\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 60, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \thanks{\copyright 2003 Southwest Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE--2003/60\hfil On $\Gamma$-convergence] {On $\Gamma$-convergence for problems of jumping type} \author[Alessandro Groli\hfil EJDE--2003/60\hfilneg] {Alessandro Groli} \address{ Alessandro Groli \hfill\break Department of Mathematics, University of Brescia, Via Valotti 9, 25133 Brescia, Italy} \email{alessandro.groli@ing.unibs.it} \date{} \thanks{Submitted February 19, 2003. Published May 23, 2003.} \subjclass[2000]{49J45, 58E05} \keywords{$\Gamma$-convergence, jumping problems, nonsmooth critical point theory} \begin{abstract} The convergence of critical values for a sequence of functionals $(f_h)$ $\Gamma$-converging to a functional $f_{\infty}$ is studied. These functionals are related to a classical ``jumping problem'', in which the position of two real parameters $\alpha,\beta$ plays a fundamental role. We prove the existence of at least three critical values for $f_h$, when $\alpha$ and $\beta$ satisfy the usual assumption with respect to $f_\infty$, but not with respect to $f_h$. \end{abstract} \maketitle \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \numberwithin{equation}{section} \allowdisplaybreaks \section{Introduction} Let $(f_h)$ be a sequence of functionals from $H_0^1(\Omega)$ to $\mathbb{R}$ and $f_{\infty}$ a functional from $H_0^1(\Omega)$ to $\mathbb{R}$. It is well known that the convergence of (possible) minima of $f_h$ to those of $f_{\infty}$ can be studied in an efficient way by the notion of $\Gamma$-convergence \cite{Dal, DegFra} (epiconvergence, in the language of \cite{Att}). The problem of the convergence of critical points, on the contrary, is much less clarified. A certain number of results is available in the literature, dealing with the case in which $f_h$ is $\Gamma$-convergent to $f_{\infty}$ and satisfies suitable uniform assumptions (see e.g. \cite{Homoto, DMT, DE} and references therein). In particular, let us remark that the applications to PDE's, so far considered, concern only functionals of the calculus of variations whose principal part is convex. We are interested in a further case, which is not covered in the literature and is particularly interesting for critical point theory: that of ``jumping problems''. It can be considered as a perturbation of the functional $f_\infty: H_0^1(\Omega)\to \mathbb{R}$ defined as \[ f_{\infty}(u) = \frac{1}{2}\int_{\Omega}\sum_{i,j=1}^n A^{(\infty)}_{ij}(x)D_iuD_ju \,dx -\frac{\alpha}{2}\int_{\Omega} (u^+)^2 \, dx -\frac{\beta}{2}\int_{\Omega} (u^-)^2 \, dx +\int_{\Omega} \phi_1 u\, dx, \] where $\beta <\alpha$ and $\phi_1$ is a positive eigenfunction of $-\sum D_j(A^{(\infty)}_{ij}D_iu)$ with homogeneous Dirichlet condition. The simplest type of perturbation, extensively considered in the literature, amounts to consider \[\begin{aligned} f_h(u) = & \frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n A^{(\infty)}_{ij}(x)D_iuD_ju \,dx -\frac{\alpha}{2}\int_{\Omega} (u^+)^2 \,dx\\ & -\frac{\beta}{2}\int_{\Omega} (u^-)^2 \,dx- \int_\Omega\frac{G_0(x,t_hu)}{t_h^2}\, dx + \int_{\Omega} \phi_1 u \, dx, \end{aligned} \] where $t_h \to +\infty$, and \[ \lim_{|s| \to +\infty} \frac{D_sG_0(x,s)}{s}=0\,. \] In such a case, very refined results have been obtained, starting from the pioneering paper \cite{ambr}, (see e.g. \cite{Hof, marmicpis, masa, solim} and references therein). \\ More recently, some results have been obtained when \begin{align*} f_h(u) = & \frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n a_{ij}(x,t_h u)D_iuD_ju \, dx -\frac{\alpha}{2}\int_{\Omega} (u^+)^2 \, dx\\ & -\frac{\beta}{2}\int_{\Omega} (u^-)^2 \, dx- \int_\Omega\frac{G_0(x,t_hu)}{t_h^2}\, dx + \int_{\Omega} \phi_1 u \, dx, \end{align*} where $t_h$ and $G_0$ are as above (see \cite{cj, cthree}). Observe that in this case the principal part is no longer convex. Here we are interested in a more general perturbation of the form \begin{align*} f_h(u) = & \frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n a^{(h)}_{ij}(x,u)D_iuD_ju \, dx -\frac{\alpha}{2}\int_{\Omega} (u^+)^2 \, dx\\ & -\frac{\beta}{2}\int_{\Omega} (u^-)^2 \, dx- \int_\Omega\frac{G_0(x,t_hu)}{t_h^2}\, dx + \int_{\Omega} \phi_1 u \, dx. \end{align*} Actually, for the sake of simplicity, we will consider only the case $G_0=0$, being the perturbation of the principal part the most interesting feature. Let us mention that the result we are interested in, namely the existence of at least three critical points for $f_h$, is well known if $\beta<\mu_1^{(h)} <\mu_2^{(h)} <\alpha$, where $\mu_1^{(h)},\mu_2^{(h)}$ are the first two eigenvalues of $-\sum D_j(A_{ij}^{(h)}D_i u)$, then \[ \lim_{s \to +\infty} a^{(h)}_{ij}(x,s)= \lim_{s \to -\infty} a^{(h)}_{ij}(x,s)= A^{(h)}_{ij}(x) \] (see \cite{cthree}). The point is that, under our assumptions, we have $\beta <\mu_{1}^{(h)}$. But it may happen that $\alpha<\mu_{2}^{(h)}$ for any $h \in \mathbb{N}$ (see Example \ref{ex}). Nevertheless, the hypothesis that $\alpha>\mu_{2}$, where $\mu_2$ is the second eigenvalue of $-\sum D_j(A_{ij}^{(\infty)}D_i u)$ combined with the $\Gamma$-convergence of $f_h$ to $f_\infty$, is sufficient to ensure, for $h$ large, the existence of at least three critical points of $f_h$. In some sense, we find a genuine effect of $\Gamma$-convergence, which cannot be deduced by the usual study of the position of $\beta$ and $\alpha$ with respect to the spectrum of $-\sum D_j(A_{ij}^{(h)}D_i u)$. Let us also mention that a relevant question, in jumping problem, is the position of $\alpha$ and $\beta$ with respect to the Fu\v{c}ik spectrum (see e.g. \cite{Dan}). However this seems to be important mainly for the verification of the Palais-Smale condition, while the persistence of the geometrical conditions on the functional under $\Gamma$-convergence is the key point in our problem. This paper is organized as follows. In section \ref{sct:2} we recall some notions of nonsmooth analysis and prove a nonsmooth version of the classical ``local saddle theorem''. In section \ref{sct:3} we present the problem and the main result. Section \ref{sct:4} is devoted to show some minmax estimates which allow us to prove the main theorem in section \ref{sct:5}. \section{Tools of nonsmooth analysis}\label{sct:2} In this section, we recall some by-products of the nonsmooth critical point theory developed in \cite{cdm,dm}. Let X be a metric space endowed with the metric $d$ and $r>0$. Let us set $B_r(u)=\{v\in X: d(u,v) < r\}$ and $S_r(u)=\{v\in X: d(u,v) =r\}$. \begin{definition}\label{pendeb} \rm Let $f: X \to \mathbb{R}$ be a continuous function and let $u \in X$. We denote by $|df|(u)$ the supremum of the $\sigma'$s in $[0,+\infty[$ such that there exist $\delta>0$ and a continuous map ${\mathcal H}: B_{\delta}(u)\times [0,\delta] \to X$ satisfying \[ d({\mathcal H}(v,t),v)\leq t, \quad f({\mathcal H}(v,t)) \leq f(v) -\sigma t, \] whenever $v \in B_{\delta}(u)$ and $t \in [0,\delta]$. The extended real number $|df|(u)$ is called the {\rm weak slope} of $f$ at $u$. \end{definition} The following two definitions are related to the notion above. \begin{definition}\label{puntocritico} \rm Let $f: X \to \mathbb{R}$ be a continuous function. An element $u \in X$ is said to be {\rm critical point } of $f$, if $|df|(u)=0$. A real number $c$ is said to be a {\rm critical value} for $f$, if there exists a critical point $u \in X$ of $f$ such that $f(u)=c$. Otherwise $c$ is said to be a {\rm regular value} of $f$. \end{definition} \begin{definition}\label{palais} \rm Let $f: X \to \mathbb{R}$ be a continuous function and $c \in \mathbb{R}$. The function $f$ is said to satisfy {\rm the Palais-Smale condition at level c} ($(PS)_c$ for short), if every sequence $(u_h)$ in $X$ with $|df|(u_h) \to 0$ and $f(u_h)\to c$ admits a subsequence converging in $X$. \end{definition} The next result is an adaptation to a continuous functional of the classical local saddle theorem (see e.g. \cite{marmicpis}). \begin{theorem}\label{selleinscala} Let $X$ be a Banach space and $f:X \to \mathbb{R}$ be a continuous function. Assume that there exist two closed subspaces $X_1,X_2$ of $X$ with $\dim X_1 <+\infty$ and $X=X_1\oplus X_2$. Let $u_0 \in X$ and $U_1,U_2$ be two bounded neighborhoods of $0$ in respectively $X_1$ and $X_2$ with $U_2$ convex. Suppose that \[ \sup f(u_0 + \partial U_1) < a= \inf f(u_0 + \overline{U_2}),\quad b= \sup f(u_0 + \overline{U_1}) < \inf f(u_0 + \partial U_2), \] and $f$ satisfies $(PS)_c$ for any $c \in [a,b]$. Then there exists at least a critical point for $f$ in $f^{-1}([a,b])$. \end{theorem} \begin{proof} Without loss of generality, we can suppose $u_0=0$. We argue by contradiction and assume that there are no critical values for $f$ in $[a,b]$. Since $f$ satisfies $(PS)_c$ for every $c \in [a,b]$, it is readily seen that, for some $\varepsilon>0$, there are no critical values for $f$ in $[a-\varepsilon,b]$ and that $f$ satisfies $(PS)_c$ for any $c \in [a-\varepsilon,b]$. By \cite[Theorem 2.15]{cdm} or \cite[Theorem 1.1.14]{cd} there exists a continuous map $ \eta:X\times~[0,1] \to X$ such that \begin{gather*} \eta(u,0)=u \quad \forall \,u \in X,\\ \eta(u,t)=u \quad \forall \,t \in [0,1],\, \forall u \,\in f^{a-\varepsilon},\\ \eta(u,1) \in f^{a-\varepsilon} \quad \forall \,u \in f^b,\\ f(\eta(u,t)) \leq f(u) \quad \forall t \in [0,1], \quad \forall u \in X. \end{gather*} Since $\overline{U_1} \subset f^b$, $\eta(\overline{U_1}\times \{1\}) \subset f^{a-\varepsilon}$. On the other hand, since $f^{a-\varepsilon} \cap \overline{U_2}=\emptyset$, it follows that \begin{equation}\label{eq:noninterseco} \eta(\overline{U_1}\times \{1\}) \cap \overline{U_2}=\emptyset. \end{equation} Now consider the continuous map \[\begin{array}{rcl} \Phi: [-1,1]\times \overline{U_1}& \to &\mathbb{R}\times X_1\\ (s,u) &\mapsto & \left(\rho_{U_2}(P_2\eta(u,1))+s,P_1\eta(u,1)\right) \end{array}\] where $P_i:X \to X_i$ ($i=1,2$) are the projections of $ X$ onto $X_i$ and $\rho_{U_2}: X_2 \to [0,+\infty[$ is the Minkowski functional associated with $U_2$. Since $(0,0) \notin \Phi\left(\partial(]-1,1[\times U_1)\right)$, the Brouwer degree (see e.g. \cite{Dei}) \[\deg\left(\Phi,]-1,1[\times U_1,(0,0)\right) \] is well defined. Moreover the continuous function defined by \[{\mathcal H}((s,u),t)=\left(\rho_{U_2}(P_2\eta(u,t))+s, P_1\eta(u,t)\right) \] is a homotopy between the identity map and $\Phi$. Since $(0,0)\notin {\mathcal H}\left(\partial(]-1,1[\times U_1)\times [0,1]\right)$, it follows that \[\deg\left(\Phi,]-1,1[\times U_1,(0,0)\right)=1. \] Therefore, there exists $(s,u) \in \,]-1,1[\times U_1$ such that $\Phi(s,u)=(0,0)$. Hence we have $\eta(u,1) \in X_2$ and $\rho_{U_2}(\eta(u,1))=-s$, namely $\rho_{U_2}(\eta(u,1)) \leq 1$. Therefore, $\eta(u,1) \in \overline{U_2}$ and we have \[\eta(U_1 \times \{1\}) \cap \overline{U_2} \neq \emptyset \] which contradicts \eqref{eq:noninterseco}. \end{proof} Let us recall the notion of $\Gamma$-convergence (epiconvergence in the language of \cite{Att}) from \cite{DegFra}. \begin{definition}\label{Gamma} \rm Consider a topological space $X$. For any $h\in\mathbb{N}\cup\{+\infty\},$ let $g_h: X \to \mathbb{R} \cup \{+\infty\}$ be a function. According to \cite{Att,DegFra}, we write that \[g_{\infty}= \Gamma(X^-)\lim_h g_h\] if the following facts hold: \begin{itemize} \item[(i)] if $(u_h)$ is a sequence in $X$ convergent to $u$, we have $ g_{\infty}(u) \leq \liminf_h g_h(u_h)$; \item[(ii)] for every $u \in X$, there exists a sequence $(u_h)$ in $X$ convergent to $u$ such that $g_{\infty}(u)=\lim_h g_h(u_h)$. \end{itemize} \end{definition} \section{Position of the problem and main result}\label{sct:3} Let $\Omega$ be a connected bounded open subset of $\mathbb{R}^{n}$ (for the sake of simplicity we suppose $n\geq 3$). We assume that, for every $h \in \mathbb{N}$, the functions $a^{(h)}_{ij}: \Omega \times \mathbb{R} \to \mathbb{R}$ and the function $A_{ij}^{(\infty)}: \Omega \to \mathbb{R}$ ($1\leq i,j \leq n$) satisfy the following conditions: \begin{itemize} \item[(A1)] For all $s \in \mathbb{R}$, $a^{(h)}_{ij}(\cdot,s)$ and $A^{(\infty)}_{ij}(\cdot)$ are measurable; for a.e. $x \in \Omega$, $a^{(h)}_{ij}(x,\cdot)$ is of class $C^1$; for a.e. $x \in \Omega$, $\forall s \in \mathbb{R}$, $a^{(h)}_{ij}(x,s)=a^{(h)}_{ji}(x,s)$, $A^{(\infty)}_{ij}(x)=A^{(\infty)}_{ji}(x)$. \item[(A2)] There exists $C>0$ such that for each $h \in \mathbb{N}$, for a.e. $x \in \Omega$, for all $s \in \mathbb{R}$, for all $\xi \in \mathbb{R}^{n}$, $1\leq i,j \leq n$, \[ |a^{(h)}_{ij}(x,s)|\leq C,\quad |A^{(\infty)}_{ij}(x)|\leq C, \quad \Big|\sum_{i,j=1}^n sD_s a^{(h)}_{ij}(x,s)\xi_i\xi_j\Big|\leq C|\xi|^2. \] \item[(A3)] There exists $\nu >0$ such that for each $h \in \mathbb{N}$, for a.e. $x \in \Omega$, for all $s \in \mathbb{R}$, for all $\xi \in \mathbb{R}^{n}$, \[ \sum_{i,j=1}^n a^{(h)}_{ij}(x,s) \xi_i \xi_j \geq \nu|\xi|^2, \quad \sum_{i,j=1}^n A^{(\infty)}_{ij}(x) \xi_i \xi_j \geq \nu|\xi|^2. \] \item[(A4)] For each $h \in \mathbb{N}$, there exists $R_h>0$ such that for a.e. $x \in \Omega$, for all $s \in \mathbb{R}$, for all $\xi \in \mathbb{R}^{n}$, \[ |s|>R_h \Rightarrow \sum_{i,j=1}^n s D_sa^{(h)}_{ij}(x,s) \xi_i \xi_j \geq 0. \] \item[(A5)] For a.e. $x \in \Omega$, assume that \[ \lim_{s \to +\infty} a^{(h)}_{ij}(x,s)= \lim_{s \to -\infty} a^{(h)}_{ij}(x,s)= A^{(h)}_{ij}(x) \] (observe that by (A4) such limits exist). \item[(A6)] For all $h \in \mathbb{N}$ there exists uniformly Lipschitz continuous bounded functions $\psi_h:\mathbb{R} \to [0,+\infty[$ such that for a.e. $x \in \Omega$, for all $s \in \mathbb{R}$ and for every $\xi \in \mathbb{R}^{n}$ \[ \sum_{i,j=1}^n sD_s a^{(h)}_{ij}(x,s) \xi_i \xi_j \leq 2s \psi_h'(s) \sum_{i,j=1}^n a^{(h)}_{ij}(x,s) \xi_i \xi_j. \] Also assume that \begin{equation} \label{H1} \begin{aligned} &\int_{\Omega} \sum_{i,j=1}^n A^{(\infty)}_{ij}(x)D_iuD_ju \, dx\\ &= \Gamma(w-H_0^1(\Omega)^-)\lim_h \int_{\Omega} \sum_{i,j=1}^n a^{(h)}_{ij}(x,u)D_iuD_ju \, dx \end{aligned} \end{equation} where $w-H_0^1(\Omega)$ denotes the space $H_0^1(\Omega)$ endowed with the weak topology. Let $\mu_k$, $\mu^{(h)}_k$ denote the eigenvalues of respectively the operators $-\sum D_j(A^{(\infty)}_{ij}D_iu)$ and $-\sum D_j(A^{(h)}_{ij}D_iu)$ with homogeneous Dirichlet condition and $\phi_k$, $\phi^{(h)}_k$ the corresponding eigenfunctions. It is well known (see \cite{gt}) that $\phi_1 \in H_0^1(\Omega) \cap L^{\infty}(\Omega) \cap C(\Omega)$ and that we can take $\phi_1(x) >0$ for every $x \in \Omega$ and $\int_{\Omega}\phi_1^2\,dx=1$. \item[(A7)] Assume that $\lim_{h }\mu_1^{(h)}= \mu_1$. \end{itemize} Our purpose in this article is to study the existence of weak solutions of the family of problems: \begin{equation} \label{Ph} \begin{gathered} -\sum_{i,j=1}^nD_j(a^{(h)}_{ij}(x,u)D_iu) + \frac{1}{2} \sum_{i,j=1}^n D_s a^{(h)}_{ij}(x,u)D_iuD_ju = \alpha u^+- \beta u^- - \phi_1, \\ u \in H_0^1(\Omega), \end{gathered} \end{equation} where $\alpha,\beta$ are two real numbers, $u^+=\max\{u,0\}$, $u^-=\max\{-u,0\}$. Under the assumptions above, we shall prove is the following result. \begin{theorem}\label{es3sol} Assume that $\beta< \mu_1$ and $\alpha >\mu_{2}$. Then there exists $\overline h$ in $\mathbb{N}$ such that for all $h \geq \overline h$, the problem $(\ref{Ph})$ has at least three weak solutions in $H_0^1(\Omega)$. \end{theorem} For $\alpha >\mu_2^{(h)}$, this result corresponds to \cite[Theorem 1.1]{cthree}; however our assumptions do not imply that $\alpha>\mu_2^{(h)}$ for large $h$. As the following example shows, it may happen that $\mu_2 <\mu_2^{(h)}$ (and hence $\alpha \in ]\mu_2, \mu_2^{(h)}[$). \begin{example}\label{ex} \rm Let $\Omega=]0,\pi[$ and define the functions $a^{(h)}_{ij}(x,s)$ such that: for $x \in ]0,\frac{\pi}{2}[$, \[a^{(h)}_{ij}(x,s)= \begin{cases} \gamma \delta_{ij}(x)& s \in ]-h,h[,\\ \delta_{ij}(x)& s \in \mathbb{R} \setminus [-2h,2h]; \end{cases} \] for $x \in ]\frac{\pi}{2}, \pi[$ \[a^{(h)}_{ij}(x,s)= \begin{cases} \eta \delta_{ij}(x)& s\in ]-h,h[,\\ \delta_{ij}(x)& s \in \mathbb{R} \setminus [-2h,2h], \end{cases} \] where $\delta_{ij}(x)=1$ if $i=j$, $\delta_{ij}(x)=0$ if $i \neq j$ and $\gamma,\eta \in \mathbb{R}$. Then, $A^{(h)}_{ij}(x)=\delta_{ij}(x)$. The eigenvalues $\mu_k^{(h)}$ of the Dirichlet problem \begin{gather*} -u''=\mu u,\\ u(0)=u(\pi)=0, \end{gather*} are $\mu_k^{(h)}= k^2$, for all $k \geq 1$. On the other hand, all the assumptions of Theorem \ref{es3sol} are satisfied with \[ A_{ij}^{(\infty)}(x)= \begin{cases} \gamma \delta_{ij}(x)& 0 \arctan \sqrt 3= \frac{\pi}{3}$, it follows that \[ \mu_2= \left(\frac{\pi-\arctan \sqrt 5}{\arctan \sqrt 5}\right)^2 < 4 = \mu_2^{(h)}. \] \end{example} \section{Minmax estimates}\label{sct:4} We introduce the functionals $f_h,f_{\infty}, \widehat f_{\infty} : H_0^1(\Omega) \to \mathbb{R}$, \[f_h(u) = \frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n a^{(h)}_{ij}(x,u)D_iuD_ju \, dx -\frac{\alpha}{2}\int_{\Omega} (u^+)^2 \, dx -\frac{\beta}{2}\int_{\Omega} (u^-)^2 \, dx+ \int_{\Omega} \phi_1 u \, dx,\] \[f_{\infty}(u) = \frac{1}{2}\int_{\Omega}\sum_{i,j=1}^n A^{(\infty)}_{ij}(x)D_iuD_ju \, dx -\frac{\alpha}{2}\int_{\Omega} (u^+)^2 \, dx -\frac{\beta}{2}\int_{\Omega} (u^-)^2 \, dx +\int_{\Omega} \phi_1 u \, dx, \\ \] \[\widehat f_{\infty}(u) = \frac{1}{2}\int_{\Omega}\sum_{i,j=1}^n A^{(\infty)}_{ij}(x)D_iuD_ju \, dx -\frac{\alpha}{2}\int_{\Omega} u^2 \, dx +\int_{\Omega} \phi_1 u \, dx. \\ \] For later use, we also introduce $g_h,g_{\infty} : H_0^1(\Omega) \to \mathbb{R}$ as the ``principal parts'' of $f_h$ and $f_\infty$: \[ g_h(u)= \frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n a^{(h)}_{ij}(x, u)D_i uD_j u\, dx,\] \[g_\infty(u)= \frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n A^{(\infty)}_{ij}(x)D_i uD_j u\, dx.\] The following theorem provides a fundamental connection between the above abstract notion of weak slope and the concrete notion related to our problem. \begin{theorem}\label{soluzpticr} Let $u\in H_0^1(\Omega)$ be a critical point of $f_h$. Then, $u$ is a weak solution of $(\ref{Ph})$. \end{theorem} The proof of this theorem can be found in \cite[Corollary 2.8]{cthree}. To apply the local saddle theorem, we shall need two ingredients: the Palais Smale condition and some minmax estimates. \begin{theorem}\label{PS} Let $\beta <\mu_1< \alpha$. Then, for all $a,b \in \mathbb{R}$ there exists $\bar h \in \mathbb{N}$ such that $f_h$ satisfies $(PS)_c$ for all $h \geq \bar h$ and every $c \in [a,b]$. \end{theorem} \begin{proof} In view of assumption (A7), $\beta <\mu^{(h)}_1< \alpha$ eventually, so we can apply \cite[Theorem 3.1]{cthree} and deduce the assertion. \end{proof} For the rest of this article, we shall consider $\beta < \mu_1$ and $\mu_k<\alpha \leq \mu_{k+1}$ with $k \geq 2$. Define \[\overline{\phi}_1=\frac{\phi_1}{\alpha- \mu_1}\,,\] \[H_k= \mathop{\rm span}\{\phi_1,\cdots,\phi_k\}, \quad H_k^{\bot}= \mathop{\rm span}\{\phi_{k+1},\cdots\}. \] Let $\psi_2,\ldots, \psi_k \in C_c^\infty(\Omega)$. Consider the space \[\widehat H_k=\mathop{\rm span}\{\phi_1, \psi_2\cdots,\psi_k\}. \] If $\psi_2,\ldots, \psi_k$ are sufficiently close in the $H_0^1-$norm to $\phi_2,\ldots,\phi_k$, then $H_0^1(\Omega)= \widehat H_k \oplus H_k^\bot$. Moreover, since $\overline{\phi}_1$ is a critical point for $\widehat f_\infty$, it is readily seen that \begin{equation}\label{eq:tt} \forall \rho>0: \sup_{\widehat H_k \cap S_{\rho}(\overline{\phi}_1)} \widehat f_\infty < \widehat f_\infty(\overline{{\phi}}_1). \end{equation} \begin{lemma}\label{segnale} There exist $\varepsilon, \rho>0$ such that for all $u \in \widehat H_k \cap B_{\rho}(\overline{\phi}_1)$ the condition $u(x) \geq \varepsilon \phi_1(x)$ holds a.e. in $\Omega$. \end{lemma} \begin{proof} It is sufficient to recall that $\inf_K\phi_1>0$ for every compact subset $K$ of $\Omega$. \end{proof} \begin{lemma}\label{simplesso} There exist $u_0,\ldots,u_m \in \widehat H_k$ such that if $S= \mathop{\rm conv}\{u_0,\ldots,u_m\}$, then $S$ is a neighborhood of $\overline{\phi}_1$ and \[\sup \left\{f_{\infty}(u): u \in S \right\} \leq f_\infty (\overline{\phi}_1),\] \[\sup \left\{f_{\infty}(u): u \in \partial_{\widehat H_k} S \right\} < f_\infty (\overline{\phi}_1).\] \end{lemma} \begin{proof} If $\rho$ is as in Lemma \ref{segnale}, recalling (\ref{eq:tt}), we have \[\sup \left\{f_{\infty}(u): u \in \overline{B_{\rho}(\overline{\phi}_1)}\cap \widehat H_k \right\} \leq f_\infty (\overline{\phi}_1),\] \[\sup \left\{f_{\infty}(u): u \in \left(\overline{B_{\rho}(\overline{\phi}_1)} \setminus B_{\frac{\rho}{2}}(\overline{\phi}_1)\right) \cap \widehat H_k \right\} < f_\infty (\overline{\phi}_1).\] The assertions follow easily. \end{proof} \begin{lemma}\label{limitatiuniformemente} Let $S$ be as in Lemma \ref{simplesso}. Then, there exists $R>0$ such that, if $u \in \widehat H_k \cap S$ and \[u_h \to u \mbox{ weakly in } H_0^1(\Omega),\quad f_h(u_h) \to f_\infty(u), \] then $\limsup_h \|u_h\|_{H_0^1(\Omega)} < R$. \end{lemma} \begin{proof} Fix $u\in \widehat H_k \cap S$. In view of $(\ref{H1})$, there exists a sequence $(u_h)$ such that $u_h \to u$ weakly in $H_0^1(\Omega)$ and $f_h(u_h) \to f_\infty(u)$. Eventually we have \[ f_h(u_{h}) < \sup\{f_{\infty}(u): u \in \widehat H_k \cap S\} + 1. \] Moreover we have \begin{align*} &\lim_h \left\{ -\frac{\alpha}{2}\int_{\Omega} (u_h^+)^2 \, dx -\frac{\beta}{2}\int_{\Omega} (u_h^-)^2 \, dx+ \int_{\Omega} \phi_1 u_h \, dx \right\}\\ &= -\frac{\alpha}{2}\int_{\Omega} (u^+)^2 \, dx -\frac{\beta}{2}\int_{\Omega} (u^-)^2 \, dx + \int_{\Omega} \phi_1 u \, dx. \end{align*} Therefore, $g_h(u_h)$, the principal part of $f_h(u_h)$, is (eventually) bounded. Hence, using (A3), we deduce the assertion. \end{proof} Let now $X_1$ be the eigenspace associated to $\mu_{k+1}$ and $X_2 = \mathop{\rm span}\{\phi_{k+2},\ldots\}$ so that \[ H_k^{\bot}= X_1 \oplus X_2.\] \begin{proposition}\label{prop:1} Let $R$ be as in Lemma \ref{limitatiuniformemente}. Then there exist a finite dimensional space $\widehat X_1 \subseteq C_c^{\infty}(\Omega)$, $\rho_1 >0$ and $\rho_2 >R$ such that \begin{gather} \label{eq:prima} H_0^1(\Omega) = \widehat H_k \oplus\widehat X_1 \oplus X_2,\\ \label{eq:seconda} \liminf_h\left[\inf \left\{f_h (\overline{\phi}_1+u): u \in \partial_{\widehat X_1 \oplus X_2} Q\right\}\right]>f_{\infty}(\overline{\phi}_1),\\ \label{prop:0} \liminf_h\left[\inf \left\{f_h (\overline{\phi}_1+u): u \in Q\right\}\right]\geq f_{\infty}(\overline{\phi}_1), \end{gather} where $Q=\left(\widehat X_1 \cap \overline{B_{\rho_1}(0)}\right) + \left(X_2 \cap \overline{B_{\rho_2}(0)}\right)$. \end{proposition} \begin{proof} Since $k+1 \geq 2$, there exists $\rho_1>0$ such that \[ \forall v \in X_1: \, \overline{\phi}_1 + v \geq 0 \Rightarrow \left\|v \right\|_{H_0^1(\Omega)}< \rho_1. \] Moreover, there exists $\rho_2 >R$ such that \begin{equation} \label{eq: minore} f_{\infty}(\overline{\phi}_1)< \frac{\nu}{4}(\rho_2)^2 - \frac{C}{2}\int_{\Omega}|D(\overline{\phi}_1+v)|^2\, dx -\frac{\alpha}{2}\int_{\Omega} (\overline{\phi}_1+ v)^2 \, dx + \int_{\Omega}\phi_1 (\overline{\phi}_1+ v)\, dx, \end{equation} for every $ v \in X_1 \cap B_{\rho_1}(0)$. We prove (\ref{eq:prima}). Let $\{\varphi_1, \ldots, \varphi_l\}$ be a $L^2-$orthonormal basis of $X_1$ and consider a sequence $\{\varphi^{(s)}_m\}$ ($m=1,\ldots,l$) in $C_c^{\infty}(\Omega)$ such that $\varphi^{(s)}_m \to \varphi_m$ in $H_0^1(\Omega)$. Let \[ \widehat X^{(s)}_1 = \mathop{\rm span}\{\varphi^{(s)}_1, \ldots, \varphi^{(s)}_l\} \] Eventually as $s \to +\infty$ we have \[ H_0^1(\Omega) = \widehat H_k \oplus\widehat X^{(s)}_1 \oplus X_2. \] For proving (\ref{eq:seconda}) we argue by contradiction. Suppose that, up to a subsequence, \[\lim_s f_{h_s}\left(\overline{\phi}_1+v_s+w_s\right)\leq f_{\infty}(\overline{\phi}_1),\] with $u_s = v_s+w_s \in \partial_{\widehat X_1^{(s)}\oplus X_2} Q$. Up to a further subsequence, $u_{s}$ weakly converges to some $u$. Then $v_{s} \to v \in X_1$, while $w_{s} \to w$ weakly in $X_2$, where $u=v+w$. Using $(\ref{H1})$ we deduce that $ \widehat f_{\infty}(\overline{\phi}_1 + v + w)\leq f_{\infty}(\overline{\phi}_1 + v + w) \leq f_{\infty}(\overline{\phi}_1). $ By definition of $X_1$ and $X_2$ we have $w=0$ and $\widehat f_{\infty}(\overline{\phi}_1 + v )= f_{\infty}(\overline{\phi}_1+v)$, namely that $ \overline{\phi}_1+v\geq 0$. By the choice of $\rho_1$, we have $\|v\|_{H_0^1(\Omega)}<\rho_1$. Therefore $\|v_s\|_{H_0^1(\Omega)} < \rho_1$ and $\|w_s\|_{H_0^1(\Omega)}=\rho_2$ eventually. Using $(A2)$ and $(A3)$, we get \begin{align*} &f_{h_s}\left(\overline{\phi}_1+u_s\right)\\ &=f_{h_s}\left(\overline{\phi}_1+v_s + w_s\right) \\ & = \frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n a^{(h_s)}_{ij}(x,\overline{\phi}_1+u_s)D_i(\overline{\phi}_1+v_s) D_j(\overline{\phi}_1+v_s)\, dx\\ &\quad +\int_{\Omega} \sum_{i,j=1}^n a^{(h_s)}_{ij}(x,\overline{\phi}_1+u_s )D_i(\overline{\phi}_1+v_s)D_j w_s \, dx \\ &\quad +\frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n a^{(h_s)}_{ij}(x,\overline{\phi}_1+u_s)D_iw_s D_jw_s\, dx \\ &\quad -\frac{\alpha}{2}\int_{\Omega} ((\overline{\phi}_1+u_s)^+)^2 \, dx -\frac{\beta}{2}\int_{\Omega} ((\overline{\phi}_1+u_s)^-)^2 \, dx + \int_{\Omega} \phi_1 (\overline{\phi}_1+u_s)\, dx\\ &\geq \frac{1}{4}\int_{\Omega} \sum_{i,j=1}^n a^{(h_s)}_{ij}(x,\overline{\phi}_1+u_s)D_iw_sD_jw_s \, dx \\ & \quad-\frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n a^{(h_s)}_{ij}(x,\overline{\phi}_1+u_s)D_i(\overline{\phi}_1+v_s)D_j (\overline{\phi}_1+v_s)\, dx\\ & \quad -\frac{\alpha}{2}\int_{\Omega} ((\overline{\phi}_1+u_s)^+)^2 \, dx -\frac{\beta}{2}\int_{\Omega} ((\overline{\phi}_1+u_s)^-)^2 \, dx+ \int_{\Omega} \phi_1 (\overline{\phi}_1+u_s) \, dx \\ &\geq \frac{\nu}{4}\int_{\Omega}|Dw_s|^2\, dx - \frac{C}{2}\int_{\Omega}|D(\overline{\phi}_1+v_s)|^2\, dx -\frac{\alpha}{2}\int_{\Omega} ((\overline{\phi}_1+u_s)^+)^2 \, dx\\ & \quad- \frac{\beta}{2}\int_{\Omega} ((\overline{\phi}_1+u_s)^-)^2 \, dx+ \int_{\Omega} \phi_1 (\overline{\phi}_1+u_s)\, dx\\ & = \frac{\nu}{4}(\rho_2)^2 - \frac{C}{2}\int_{\Omega}|D(\overline{\phi}_1+v_s)|^2\, dx -\frac{\alpha}{2}\int_{\Omega} ((\overline{\phi}_1+u_s)^+)^2 \, dx\\ & \quad-\frac{\beta}{2}\int_{\Omega} ((\overline{\phi}_1+u_s)^-)^2 \, dx+ \int_{\Omega} \phi_1 (\overline{\phi}_1+u_s) \, dx. \end{align*} Hence, as $s \to +\infty$ we have \[ f_{\infty}(\overline{\phi}_1)\geq \frac{\nu}{4}(\rho_2)^2 - \frac{C}{2}\int_{\Omega}|D(\overline{\phi}_1+v)|^2\, dx -\frac{\alpha}{2}\int_{\Omega} (\overline{\phi}_1+ v)^2 \, dx + \int_{\Omega}\phi_1 (\overline{\phi}_1+ v)\, dx, \] which contradicts (\ref{eq: minore}). Finally let us prove (\ref{prop:0}). Since \begin{equation}\label{altola} f_\infty({\overline{\phi}_1})=\inf_{{\overline{\phi}_1}\oplus (\widehat X_1 \oplus X_2)} f_\infty, \end{equation} the assertion follows. \end{proof} \begin{lemma}\label{lemma:conv} For any $u \in \widehat H_k\setminus\{0\}$ there exists a sequence $(u_h) \subset H_0^1(\Omega)$ such that \begin{gather}\label{eq:0} (u_h -u) \in \widehat H_k\oplus X_2, \\ \label{eq:1} u_h \to u \mbox{ weakly in } H_0^1(\Omega), \quad f_h(u_h) \to f_\infty(u),\\ \label{eq:2} \forall h \in \mathbb{N}: \,\, \frac{u_h-u}{\overline{\phi}_1} \in L^\infty(\Omega), \quad \frac{u_h-u}{\overline{\phi}_1} \to 0 \mbox{ in } L^\infty(\Omega). \end{gather} \end{lemma} \begin{proof} Fix $u \in \widehat H_k\setminus\{0\}$. In view of $(\ref{H1})$, there exists $(\tilde u_h)$ such that \begin{equation}\label{eq:pi} \tilde u_h \to u \mbox { weakly in } H_0^1(\Omega), \quad \lim_{h} f_h(\tilde u_{h})= f_{\infty}(u). \end{equation} Consider a strictly increasing sequence $(h_k) \subset \mathbb{N}$ such that \[ \forall h \geq h_k: \; {\mathcal L}^n\big(\big\{x \in \Omega: \left|\tilde u_h -u \right|> \frac{1}{k}\overline{\phi}_1\big\}\big) < \frac{1}{k}, \] where $\mathcal L^n$ denotes the Lebesgue measure. Set \[\varepsilon_h= \begin{cases} 2 & \mbox{if } h < h_1,\\ \frac{1}{k} & \mbox{if } h_k\leq h < h_{k+1}. \end{cases} \] Then $\varepsilon_h>0$, $\varepsilon_h \to 0$ and ${\mathcal L}^n\left(\left\{x \in\Omega :\left|\tilde u_h -u \right|> \varepsilon_h\overline{\phi}_1\right\}\right) < \frac{1}{k}$ if $ h_k\leq h < h_{k+1}$. In particular \begin{equation}\label{misura0} \lim_{h}{\mathcal L}^n\left(\left\{x \in \Omega: \left|\tilde u_h -u \right|> \varepsilon_h\overline{\phi}_1\right\}\right)=0. \end{equation} Consider now \[ \check u_h= u + \left[\left((\tilde u_h-u) \vee (-\varepsilon_h\overline{\phi}_1) \right)\wedge (\varepsilon_h\overline{\phi}_1)\right],\] and denote by $\Pi_{\widehat X_1}$ the projection on $\widehat X_1$ associated to the decomposition (\ref{eq:prima}). Let $v_h= -\Pi_{\widehat X_1}(\check u_h -u)$, then \[u_h= \check u_h - \Pi_{\widehat X_1}(\check u_h -u)= \check u_h +v_h.\] satisfies all the requirements (\ref{eq:0})-(\ref{eq:2}).\\ Requirement (\ref{eq:0}) is straightforward. Furthermore, since $|\check u_h-u|\leq \varepsilon_h\overline{\phi}_1$ a.e. in $\Omega$, $(\ref{eq:2})$ follows. Since $\check u_h \to u$ weakly in $ H_0^1(\Omega)$, then $v_h \to 0$ strongly and $u_h \to u$ weakly in $ H_0^1(\Omega)$. To show that $f_h(u_h) \to f_\infty(u)$, it suffices to prove that $g_h(u) \to g_\infty(u)$, namely that \begin{equation}\label{tildehat} \lim_{h} \frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n a^{(h)}_{ij}(x,u_h)D_iu_hD_ju_h \, dx = \frac{1}{2}\int_{\Omega}\sum_{i,j=1}^n A^{(\infty)}_{ij}(x)D_iuD_ju \, dx. \end{equation} We obtain (\ref{tildehat}) by combining the two following facts: \begin{equation}\label{tildehat2} \lim_{h}\frac{1}{2} \int_{\Omega} \Big[\sum_{i,j=1}^n a^{(h)}_{ij}(x,u_h)D_iu_hD_ju_h - \sum_{i,j=1}^n a^{(h)}_{ij}(x,\check u_h)D_i\check u_hD_j\check u_h\Big] \, dx = 0 \end{equation} and \begin{equation}\label{tildehat3} \lim_{h} \frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n a^{(h)}_{ij}(x,\check u_h)D_i\check u_hD_j\check u_h \, dx = \frac{1}{2}\int_{\Omega}\sum_{i,j=1}^n A^{(\infty)}_{ij}(x)D_iuD_ju \, dx. \end{equation} Now we prove (\ref{tildehat2}). We have \begin{align*} & a^{(h)}_{ij}(x,u_h)D_iu_hD_ju_h - a^{(h)}_{ij}(x,\check u_h)D_i\check u_hD_j\check u_h \\ & = a^{(h)}_{ij}(x,u_h)D_i(\check u_h +v_h) D_j(\check u_h +v_h) - a^{(h)}_{ij}(x,\check u_h)D_i\check u_hD_j\check u_h = \\ & = [a^{(h)}_{ij}(x,u_h)- a^{(h)}_{ij}(x,\check u_h)]D_i\check u_hD_j\check u_h + 2 a^{(h)}_{ij}(x,u_h)D_i\check u_hD_jv_h \\ &\quad +a^{(h)}_{ij}(x,u_h)D_i v_h D_jv_h. \end{align*} Clearly, by assumption (A2), \begin{gather*} \lim_{h}\int_{\Omega} \sum_{i,j=1}^na^{(h)}_{ij}(x,u_h)D_i\check u_hD_jv_h dx=0,\\ \lim_{h}\int_{\Omega} \sum_{i,j=1}^na^{(h)}_{ij}(x,u_h)D_i v_hD_jv_h dx=0. \end{gather*} On the other hand, there exists $\vartheta \in ]0,1[$ such that \[ [a^{(h)}_{ij}(x,u_h)- a^{(h)}_{ij}(x,\check u_h)] = D_s a^{(h)}_{ij}(x, u_h + \vartheta v_h)v_h = D_s a^{(h)}_{ij}(x,u + \eta \overline{\phi}_1 + \vartheta v_h)v_h, \] where $\eta \in \mathbb{R}$ and we have used (\ref{eq:2}) in the last identity. Since there exists $\delta_h >0$ ($\delta_h \to 0^+$) such that \[ |v_h|\leq \delta_h |u + \eta \overline{\phi}_1 + \vartheta v_h| \] using (A2), we deduce that \[ \lim_{h}\int_{\Omega} \sum_{i,j=1}^n [a^{(h)}_{ij}(x,u_h) - a^{(h)}_{ij}(x,\check u_h)]D_i\check u_hD_j\check u_h\,dx =0; \] hence (\ref{tildehat2}) holds. To prove (\ref{tildehat3}) denote by $\chi_F$ the characteristic function of a set $F$. We have \begin{align*} & \frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n a^{(h)}_{ij}(x,\check u_h)D_i\check u_hD_j \check u_h \, dx \\ & = \frac{1}{2}\int_{\{x:|\tilde u_h -u|\leq \varepsilon_h\overline{\phi}_1\}} \sum_{i,j=1}^n a^{(h)}_{ij}(x,\tilde u_h)D_i\tilde u_hD_j\tilde u_h \, dx \\ & \quad + \frac{1}{2}\int_{\{x:(\tilde u_h -u)> \varepsilon_h\overline{\phi}_1\}} \sum_{i,j=1}^n a^{(h)}_{ij}(x, u + \varepsilon_h \overline{\phi}_1)D_i(u + \varepsilon_h \overline{\phi}_1)D_j(u + \varepsilon_h \overline{\phi}_1) \, dx \\ & \quad+ \frac{1}{2}\int_{\{x:(\tilde u_h -u)<-\varepsilon_h\overline{\phi}_1\}} \sum_{i,j=1}^n a^{(h)}_{ij}(x, u - \varepsilon_h \overline{\phi}_1)D_i(u - \varepsilon_h \overline{\phi}_1)D_j(u - \varepsilon_h \overline{\phi}_1) \, dx \\ & \leq \frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n a^{(h)}_{ij}(x,\tilde u_h)D_i\tilde u_hD_j\tilde u_h \, dx \\ & \quad + \frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n a^{(h)}_{ij}(x, u + \varepsilon_h \overline{\phi}_1)D_i(u + \varepsilon_h \overline{\phi}_1)D_j(u + \varepsilon_h \overline{\phi}_1)\chi_{ \{x:(\tilde u_h -u)> \varepsilon_h\overline{\phi}_1\} } \, dx \\ & \quad + \frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n a^{(h)}_{ij}(x, u - \varepsilon_h \overline{\phi}_1)D_i(u - \varepsilon_h \overline{\phi}_1)D_j(u - \varepsilon_h \overline{\phi}_1)\chi_{ \{x:(\tilde u_h -u)<- \varepsilon_h\overline{\phi}_1\} } \, dx. \end{align*} Using (\ref{eq:pi}) and (\ref{misura0}) we deduce \[ \limsup_h\frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n a^{(h)}_{ij}(x,\check u_h)D_i\check u_hD_j\check u_h \, dx \leq \frac{1}{2}\int_{\Omega}\sum_{i,j=1}^n A^{(\infty)}_{ij}(x)D_iuD_ju \, dx. \] Assumption (\ref{H1}) gives us the conclusion. \end{proof} \begin{theorem}\label{th:1} Let $m\in \mathbb{Z}^+$. For all $r, \varepsilon >0$ there exists $\delta >0$ such that if $u_0,\ldots,u_m \in \widehat H_k \cap B_r(\overline{\phi}_1)$ and \begin{equation}\label{la:1} \begin{gathered} \forall j=0,\ldots,m: \quad \mathop{\rm essinf}{}_{\Omega}\frac{u_j}{\overline{\phi}_1\,}\geq \varepsilon, \\ u_j^{(h)} \to u_j \quad\mbox{(as in Lemma \ref{lemma:conv}),}\\ \sup\big\{\|\frac{u-v}{\overline{\phi}_1}\|_\infty: u,v \in \mathop{\rm conv}\{u_0,\ldots,u_m\}\big\}<\delta, \end{gathered} \end{equation} then \begin{equation}\label{eq:cocco} \begin{aligned} & \limsup_h \left\{\sup\left\{f_h(v_h): v_h \in \mathop{\rm conv}\{u^{(h)}_0, \ldots,u^{(h)}_m\}\right\}\right\} \\ &\leq \sup\{f_{\infty}(u): u \in \mathop{\rm conv}\{u_0,\ldots,u_m\}\} +\varepsilon. \end{aligned} \end{equation} \end{theorem} \begin{proof} Let $r,\varepsilon >0$, $u_0,\ldots,u_m$, $(u_j^{(h)})$ be as in (\ref{la:1}). Since $u_j^{(h)} \to u_j$ strongly in $L^2(\Omega)$, then it is sufficient to prove that \begin{equation} \begin{aligned} & \limsup_h \left\{\sup\left\{g_h(v_h): v_h \in \mathop{\rm conv}\{u^{(h)}_0,\ldots,u^{(h)}_m\}\right\}\right\} \\ &\leq \sup\left\{g_{\infty}(u): u \in \mathop{\rm conv}\{u_0,\ldots,u_m\}\right\} +\varepsilon. \end{aligned} \label{eq:cocco2} \end{equation} where $g_h$ and $g_\infty$ are respectively the ``principal parts'' of $f_h$, $f_\infty$.\\ Consider $\widetilde f_h: H_0^1(\Omega) \to \mathbb{R}$ defined by \[ \widetilde f_h(u)=\frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n a^{(h)}_{ij}(x, u_0)D_i u D_j u\, dx. \] It is readily seen that $\widetilde f_h$ is convex. Therefore to prove (\ref{eq:cocco2}) it suffices to verify that \begin{equation}\label{sonno} \limsup_h \left\{\sup\left\{|g_h(v_h)- \widetilde f_h(v_h)|: v_h \in \mbox{\rm conv}\{u^{(h)}_0,\ldots,u^{(h)}_m\}\right\}\right\} <\frac{\varepsilon}{2}. \end{equation} Of course, if $v_h \in \mathop{\rm conv}\{u^{(h)}_0,\ldots,u^{(h)}_m\}$, we have \begin{equation}\label{eq:lala} g_h(v_h)- \widetilde f_h(v_h)= \frac{1}{2}\int_{\Omega} \sum_{i,j=1}^n \left[a^{(h)}_{ij}(x, v_h)-a^{(h)}_{ij}(x, u_0)\right] D_i v_hD_j v_h\, dx. \end{equation} It is not difficult to see that, if $v_h \in \mathop{\rm conv}\{u^{(h)}_0,\ldots,u^{(h)}_m\}$, then there exist $\delta>0$, $c,d,e_h \in L^\infty(\Omega)$, with $\mbox{essinf}_\Omega c \geq \varepsilon$, $\|d\|_\infty<\delta$ and $\|e_h\|_\infty \to 0$ such that \[ v_h= u_0 + (d+e_h)\overline{\phi}_1= (c+d+e_h)\overline{\phi}_1. \] By Lagrange Theorem, there exists $0<\eta<1$ such that \begin{align*} & a^{(h)}_{ij}(x, v_h)-a^{(h)}_{ij}(x, u_0) \\ & = \overline{\phi}_1(d+e_h)D_s a^{(h)}_{ij}\left(x, (c +\eta(d+e_h))\overline{\phi}_1\right) \\ & = \frac{(d+e_h)}{c +\eta(d+e_h)}\left((c +\eta(d+e_h))\overline{\phi}_1\right) D_s a^{(h)}_{ij}\left(x,\left(c+\eta(d+e_h)\right)\overline{\phi}_1\right). \end{align*} Therefore, if $\delta$ is small enough, by using (A2), we deduce that \[ \limsup_h\|a^{(h)}_{ij}(x, v_h)-a^{(h)}_{ij}(x, u_0)\|_\infty \] is also small. Since $f_\infty$ is bounded in $\widehat H_k \cap B_r(\overline{\phi}_1)$, we can assume without loss of generality that (eventually) \[ f_h(u_j^{(h)}) < \sup\{f_{\infty}(u): u \in \widehat H_k \cap B_r(\overline{\phi}_1)\}+ 1. \] So, in view of $(A3)$ we may deduce that $\|u_j^{(h)}\|_{H_0^1}$ is bounded; hence also $\|v_h\|_{H_0^1}$ is bounded. By using all these facts in (\ref{eq:lala}) we obtain that, for $\delta$ small enough, (\ref{sonno}) holds. \end{proof} \begin{remark}\label{rmk:1} \rm We point out that Theorem \ref{th:1} is still valid if, in (\ref{la:1}), we replace assumption $\mathop{\rm essinf}_{\Omega}\frac{u_j}{\,\overline{\phi}_1\,} \geq \varepsilon$ with $\mathop{\rm esssup}_{\Omega}\frac{u_j}{\,\overline{\phi}_1\,} \leq -\varepsilon$. \end{remark} Now, let $S$ be as in Lemma \ref{simplesso} and $Q$ be as in Proposition \ref{prop:1}. Let also $\varepsilon >0$. We can suppose that \begin{equation}\label{eq:1a} \sup \left\{f_{\infty}(u): u \in \partial_{\widehat H_k} S \right\} < f_\infty (\overline{\phi}_1) - 2\varepsilon, \end{equation} \begin{equation}\label{eq:2a} \liminf_h\left[\inf \left\{f_h (\overline{\phi}_1+u): u \in \partial_{\widehat X_1 \oplus X_2} Q\right\}\right]>f_{\infty}(\overline{\phi}_1) + 2 \varepsilon. \end{equation} For $r=\rho$ where $\rho$ is introduced in Lemma \ref{segnale} and $\varepsilon$ given as above, take $\delta>0$ as in Theorem \ref{th:1}. Let now \[ S= \bigcup_{j=1}^NS_j, \] where $S_j$ are the convex sets generated by the points $u_0^{(j)}, \ldots,u_m^{(j)} \in \widehat H_k \cap B_r(\overline{\phi}_1)$, such that \[ \sup\big\{\|\frac{u-v}{\overline{\phi}_1}\|_\infty: u,v \in S_j\big\}<\delta. \] For $k=0,\ldots,m$, we consider $(u^{(j)}_{k,h})_h$ the approximating sequence introduced in Theorem \ref{th:1} and let \[ P_h= \bigcup_{j=1}^N \mathop{\rm conv }\{u^{(j)}_{0,h}, \ldots, u^{(j)}_{m,h}\}. \] \begin{proposition}\label{disusella} Take $\varepsilon$ as above, then there exists $\overline h \in \mathbb{N}$ such that for every $h \geq \overline h$ we have \begin{gather*} \sup_{P_h} f_h < \inf_{\overline{\phi}_1 + \partial Q} f_h, \quad b_1=\sup_{P_h} f_h < f_\infty (\overline{\phi}_1)+ \varepsilon, \\ \sup_{\partial P_h} f_h <\inf_{\overline{\phi}_1 + Q} f_h, \quad a_1= \inf_{\overline{\phi}_1 + Q} f_h > f_\infty (\overline{\phi}_1)- \varepsilon. \end{gather*} \end{proposition} \begin{proof} By (\ref{eq:2a}) and (\ref{prop:0}) we deduce that there exists $\overline h_1 \in \mathbb{N}$ such that for every $h \geq \overline h_1$ \[\inf_{\overline{\phi}_1 + \partial Q} f_h > f_\infty (\overline{\phi}_1) + \varepsilon, \quad \inf_{\overline{\phi}_1 + Q} f_h > f_\infty (\overline{\phi}_1) -\varepsilon.\] Using Lemma \ref{simplesso}, Theorem \ref{th:1} and (\ref{eq:1a}) we see that there exists $\overline h_2 \in \mathbb{N}$ such that for every $h \geq \overline h_2$ we have \[ \sup _{P_h} f_h < f_\infty (\overline{\phi}_1) +\varepsilon, \quad \sup _{\partial P_h} f_h < f_\infty (\overline{\phi}_1) - \varepsilon. \] The assertions follow, taking $\overline h =\max\{\overline h_1, \overline h_2\}$. \end{proof} \begin{theorem}\label{terzopunto} For every $\varepsilon >0$, there exists $\overline h \in \mathbb{N} $ such that for all $h \geq \overline h$, the functional $f_h$ has a critical point $ u_3^{(h)}$ with \begin{equation}\label{stimaeps} \big|f_h(u_3^{(h)}) - f_\infty(\overline{\phi}_1)\big| <\varepsilon. \end{equation} \end{theorem} \begin{proof} Let $\Pi_{1} : H_0^1(\Omega) \to \widehat H_k$ be projection induced by the decomposition $H_0^1(\Omega) = \widehat H_k \oplus (\widehat X_1 \oplus X_2)$. Then, for $h$ large, the restriction of $\Pi_1$ to $P_h$ is an injective map with inverse Lipschitz continuous and such that $x - \Pi_1(x) \in \widehat X_1 \oplus X_2$. Let $\varphi_h: \widehat H_k \to \widehat X_1 \oplus X_2$ be a Lipschitz continuous function such that \[ \Pi_1(x) + \varphi_h(\Pi_{1}(x)) = x \quad \forall x \in P_h. \] If $\Phi_h: H_0^1(\Omega) \to H_0^1(\Omega)$ is defined by $\Phi_h(x)= \varphi_h(\Pi_{1}(x)) +x$, then $\Phi_h$ is a Lipschitz homeomorphism with inverse Lipschitz continuous. Moreover, \[\Phi_h(\Pi_{1}(x))=x \quad \forall x \in P_h. \] Define $\widetilde f_h = f_h \circ \Phi_h$. Clearly, $f_h$ satisfies $(PS)_c$ if and only if $\widetilde f_h$ satisfies $(PS)_{c}$; furthermore $u^{(h)}$ is a critical point of $\widetilde f_h$ if and only if $\Phi_h(u^{(h)})$ is a critical point of $f_h$. Using Proposition \ref{disusella}, it follows that \begin{gather*} \sup_{\Pi_1(P_h)} \widetilde f_h < \inf_{\overline{\phi}_1 - \varphi_h(\overline{\phi}_1) + \partial Q} \widetilde f_h,\\ \sup_{\Pi_1(\partial P_h)} \widetilde f_h <\inf_{\overline{\phi}_1 - \varphi_h(\overline{\phi}_1)+ Q} \widetilde f_h. \end{gather*} We have \[ a_1= \inf_{\overline{\phi}_1 + Q} f_h= \inf_{\overline{\phi}_1 - \varphi_h(\overline{\phi}_1)+ Q} \widetilde f_h, \quad b_1=\sup_{P_h} f_h = \sup_{\Pi_1(P_h)} \widetilde f_h. \] By Theorem \ref{selleinscala}, we deduce that there exists a critical point $\widetilde u^{(h)}_3$ for $\widetilde f_h$ with $\widetilde f_h(\widetilde u^{(h)}_3) \in [a_1,b_1]$. Therefore, there exists a critical point $u^{(h)}_3$ for $f_h$ with $f_h(u^{(h)}_3) \in [a_1,b_1]$. Proposition \ref{disusella} now gives (\ref{stimaeps}). \end{proof} \section{Proof of the main result}\label{sct:5} \begin{theorem}\label{2livelli} Let $\beta <\mu_1$ and $\alpha>\mu_2$. Then, there exist $\overline h \in \mathbb{N} $, $\varepsilon >0$ such that for all $h \geq \overline h$, the functional $f_h$ has at least two critical points $u_1^{(h)}$, $u_2^{(h)}$ with \[ f_h(u_1^{(h)}) 0$ such that \[ \inf_{S_{\rho}\left(\frac{\phi_1}{\beta -\mu_1}\right)} f_\infty> f_{\infty}\big(\frac{\phi_1}{\beta -\mu_1}\big). \] By \cite[Lemma 4.1]{cthree}, there exist a continuous curve $\gamma:[0,1] \to H_0^1(\Omega)$, $\varepsilon >0$ such that \[ \gamma(0)= \frac{\phi_1}{\beta -\mu_1}, \quad \gamma(1) \notin \overline{B_{\rho}\big(\frac{\phi_1}{\beta -\mu_1}\big)} , \quad \sup_{s \in [0,1]} f_\infty(\gamma(s))< f_\infty(\overline{\phi}_1)-\varepsilon. \] The same argument of \cite[Theorem 4.2]{cj} shows that there exists $\overline h \in \mathbb{N}$ such that for all $h \geq \overline h$ \[ \inf_{S_{\rho}\left(\frac{\phi_1}{\beta -\mu_1}\right)} f_{h} > f_{\infty}\left(\frac{\phi_1}{\beta -\mu_1}\right). \] On the other hand, the argument used in the proof of Theorem \ref{th:1} allows us to build a polygonal curve $\gamma_h$ with \[ \gamma_h(0) \in B_{\rho}\left(\frac{\phi_1}{\beta -\mu_1}\right), \quad \gamma_h(1) \notin \overline{B_{\rho}\big(\frac{\phi_1}{\beta -\mu_1}\big)} , \quad \sup_{s \in [0,1]} f_h(\gamma_h(s))< f_\infty(\overline{\phi}_1)-\varepsilon. \] In view of $(A7)$ we can follow the same argument used in the proof of \cite[Theorem 4.2]{cthree} and deduce the assertion. \end{proof} \begin{proof}[Proof of Theorem \ref{es3sol}] By Theorem \ref{2livelli} and Theorem \ref{terzopunto} we deduce that for $h \geq \overline h$ the functional $f_h$ has at least three critical points. 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