1 Introduction↩︎

In this paper we consider the following Dirichlet problem with a small parameter \(\varepsilon>0\): \[\label{eq:1} \left\{ \begin{align} -\varepsilon \Delta_p u &= a(x)|u|^{q-2}u - b(x)|u|^{\gamma-2}u, && x\in\Omega,\\ u&=0, && x\in\partial\Omega . \end{align} \right.\tag{1}\] Here \[\Delta_p u = \operatorname{div}\!\left(|\nabla u|^{p-2}\nabla u\right)\] is the \(p\)-Laplacian, \(\Omega\subset\mathbb{R}^N\) is a bounded connected domain with \(C^1\) boundary, and \(1<p<q<\gamma<p^*\), where \[p^* = \begin{cases} \dfrac{pN}{N-p}, & \text{if } p<N, \\[4pt] +\infty, & \text{if } p\ge N . \end{cases}\] We assume that \[\label{eq:Condab} a,b\in L^\infty(\Omega),\qquad a(x)\ge0,\qquad b(x)\ge\sigma_b>0 \quad\text{for a.e. }x\in\Omega ,\tag{2}\] where \(\sigma_b>0\) is a constant. Throughout the paper we also assume that \(a\not\equiv0\).

A function \(u\in W^{1,p}_0(\Omega)\) is called a weak solution of problem 1 if it is a critical point of the energy functional \[\label{eq:energy} \Phi_\varepsilon(u) = \frac{\varepsilon}{p}\int_\Omega |\nabla u|^p\,dx - \frac{1}{q}\int_\Omega a|u|^q\,dx + \frac{1}{\gamma}\int_\Omega b|u|^\gamma\,dx .\tag{3}\] Here \(W^{1,p}_0:=W^{1,p}_0(\Omega)\) is the closure of \(C^\infty_0(\Omega)\) in the Sobolev space \(W^{1,p}(\Omega)\). A weak solution \(u\in W^{1,p}_0(\Omega)\) is called a ground state of \(\Phi_\varepsilon\) if \[\Phi_\varepsilon(u)\le \Phi_\varepsilon(w)\] for every weak solution \(w\in W^{1,p}_0(\Omega)\) of problem 1 .

Problem 1 is singularly perturbed: as \(\varepsilon\to0^+\), the differential order disappears in the limit, and boundary layers may form near \(\partial\Omega\). Already in the one-dimensional case \(p=2\), \(a=b=1\), the problem becomes \[\label{eq:11} \left\{ \begin{align} -\varepsilon u''(x) &= u^{q-1}(x)-u^{\gamma-1}(x), && x\in(0,1),\\ u(0)&=u(1)=0 . \end{align} \right.\tag{4}\] The formal method of matched asymptotic expansions (see, for instance, [1], [2]) leads to the approximation \[u(x) \sim U\!\left(\frac{x}{\sqrt{\varepsilon}}\right) + U\!\left(\frac{1-x}{\sqrt{\varepsilon}}\right) - 1, \qquad 0\le x\le1,\] where the boundary-layer profile \(U\) is determined by \[-U''=U^{q-1}-U^{\gamma-1}, \qquad U(0)=0,\qquad U(+\infty)=1 .\] This profile admits the integral representation \[\int_0^{U(\xi)} \frac{dt}{ \sqrt{ 2\left( \frac{t^\gamma}{\gamma} - \frac{t^q}{q} + \frac{1}{q} - \frac{1}{\gamma} \right) } } = \xi .\] The approximation is consistent with the boundary conditions in the leading order: in the interior of the interval, both arguments \(x/\sqrt{\varepsilon}\) and \((1-x)/\sqrt{\varepsilon}\) are large, so that \(U\approx1\) and \(u(x)\approx1\), whereas near the endpoints the corresponding boundary-layer profile provides the transition to the zero boundary value.

Extending such a description to multidimensional domains, especially in the presence of non-symmetric geometry or variable coefficients, is a substantial mathematical difficulty. At the same time, singularly perturbed nonlinear boundary value problems form a central topic in modern elliptic theory because they combine delicate asymptotic analysis with a wide range of applications. They arise, for instance, in population dynamics, models of infection spread, control theory, mechanics of materials with memory, nonlinear optics, physiological models, and predator–prey systems; see, for example, [3]–[6].

A large part of the mathematical literature on singularly perturbed elliptic problems is devoted to concentration phenomena. In this direction one usually studies families of solutions which, as the perturbation parameter tends to zero, develop sharply localized profiles: interior or boundary spikes, spike-layer solutions, multi-peak solutions, or bubble-type solutions. The main questions concern the existence of such concentrating solutions, the number and location of their concentration points, the influence of the geometry of \(\Omega\), and the role of the coefficients and lower-order terms in selecting the limiting profiles. This circle of problems has been extensively studied for semilinear and quasilinear elliptic equations and is closely related to the analysis of concentration near critical points of auxiliary potentials, mean-curvature type quantities, or other effective variational landscapes; see, among many others, [7]–[15].

The problem considered in the present paper has a different emphasis. Rather than constructing spike or bubble solutions and describing the fine geometry of their concentration sets, we study the asymptotic behaviour of ground states for a competing-superlinear Dirichlet problem with small diffusion. In this setting the formal limiting equation is no longer an elliptic boundary value problem, but a pointwise algebraic balance between the two nonlinear terms. Our main result shows that, under the positivity assumption on \(a\), ground states converge to the explicit profile \[\bar u_0(x) = \left(\frac{a(x)}{b(x)}\right)^{1/(\gamma-q)},\] with convergence in measure and strong convergence in \(L^r(\Omega)\) for \(1\le r<\gamma\). Thus the singular limit is described not by localization at isolated points, but by convergence to a spatially distributed equilibrium profile determined by the variable coefficients.

A further difficulty is caused by the loss of differential order in the singular limit. Formally setting \(\varepsilon=0\) reduces the elliptic problem to the algebraic equation \[a(x)|u|^{q-2}u-b(x)|u|^{\gamma-2}u=0,\] so that the boundary condition is no longer encoded in the limiting equation. Moreover, this equation does not select a unique limit. In the nonnegative class, for example, every function of the form \[u(x)=\bar u_0(x)\chi_E(x), \qquad \bar u_0(x)=\left(\frac{a(x)}{b(x)}\right)^{1/(\gamma-q)},\] where \(E\subset\Omega\) is measurable, is a formal solution of the limiting equation. Thus the limit problem has a continuum of possible solutions.

For this reason, the main asymptotic issue is not merely to identify the algebraic balance between the nonlinearities, but to determine which of the many possible limiting profiles is selected by ground states of the original elliptic problem. The result proved below shows that, under the positivity assumption on \(a\), the selected profile is the full positive branch \(\bar u_0\). This also explains why the natural convergence statement is formulated first in measure and then in \(L^r(\Omega)\), \(1\le r<\gamma\), rather than as a classical convergence of solutions of elliptic boundary value problems.

In the present paper, problem 1 is studied by means of the nonlinear Rayleigh quotient method developed in [16], [17]. Applying this method to the parameter \(\varepsilon\) leads to the nonlinear generalized Rayleigh quotient \[\Upsilon(u) := \frac{ \left(\displaystyle\int_\Omega a|u|^q\,dx\right)^{ \frac{\gamma-p}{\gamma-q}} }{ \displaystyle \int_\Omega |\nabla u|^p\,dx \left(\displaystyle\int_\Omega b|u|^\gamma\,dx\right)^{ \frac{q-p}{\gamma-q}} }.\] Since in the general setting we assume only \(a\ge0\), there may exist nonzero functions \(u\in W^{1,p}_0(\Omega)\) such that \[\int_\Omega a|u|^q\,dx=0.\] Thus the extremal quantities are naturally defined on the set \[\mathcal{D} := \left\{ u\in W^{1,p}_0(\Omega)\setminus\{0\}: \int_\Omega a|u|^q\,dx>0 \right\}.\] In what follows, \(\Upsilon\) is considered on \(\mathcal{D}\).

Two extremal parameter values are associated with this quotient: \[\varepsilon^* = c_{p,q,\gamma} \sup_{u\in\mathcal{D}}\Upsilon(u), \qquad \varepsilon_e^* = c_{e,p,q,\gamma} \sup_{u\in\mathcal{D}}\Upsilon(u),\] where \[\label{eq:c-pq-intro} c_{p,q,\gamma} = \frac{\gamma-q}{\gamma-p} \left( \frac{q-p}{\gamma-p} \right)^{\frac{q-p}{\gamma-q}}, \qquad c_{e,p,q,\gamma} = \frac{p(\gamma-q)}{q(\gamma-p)} \left( \frac{\gamma(q-p)}{q(\gamma-p)} \right)^{\frac{q-p}{\gamma-q}} .\tag{5}\] Moreover, \[0<\varepsilon_e^*<\varepsilon^*<+\infty .\]

The first main result describes existence and nonexistence of positive solutions of problem 1 depending on the parameter \(\varepsilon\).

The next result describes the asymptotic behaviour of ground states as \(\varepsilon\to0^+\).

By the simple scaling \(u\mapsto tu\), problem 1 with \(\varepsilon>0\) can be transformed into either of the following equivalent forms: \[\label{eq:1lam} \left\{ \begin{align} -\Delta_p u &= \lambda a(x)|u|^{q-2}u - b(x)|u|^{\gamma-2}u, && x\in\Omega,\\ u&=0, && x\in\partial\Omega, \end{align} \right.\tag{6}\] and \[\label{eq:1nu} \left\{ \begin{align} -\Delta_p u &= a(x)|u|^{q-2}u - \nu b(x)|u|^{\gamma-2}u, && x\in\Omega,\\ u&=0, && x\in\partial\Omega, \end{align} \right.\tag{7}\] where \(\lambda,\nu>0\).

These problems are in one-to-one correspondence with problem 1 whenever \(\varepsilon\lambda\nu\neq0\). Namely, if \(u_\varepsilon\in W^{1,p}_0(\Omega)\) is a weak solution of 1 , then the functions \[\begin{align} u_\lambda &:= \varepsilon^{-1/(\gamma-p)}u_\varepsilon, & \lambda &= \varepsilon^{-(\gamma-q)/(\gamma-p)}, \tag{8} \\ u_\nu &:= \varepsilon^{-1/(q-p)}u_\varepsilon, & \nu &= \varepsilon^{(\gamma-q)/(q-p)} \tag{9} \end{align}\] are weak solutions of problems 6 and 7 , respectively. The inverse transformations are \[u_\varepsilon = \lambda^{-1/(\gamma-q)}u_\lambda, \qquad u_\varepsilon = \nu^{1/(\gamma-q)}u_\nu .\] Hence Theorems 1 and 2 imply the following assertion.

2 Preliminaries↩︎

In what follows, \(L^r:=L^r(\Omega)\), \(1<r<+\infty\), denotes the space of measurable functions with finite norm \[\|u\|_r = \left(\int_\Omega |u|^r\,dx\right)^{1/r}.\] By the Poincaré–Friedrichs inequality, the norm in \(W^{1,p}_0(\Omega)\) can be equivalently defined by \[\|u\|_{1,p} = \left(\int_\Omega |\nabla u|^p\,dx\right)^{1/p}.\]

We set \[T(u):=\int_\Omega |\nabla u|^p\,dx, \qquad A(u):=\int_\Omega a(x)|u|^q\,dx, \qquad B(u):=\int_\Omega b(x)|u|^\gamma\,dx .\] Since \(b(x)\ge\sigma_b>0\) for a.e. \(x\in\Omega\), we have \[B(u)>0 \quad \text{for all }u\in W^{1,p}_0(\Omega)\setminus\{0\}.\] We also assume throughout that \(a\not\equiv0\). Hence the set \[\mathcal{D} := \left\{ u\in W^{1,p}_0(\Omega)\setminus\{0\}: A(u)>0 \right\}\] is nonempty.

The Nehari manifold associated with problem 1 is given by \[\mathcal{N}_\varepsilon := \left\{ u\in W^{1,p}_0(\Omega)\setminus\{0\}: \varepsilon T(u)-A(u)+B(u)=0 \right\}.\] Equivalently, \[D_u\Phi_\varepsilon(u)(u)=0 .\] Observe that, for \(\varepsilon>0\), every function \(u\in\mathcal{N}_\varepsilon\) satisfies \[A(u)=\varepsilon T(u)+B(u)>0.\] Therefore, when positive values of the parameter are considered, it is natural to restrict attention to directions \(u\in\mathcal{D}\). Only such directions can generate an intersection of the ray \(\{su:s>0\}\) with the Nehari manifold for \(\varepsilon>0\).

Following the approach based on nonlinear generalized Rayleigh quotients [17], we introduce two Rayleigh quotients associated, respectively, with the Nehari manifold and with the zero energy level. The first quotient is defined by \[R_{\mathcal{N}}(u) := \frac{A(u)-B(u)}{T(u)}, \qquad u\in W^{1,p}_0(\Omega)\setminus\{0\}.\] Then \[\mathcal{N}_\varepsilon = \left\{ u\in W^{1,p}_0(\Omega)\setminus\{0\}: R_{\mathcal{N}}(u)=\varepsilon \right\}.\]

Fix \(u\in\mathcal{D}\) and consider the function \[s\mapsto R_{\mathcal{N}}(su), \qquad s>0.\] We have \[R_{\mathcal{N}}(su) = \frac{s^qA(u)-s^\gamma B(u)}{s^pT(u)} = \frac{A(u)s^{q-p}-B(u)s^{\gamma-p}}{T(u)} .\] Since \(1<p<q<\gamma\), this function has a unique positive critical point, which is the point of its global maximum. A direct computation gives \[s_{\mathcal{N}}(u) = \left( \frac{(q-p)A(u)}{(\gamma-p)B(u)} \right)^{1/(\gamma-q)} . \label{eq:smu}\tag{10}\]

Substituting \(s=s_{\mathcal{N}}(u)\) into \(R_{\mathcal{N}}(su)\), we obtain the nonlinear generalized Rayleigh quotient \[\varepsilon(u) := R_{\mathcal{N}}(s_{\mathcal{N}}(u)u) = c_{p,q,\gamma} \frac{ A(u)^{\frac{\gamma-p}{\gamma-q}} }{ T(u)B(u)^{\frac{q-p}{\gamma-q}} }, \qquad u\in\mathcal{D}, \label{eq:mu95u}\tag{11}\] where \[c_{p,q,\gamma} = \frac{\gamma-q}{\gamma-p} \left( \frac{q-p}{\gamma-p} \right)^{\frac{q-p}{\gamma-q}} . \label{eq:c95pq}\tag{12}\]

We define the extremal value \[\varepsilon^* := \sup_{u\in\mathcal{D}}\varepsilon(u). \label{eq:mustar}\tag{13}\] Since \(a\not\equiv0\), the set \(\mathcal{D}\) is nonempty; hence \[\varepsilon^*>0.\] We now show that \(\varepsilon^*<+\infty\). By the Hölder, Poincaré–Friedrichs, and Sobolev inequalities, there exists a constant \(C>0\), independent of \(u\), such that \[A(u) = \int_\Omega a(x)|u|^q\,dx \le C T(u)^{\frac{\gamma-q}{\gamma-p}} B(u)^{\frac{q-p}{\gamma-p}} .\] Indeed, since \(q\in(p,\gamma)\), the \(L^q\)-norm is interpolated between the \(L^p\)- and \(L^\gamma\)-norms, while the \(L^p\)-norm is controlled by \(\|\nabla u\|_p\). The condition \(b(x)\ge\sigma_b>0\) allows us to replace the \(L^\gamma\)-norm by the corresponding \(B(u)\)-term. Substituting this estimate into 11 , we obtain \[0<\varepsilon^*<+\infty . \label{eq:214}\tag{14}\]

The second Rayleigh quotient corresponds to the zero energy level (cf. [16], [17], [25], [26]): \[R_e(u) := \frac{ \frac{1}{q}A(u)-\frac{1}{\gamma} B(u) }{ \frac{1}{p}T(u) }, \qquad u\in W^{1,p}_0(\Omega)\setminus\{0\}.\] Then \[R_e(u)=\varepsilon \quad \Longleftrightarrow \quad \Phi_\varepsilon(u)=0 .\]

For fixed \(u\in\mathcal{D}\), the function \[s\mapsto R_e(su), \qquad s>0,\] has a unique positive critical point. It is given by \[s_e(u) = \left( \frac{\gamma(q-p)A(u)}{q(\gamma-p)B(u)} \right)^{1/(\gamma-q)} . \label{eq:smu95e}\tag{15}\] Substituting \(s=s_e(u)\) into \(R_e(su)\), we obtain \[\varepsilon_e(u) := R_e(s_e(u)u) = c_{e,p,q,\gamma} \frac{ A(u)^{\frac{\gamma-p}{\gamma-q}} }{ T(u)B(u)^{\frac{q-p}{\gamma-q}} }, \qquad u\in\mathcal{D}, \label{eq:mu95e95u}\tag{16}\] where \[c_{e,p,q,\gamma} = \frac{p(\gamma-q)}{q(\gamma-p)} \left( \frac{\gamma(q-p)}{q(\gamma-p)} \right)^{\frac{q-p}{\gamma-q}} . \label{eq:c95pq95e}\tag{17}\] Set \[\varepsilon_e^* := \sup_{u\in\mathcal{D}}\varepsilon_e(u).\] The same estimate as above yields \[0<\varepsilon_e^*<+\infty .\]

3 Proof of Theorem 1↩︎

We first prove (1). Suppose, arguing by contradiction, that for some \(\varepsilon>\varepsilon^*\) problem 1 has a nontrivial weak solution \(\bar u\in W^{1,p}_0(\Omega)\), \(\bar u\not\equiv0\). Then \[D_u\Phi_\varepsilon(\bar u)(\bar u)=0,\] that is, \[\varepsilon\int_\Omega |\nabla \bar u|^p\,dx - \int_\Omega a(x)|\bar u|^q\,dx + \int_\Omega b(x)|\bar u|^\gamma\,dx =0.\] Since \(\varepsilon>0\), \(\bar u\not\equiv0\), and \(b\ge\sigma_b>0\), it follows that \[\int_\Omega a(x)|\bar u|^q\,dx>0.\] Hence \(\bar u\in\mathcal{D}\). Moreover, \[R_{\mathcal{N}}(\bar u)=\varepsilon .\] By the definition of \(\varepsilon(\bar u)\) as the maximum of \(s\mapsto R_{\mathcal{N}}(s\bar u)\), we have \[\varepsilon = R_{\mathcal{N}}(\bar u) \le \varepsilon(\bar u) \le \varepsilon^* < \varepsilon ,\] which is impossible. Thus, for \(\varepsilon>\varepsilon^*\), problem 1 has no nontrivial weak solutions.

We now prove (2).

Let \(0<\varepsilon<\varepsilon_e^*\). Consider \[\hat{\Phi}_\varepsilon := \inf_{u\in W^{1,p}_0(\Omega)}\Phi_\varepsilon(u).\] By the definition of \(\varepsilon_e^*\), there exists \(u_0\in\mathcal{D}\) such that \[\varepsilon<\varepsilon_e(u_0).\] Set \[w_0:=s_e(u_0)u_0 .\] Then, by the definition of \(\varepsilon_e(u_0)\), \[R_e(w_0)=\varepsilon_e(u_0)>\varepsilon .\] Since \[R_e(w_0) = \frac{ \frac{1}{q}\int_\Omega a(x)|w_0|^q\,dx - \frac{1}{\gamma}\int_\Omega b(x)|w_0|^\gamma\,dx }{ \frac{1}{p}\int_\Omega |\nabla w_0|^p\,dx },\] the inequality \(R_e(w_0)>\varepsilon\) is equivalent to \[\Phi_\varepsilon(w_0)<0 .\] Consequently, \[\label{eq:negative-min-level} \hat{\Phi}_\varepsilon<0 .\tag{18}\]

Let \((u_m)\subset W^{1,p}_0(\Omega)\) be a minimizing sequence: \[\Phi_\varepsilon(u_m)\to\hat{\Phi}_\varepsilon .\] By Lemma 3, the sequence \((u_m)\) is bounded in \(W^{1,p}_0(\Omega)\). Hence, passing to a subsequence, we may assume that there exists \(u_\varepsilon\in W^{1,p}_0(\Omega)\) such that \[u_m\rightharpoonup u_\varepsilon \quad\text{weakly in }W^{1,p}_0(\Omega),\] and, by the compact Sobolev embedding, \[u_m\to u_\varepsilon \quad\text{strongly in }L^r(\Omega), \qquad 1<r<p^* .\] In particular, \[u_m\to u_\varepsilon \quad\text{strongly in }L^q(\Omega) \quad\text{and}\quad u_m\to u_\varepsilon \quad\text{strongly in }L^\gamma(\Omega).\] Using the weak lower semicontinuity of the norm \(\|\nabla u\|_p\) and the strong convergence of the nonlinear terms, we obtain \[\Phi_\varepsilon(u_\varepsilon) \le \liminf_{m\to +\infty}\Phi_\varepsilon(u_m) = \hat{\Phi}_\varepsilon .\] Therefore \[\Phi_\varepsilon(u_\varepsilon)=\hat{\Phi}_\varepsilon .\] In particular, by 18 , \[\Phi_\varepsilon(u_\varepsilon)<0=\Phi_\varepsilon(0),\] and hence \(u_\varepsilon\not\equiv0\).

Since \[\Phi_\varepsilon(|u|)=\Phi_\varepsilon(u), \qquad |u|\in W^{1,p}_0(\Omega),\] we may replace \(u_\varepsilon\) by \(|u_\varepsilon|\) and assume that \[u_\varepsilon\ge0 \quad\text{in }\Omega .\] Since \(u_\varepsilon\) is a global minimizer of \(\Phi_\varepsilon\) on \(W^{1,p}_0(\Omega)\), we have \[D_u\Phi_\varepsilon(u_\varepsilon)=0 .\] Thus \(u_\varepsilon\) is a weak solution of problem 1 .

Let us show that this solution is positive. Standard a priori estimates for weak solutions of quasilinear elliptic equations with subcritical growth imply that \[u_\varepsilon\in L^\infty(\Omega).\]

Let us show that this solution is positive. We first recall that \(u_\varepsilon\) is bounded. Indeed, since \(a,b\in L^\infty(\Omega)\) and \[1<p<q<\gamma<p^*,\] the right-hand side of equation 1 has subcritical growth in \(u_\varepsilon\). Therefore the standard \(L^\infty\)-estimates for weak solutions of quasilinear elliptic equations, obtained for instance by the Moser iteration scheme, imply that \[u_\varepsilon\in L^\infty(\Omega);\] see, e.g., [27]. Hence \[a(x)u_\varepsilon^{q-1} - b(x)u_\varepsilon^{\gamma-1} \in L^\infty(\Omega).\] By the interior regularity theory for degenerate quasilinear elliptic equations [28], [29], we obtain \[u_\varepsilon\in C^{1,\kappa}_{\mathrm{loc}}(\Omega)\] for some \(\kappa\in(0,1)\).

We now prove that \(u_\varepsilon>0\) in \(\Omega\). Since \(u_\varepsilon\ge0\), \(u_\varepsilon\not\equiv0\), and \(u_\varepsilon\in L^\infty(\Omega)\), there exists \(M>0\) such that \[0\le u_\varepsilon(x)\le M \quad\text{for a.e. }x\in\Omega .\] Since \(\gamma>p\), we have \[u_\varepsilon^{\gamma-1} = u_\varepsilon^{\gamma-p}u_\varepsilon^{p-1} \le M^{\gamma-p}u_\varepsilon^{p-1}.\] Consequently, \[a(x)u_\varepsilon^{q-1} - b(x)u_\varepsilon^{\gamma-1} \ge -\|b\|_\infty M^{\gamma-p}u_\varepsilon^{p-1} \quad\text{for a.e. }x\in\Omega .\] It follows from the equation for \(u_\varepsilon\) that, with some constant \(C>0\), \[-\varepsilon\Delta_p u_\varepsilon + C u_\varepsilon^{p-1} \ge0 \quad\text{in the weak sense in }\Omega .\] Equivalently, \[\Delta_p u_\varepsilon - \frac{C}{\varepsilon}u_\varepsilon^{p-1} \le0 \quad\text{in the weak sense in }\Omega .\] By the strong maximum principle for the \(p\)-Laplacian (see, for instance, [30]) and by the connectedness of \(\Omega\), we conclude that \[u_\varepsilon>0 \quad\text{in }\Omega .\]

Thus \(u_\varepsilon\) is a positive weak solution of problem 1 . Moreover, since \(u_\varepsilon\) is a global minimizer of \(\Phi_\varepsilon\), this solution is a ground state.

It remains to verify the sign of the second derivative of the function \(t\mapsto\Phi_\varepsilon(tu_\varepsilon)\) at \(t=1\). Put \[T_\varepsilon:=\int_\Omega |\nabla u_\varepsilon|^p\,dx, \qquad A_\varepsilon:=\int_\Omega a(x)u_\varepsilon^q\,dx, \qquad B_\varepsilon:=\int_\Omega b(x)u_\varepsilon^\gamma\,dx .\] Since \(u_\varepsilon\) is a critical point, we have \[\varepsilon T_\varepsilon-A_\varepsilon+B_\varepsilon=0.\] Moreover, \[\Phi_\varepsilon(u_\varepsilon)<0 .\] Using \(A_\varepsilon=\varepsilon T_\varepsilon+B_\varepsilon\), we obtain \[0> \Phi_\varepsilon(u_\varepsilon) = \varepsilon T_\varepsilon\left(\frac{1}{p}-\frac{1}{q}\right) + B_\varepsilon\left(\frac{1}{\gamma}-\frac{1}{q}\right).\] Hence \[B_\varepsilon> \frac{\gamma(q-p)}{p(\gamma-q)}\,\varepsilon T_\varepsilon .\] Consequently, \[\left.\frac{d^2}{dt^2}\Phi_\varepsilon(tu_\varepsilon)\right|_{t=1} = (p-q)\varepsilon T_\varepsilon+(\gamma-q)B_\varepsilon > 0 .\] Assertion (2) is proved.

We now prove (3). Introduce the truncated functional \[\Phi_\varepsilon^+(u) = \frac{\varepsilon}{p}\int_\Omega |\nabla u|^p\,dx - \frac{1}{q}\int_\Omega a(x)(u^+)^q\,dx + \frac{1}{\gamma}\int_\Omega b(x)(u^+)^\gamma\,dx, \qquad u^+:=\max\{u,0\}.\] The functional \(\Phi_\varepsilon^+\) satisfies the same coercivity estimates as \(\Phi_\varepsilon\) and satisfies the Palais–Smale condition.

Indeed, let \((u_n)\subset W^{1,p}_0(\Omega)\) be a Palais–Smale sequence for \(\Phi_\varepsilon^+\), that is, \[\Phi_\varepsilon^+(u_n)\;\text{is bounded}, \qquad D\Phi_\varepsilon^+(u_n)\to0 \quad\text{in }(W^{1,p}_0(\Omega))^* .\] By coercivity, \((u_n)\) is bounded in \(W^{1,p}_0(\Omega)\). Hence, passing to a subsequence, we may assume that \[u_n\rightharpoonup u \quad\text{weakly in }W^{1,p}_0(\Omega),\] and \[u_n\to u \quad\text{strongly in }L^r(\Omega), \qquad 1<r<p^* .\] In particular, \[u_n\to u \quad\text{strongly in }L^q(\Omega) \quad\text{and}\quad u_n\to u \quad\text{strongly in }L^\gamma(\Omega).\] Since the map \(\mathbb{R}\ni z\mapsto z^+\) is Lipschitz continuous, we also have \[u_n^+\to u^+ \quad\text{strongly in }L^q(\Omega) \quad\text{and}\quad u_n^+\to u^+ \quad\text{strongly in }L^\gamma(\Omega).\]

Substituting \(u_n-u\) into the derivative \(D\Phi_\varepsilon^+(u_n)\), we obtain \[\begin{align} o(1) &= D\Phi_\varepsilon^+(u_n)(u_n-u) \\ &= \varepsilon \int_\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot (\nabla u_n-\nabla u)\,dx \\ &\quad - \int_\Omega a(x)(u_n^+)^{q-1}(u_n-u)\,dx + \int_\Omega b(x)(u_n^+)^{\gamma-1}(u_n-u)\,dx . \end{align}\] The last two integrals tend to zero. For example, \[\left| \int_\Omega a(x)(u_n^+)^{q-1}(u_n-u)\,dx \right| \le \|a\|_\infty \|u_n^+\|_q^{q-1} \|u_n-u\|_q \to0,\] and, similarly, \[\int_\Omega b(x)(u_n^+)^{\gamma-1}(u_n-u)\,dx \to0 .\] Therefore \[\int_\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot (\nabla u_n-\nabla u)\,dx \to0 .\] We use the standard \(S^+\)-type property of the \(p\)-Laplacian: if \[u_n\rightharpoonup u \quad\text{weakly in }W^{1,p}_0(\Omega)\] and \[\limsup_{n\to+\infty} \int_\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot (\nabla u_n-\nabla u)\,dx \le0,\] then \[u_n\to u \quad\text{strongly in }W^{1,p}_0(\Omega).\] This property is a standard consequence of the monotonicity of the \(p\)-Laplacian and the uniform convexity of \(W^{1,p}_0(\Omega)\); see, for example, the results on monotone operators of type \(S^+\) in [31]. Applying this property to \((u_n)\), we obtain \[u_n\to u \quad\text{strongly in }W^{1,p}_0(\Omega).\] Thus \(\Phi_\varepsilon^+\) satisfies the Palais–Smale condition.

We next verify the mountain-pass geometry. By the Sobolev embedding, \[\Phi_\varepsilon^+(u) \ge \left( \frac{\varepsilon}{p} - C\|u\|_{W^{1,p}_0}^{q-p} \right) \|u\|_{W^{1,p}_0}^{p}.\] Choosing \(\rho>0\) sufficiently small, we obtain \[\Phi_\varepsilon^+(u)>0 \quad\text{for}\quad 0<\|u\|_{W^{1,p}_0}\le \rho .\] In particular, there exists \(\delta>0\) such that \[\Phi_\varepsilon^+(u)\ge\delta \quad\text{whenever}\quad \|u\|_{W^{1,p}_0}=\rho .\] On the other hand, the ground state \(u_\varepsilon\) found above is nonnegative and satisfies \[\Phi_\varepsilon^+(u_\varepsilon) = \Phi_\varepsilon(u_\varepsilon) <0 .\] Therefore \(\|u_\varepsilon\|_{W^{1,p}_0}>\rho\), and \(\Phi_\varepsilon^+\) has the mountain-pass geometry.

By the mountain pass theorem [32], there exists a critical point \(v_\varepsilon\in W^{1,p}_0(\Omega)\) such that \[\Phi_\varepsilon^+(v_\varepsilon) = c_\varepsilon := \inf_{\xi\in\Gamma} \max_{t\in[0,1]} \Phi_\varepsilon^+(\xi(t)) \ge \delta>0,\] where \[\Gamma = \left\{ \xi\in C([0,1];W^{1,p}_0(\Omega)): \xi(0)=0,\;\xi(1)=u_\varepsilon \right\}.\] In particular, \[v_\varepsilon\not\equiv0 .\]

We show that \(v_\varepsilon\ge0\). Since \[D\Phi_\varepsilon^+(v_\varepsilon)=0,\] we take \(-v_\varepsilon^-\) as a test function, where \[v_\varepsilon^-:=\max\{-v_\varepsilon,0\}.\] The nonlinear terms involving \(v_\varepsilon^+\) vanish, and we obtain \[\varepsilon \int_\Omega |\nabla v_\varepsilon^-|^p\,dx = 0 .\] Consequently, \[v_\varepsilon^-=0, \qquad v_\varepsilon\ge0 \quad\text{in }\Omega .\] Thus \(v_\varepsilon^+=v_\varepsilon\), and \(v_\varepsilon\) is a critical point of the original functional \(\Phi_\varepsilon\). Hence \(v_\varepsilon\) is a weak solution of problem 1 .

Since \[\Phi_\varepsilon(v_\varepsilon) = \Phi_\varepsilon^+(v_\varepsilon) = c_\varepsilon > 0,\] this solution is distinct from the ground state \(u_\varepsilon\), because \[\Phi_\varepsilon(u_\varepsilon)<0 .\]

As above, the a priori estimates and interior regularity results for the \(p\)-Laplacian imply that \[v_\varepsilon\in C^{1,\kappa}_{\mathrm{loc}}(\Omega)\] for some \(\kappa\in(0,1)\). Finally, since \(v_\varepsilon\ge0\), \(v_\varepsilon\not\equiv0\), and \[-\varepsilon\Delta_p v_\varepsilon + C v_\varepsilon^{p-1} \ge0 \quad\text{in the weak sense in }\Omega\] with some constant \(C>0\), the strong maximum principle for the \(p\)-Laplacian (see, for instance, [30]) gives \[v_\varepsilon>0 \quad\text{in }\Omega .\] Assertion (3) is proved.

4 Proof of Theorem 2↩︎

Set \[J(u):= -\frac{1}{q}\int_\Omega a(x)|u|^q\,dx + \frac{1}{\gamma}\int_\Omega b(x)|u|^\gamma\,dx .\] For \(s\ge0\), denote \[j_x(s):= -\frac{a(x)}{q} s^q+\frac{b(x)}{\gamma} s^\gamma .\] Then \[J(u)=\int_\Omega j_x(|u(x)|)\,dx .\]

Since \(a(x)\ge\sigma_a>0\) and \(b(x)\ge\sigma_b>0\) for a.e. \(x\in\Omega\), the function \(s\mapsto j_x(s)\) has a unique positive global minimizer on \([0,+\infty)\), namely \[\bar u_0(x) = \left(\frac{a(x)}{b(x)}\right)^{1/(\gamma-q)} .\] Hence \[\label{eq:J-min} J(u)\ge J(\bar u_0)\tag{19}\] for every \(u\in L^\gamma(\Omega)\). Moreover, the assumptions \(a,b\in L^\infty(\Omega)\), \(a\ge\sigma_a>0\), and \(b\ge\sigma_b>0\) imply that \[\bar u_0\in L^\infty(\Omega), \qquad \bar u_0>0 \quad\text{for a.e. }x\in\Omega .\]

Let \(u_\varepsilon\) be a positive ground state of problem 1 , \(0<\varepsilon<\varepsilon_e^*\). We first note that \(u_\varepsilon\) is a global minimizer of \(\Phi_\varepsilon\) on \(W^{1,p}_0(\Omega)\). Indeed, by the proof of Theorem 1, the functional \(\Phi_\varepsilon\) has a global minimizer \(z_\varepsilon\), and this minimizer is a weak solution of 1 . Since \(u_\varepsilon\) is a ground state, we have \[\Phi_\varepsilon(u_\varepsilon) \le \Phi_\varepsilon(z_\varepsilon).\] On the other hand, by the global minimality of \(z_\varepsilon\), \[\Phi_\varepsilon(z_\varepsilon) \le \Phi_\varepsilon(u_\varepsilon).\] Thus \[\Phi_\varepsilon(u_\varepsilon) = \inf_{u\in W^{1,p}_0(\Omega)}\Phi_\varepsilon(u).\] Therefore, from 19 , we obtain \[\label{eq:lower-energy} \Phi_\varepsilon(u_\varepsilon) \ge J(u_\varepsilon) \ge J(\bar u_0).\tag{20}\]

On the other hand, since \(C_0^\infty(\Omega)\) is dense in \(L^\gamma(\Omega)\), for every \(\delta>0\) one can choose a nonnegative function \(w_\delta\in C_0^\infty(\Omega)\) such that \[\|w_\delta-\bar u_0\|_\gamma<\delta .\] Then, by the global minimality of \(u_\varepsilon\), \[\Phi_\varepsilon(u_\varepsilon) \le \Phi_\varepsilon(w_\delta) = \frac{\varepsilon}{p}\int_\Omega |\nabla w_\delta|^p\,dx + J(w_\delta).\] Consequently, \[\limsup_{\varepsilon\to0^+} \Phi_\varepsilon(u_\varepsilon) \le J(w_\delta).\] Since \(q<\gamma\) and \(\Omega\) is bounded, the functional \(J\) is continuous with respect to strong convergence in \(L^\gamma(\Omega)\). Passing to the limit as \(\delta\to0\), we get \[\label{eq:upper-energy} \limsup_{\varepsilon\to0^+} \Phi_\varepsilon(u_\varepsilon) \le J(\bar u_0).\tag{21}\] It follows from 20 and 21 that \[\label{eq:energy-limit} \Phi_\varepsilon(u_\varepsilon) \to J(\bar u_0) \quad\text{as } \varepsilon\to0^+ .\tag{22}\] Moreover, \[0 \le J(u_\varepsilon)-J(\bar u_0) \le \Phi_\varepsilon(u_\varepsilon)-J(\bar u_0),\] and therefore \[\label{eq:J-limit} J(u_\varepsilon) \to J(\bar u_0) \quad\text{as } \varepsilon\to0^+ .\tag{23}\]

We now prove that \(u_\varepsilon\to\bar u_0\) in measure. First observe that the family \(\{u_\varepsilon\}\) is bounded in \(L^\gamma(\Omega)\) as \(\varepsilon\to0^+\). Indeed, since \(a,b\in L^\infty(\Omega)\), \(b\ge\sigma_b>0\), and \(q<\gamma\), there exist constants \(c_1,c_2>0\), independent of \(x\) and \(s\ge0\), such that \[j_x(s) = -\frac{a(x)}{q} s^q+\frac{b(x)}{\gamma} s^\gamma \ge c_1s^\gamma-c_2 .\] This estimate and 23 imply that \(\{u_\varepsilon\}\) is bounded in \(L^\gamma(\Omega)\) for all sufficiently small \(\varepsilon>0\).

Next, since \(u_\varepsilon\ge0\), from 23 we have \[\label{eq:integral-gap} \int_\Omega \bigl( j_x(u_\varepsilon(x))-j_x(\bar u_0(x)) \bigr)\,dx \to0 .\tag{24}\] The integrand is nonnegative by the definition of \(\bar u_0(x)\).

Put \[\hat{a}:=\|a\|_\infty, \qquad \hat{b}:=\|b\|_\infty .\] Then, for a.e. \(x\in\Omega\), \[0<\rho_- := \left(\frac{\sigma_a}{\hat{b}}\right)^{1/(\gamma-q)} \le \bar u_0(x) \le \left(\frac{\hat{a}}{\sigma_b}\right)^{1/(\gamma-q)} =:\rho_+ <+\infty .\]

Fix \(\eta>0\). We claim that there exists \(\kappa_\eta>0\) such that \[\label{eq:uniform-separation} j_x(s)-j_x(\bar u_0(x)) \ge \kappa_\eta\tag{25}\] for all \(s\ge0\) satisfying \[|s-\bar u_0(x)|\ge\eta,\] and for a.e. \(x\in\Omega\).

Consider the family of functions \[j_{\alpha,\beta}(s) := -\frac{\alpha}{q}s^q+\frac{\beta}{\gamma}s^\gamma, \qquad (\alpha,\beta)\in[\sigma_a,\hat{a}]\times[\sigma_b,\hat{b}].\] For every pair \((\alpha,\beta)\), the function \(j_{\alpha,\beta}\) has a unique positive global minimizer \[\rho(\alpha,\beta) = \left(\frac{\alpha}{\beta}\right)^{1/(\gamma-q)} ,\] and \[\rho(\alpha,\beta)\in[\rho_-,\rho_+].\]

First observe that \[j_{\alpha,\beta}(s)-j_{\alpha,\beta}(\rho(\alpha,\beta)) \to+\infty \quad\text{as }s\to+\infty\] uniformly with respect to \[(\alpha,\beta)\in[\sigma_a,\hat{a}]\times[\sigma_b,\hat{b}].\] Indeed, \[j_{\alpha,\beta}(s) \ge -\frac{\hat{a}}{q}s^q+\frac{\sigma_b}{\gamma}s^\gamma ,\] whereas the function \[(\alpha,\beta) \mapsto j_{\alpha,\beta}(\rho(\alpha,\beta))\] is continuous on the compact set \[[\sigma_a,\hat{a}]\times[\sigma_b,\hat{b}].\] Thus there exists \(M_0>0\) such that \[\bigl|j_{\alpha,\beta}(\rho(\alpha,\beta))\bigr| \le M_0\] for all \((\alpha,\beta)\in[\sigma_a,\hat{a}]\times[\sigma_b,\hat{b}]\). Hence \[j_{\alpha,\beta}(s)-j_{\alpha,\beta}(\rho(\alpha,\beta)) \ge \frac{\sigma_b}{\gamma}s^\gamma - \frac{\hat{a}}{q}s^q - M_0 .\] Since \(\gamma>q\), the right-hand side tends to \(+\infty\) as \(s\to+\infty\), uniformly with respect to \((\alpha,\beta)\). Therefore, one can choose \(M>0\) such that \[\label{eq:large-s-separation} j_{\alpha,\beta}(s)-j_{\alpha,\beta}(\rho(\alpha,\beta)) \ge1\tag{26}\] for all \(s\ge M\) and all \[(\alpha,\beta)\in[\sigma_a,\hat{a}]\times[\sigma_b,\hat{b}].\]

It remains to consider the region \(0\le s\le M\). Let \[K_\eta := \left\{ (\alpha,\beta,s): \alpha\in[\sigma_a,\hat{a}],\; \beta\in[\sigma_b,\hat{b}],\; s\in[0,M],\; |s-\rho(\alpha,\beta)|\ge\eta \right\}.\] If \(K_\eta\ne\varnothing\), then \(K_\eta\) is compact. The function \[(\alpha,\beta,s) \mapsto j_{\alpha,\beta}(s)-j_{\alpha,\beta}(\rho(\alpha,\beta))\] is continuous and strictly positive on \(K_\eta\), since equality to zero is possible only when \(s=\rho(\alpha,\beta)\), and such points are excluded by the condition \(|s-\rho(\alpha,\beta)|\ge\eta\). Therefore \[\kappa_\eta' := \min_{K_\eta} \bigl[ j_{\alpha,\beta}(s)-j_{\alpha,\beta}(\rho(\alpha,\beta)) \bigr] >0 .\] If \(K_\eta=\varnothing\), set, for instance, \(\kappa_\eta'=1\). In both cases, define \[\kappa_\eta:=\min\{\kappa_\eta',1\}>0.\] Then 26 and the definition of \(\kappa_\eta'\) imply that \[j_{\alpha,\beta}(s)-j_{\alpha,\beta}(\rho(\alpha,\beta)) \ge \kappa_\eta\] for all \(s\ge0\) and all \((\alpha,\beta)\in[\sigma_a,\hat{a}]\times[\sigma_b,\hat{b}]\) such that \[|s-\rho(\alpha,\beta)|\ge\eta .\] Applying this assertion with \[\alpha=a(x), \qquad \beta=b(x), \qquad \rho(\alpha,\beta)=\bar u_0(x),\] we obtain 25 .

Consequently, \[\kappa_\eta \operatorname{meas} \{x\in\Omega:\;|u_\varepsilon(x)-\bar u_0(x)|\ge\eta\} \le \int_\Omega \bigl( j_x(u_\varepsilon(x))-j_x(\bar u_0(x)) \bigr)\,dx .\] By 24 , the right-hand side tends to zero. Hence, for every \(\eta>0\), \[\operatorname{meas} \{x\in\Omega:\;|u_\varepsilon(x)-\bar u_0(x)|\ge\eta\} \to0,\] that is, \[\label{eq:measure-convergence} u_\varepsilon\to\bar u_0 \quad\text{in measure in }\Omega .\tag{27}\]

We now prove strong convergence in \(L^r(\Omega)\) for \(1\le r<\gamma\). Since \(\{u_\varepsilon\}\) is bounded in \(L^\gamma(\Omega)\) and \(\bar u_0\in L^\gamma(\Omega)\), the family \(\{u_\varepsilon-\bar u_0\}\) is bounded in \(L^\gamma(\Omega)\). Therefore, for every \(1\le r<\gamma\), the family \(\{|u_\varepsilon-\bar u_0|^r\}\) is bounded in \(L^{\gamma/r}(\Omega)\), where \(\gamma/r>1\). Hence it is uniformly integrable. Together with 27 , Vitali’s convergence theorem gives \[\int_\Omega |u_\varepsilon-\bar u_0|^r\,dx\to0 .\] Thus \[u_\varepsilon\to\bar u_0 \quad\text{strongly in }L^r(\Omega), \qquad 1\le r<\gamma .\]

Moreover, \[u_\varepsilon\rightharpoonup\bar u_0 \quad\text{weakly in }L^\gamma(\Omega).\] Indeed, the family \(\{u_\varepsilon\}\) is bounded in the reflexive space \(L^\gamma(\Omega)\). Hence every sequence \(\varepsilon_m\to0^+\) contains a subsequence converging weakly in \(L^\gamma(\Omega)\). By 27 , any such weak limit must coincide with \(\bar u_0\). Therefore the whole family \(u_\varepsilon\) converges weakly to \(\bar u_0\) in \(L^\gamma(\Omega)\). Since \(\Omega\) is bounded, it follows in particular that \[u_\varepsilon-\bar u_0\rightharpoonup0 \quad\text{weakly in }L^r(\Omega), \qquad 1<r\le\gamma .\]

Finally, by the definition of \(\bar u_0\), we have \[a(x)\bar u_0^{q-1}(x) - b(x)\bar u_0^{\gamma-1}(x) = 0 \quad\text{for a.e. }x\in\Omega .\] Thus \(\bar u_0\) satisfies the limiting equation 1 with \(\varepsilon=0\): \[a(x)\bar u_0^{q-1} = b(x)\bar u_0^{\gamma-1} \quad\text{a.e. in }\Omega .\] The theorem is proved.

References↩︎