi ^2\,{\rm d}x\right),\,\quad m\geq1, k\geq0.\] Since \(u^*_{n_k,k}\to u_0\) in \(X_0(\Omega)\), \(\lambda_m(S_k)\to \lambda_m(S_0)\) as \(k\to\infty\) for every \(m\). On the other hand, we have for any \(k\geq0\), \(\lambda_1(S_k)\leq\lambda_2(S_k)\leq\cdots\to\infty\). This implies that\[m^*(u^*_{n_k,k})\leq m^*(u_0),\;\text{for large }k.\]This contradicts the fact that \(m^*(u^*_{n_k,k})\geq k\) for every \(k\). Hence, \(C^*(\lambda,k)\to\infty\) as \(k\to\infty\). 0◻

Acknowledgement: We sincerely thank Prof. Debabrata Karmakar (TIFR-CAM) for many insightful discussions and for proposing the use of the Caffarelli-Silvestre extension in estimating certain integrals. P. Das also expresses his gratitude to TIFR-CAM for its hospitality during his visit in February 2026.

Funding: The research of M. Bhakta is supported by the Swarnajaynti Fellowship (SB/SJF/2021-22/09) and the ANRF-MATRICS grant (ANRF/ARGM/2025/000126/MTRDST). The research of N. Biswas is supported by the National Board for Higher Mathematics Postdoctoral Fellowship (0204/16(9)/2024/RD-II/6761). The research of P. Das is supported by the National Board for Higher Mathematics PhD Fellowship (0203/5(38)/ 2024-R&D-II/11224).

Competing interests: The authors have no competing interests to declare that are relevant to the content of this article.

Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analysed during the current study.

4 ↩︎

This section accumulates several key estimates needed for the proof of the compactness theorem. In the following lemma, we prove some integral estimates for the weak solution to a certain linear mixed local-nonlocal problem with a potential.

In the following lemma, we prove a Moser iteration for the weak solution to the extension problem.

References↩︎