\documentclass[reqno]{amsart} \usepackage{amssymb,mathrsfs} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2001(2001), No. 63, pp. 1--10.\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 2001 Southwest Texas State University.} \vspace{1cm}} \begin{document} \title[\hfilneg EJDE--2001/63\hfil A generalized Emden--Fowler equation] {On the multiplicity of solutions for a fully nonlinear Emden--Fowler equation} \author[Marco Squassina\hfil EJDE--2001/63\hfilneg] {Marco Squassina} \address{Marco Squassina\hfill\break Universit\'a Cattolica del S.C., Dipartimento di Matematica \& Fisica, Via Musei 41, 25121 Brescia, Italy} \email{squassin@dmf.unicatt.it} \date{} \thanks{Submitted April 4, 2001. Published October 3, 2001.} \thanks{Partially supported by M.U.R.S.T. (40\% -- project 1999) and by G.N.A.F.A} \subjclass[2000]{35D05, 58E05} \keywords{Trudinger-Moser inequality, Euler's equations, exponential growth, \hfill\break\indent Palais-Smale condition} \begin{abstract} We are concerned with the existence of two solutions for a fully nonlinear Emden--Fowler type equation. One solution is obtained via local minimization while the second solution follows by a mountain pass argument. A non-existence result in strictly star-shaped domains is also proven. \end{abstract} \maketitle \renewcommand{\leq}{\leqslant} \renewcommand{\geq}{\geqslant} \renewcommand{\le}{\leqslant} \renewcommand{\ge}{\geqslant} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} Let ${\mathcal M}$ be a $C^\infty$ compact connected manifold of dimension two and $g$ a metric on ${\mathcal M}$. As known, the problem of finding a conformal metric $g'$ such that the scalar curvature of $({\mathcal M},g')$ is equal to a given function ${\mathcal K}(x)$, gives rise to the following problem (scalar curvature problem, Nirenberg 1974) $$\label{pb} -\varDelta_gu+R_gu={\mathcal K}(x)e^{2u}\,,\quad g'=e^{2u}g\,,$$ where $R_g$ denotes the curvature of ${\mathcal M}$ and $$\varDelta_g=\sum_{i,j=1}^n\frac{1}{\sqrt{\det(g_{ij})}} \frac{\partial}{\partial x_i} \left(\sqrt{\det(g_{ij})}g^{ij}\frac{\partial }{\partial x_j}\right)$$ is the Laplace--Beltrami operator. Problems like (\ref{pb}) are also involved in the study of stellar structure and in nonlinear diffusion and heat transfer in chemical kinetic. \par Starting from these geometrical and physical motivations, let us consider a smooth bounded domain $\varOmega$ in ${\mathbb R}^n$ with $n\geq 2$ and a possibly changing sign function ${\mathcal K}$ in $L^q(\varOmega)$ for some $q>1$ with ${\mathcal K}>0$ a.e. in an open ball $B$ of $\varOmega$. When $n=2$, the semilinear elliptic equation with exponential growth \begin{gather*} -\varDelta u={\mathcal K}(x)\,e^u \quad\text{in }\varOmega \\ u=0 \quad\text{on }\partial\varOmega, \end{gather*} has been studied in 1974 by Kazdan and Warner in \cite{kazdan} and in 1992 by Br\'ezis and Merle in \cite{bm}. In the case of the $p-$Laplacian problem ($p>1$ and $\lambda>0$) $$\label{Q_lambda} \begin{gathered} -\varDelta_pu=\lambda {\mathcal K}(x)e^u \quad\text{in }\varOmega \\ u=0 \quad\text{on }\partial\varOmega, \end{gathered}$$ a complete picture has been given by Aguilar Crespo and Peral Alonso in 1996 in \cite{crespo}. In particular, it was shown existence of solutions of (\ref{Q_lambda}) for $\lambda$ small (one solution for $pn$ by the Morrey embedding $$W^{1,p}_0(\varOmega)\hookrightarrow L^{\infty}(\varOmega),$$ but fails to be regular when $p=n$ unless ${\mathcal L}$ does not depend on $u$ or it is subjected to some very restrictive growth conditions. Indeed, with natural growth conditions (see (\ref{L}) and (\ref{xi}) below), in general, being $$\forall s<+\infty:\,\,\, W^{1,n}_0(\varOmega) \hookrightarrow L^{s}(\varOmega)\,\,\quad \quad\text{but}\,\,\quad W^{1,n}_0(\varOmega)\not\hookrightarrow L^{\infty}(\varOmega)\,,$$ if $u\in W^{1,n}_0(\varOmega)$ it may happen that $$D_s{\mathcal L}(x,u,\nabla u) \not\in W^{-1,n'}(\varOmega)\,,$$ so that $f_\lambda$ is not even locally Lipschitzian. Therefore we focus on the case $p=n$ and prove the existence of at least two nontrivial solutions in $W^{1,n}_0(\varOmega)$ of (\ref{P_lambda}) for $\lambda$ positive and small. To solve (\ref{P_lambda}) we look for critical points of (\ref{funct}) in the sense of non-smooth critical point theory (see \cite{cd,toulouse} and references therein). The case $p>n$ may be treated in a similar fashion via classical critical point theory. \vskip6pt We assume that ${\mathcal L}:\varOmega\times{\mathbb R}\times{\mathbb R}^{n}\to{\mathbb R}$ is measurable in $x$ for all $(s,\xi)\in{\mathbb R}\times{\mathbb R}^{n}$ and of class $C^1$ in $(s,\xi)$ a.e. in $\varOmega$.\ Moreover ${\mathcal L}(x,s,\cdot)$ is strictly convex, $n-$homogeneous with ${\mathcal L}(x,s,0)=0$ and the following conditions hold: \vskip4pt \begin{enumerate} \item[$({\mathscr H}_1)$] there exist $a_1\in L^1(\varOmega)$, $r>1$ and $b_0,b_1,b_2,\nu>0$ such that: $$\label{L} \nu |\xi|^n\leq {\mathcal L}(x,s,\xi) \leq b_0|s|^r+b_0|\xi|^n,$$ $$\label{xi} \left|D_s{\mathcal L}(x,s,\xi)\right|\leq a_1(x)+b_1|\xi|^n,\quad \left|\nabla_\xi {\mathcal L}(x,s,\xi)\right|\leq b_2|\xi|^{n-1}$$ \vskip2pt a.e. in $\varOmega$ and for all $(s,\xi)\in{\mathbb R}\times{\mathbb R}^n$\,; \vskip4pt \item[$({\mathscr H}_2)$] there exist $R>0$ and $\gamma>0$ such that: $$\label{posit} |s|\geq R\,\,\Longrightarrow\,\, D_s{\mathcal L}(x,s,\xi)s\geq 0,$$ $$\label{boundps} \gamma {\mathcal L}(x,s,\xi)-D_s {\mathcal L}(x,s,\xi)s \geq \nu |\xi|^n$$ \vskip2pt a.e. in $\varOmega$ and for all $(s,\xi)\in{\mathbb R}\times{\mathbb R}^n$. \end{enumerate} \vskip4pt The growth conditions of $({\mathscr H}_1)$ and the assumptions in $({\mathscr H}_2)$ are natural in the fully nonlinear setting and were considered in \cite{toulouse} and in a stronger form in \cite{ab,pell} also. \vskip4pt Under the preceding assumptions, the following is our main result. \begin{theorem} \label{mainth} There exists $\lambda_0>0$ such that (\ref{P_lambda}) admits at least two nontrivial solutions in $W^{1,n}_0(\varOmega)$ for each $\lambda<\lambda_0$. \end{theorem} This result extends \cite[Theorem 4.1]{crespo} to a more general class of nonlinear elliptic equations. In particular, quasilinear $n-$Laplacian problems of the type \begin{gather*} -\mathop{\rm div}(a(u)|\nabla u|^{n-2}\nabla u)+a'(u)|\nabla u|^n =\lambda {\mathcal K}(x)e^u \quad\text{in }\varOmega \\ u=0 \quad\text{on }\partial\varOmega \end{gather*} admit a pair of solutions for $\lambda$ small and $a\in C^1({\mathbb R})$ satisfying suitable assumptions. \vskip4pt \begin{remark} \rm In general, problems (\ref{P_lambda}) are expected to have no bounded solution when ${\mathcal K}\geq 0$ and $\lambda>\lambda^*$ for a suitable $\lambda^*>0$.\ See Theorem 5.8 of \cite{crespo} where this is showed for problem (\ref{Q_lambda}) with $p=n$ and $$\lambda^*=\max\left\{\lambda_1,\left(\frac{n-1}{e}\right)^{n-1}\!\!\!\lambda_1\right\}\!,$$ being $\lambda_1$ the first eigenvalue of $-\varDelta_n$ weighted by ${\mathcal K}$. See also Proposition~\ref{reg}. \end{remark} \vskip2pt \begin{remark}\rm In general, problems (\ref{P_lambda}) have no solution if $\varOmega$ is an unbounded domain of ${\mathbb R}^n$.\ See Theorem 3.3 of \cite{crespo3} where this is proved for problems (\ref{Q_lambda}). \end{remark} \vskip2pt \begin{remark}\rm Condition ${\mathcal K}^+\not\equiv 0$ is crucial for the multiplicity result to hold. For example for (\ref{Q_lambda}), if ${\mathcal K}<0$ one finds only one solution.\ See \cite[Section 4]{crespo}. \end{remark} \vskip2pt In particular $\lambda$ small, $\varOmega$ bounded and ${\mathcal K}^+\not\equiv 0$ seem to be natural assumptions in order to get the multiplicity result. \begin{remark}\rm By the regularity result of Tolksdorf \cite{tolk}, each bounded weak solution of problem (\ref{P_lambda}) belongs to $C^{1,\alpha}(\varOmega)$ for some $\alpha>0$. \end{remark} \section{The concrete Palais--Smale condition} Let us now recall two basic definitions of abstract critical point theory (see \cite{cd}). \begin{definition} \rm Let $(X,d)$ be a metric space, $f:X\to{\mathbb R}$ a continuous function and $u\in X$. We denote by $|df|(u)$ the supremum of $\sigma\geq 0$ such that there exist $\delta>0$ and a continuous map $$\mathcal{H}:B_{{\delta}}(u)\times[0,\delta]\to X$$ such that for all $(v,t)\in B_{{\delta}}(u)\times[0,\delta]$ $$d(\mathcal{H}(v,t),v)\leq t,\quad f(\mathcal{H}(v,t))\leq f(v)-\sigma t.$$ We say that the extended real number $|df|(u)$ is the weak slope of $f$ at $u$. \end{definition} \begin{definition} \rm Let $(X,d)$ be a metric space, $f:X\to{\mathbb R}$ a continuous function and $u\in X$. We say that $u$ is a critical point of $f$ if $|df|(u)=0$. \end{definition} \begin{definition} \rm We say that $f:X\to{\mathbb R}$ satisfies the Palais--Smale condition at level $c$ (in short $(PS)_c$) if each sequence $(u_h)$ in $X$ with $f(u_h)\to c$ and $|df|(u_h)\to 0$ admits a convergent subsequence in $X$. \end{definition} We now return to the concrete case and set $f=f_\lambda$ and $X=W^{1,n}_0(\varOmega)$ endowed with the standard norm $\|u\|_{1,n}^n=\int_{\varOmega}|\nabla u|^n\,dx$. \begin{definition} \rm \label{definitioncpsc} A sequence $(u_h)\subset W^{1,n}_0(\varOmega)$ is said to be a concrete Palais--Smale sequence at level $c\in{\mathbb R}$ ($(CPS)_c-$sequence, in short) for $f_\lambda$, if $f_{\lambda}(u_h)\to c$, $$-\mathop{\rm div}\left(\nabla_{\xi}{\mathcal L}(x,u_h,\nabla u_h)\right)+ D_s{\mathcal L}(x,u_h,\nabla u_h) \in W^{-1,n'}(\varOmega)$$ eventually as $h\to+\infty$ and $$-\mathop{\rm div}\left(\nabla_{\xi}{\mathcal L}(x,u_h,\nabla u_h)\right)+ D_s{\mathcal L}(x,u_h,\nabla u_h)-\lambda {\mathcal K}(x)e^{u_h}\to 0$$ strongly in $W^{-1,n'}(\varOmega)$. We say that $f_{\lambda}$ satisfies the concrete Palais--Smale condition at level $c$ ($(CPS)_c$ in short), if every $(CPS)_c-$sequence for $f_{\lambda}$ admits a strongly convergent subsequence. \end{definition} \begin{lemma} \label{theorcomp} Let $u\in W^{1,n}_0(\varOmega)$ be such that $|df_{\lambda}|(u)<+\infty$. Then $$-\mathop{\rm div}\left(\nabla_{\xi}{\mathcal L}(x,u,\nabla u)\right)+ D_s{\mathcal L}(x,u,\nabla u)-\lambda {\mathcal K}(x)e^u\in W^{-1,n'}(\varOmega)$$ and $$\left\|-\mathop{\rm div}\left(\nabla_{\xi}{\mathcal L}(x,u,\nabla u)\right)+ D_s{\mathcal L}(x,u,\nabla u)-\lambda {\mathcal K}(x)e^u\right\|_{-1,n'} \leq |df_\lambda|(u).\$$ In particular, if $|df_{\lambda}|(u)=0$ then $u$ solves (\ref{P_lambda}) in the distributional space ${\mathscr D}'(\varOmega)$. \end{lemma} For the proof of the above lemma, see \cite[Theorem 2.3]{toulouse}.\ It is readily seen that if $f_\lambda$ satisfies $(CPS)_c$, then it satisfies $(PS)_c$. \vskip3pt \noindent Let us now recall a very useful consequence of Brezis-Browder's Theorem \cite{bb}. \begin{proposition} \label{bb} Let $u,v\in W^{1,n}_0(\varOmega)$ be such that $D_s{\mathcal L}(x,u,\nabla u)v\geq 0$ and assume that $w\in W^{-1,n^{\prime}}(\varOmega)$ is defined by $$\forall\varphi\in C^\infty_c(\varOmega):\,\,\langle w,\varphi\rangle=\int_\varOmega \nabla_\xi {\mathcal L}(x,u,\nabla u) \cdot \nabla \varphi\,dx+\int_\varOmega D_s{\mathcal L}(x,u,\nabla u)\varphi\,dx.\$$ Then $D_s {\mathcal L}(x,u,\nabla u)v\in L^1(\varOmega)$ and \begin{equation*} \langle w,v\rangle=\int_\varOmega \nabla_\xi {\mathcal L}(x,u,\nabla u) \cdot \nabla v\,dx+\int_\varOmega D_s {\mathcal L}(x,u,\nabla u)v\,dx.\ \end{equation*} \end{proposition} For the proof of this proposition, see \cite[Proposition 3.1]{toulouse}. \vskip3pt \noindent The next result will provide compactness of concrete Palais--Smale sequences. \begin{lemma} \label{compattezza} Let $(u_h)$ be a bounded sequence in $W^{1,n}_0(\varOmega)$ and set $$\forall\varphi\in C^\infty_c(\varOmega):\, \langle w_h,\varphi\rangle=\int_\varOmega \nabla_\xi {\mathcal L}(x,u_h,\nabla u_h) \cdot\nabla\varphi\,dx+\int_\varOmega D_s{\mathcal L}(x,u_h,\nabla u_h)\varphi\,dx.$$ If $(w_h)$ is strongly convergent to some $w$ in $W^{-1,n'}(\varOmega),$ then $(u_h)$ admits a strongly convergent subsequence in $W^{1,n}_0(\varOmega)$. \end{lemma} For the proof of this lemma, see \cite[Theorem 3.4]{toulouse}. \vskip5pt \noindent Let us prove that $f_\lambda$ satisfies the concrete Palais--Smale condition. \begin{theorem} \label{palsma} $f_{\lambda}$ satisfies $(CPS)_c$ for each $c\in{\mathbb R}$. \end{theorem} \begin{proof} Let $(u_h)$ be a concrete Palais--Smale sequence for $f_{\lambda}$ at level $c\in{\mathbb R}$. We shall divide the proof into two steps: \vskip4pt \noindent I) Let us first show that $(u_h)$ is bounded in $W^{1,n}_0(\varOmega)$. Note that in view of (\ref{posit}), by Proposition~\ref{bb} one can take $u_h$ as test functions in $f_\lambda'(u_h)$. Therefore, since $f_{\lambda}^{\prime}(u_h)(u_h)=o(1)$ as $h\to+\infty$, by (\ref{boundps}) one obtains \begin{align*} f_\lambda(u_h)=&f_\lambda(u_h)-\frac{1}{n} f_{\lambda}^{\prime}(u_h)(u_h)+o(1)\\\ =&-\frac{1}{n}\int_{\varOmega}D_s {\mathcal L}(x,u_h,\nabla u_h)u_h\,dx+ \lambda\int_{\varOmega}{\mathcal K}(x)e^{u_h} \left\{\frac{u_h}{n}-1\right\}\,dx+o(1) \\ \geq &-\frac{\gamma}{n}\int_{\varOmega}{\mathcal L}(x,u_h,\nabla u_h)\,dx+ \lambda\int_{\varOmega}{\mathcal K}(x)e^{u_h} \left\{\frac{u_h}{n}-1\right\}\,dx \\ &+\frac{\nu}{n}\int_{\varOmega}|\nabla u_h|^n\,dx+o(1)\\ =&-\frac{\gamma}{n}\left\{f_\lambda(u_h)+\lambda \int_{\varOmega}{\mathcal K}(x)e^{u_h}\,dx\right\} +\lambda\int_{\varOmega}{\mathcal K}(x)e^{u_h} \left\{\frac{u_h}{n}-1\right\}\,dx\\ &+\frac{\nu}{n} \int_{\varOmega}|\nabla u_h|^n\,dx+o(1) \end{align*} as $h\to+\infty$, which yields \begin{align*} (n+\gamma) f_\lambda(u_h)\geq \lambda\int_{\varOmega}{\mathcal K}(x)e^{u_h}(u_h- \gamma-n)\,dx+\nu\int_{\varOmega}|\nabla u_h|^n\,dx+o(1) \end{align*} as $h\to +\infty$. Since $$\lim_{\xi\to-\infty} e^\xi\big\{\xi- \gamma-n\big\}=0^-,\qquad \lim_{\xi\to+\infty} e^\xi\big\{\xi- \gamma-n\big\}=+\infty\,,$$ if we set $$C=-\min_{\xi\in{\mathbb R}}e^\xi\big\{\xi-\gamma-n\big\}\,,$$ it results $C>0$ and \begin{align*} (n+\gamma) f_\lambda(u_h)\geq -\lambda C\|{\mathcal K}\|_q{\mathscr L}^n(\varOmega)^{1/{q^{\prime}}} +\nu\int_{\varOmega}|\nabla u_h|^n\,dx+o(1) \end{align*} as $h\to+\infty$, where ${\mathscr L}^n$ denotes the $n-$dimensional Lebesgue measure.\ Being $f_\lambda(u_h)\to c$, we conclude that \begin{align*} \nu\int_{\varOmega}|\nabla u_h|^n\,dx\leq (n+\gamma)c+\lambda C\|{\mathcal K}\|_q{\mathscr L}^n (\varOmega)^{1/{q^{\prime}}}+o(1) \end{align*} as $h\to+\infty$, which implies the boundedness of $(u_h)$ in $W^{1,n}_0(\varOmega)$. \vskip4pt \noindent II) By step I, up to a subsequence, one has $u_h\rightharpoonup u$ in $W^{1,n}_0(\varOmega)$ for some $u$ and \label{conv} u_h\to u\quad \quad\text{in}\,\, L^{s}(\varOmega),\quad 10$. Since by Trudinger inequality there exist$c_{1,n},c_{2,n}>0$so that: $$\forall\beta>0,\,\forall w\in W^{1,n}_0(\varOmega):\,\, \int_{\varOmega}e^{\beta|w|}\,dx\leq c_{1,n}{\mathscr L}^n(\varOmega)e^{c_{2,n}\beta^n\|\nabla w\|_n^n},$$ the exponential terms in (\ref{compineq}) are bounded by step I and the last term goes to zero in view of (\ref{conv}). Thus, $$\sup_{\|\eta\|_{1,n}=1} \Big|\int_{\varOmega}{\mathcal K}(x) (e^{u_h}-e^{u})\eta\,dx\Big|=o(1),$$ as$h\to+\infty$, which shows that $${\mathcal K}(x)e^{u_h}\to {\mathcal K}(x)e^{u}\,\,\, \quad\text{in}\,\,\, W^{-1,n'}(\varOmega).\$$ Therefore, since we have that for all$\varphi\in C^\infty_c(\varOmega): \begin{align*} &\int_\varOmega\nabla_\xi {\mathcal L}(x,u_h,\nabla u_h) \cdot \nabla\varphi\,dx +\int_\varOmega D_s{\mathcal L}(x,u_h,\nabla u_h)\varphi\,dx\\ &=\int_{\varOmega}{\mathcal K}(x)e^{u}\varphi\,dx+ \langle w_h,\varphi\rangle+o(1)\,, \end{align*} withw_h\to 0$in$W^{-1,n'}(\varOmega)$as$h\to+\infty$, by Lemma~\ref{compattezza} up to a further subsequence$(u_h)$strongly converges to$u$in$W^{1,n}_0(\varOmega)$. \end{proof} \section{Mountain pass critical point and local minimum} \begin{proposition} \label{mountain} There exist$\lambda_0>0$and$R_2>R_1>0$such that \begin{gather} \label{mp1} \forall u\in W^{1,n}_0(\varOmega):\,\, \|u\|_{1,n}=R_1\,\,\Longrightarrow\,\, f_{\lambda}(u)>f_\lambda(0) \\ \label{mp2} \exists w\in W^{1,n}_0(\varOmega):\,\, \|w\|_{1,n}=R_2\,\,\,\text{and}\,\,\, f_{\lambda}(w)0$ we have \begin{align} f_\lambda(u)= \int_\varOmega{\mathcal L}(x,u,\nabla u)\,dx -\lambda\int_\varOmega {\mathcal K}(x)e^{u}\,dx \geq\varphi_{a,b}(\|\nabla u\|_n) \end{align} where $\varphi_{a,b}:[0,+\infty[\to{\mathbb R}$ is such that \begin{equation*} \varphi_{a,b}(\|\nabla u\|_n)=\nu\|\nabla u\|_n^n -\lambda a\|{\mathcal K}\|_qe^{b\|\nabla u\|_n^n}\,, \end{equation*} for each $u\in W^{1,n}_0(\varOmega)$, with $$\label{abs} a=c_1^{\frac{q-1}{q}} {\mathscr L}^n(\varOmega)^{\frac{q-1}{q}},\quad b=c_2\Big({\frac{q-1}{q}}\Big)^{n-1} {\mathscr L}^n(\varOmega)^{\frac{n-1}{n}},$$ being $c_1,c_2>0$ such that $$\int_{\varOmega}\exp\Big\{\frac{c_1|u|}{\|\nabla u\|_n}\Big\} ^{\frac{n}{n-1}}\,dx\leq c_2{\mathscr L}^n(\varOmega)\,,$$ (cf. \cite[Theorem 7.15]{gilbarg}). In particular, (\ref{mp1}) follows arguing as in \cite{crespo}. To prove (\ref{mp2}), fix $\phi\in C^\infty_c(B)$ with $\phi\geq 0$, where $B\subset\varOmega$ is the set where ${\mathcal K}$ is positive. For each $\tau>0$ it results \begin{align*} f_\lambda(\tau\phi)=&\int_{\varOmega} {\mathcal L}(x,\tau\phi,\tau\nabla\phi)\,dx- \lambda\int_{\varOmega}{\mathcal K}(x)e^{\tau\phi}\,dx \\ \leq& b_0\tau^r\int_{\varOmega}|\phi|^r\,dx+ b_0\int_{\varOmega}\tau^n|\nabla\phi|^n\,dx -\lambda\tau^{2\max\{r,n\}}\int_{\varOmega}{\mathcal K}^+(x)\phi^{2\max\{r,n\}} \,dx\\ & +\lambda\int_{\varOmega}{\mathcal K}^-(x)\,dx.\ \end{align*} Then, since ${\mathcal K}^+\not\equiv 0$, we deduce that $f_\lambda(\tau\phi)\to-\infty$ as $\tau\to+\infty$, thus yielding the second assertion. \end{proof} \noindent We now come to the proof of the main result of the paper. \vskip2pt \noindent {\em Proof of Theorem}~\ref{mainth}.\,\, If we set $$\varTheta=\left\{\gamma\in C([0,1],W^{1,n}_0(\varOmega)): \,\,\gamma(0)=0,\,\,\gamma(1)=w\right\}$$ for some $w\in W^{1,n}_0(\varOmega)$ with $f_\lambda(w)f_\lambda(0)$. To get a second solution, argue on the truncated functional $f^\tau_{\lambda}$ given by \begin{equation*} f^\tau_{\lambda}(u)= \int_\varOmega {\mathcal L}(x,u,\nabla u)\,dx- \lambda\int_\varOmega {\mathcal K}(x)\tau(\|u\|_{1,n})e^u\,dx\,, \end{equation*} where $\tau\in C^{\infty}({\mathbb R})$ is nonincreasing and $$\tau(x)= \begin{cases} 1 & \text{if } x\leq R_1 \\ 0 & \text{if } x\geq R_2 \\ \end{cases}$$ being $R_1$ and $R_2$ as in Proposition \ref{mountain}.\ Note that since for all $u\in W^{1,n}_0(\varOmega)$: $$\|u\|_{1,n}\geq R_2\,\,\Longrightarrow\,\, f^\tau_{\lambda}(u)\geq\nu \int_{\varOmega}|\nabla u|^n\,dx\,,$$ there results $$\lim_{\|u\|_{1,n}\to+\infty}f^\tau_{\lambda}(u)=+\infty.\$$ Observe that by Lemma~\ref{palsma} and the definition of $\tau$, $f^\tau_\lambda$ satisfies the concrete Palais--Smale condition at each level $c'$ such that $$c'<-\lambda\int_{\varOmega} {\mathcal K}(x)\,dx.\$$ If we fix $\phi\in C^\infty_c(B)$ (recall that ${\mathcal K}>0$ a.e. in $B$) with $\phi\geq 0$, $\|\phi\|_{1,n}=1$ and $\varrho0$ such that \begin{align*} f^\tau_\lambda(\varrho\phi)=&f_\lambda(\varrho\phi)= \int_{\varOmega}{\mathcal L}(x,\varrho\phi,\varrho\nabla\phi)\,dx- \lambda\int_{\varOmega}{\mathcal K}(x)e^{\varrho\phi}\,dx \\ \leq &c_0\varrho^{\max\{r,n\}}-\lambda\int_{\varOmega} {\mathcal K}^+(x)(1+\varrho\phi)\,dx +\lambda\int_{\varOmega}{\mathcal K}^-(x)\,dx \\ =&\varrho\left\{c_0\varrho^{\max\{r,n\}-1} -\lambda\int_{\varOmega}{\mathcal K}^+(x)\phi\,dx\right\} -\lambda\int_{\varOmega}{\mathcal K}^+(x)\,dx \\ &+\lambda\int_{\varOmega}{\mathcal K}^-(x)\,dx<-\lambda\int_{\varOmega} {\mathcal K}(x)\,dx \end{align*} provided that $\varrho>0$ is sufficiently small.\ Then $$c=\inf_{B_{W^{1,n}_0}(0,R_1)}f^\tau_\lambda<-\lambda\int_{\varOmega} {\mathcal K}(x)\,dx.\$$ Let us note that there exists a $(CPS)_c-$sequence for $f^\tau_\lambda$ in $B_{W^{1,n}_0}(0,R_1)$. Indeed, since $f_\lambda^\tau$ is bounded from below, we find a minimizing sequence $(u_h)$ for $f^\tau_\lambda$ in $B_{W^{1,n}_0}(0,R_1)$. Of course we have $f^\tau_\lambda(u_h)\to c$. Moreover, if it was $|df^\tau_\lambda|(u_h)\not\to 0,$ we would find $\sigma>0$ such that $|df^\tau_\lambda|(u_h)\geq\sigma$. Then by \cite[Theorem 1.1.11]{cd} one would get a continuous deformation $$\eta: B_{W^{1,n}_0}(0,R_1)\times [0,\delta] \to B_{W^{1,n}_0}(0,R_1)$$ for some $\delta>0$ such that for all $t\in[0,\delta]$ and $h\in{\mathbb N}$ $$f^\tau_\lambda(\eta(u_h,t))\leq f^\tau_\lambda(u_h)-\sigma t.\$$ This yields the contradiction $f^\tau_\lambda(\eta(u_h,t))f_\lambda(0)$, one obtains $u_2\not\equiv u_1$.\qed \section{A non-existence result} Assume now that ${\mathcal L}$ does not depend on $x$, ${\mathcal L}(s,\xi)$ is of class $C^1$ in ${\mathbb R}\times{\mathbb R}^n$ and, additionally, that the vector valued function $$\nabla_\xi{\mathcal L}(s,\xi)=\Big(\frac{\partial {\mathcal L}} {\partial\xi_1}(s,\xi),\cdots, \frac{\partial {\mathcal L}}{\partial\xi_n}(s,\xi)\Big)$$ is of class $C^1$ in ${\mathbb R}\times{\mathbb R}^n$ (see \cite{pucci}). We recall that a smooth bounded domain $\varOmega\subset {\mathbb R}^n$ is said to be strictly star--shaped with respect to the origin if $x\cdot \nu>0$ for a.e. $x\in\partial\varOmega$, where $\nu$ denotes the outer normal to $\partial\varOmega$. \begin{proposition} \label{reg} Assume that $\varOmega$ is strictly star--shaped with respect to the origin, $D_s{\mathcal L}\leq 0$ and ${\mathcal K}$ is constant and positive. Then there exists $\lambda^*>0$ such that (\ref{P_lambda}) admits no solution $u\in C^2(\varOmega)\cap C^1(\overline{\varOmega})$ for each $\lambda>\lambda^*$. \end{proposition} \begin{proof} Assume by contradiction that (\ref{P_lambda}) has a solution $u\in C^2(\varOmega)\cap C^1(\overline{\varOmega})$. By applying the Pucci-Serrin identity \cite[formula 5]{pucci} to $${\mathcal F}(u,\nabla u)={\mathcal L}(u,\nabla u)-\lambda {\mathcal K}e^u,$$ with $h(x)=x$ and $a=0$, since ${\mathcal L}(s,\cdot)$ is $n-$homogeneous, we obtain \label{pohozaev} \begin{aligned} &(n-1)\int_{\partial\varOmega}{\mathcal L}(0,\nabla u) (x\cdot\nu)\,d{\mathscr H}^{n-1} +\lambda\int_{\partial\varOmega}{\mathcal K} \,x\cdot\nu\,d{\mathscr H}^{n-1}\\ &=\lambda n\int_{\varOmega}{\mathcal K}e^u\,dx\,, \end{aligned} where ${\mathscr H}^{n-1}$ denotes the Hausdorff measure.\ Integrating (\ref{P_lambda}), since $x\cdot\nu>0$ on the boundary $\partial\varOmega$, by (\ref{L}) and (\ref{xi}) we obtain \begin{align*} \int_{\varOmega}\lambda{\mathcal K}e^u\,dx =& \int_{\varOmega}-\mathop{\rm div}\left(\nabla_{\xi}{\mathcal L}( u,\nabla u)\right)\,dx +\int_{\varOmega} D_s{\mathcal L}(u,\nabla u)\,dx \\ \leq&-\int_{\partial\varOmega}\nabla_{\xi}{\mathcal L}(0,\nabla u) \cdot\nu\,d{\mathscr H}^{n-1}\\ \leq&\int_{\partial\varOmega}\left|\nabla_{\xi}{\mathcal L}(0,\nabla u)\right| \,d{\mathscr H}^{n-1}\\ \leq&\Big(\int_{\partial\varOmega}\left|\nabla_{\xi} {\mathcal L}(0,\nabla u)\right|^{n'}(x\cdot\nu)\,d{\mathscr H}^{n-1} \Big)^{\frac{n-1}{n}}\! \Big(\int_{\partial\varOmega}(x\cdot\nu)^{-(n-1)}\,d{\mathscr H}^{n-1} \Big)^{\frac{1}{n}}\\ \leq& c_{1,n}\Big(\int_{\partial\varOmega}|\nabla u|^{n}(x\cdot\nu) \,d{\mathscr H}^{n-1}\Big)^{\frac{n-1}{n}} \Big(\int_{\partial\varOmega}(x\cdot\nu)^{-(n-1)}\,d{\mathscr H}^{n-1} \Big)^{\frac{1}{n}}\\ \leq& c_{2,n}\Big(\int_{\partial\varOmega}{\mathcal L} (0,\nabla u)(x\cdot\nu)\,d{\mathscr H}^{n-1}\Big)^{\frac{n-1}{n}} \Big(\int_{\partial\varOmega}(x\cdot\nu)^{-(n-1)}\,d{\mathscr H}^{n-1} \Big)^{\frac{1}{n}} \end{align*} which implies $$\Big(\lambda\int_{\varOmega}{\mathcal K}e^u\,dx\Big)^{\frac{n}{n-1}}\leq A \int_{\partial\varOmega}{\mathcal L}(0,\nabla u)(x\cdot\nu)\,d{\mathscr H}^{n-1}\,,$$ where we set $$A= c_{2,n}^{n/n-1}\Big(\int_{\partial\varOmega}(x\cdot\nu)^{-(n-1)} \,d{\mathscr H}^{n-1}\Big)^{\frac{1}{n-1}}.\$$ In particular, by (\ref{pohozaev}) we obtain $$\frac{n-1}{A}\Big(\lambda\int_{\varOmega}{\mathcal K}e^u\,dx\Big)^{\frac{n}{n-1}} +\lambda B-n\lambda\int_{\varOmega}{\mathcal K}e^u\,dx\leq 0\,,$$ where $B=\int_{\partial\varOmega}{\mathcal K}\,x\cdot\nu\,d{\mathscr H}^{n-1}$.\ Since the map $\varphi:[0,+\infty[\to{\mathbb R}$ given by $$\varphi(x)=\frac{n-1}{A}x^{\frac{n}{n-1}}+\lambda B-nx$$ achieves its absolute minimum at $x_0=A^{n-1}$ with $\varphi(x_0)=\lambda B-A^{n-1}$, we get $$\lambda\leq c_{2,n}^n\frac{\int_{\partial\varOmega}(x\cdot\nu)^{-(n-1)} \,d{\mathscr H}^{n-1}} {\int_{\partial\varOmega}{\mathcal K}\,x\cdot\nu\,d{\mathscr H}^{n-1}}.\$$ In particular, by setting $$\lambda^*=c_{2,n}^n\frac{\int_{\partial\varOmega}(x\cdot\nu)^{-(n-1)} \,d{\mathscr H}^{n-1}}{\int_{\partial\varOmega} {\mathcal K}\,x\cdot\nu\,d{\mathscr H}^{n-1}},$$ the assertion follows. \end{proof} \begin{remark}\rm For the lagrangian $${\mathcal L}(x,s,\xi)=\frac{1}{n}|\xi|^n,$$ it was shown in \cite{crespo3} by an approximation procedure that the non-existence of smooth solutions ($C^2$) implies the non-existence of bounded solutions. \end{remark} \vskip2pt \begin{remark}\rm By Proposition~\ref{reg} assumption (\ref{posit}) seems to be natural in order to get existence of solutions of (\ref{P_lambda}).\ \end{remark} \begin{thebibliography}{99} \bibitem{crespo} {\sc J.A. 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