\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 266, pp. 1--19.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/266\hfil Limit of minimax values] {Limit of minimax values under $\Gamma$-convergence} \author[M. Degiovanni, M. Marzocchi \hfil EJDE-2014/266\hfilneg] {Marco Degiovanni, Marco Marzocchi} % in alphabetical order \address{Marco Degiovanni \newline Dipartimento di Matematica e Fisica\\ Universit\`a Cattolica del Sacro Cuore\\ Via dei Musei 41\\ 25121 Brescia, Italy} \email{marco.degiovanni@unicatt.it} \address{Marco Marzocchi \newline Dipartimento di Matematica e Fisica\\ Universit\`a Cattolica del Sacro Cuore\\ Via dei Musei 41\\ 25121 Brescia, Italy} \email{marco.marzocchi@unicatt.it} \thanks{Submitted November 15, 2014. Published December 25, 2014.} \subjclass[2000]{35P30, 49R05, 58E05} \keywords{Nonlinear eigenvalues; variational convergence; $p$-Laplace operator; \hfill\break\indent total variation} \begin{abstract} We consider a sequence of minimax values related to a class of even functionals. We show the continuous dependence of these values under the $\Gamma$-convergence of the functionals. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} Let $X$ be a Banach space and $f,g:X\to\mathbb{R}$ two functions of class $C^1$. Assume also that $f$ and $g$ are even and positively homogeneous of the same degree. Several results of critical point theory (see \cite{bartsch1993, krasnoselskii1964,rabinowitz1986, willem1996}) are based on the construction of a sequence of minimax values $(c_m)$ given by \[ c_m = \inf_{K\in \mathcal{K}_s^{(m)}}\, \max_{u\in K} f(u)\,, \] where $\mathcal{K}_s^{(m)}$ is the family of compact and symmetric subsets $K$ of \[ \{u\in X:g(u)=1\} \] such that $\operatorname{i}(K)\geq m$ and $\operatorname{i}$ is a topological index which takes into account the symmetry of $f$ and $g$. Typical examples are the Krasnosel'ski\u{\i} genus (see e.g.~\cite{krasnoselskii1964, rabinowitz1986, willem1996}) and the $\mathbb{Z}_2$-cohomological index (see~\cite{fadell_rabinowitz1977, fadell_rabinowitz1978}). More general examples are contained in \cite{bartsch1993}. A natural question concerns the behavior of the minimax values $c_m$ when $f$ and $g$ are substituted by two sequences $(f_h)$ and $(g_h)$ converging in a suitable sense. This problem has been recently treated (see \cite{champion_depascale2007, littig_schuricht2014, parini2011} and references therein) in the setting of homogenization problems and limit behavior of the $p$-Laplace operator. As pointed out in \cite{champion_depascale2007}, one has \[ c_m = \inf_{K\in\mathcal{K}} \mathcal{F}^{(m)}(K)\,, \] where $\mathcal{K}$ is the family of nonempty compact subsets $K$ of $X$ and $\mathcal{F}^{(m)}:\mathcal{K}\to\overline{\mathbb{R}}$ is defined as \[ \mathcal{F}^{(m)}(K) = \begin{cases} \max_{u\in K} f(u) &\text{if } K\in \mathcal{K}_s^{(m)}\,,\\ +\infty &\text{otherwise}\,. \end{cases} \] In this way the behavior of minimax values of $f$ is reduced to that of infimum values for the related functionals $\mathcal{F}^{(m)}$ and the convergence of infima has been extensively studied in the setting of $\Gamma$-convergence of functionals (see e.g.~\cite{attouch1984, dalmaso1993}). Let us mention that the behavior of critical values under $\Gamma$-convergence has been already studied also in \cite{ambrosetti_sbordone1976, degiovanni_eisner2000, groli2003, jerrard_sternberg2009}. A goal of this article is to answer a question raised in \cite[Remark~5.2]{champion_depascale2007}, concerning the relation between the $\Gamma$-convergence of the functionals $(f_h)$ and that of the related functionals $(\mathcal{F}_h^{(m)})$ (see the next Corollaries~\ref{cor:limval} and \ref{cor:limconcr}). By the way, \cite[Remark~5.2]{champion_depascale2007} seemed to suggest a negative answer, while we will show that it is affirmative. In particular, our results allow to treat the convergence of the minimax eigenvalues $\lambda$ associated to nonlinear problems of the form \begin{gather*} - \Delta_p u= \lambda V_p |u|^{p-2}u \quad\text{in $\Omega$}\,,\\ u=0 \quad\text{on $\partial\Omega$}\,, \end{gather*} where $\Omega$ is a (possibly unbounded) open subset on $\mathbb{R}^N$, $1\leq p \liminf_{h\to\infty} \Big(\inf_{X} f_h\Big) \] and let $(f_{h_n})$ be a subsequence such that \[ \sup_{n\in\mathbb{N}} \Big(\inf_{X} f_{h_n}\Big) < b\,. \] Let $u_n\in X$ be such that \[ f_{h_n}(u_n) < b\,. \] Then a subsequence $(u_{n_j})$ is convergent to some $u$ in $X$. We infer that \[ \inf_X \Big(\Gamma-\liminf_{h\to\infty} f_h\Big) \leq \Big(\Gamma-\liminf_{h\to\infty} f_h\Big)(u) \leq \liminf_{j\to\infty} f_{h_{n_j}}(u_{n_j}) \leq b \] and the assertion follows by the arbitrariness of $b$. \end{proof} In the following, we denote by $\mathcal{K}$ be the family of nonempty compact subsets of $X$. If $d$ is a compatible distance on $X$, the associated \emph{Hausdorff distance} $d_{\mathcal{H}}$ is defined on $\mathcal{K}$ as \[ d_{\mathcal{H}}(K_1,K_2) = \max\Big\{ \max_{u\in K_1}\,d(u,K_2)\,,\, \max_{v\in K_2}\,d(v,K_1)\Big\}\,. \] The \emph{$\mathcal{H}$-topology} is the topology on $\mathcal{K}$ induced by $d_{\mathcal{H}}$. Recall that the $\mathcal{H}$-topology just depends on the topology of $X$, not on the distance $d$. Therefore~$\mathcal{K}$ has an intrinsic structure of metrizable topological space. \begin{proposition} \label{prop:coerchausd} Let $(f_h)$ be a sequence of functions from $X$ to $\overline{\mathbb{R}}$ and define $\mathcal{F}_h:\mathcal{K}\to\overline{\mathbb{R}}$ as \[ \mathcal{F}_h(K) = \sup_K f_h\,. \] Then $(f_h)$ is asymptotically equicoercive if and only if $(\mathcal{F}_h)$ is asymptotically equicoercive with respect to the $\mathcal{H}$-topology. \end{proposition} \begin{proof} Assume that $(f_h)$ is asymptotically equicoercive and let $(h_n)$ be a strictly increasing sequence in $\mathbb{N}$ and $(K_n)$ a sequence in $\mathcal{K}$ such that \[ \sup_{n\in\mathbb{N}} \mathcal{F}_{h_n}(K_n) < +\infty\,. \] We claim that $\overline{\cup_{n\in\mathbb{N}} K_n}$ is compact. Actually, given a compatible distance $d$ on $X$, let $(u_j)$ be a sequence in this set and let $v_j\in K_{n_j}$ be such that $d(v_j,u_j)\to 0$. Up to a subsequence, either $(n_j)$ is constant or $(n_j)$ is strictly increasing. In the former case it is obvious that $(v_j)$ admits a convergent subsequence, while in the latter case this is due to the asymptotic equicoercivity of $(f_h)$. In any case, $(u_j)$ also admits a convergent subsequence. By Blaschke's theorem (see e.g.~\cite[Theorem~4.4.15]{ambrosio_tilli2004}) we infer that the image of the sequence $(K_n)$ is included in a compact subset of $\mathcal{K}$ and the assertion follows. Conversely, assume that $(\mathcal{F}_h)$ is asymptotically equicoercive and let $(h_n)$ and $(u_n)$ be such that \[ \sup_{n\in\mathbb{N}} f_{h_n}(u_n) < +\infty\,. \] If we set $K_n=\{u_n\}$, then $(K_n)$ is a sequence in $\mathcal{K}$ with \[ \sup_{n\in\mathbb{N}} \mathcal{F}_{h_n}(K_n) < +\infty\,. \] If $(K_{n_j})$ is convergent in $\mathcal{K}$, then $(u_{n_j})$ is convergent in $X$. \end{proof} \section{Index theory and minimax values} \label{sect:minimax} In this article, we consider an index $\operatorname{i}$ with the following properties: \begin{itemize} \item[(i)] $\operatorname{i}(K)$ is an integer greater or equal than $1$ and is defined whenever $K$ is a nonempty, compact and symmetric subset of a topological vector space such that $0\not\in K$; \item[(ii)] if $X$ is a topological vector space and $K\subseteq X\setminus\{0\}$ is compact, symmetric and nonempty, then there exists an open subset $U$ of $X\setminus\{0\}$ such that $K\subseteq U$ and \[ \operatorname{i}(\widehat{K}) \leq \operatorname{i}(K) \text{ for any compact, symmetric and nonempty $\widehat{K}\subseteq U$}\,; \] \item[(iii)] if $X, Y$ are two topological vector spaces, $K\subseteq X\setminus\{0\}$ is compact, symmetric and nonempty and $\pi:K\to Y\setminus\{0\}$ is continuous and odd, we have \[ \operatorname{i}(\pi(K)) \geq \operatorname{i}(K)\,. \] \end{itemize} Well known examples are the Krasnosel'ski\u{\i} genus (see e.g.~\cite{krasnoselskii1964, rabinowitz1986}) and the $\mathbb{Z}_2$-cohomo\-logical index (see \cite{fadell_rabinowitz1977, fadell_rabinowitz1978}). More general examples are contained in \cite{bartsch1993}. In the following, if $X$ is a topological vector space we will denote by $\mathcal{K}_s$ the family of nonempty, compact and symmetric subsets of $X\setminus\{0\}$. If $X$ is just a vector space, we denote by $\mathcal{K}_{s,F}$ the family of nonempty, compact and symmetric subsets $K$ of some finite dimensional subspace of $X$ such that $0\not\in K$. Of course, we mean that the subspace is endowed with the unique topology which makes it a topological vector space. Let us point out a situation in which the behavior of $\operatorname{i}$ on $\mathcal{K}_s$ is completely determined by that on $\mathcal{K}_{s,F}$. \begin{proposition} \label{prop:iusc} If $X$ is a metrizable and locally convex topological vector space, the following facts hold: \begin{itemize} \item[(a)] for every $K\in \mathcal{K}_s$ and every sequence $(K_h)$ in $\mathcal{K}_s$ converging to $K$ with respect to the $\mathcal{H}$-topology, it holds \[ \operatorname{i}(K) \geq \limsup_{h\to\infty} \operatorname{i}(K_h)\,; \] \item[(b)] for every $K\in \mathcal{K}_s$ there exists a sequence $(K_h)$ in $\mathcal{K}_{s,F}$ converging to $K$ with respect to the $\mathcal{H}$-topology such that \[ \operatorname{i}(K) = \lim_{h\to\infty} \operatorname{i}(K_h)\,. \] \end{itemize} \end{proposition} \begin{proof} Assertion (a) easily follows from property (ii) of the index $\operatorname{i}$. To prove (b), consider a compatible distance $d$ on $X$ such that $d(-u,-v) = d(u,v)$ and such that $B_r(u)$ is convex for any $u\in X$ and $r>0$ (see e.g.~\cite{rudin1991}). Given $K\in\mathcal{K}_s$, let $r>0$ with $K\cap B_r(0)=\emptyset$ and let $F\subseteq K$ be a finite set such that \[ K\subseteq \cup_{v\in F} B_r(v)\,. \] By substituting $F$ with $F\cup(-F)$, we may assume that $F$ is symmetric. For every $v\in F$, let $\vartheta_v:X\to [0,1]$ be a continuous function such that \begin{gather*} \vartheta_v(u)=0 \quad\text{whenever $u\not\in B_r(v)$}\,,\\ \sum_{v\in F} \vartheta_v(u) = 1\quad\text{for all $u\in K$}\,,\\ \sum_{v\in F}\, \vartheta_v(u) \leq 1 \quad\text{for all $u\in X$}\,,\\ \vartheta_{-v}(u) = \vartheta_v(-u) \quad\text{for all $v\in F$ and $u \in X$}\,. \end{gather*} Since $0\in\operatorname{conv}(F)$, we can define an odd and continuous map $\pi:X\to \operatorname{conv}(F)$ as \[ \pi(u)=\sum_{v\in F}\,\vartheta_v(u)\, v \,. \] For every $u\in K$ and $v\in F$, we have either $\vartheta_v(u)=0$ or $d(v,u) \sup_K f \,. \] Consider a compatible distance $d$ on $X$ as in the proof of Proposition~\ref{prop:iusc}. By assumption~$(c)$ we can find $r>0$ such that $K\cap B_r(0)=\emptyset$ and \begin{equation}\label{eq:minimaxfin} \begin{aligned} &g(w)>0\,,\quad \sup_K f < b\,g(w) \\ &\text{whenever $w\in X$ with $d(w,K)0$ and \[ \frac{f(\pi(u))}{g(\pi(u))} \leq \frac{f(v)}{g(\pi(u))} < b \] by \eqref{eq:minimaxfin}. Since $g$ is even and continuous on $\pi(K)$ by assumption~(c), if we set \[ \widehat{K} = \big\{\frac{\,\pi(u)}{g(\pi(u))}: u\in K\big\}\,, \] we have $\widehat{K}\in\mathcal{K}_{s,F}^{(m)}$ with \[ \sup_{\widehat{K}} f \leq b \] and the assertion follows by the arbitrariness of $b$. \end{proof} \begin{corollary}\label{cor:norm} Under the assumptions of Theorem~\ref{thm:minimaxfin}, let $Y$ be a vector subspace of $X$ such that \[ \left\{u\in X\setminus\{0\}: \text{$g(u)>0$ and $f(u)<+\infty$}\right\} \subseteq Y \] and let $\tau_Y$ be any topology on $Y$ which makes $Y$ a metrizable and locally convex topological vector space such that, for every $b\in \mathbb{R}$, the restriction of $g$ to \[ \{u\in Y\setminus\{0\}:f(u)\leq b\} \] is $\tau_Y$-continuous. Then the minimax values defined in the space $Y$ agree with those defined in the originary space $X$. \end{corollary} \begin{proof} First of all, there is no change if $X$ is substituted by $Y$ endowed with the topology of~$X$. By Theorem~\ref{thm:minimaxfin} it is equivalent to consider the classes $\mathcal{K}_{s,F}^{(m)}$ which do not change, when passing from the topology of $X$ to $\tau_Y$. \end{proof} \section{Variational convergence of functions and sup-functions} \label{sect:est} Let $X$ be a metrizable and locally convex topological vector space and, for every $h\in \mathbb{N}$, let $f_h:X\to[0,+\infty]$ and $g_h:X\setminus\{0\}\to \mathbb{R}$ be two functions such that: \begin{itemize} \item[(a)] $f_h$ and $g_h$ are both even and positively homogeneous of degree $1$; \item[(b)] $f_h$ is convex; \item[(c)] for every $b\in \mathbb{R}$, the restriction of $g_h$ to $\left\{u\in X\setminus\{0\}:f_h(u) \leq b\right\}$ is continuous. \end{itemize} For any integer $m\geq 1$, denote by $\mathcal{K}_{s,h}^{(m)}$ the family of nonempty, compact and symmetric subsets~$K$ of \[ \left\{u\in X\setminus\{0\}:g_h(u)=1\right\} \] such that $\operatorname{i}(K)\geq m$ and define $\mathcal{F}_h^{(m)}:\mathcal{K}\to[0,+\infty]$ as \[ \mathcal{F}_h^{(m)}(K) =\begin{cases} \sup_K f_h &\text{if $K\in \mathcal{K}_{s,h}^{(m)}$}\,,\\ +\infty &\text{otherwise}\,. \end{cases} \] The set $\mathcal{K}$ will be endowed with the $\mathcal{H}$-topology. Let also $f:X\to[0,+\infty]$ and $g:X\to \mathbb{R}$ be two even functions such that $g(0)=0$ and define $\mathcal{K}_s^{(m)}\subseteq\mathcal{K}$ and $\mathcal{F}^{(m)}:\mathcal{K}\to[0,+\infty]$ in an analogous way. \begin{theorem} \label{thm:limsup} Assume that \[ f(u) \geq \Big(\Gamma-\limsup_{h\to\infty} f_h\Big)(u) \quad\text{for all $u\in X$} \] and that, for every strictly increasing sequence $(h_n)$ in $\mathbb{N}$ and every sequence $(u_n)$ in $X\setminus\{0\}$ converging to $u\neq 0$ such that \[ \sup_{n\in\mathbb{N}} f_{h_n}(u_n) < +\infty\,, \] it holds \[ g(u) = \lim_{n\to\infty} g_{h_n}(u_n) \,. \] Then, for every $m\geq 1$, we have \begin{gather*} \mathcal{F}^{(m)}(K)\geq \Big(\Gamma-\limsup_{h\to\infty} \mathcal{F}_h^{(m)}\Big)(K) \quad\text{for all $K\in\mathcal{K}$}\,,\\ \inf_{K\in\mathcal{K}} \mathcal{F}^{(m)}(K) \geq \limsup_{h\to\infty} \Big( \inf_{K\in \mathcal{K}} \mathcal{F}_h^{(m)}(K)\Big)\,,\\ \inf_{K\in\mathcal{K}_s^{(m)}} \sup_K f \geq \limsup_{h\to\infty} \Big(\inf_{K\in\mathcal{K}_{s,h}^{(m)}} \sup_K f_h\Big)\,. \end{gather*} \end{theorem} \begin{proof} Let $m\geq 1$ and let $K\in\mathcal{K}$ with $\mathcal{F}^{(m)}(K)<+\infty$. Then $K$ is a nonempty, compact and symmetric subset of $\{u\in X\setminus\{0\}:g(u)=1\}$ with $\operatorname{i}(K)\geq m$. Consider a compatible distance $d$ on $X$ as in the proof of Proposition~\ref{prop:iusc}. Now, let $b\in \mathbb{R}$ with \[ b > \mathcal{F}^{(m)}(K) = \sup_K f \] and let $\delta>0$. Let $\sigma\in]0,1[$ be such that \begin{gather} \label{eq:e1} \sup_K f+\sigma < bs\quad\text{whenever $|s-1|<\sigma$}\,,\\ \label{eq:e2} d\big(s^{-1}\,w,u\big) < \delta \quad\text{whenever $u\in K$, $w\in X$ with $d(w,u)<\sigma$ and $|s-1|<\sigma$}\,. \end{gather} Then let $\overline{h}\in\mathbb{N}$ and $r\in]0,\sigma/2]$ be such that $K\cap B_{2r}(0)=\emptyset$ and \begin{equation} \label{eq:r} |g_h(w) - 1| < \sigma \end{equation} for any $h\geq\overline{h}$ and any $w\in X$ with $d(w,K)<2r$ and $f_h(w) < b+\sigma$. Again, let $F$ and $\vartheta_v$ be as in the proof of Proposition~\ref{prop:iusc}. Since $F$ is a finite set, by (d) of Proposition~\ref{prop:gammaseq} we can define, for every $h\in\mathbb{N}$, an odd map $\psi_h:F\to X$ such that \begin{gather*} \lim_{h\to\infty} \psi_h(v) = v \quad\text{for all $v \in F$}\,,\\ f(v) \geq \limsup_{h\to\infty} f_h(\psi_h(v)) \quad\text{for all $v \in F$}\,. \end{gather*} Without loss of generality, we assume that \[ \text{$d(\psi_h(v),v)0$}\,, \] it holds \[ \text{$u\neq 0$ and $g(u)=c$}\,. \] Then, for every $m\geq 1$, we have \[ \mathcal{F}^{(m)}(K) \leq \Big(\Gamma-\liminf_{h\to\infty} \mathcal{F}_h^{(m)}\Big)(K) \quad\text{for all $K\in\mathcal{K}$}\,. \] \end{theorem} \begin{proof} Let $m\geq 1$, let $K\in\mathcal{K}$ and let $(K_h)$ be a sequence converging to $K$ in $\mathcal{K}$ such that \[ \Big(\Gamma-\liminf_{h\to\infty} \mathcal{F}_h^{(m)}\Big)(K) = \liminf_{h\to\infty} \mathcal{F}_h^{(m)}(K_h) \,. \] Without loss of generality, we may assume that this value is not $+\infty$. Let $b\in \mathbb{R}$ with \[ b > \liminf_{h\to\infty} \mathcal{F}_h^{(m)}(K_h)\,. \] Then there exists a subsequence $(K_{h_n})$ such that \[ \sup_{n\in\mathbb{N}} \,\sup_{K_{h_n}} \,f_{h_n} = \sup_{n\in \mathbb{N}}\,\mathcal{F}_{h_n}^{(m)}(K_{h_n}) < b \,. \] In particular, $K_{h_n}\in\mathcal{K}_{s,h_n}^{(m)}$ so that $K$ also is symmetric. On the other hand, for every $u\in K$, there exists $u_h\in K_h$ with $u_h\to u$. Since $f_{h_n}(u_{h_n})0$}\,, \] there exists a subsequence $(u_{n_j})$ such that \[ \lim_{j\to\infty} u_{n_j} = u\quad\text{with $u\neq 0$ and $g(u)=c$}\,. \] Then, for every $m\geq 1$, the sequence $(\mathcal{F}_h^{(m)})$ is asymptotically equicoercive and \begin{gather*} \mathcal{F}^{(m)}(K)\leq \Big(\Gamma-\liminf_{h\to\infty} \mathcal{F}_h^{(m)}\Big)(K) \quad\text{for all $K\in\mathcal{K}$}\,,\\ \inf_{K\in\mathcal{K}} \mathcal{F}^{(m)}(K) \leq \liminf_{h\to\infty} \Big(\inf_{K\in \mathcal{K}} \mathcal{F}_h^{(m)}(K)\Big)\,,\\ \inf_{K\in\mathcal{K}_s^{(m)}} \sup_K f \leq \liminf_{h\to\infty} \Big( \inf_{K\in\mathcal{K}_{s,h}^{(m)}} \sup_K f_h\Big)\,. \end{gather*} \end{corollary} \begin{proof} If we define $\tilde{f}_h:X\to[0,+\infty]$ and $\widetilde{\mathcal{F}}_h:\mathcal{K}\to[0,+\infty]$ as \begin{gather*} \tilde{f}_h(u) = \begin{cases} f_h(u) &\text{if $g_h(u)=1$}\,,\\ +\infty &\text{otherwise}\,, \end{cases} \\ \widetilde{\mathcal{F}}_h(K)= \sup_K \tilde{f}_h\,, \end{gather*} it is easily seen that $(\tilde{f}_h)$ is asymptotically equicoercive. By Proposition~\ref{prop:coerchausd} $(\widetilde{\mathcal{F}}_h)$ also is asymptotically equicoercive. In turn, from $\mathcal{F}_h^{(m)}\geq \widetilde{\mathcal{F}}_h$ it follows that $(\mathcal{F}_h^{(m)})$ is asymptotically equicoercive. From Theorem~\ref{thm:liminf} we infer that \[ \mathcal{F}^{(m)}(K) \leq \Big(\Gamma-\liminf_{h\to\infty} \mathcal{F}_h^{(m)}\Big)(K) \quad\text{for all $K\in\mathcal{K}$} \] and the other assertions follow from Proposition~\ref{prop:liminfinf}. \end{proof} \begin{corollary} \label{cor:limval} Assume that \[ f(u) = \Big(\Gamma-\lim_{h\to\infty} f_h\Big)(u) \quad\text{for all $u\in X$} \] and that, for every strictly increasing sequence $(h_n)$ in $\mathbb{N}$ and every sequence $(u_n)$ in $X\setminus\{0\}$ such that \[ \sup_{n\in\mathbb{N}} f_{h_n}(u_n) < +\infty\,, \] there exists a subsequence $(u_{n_j})$ converging to some $u$ in $X$ with \[ \lim_{j\to\infty} g_{h_{n_j}}(u_{n_j}) = g(u)\,. \] Then, for every $m\geq 1$, the sequence $(\mathcal{F}_h^{(m)})$ is asymptotically equicoercive and \begin{gather*} \mathcal{F}^{(m)}(K)= \Big(\Gamma-\lim_{h\to\infty} \mathcal{F}_h^{(m)}\Big)(K) \quad\text{for all $K\in\mathcal{K}$}\,,\\ \inf_{K\in\mathcal{K}} \mathcal{F}^{(m)}(K) = \lim_{h\to\infty} \Big(\inf_{K\in \mathcal{K}} \mathcal{F}_h^{(m)}(K)\Big)\,,\\ \inf_{K\in\mathcal{K}_s^{(m)}} \sup_K f = \lim_{h\to\infty} \Big(\inf_{K\in\mathcal{K}_{s,h}^{(m)}} \sup_K f_h\Big)\,. \end{gather*} \end{corollary} \begin{proof} Since $g(0)=0$, if $(u_{n_j})$ is convergent to some $u$ in $X$ with \[ \sup_{n\in\mathbb{N}} f_{h_n}(u_n) < +\infty\,,\quad \lim_{n\to\infty} g_{h_n}(u_n) = c >0\,, \] it follows that $u\neq 0$ and $g(u)=c$. Then the assertion is just a combination of Theorem~\ref{thm:limsup} and Corollary~\ref{cor:liminfval}. \end{proof} \section{Minimax values and functionals of calculus of variations} \label{sect:calcvar} Throughout this section, $\Omega$ denotes an open subset of $\mathbb{R}^N$ with $N\geq 2$ and, for any $q\in[1,\infty]$, $\|\cdot\|_q$ the usual norm in $L^q$. Since $\Omega$ is allowed to be unbounded, for any $p\in]1,N[$ we will consider the Banach space $D^{1,p}_0(\Omega)$ (see e.g.~\cite{lucia_schuricht2013}) endowed with the norm \[ \|u\| = \|\nabla u\|_p = \Big(\int_\Omega |\nabla u|^p\,dx\Big)^{1/p}\,. \] Recall that $D^{1,p}_0(\Omega)$ is continuously embedded in $L^{p^*}(\Omega)$, where $p^*=Np/(N-p)$, and contains $C^{\infty}_c(\Omega)$ as a dense vector subspace. For any $p\in]1,N[$, define $\mathcal{E}_p:L^1_{\rm loc}(\Omega)\to[0,+\infty]$ as \[ \mathcal{E}_p(u) =\begin{cases} \|\nabla u\|_p &\text{if $u\in D^{1,p}_0(\Omega)$}\,,\\ +\infty &\text{otherwise}\,. \end{cases} \] In the case $p=1$, define first $\widehat{\mathcal{E}}_1: L^1_{\rm loc}(\Omega)\to[0,+\infty]$ as \[ \widehat{\mathcal{E}}_1(u) =\begin{cases} \int_\Omega |\nabla u|\,dx &\text{if $u\in C^1_c(\Omega)$}\,,\\ +\infty &\text{otherwise}\,, \end{cases} \] then denote by $\mathcal{E}_1:L^1_{\rm loc}(\Omega)\to[0,+\infty]$ the lower semicontinuous envelope of $\widehat{\mathcal{E}}_1$ with respect to the $L^1_{\rm loc}(\Omega)$-topology. If $\Omega$ is bounded and has Lipschitz boundary, then $\mathcal{E}_1$ has a well known integral representation (see e.g.~\cite[Example~3.14]{dalmaso1993}). In any case, $\mathcal{E}_1$ is convex, even and positively homogeneous of degree $1$. Moreover, \[ X_1 = \{u\in L^1_{\rm loc}(\Omega): \mathcal{E}_1(u)<+\infty\} \] is a vector subspace of $L^1_{\rm loc}(\Omega)$ and $\mathcal{E}_1$ is a norm on $X_1$ which makes $X_1$ a normed space continuously embedded in $L^{1^*}(\Omega)=L^{\frac{N}{N-1}}(\Omega)$. More precisely, if we set \[ S(N,p) = \inf\Big\{ \frac{{\int_{\mathbb{R}^N} |\nabla u|^p\,dx}}{ {\big(\int_{\mathbb{R}^N} |u|^{p^*}\,dx\big)^{p/p^*}}}: u\in C^1_c(\mathbb{R}^N)\setminus\{0\}\Big\} \quad\text{whenever $1\leq p 0 \quad\text{for all $q\in]1,N[$}\,,\\ S(N,p)^{1/p}\,\|u\|_{p^*} \leq\mathcal{E}_p(u) \quad\text{whenever $1\leq p0$ such that \[ f_p(u) \geq \nu \,\mathcal{E}_p(u) \quad\text{for all $u\in L^1_{\rm loc}(\Omega)$}\,. \] Then, for every $m\geq 1$, we have \[ \inf_{K\in\mathcal{K}_{s,p}^{(m)}} \sup_K f_p = \inf_{K\in\mathcal{V}_p^{(m)}} \sup_K f_p\,. \] \end{theorem} \begin{proof} From Proposition \ref{prop:gV} and the lower estimate on $f_p$ we infer that, for every $b\in \mathbb{R}$, the restriction of $g_p$ to $\{u\in L^1_{\rm loc}(\Omega):f_p(u)\leq b\}$ is $L^1_{\rm loc}(\Omega)$-continuous. Of course, the same is true if we consider a stronger topology. Then the assertion follows from Corollary~\ref{cor:norm}. \end{proof} Now, in view of the convergence results of the next section, let us prove some further basic facts concerning $\mathcal{E}_p$ and $g_p$. The authors want to thank Lorenzo Brasco for pointing out that a previous version of this theorem was incorrect. \begin{theorem}\label{thm:gammaE} For every sequence $(p_h)$ decreasing to $p$ in $[1,N[$, we have \[ \mathcal{E}_p(u) = \Big(\Gamma-\lim_{h\to\infty} \mathcal{E}_{p_h}\Big)(u) \quad\text{for all $u\in L^1_{\rm loc}(\Omega)$}\,. \] \end{theorem} \begin{proof} Let us prove only the case $p=1\Big(\Gamma-\liminf_{h\to\infty} \mathcal{E}_{p_h}\Big)(u) \] and let $(u_h)$ be a sequence converging to $u$ in $L^1_{\rm loc}(\Omega)$ such that \[ \Big(\Gamma-\liminf_{h\to\infty} \mathcal{E}_{p_h}\Big)(u) = \liminf_{h\to\infty} \mathcal{E}_{p_h}(u_h)\,. \] Let $(\mathcal{E}_{p_{h_n}})$ be such that \[ \sup_{n\in\mathbb{N}}\,\mathcal{E}_{p_{h_n}}(u_{h_n}) \|\nabla v_n\|_{p_{h_n}} \geq \|\vartheta\nabla v_n\|_{p_{h_n}}\\ & \geq \|\nabla(\vartheta v_n)\|_{p_{h_n}} - \|v_n \nabla\vartheta\|_{p_{h_n}} \\ & \geq \mathcal{L}^n(\operatorname{supp}(\vartheta))^{\frac{1-p_{h_n}}{p_{h_n}}} \,\|\nabla(\vartheta v_n)\|_1 - \|v_n \nabla\vartheta\|_{p_{h_n}} \\ &\geq \mathcal{L}^n(\operatorname{supp}(\vartheta))^{\frac{1-p_{h_n}}{p_{h_n}}} \,\mathcal{E}_1(\vartheta v_n) - \|v_n \nabla\vartheta\|_{p_{h_n}}\,, \end{align*} where $\mathcal{L}^n$ denotes the Lebesgue measure. Passing to the lower limit as $n\to\infty$, we obtain \[ b\geq \mathcal{E}_1(\vartheta u) - \|u \nabla\vartheta\|_1\,. \] Let $\vartheta:\mathbb{R}^N\to[0,1]$ be a $C^1$-function such that $\vartheta(x)=1$ if $|x|\leq 1$ and $\vartheta(x)=0$ if $|x|\geq 2$ and let $\vartheta_k(x) = \vartheta(x/k)$. Then \[ b\geq \mathcal{E}_1(\vartheta_k u) - \int_\Omega |u|\,|\nabla\vartheta_k|\,dx\,. \] It is easily seen that $(\vartheta_k u)$ is convergent to $u$ in $L^1_{\rm loc}(\Omega)$, while $(|\nabla\vartheta_k|)$ is bounded in $L^N(\Omega)$ and convergent to $0$ a.e. in $\Omega$. Passing to the lower limit as $k\to\infty$, we obtain $b\geq \mathcal{E}_1(u)$, hence \[ \mathcal{E}_1(u) \leq \Big(\Gamma-\liminf_{h\to\infty} \mathcal{E}_{p_h}\Big)(u) \] by the arbitrariness of $b$. Now let $u\in L^1_{\rm loc}(\Omega)$, let $b\in \mathbb{R}$ with $b>\mathcal{E}_1(u)$ and let $\delta>0$. Let $w\in C^1_c(\Omega)$ with $d(w,u)<\delta$ and $\|\nabla w\|_1 \lim_{h\to\infty} \mathcal{E}_{p_h}(w)\,, \] whence \[ b> \limsup_{h\to\infty}\bigl( \inf\{\mathcal{E}_{p_h}(v):d(v,u)<\delta\}\bigr)\,. \] By the arbitrariness of $\delta$, it follows that \[ b\geq \Big(\Gamma-\limsup_{h\to\infty} \mathcal{E}_{p_h}\Big)(u)\,, \] hence \[ \mathcal{E}_1(u) \geq \Big(\Gamma-\limsup_{h\to\infty} \mathcal{E}_{p_h}\Big)(u) \] by the arbitrariness of $b$. \end{proof} \begin{theorem}\label{thm:convg} Let $(p_h)$ be a sequence converging to $p$ in $[1,N[$ and let $V_h\in L^{N/p_h}(\Omega)$ and $V\in L^{N/p}(\Omega)$ be such that \begin{gather*} \lim_{h\to\infty} V_h(x) = V(x) \quad\text{for a.e. $x\in\Omega$}\,,\\ \lim_{h\to\infty} \|V_h\|_{N/p_h} = \|V\|_{N/p}\,. \end{gather*} Define $g_h, g:L^1_{\rm loc}(\Omega)\to \mathbb{R}$ according to \eqref{eq:gV}. Then, for every strictly increasing sequence $(h_n)$ in $\mathbb{N}$ and $(u_n)$ in $L^1_{\rm loc}(\Omega)$ such that \[ \sup_{n\in\mathbb{N}} \mathcal{E}_{p_{h_n}}(u_n) <+\infty\,, \] there exists a subsequence $(u_{n_j})$ such that \begin{gather*} \lim_{j\to\infty} u_{n_j} = u \quad\text{in $L^1_{\rm loc}(\Omega)$}\,,\\ \lim_{j\to\infty} g_{h_{n_j}}(u_{n_j}) = g(u)\,. \end{gather*} \end{theorem} \begin{proof} Up to a subsequence, $(u_n)$ is convergent to some $u$ in $L^1_{\rm loc}(\Omega)$ and a.e. in $\Omega$. Moreover, for every $\varepsilon>0$ there exists $C_\varepsilon>0$ independent of $n$ such that \[ \big|V_{h_n}\,|u_n|^{p_{h_n}} - V\,|u|^p\big| \leq C_\varepsilon |V_{h_n}|^{N/p_{h_n}} + \varepsilon |u_n|^{p_{h_n}^*} + |V|\,|u|^p\,, \] whence \[ C_\varepsilon |V_{h_n}|^{N/p_{h_n}} + \varepsilon |u_n|^{p_{h_n}^*} - \big|V_{h_n}\,|u_n|^{p_{h_n}} - V\,|u|^p\big| \geq - |V|\,|u|^p\,. \] From Fatou's lemma it follows that \begin{align*} & C_\varepsilon \int_\Omega |V|^{N/p}\,dx \\ & \leq C_\varepsilon \int_\Omega |V|^{N/p}\,dx + \varepsilon \Big(\sup_{n\in\mathbb{N}}\|u_n\|_{p_{h_n}^*}^{p_{h_n}^*}\Big) - \limsup_{n\to\infty} \int_\Omega \big|V_{h_n}\,|u_n|^{p_{h_n}} - V\,|u|^p\big|\,dx\,, \end{align*} whence \[ \limsup_{n\to\infty}\int_\Omega \big|V_{h_n}\,|u_n|^{p_{h_n}} - V\,|u|^p\big|\,dx \leq \varepsilon \Big(\sup_{n\in\mathbb{N}}\|u_n\|_{p_{h_n}^*}^{p_{h_n}^*}\Big)\,. \] Since $(\mathcal{E}_{p_{h_n}}(u_n))$ is bounded, we infer that \[ \sup_{n\in\mathbb{N}}\|u_n\|_{p_{h_n}^*}^{p_{h_n}^*}< +\infty \] and the assertion follows by the arbitrariness of $\varepsilon$. \end{proof} \section{Convergence of minimax values for functionals of calculus of variations} \label{sect:convcalcvar} In this section, $\Omega$ still denotes an open subset of $\mathbb{R}^N$ with $N\geq 2$ and, for any $p\in[1,N[$, $\mathcal{E}_p:L^1_{\rm loc}(\Omega)\to[0,+\infty]$ the functional introduced in the previous section. Assume that $(p_h)$ is a sequence converging to $p$ in $[1,N[$, $f:L^1_{\rm loc}(\Omega)\to[0,+\infty]$ is a functional, $(f_h)$ is a sequence of functionals from $L^1_{\rm loc}(\Omega)$ to $[0,+\infty]$, $V\in L^{N/p}(\Omega)$ and $(V_h)$ is a sequence with $V_h\in L^{N/p_h}(\Omega)$. Also suppose that: \begin{itemize} \item[(H1)] $f$ is even; \item[(H2)] each $f_h$ is convex, even and positively homogeneous of degree $1$; moreover, there exists $\nu>0$ such that \[ f_h(u) \geq \nu \mathcal{E}_{p_h}(u) \quad\text{for all $h\in\mathbb{N}$ and $u\in L^1_{\rm loc}(\Omega)$}\,; \] \item[(H3)] we have \begin{gather*} \lim_{h\to\infty} V_h(x) = V(x) \quad\text{for a.e. $x\in\Omega$}\,,\\ \lim_{h\to\infty} \|V_h\|_{N/p_h} = \|V\|_{N/p}\,. \end{gather*} \end{itemize} Let $\mathcal{K}$ be the family of nonempty compact subsets of $L^1_{\rm loc}(\Omega)$ endowed with the $\mathcal{H}$-topology and define $g_h, g:L^1_{\rm loc}(\Omega)\to \mathbb{R}$ according to \eqref{eq:gV}. Then define $\mathcal{K}_{s,h}^{(m)}, \mathcal{K}_s^{(m)} \subseteq\mathcal{K}$ and $\mathcal{F}_h^{(m)}, \mathcal{F}^{(m)}:\mathcal{K}\to[0,+\infty]$ as in Section~\ref{sect:est}. \begin{theorem} \label{thm:limsupconcr} Assume that \[ f(u) \geq \Big(\Gamma-\limsup_{h\to\infty} f_h\Big)(u) \quad\text{for all $u\in L^1_{\rm loc}(\Omega)$}\,. \] Then, for every $m\geq 1$, we have \begin{gather*} \mathcal{F}^{(m)}(K) \geq \Big(\Gamma-\limsup_{h\to\infty} \mathcal{F}_h^{(m)}\Big)(K) \quad\text{for all $K\in\mathcal{K}$}\,,\\ \inf_{K\in\mathcal{K}} \mathcal{F}^{(m)}(K)\geq \limsup_{h\to\infty} \Big(\inf_{K\in \mathcal{K}} \mathcal{F}_h^{(m)}(K)\Big)\,,\\ \inf_{K\in\mathcal{K}_s^{(m)}} \sup_K f \geq \limsup_{h\to\infty} \Big(\inf_{K\in\mathcal{K}_{s,h}^{(m)}} \sup_K f_h\Big)\,. \end{gather*} \end{theorem} The proof of the above theorem follows from Theorem~\ref{thm:limsup}, Proposition~\ref{prop:gV} and Theorem~\ref{thm:convg}. \begin{theorem} \label{thm:liminfconcr} Assume that \[ f(u) \leq \Big(\Gamma-\liminf_{h\to\infty} f_h\Big)(u) \quad\text{for all $u\in L^1_{\rm loc}(\Omega)$}\,. \] Then, for every $m\geq 1$, the sequence $(\mathcal{F}_h^{(m)})$ is asymptotically equicoercive and we have \begin{gather*} \mathcal{F}^{(m)}(K) \leq \Big(\Gamma-\liminf_{h\to\infty} \mathcal{F}_h^{(m)}\Big)(K) \quad\text{for all $K\in\mathcal{K}$}\,,\\ \inf_{K\in\mathcal{K}} \mathcal{F}^{(m)}(K) \leq \liminf_{h\to\infty} \Big(\inf_{K\in \mathcal{K}} \mathcal{F}_h^{(m)}(K)\Big)\,,\\ \inf_{K\in\mathcal{K}_s^{(m)}} \sup_K f \leq \liminf_{h\to\infty} \Big(\inf_{K\in\mathcal{K}_{s,h}^{(m)}} \sup_K f_h\Big)\,. \end{gather*} \end{theorem} The proof of the above theorem follows from Corollary~\ref{cor:liminfval}, Proposition~\ref{prop:gV} and Theorem~\ref{thm:convg}. \begin{corollary} \label{cor:limconcr} Assume that \[ f(u) = \Big(\Gamma-\lim_{h\to\infty} f_h\Big)(u) \quad\text{for all $u\in L^1_{\rm loc}(\Omega)$}\,. \] Then, for every $m\geq 1$, the sequence $(\mathcal{F}_h^{(m)})$ is asymptotically equicoercive and we have \begin{gather*} \mathcal{F}^{(m)}(K) = \Big(\Gamma-\lim_{h\to\infty} \mathcal{F}_h^{(m)}\Big)(K) \quad\text{for all $K\in\mathcal{K}$}\,,\\ \inf_{K\in\mathcal{K}} \mathcal{F}^{(m)}(K) = \lim_{h\to\infty} \Big(\inf_{K\in \mathcal{K}} \mathcal{F}_h^{(m)}(K)\Big)\,,\\ \inf_{K\in\mathcal{K}_s^{(m)}} \sup_K f =\lim_{h\to\infty} \Big(\inf_{K\in\mathcal{K}_{s,h}^{(m)}} \sup_K f_h\Big)\,. \end{gather*} \end{corollary} The proof of the above corollary follows from Corollary~\ref{cor:limval}, Proposition~\ref{prop:gV} and Theorem~\ref{thm:convg}. As an example, whenever $1\leq p