\documentclass[reqno]{amsart}
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\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}
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\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)
\begin{equation}
\label{pb}
-\varDelta_gu+R_gu={\mathcal K}(x)e^{2u}\,,\quad g'=e^{2u}g\,,
\end{equation}
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 $\lambda>0$)
\begin{equation}
\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}
\end{equation}
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 $p<n$ and two solutions for $p\geq n$) and
nonexistence for values of $\lambda$ sufficiently large.
The case $p<n$ was treated by a fixed point argument, while the case
$p\geq n$ was investigated by classical critical
point methods. (see also \cite{bpv}).\
In this note, we are concerned with the following more general problem
at exponential growth
\vskip1pt
\begin{equation}
\label{P_lambda}
\begin{gathered}
-\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 \quad\text{in }\varOmega \\
u=0  \quad\text{on }\partial\varOmega.\
\end{gathered}
\end{equation}
For nonlinearities of power type
problems like (\ref{P_lambda})
have been studied in \cite{ab} and recently in \cite{pell,toulouse}
by techniques of non-smooth analysis.
Analogously, in our case the functional
$f_{\lambda}:W^{1,p}_0(\varOmega)\to{\mathbb R}$
associated with (\ref{P_lambda})
\begin{equation}
\label{funct}
f_{\lambda}(u)=
\int_\varOmega {\mathcal L}(x,u,\nabla u)\,dx-
\lambda\int_\varOmega {\mathcal K}(x)e^u\,dx\,,
\end{equation}
is well defined for $p\geq n$, smooth if $p>n$ 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:
\begin{equation}
\label{L}
\nu |\xi|^n\leq {\mathcal L}(x,s,\xi)
\leq b_0|s|^r+b_0|\xi|^n,
\end{equation}
\begin{equation}
\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}
\end{equation}
\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:
\begin{equation}
\label{posit}
|s|\geq R\,\,\Longrightarrow\,\, D_s{\mathcal L}(x,s,\xi)s\geq 0,
\end{equation}
\begin{equation}
\label{boundps}
\gamma {\mathcal L}(x,s,\xi)-D_s {\mathcal L}(x,s,\xi)s
\geq \nu |\xi|^n
\end{equation}
\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
\begin{equation}
\label{conv}
u_h\to u\quad \quad\text{in}\,\, L^{s}(\varOmega),\quad 1<s<\infty.\
\end{equation}
If we now fix $\eta\in W^{1,n}_0(\varOmega)$ with
$\|\nabla \eta\|_n=1$, H\"older and Sobolev
inequalities yield
\begin{equation}
\label{compineq} \begin{aligned}
&\Big|\int_{\varOmega}{\mathcal K}(x)(e^{u_h}-e^{u})\eta\,dx\Big|\\
&\leq\int_{\varOmega}|{\mathcal K}(x)|e^u|e^{u_h-u}-1||\eta|\,dx \\
&\leq\int_{\varOmega}|{\mathcal K}(x)|e^u|u_h-u|e^{|u_h-u|}|\eta|\,dx \\
&\leq\|{\mathcal K}\|_q\Big(\int_{\varOmega}e^{\beta_1u}
\,dx\Big)^{1/\beta_1}\Big(\int_{\varOmega}e^{\beta_2|u_h-u|}
\,dx\Big)^{1/\beta_2}
\|\eta\|_{\beta_3}\|u_h-u\|_{\beta_4} \\
&\leq c\|{\mathcal K}\|_q\Big(\int_{\varOmega}e^{\beta_1u}
\,dx\Big)^{1/\beta_1}\Big(\int_{\varOmega}e^{\beta_2|u_h-u|}
\,dx\Big)^{1/\beta_2}\|u_h-u\|_{\beta_4}
\end{aligned}\end{equation}
with $1/q+1/{\beta_1}+1/{\beta_2}+1/{\beta_3}+1/{\beta_4}=1$ and $c>0$.
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*}
with $w_h\to 0$ 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)<f_\lambda(0)
\end{gather}
for each $\lambda\in]0,\lambda_0[$.
\end{proposition}
\begin{proof}
Note that by (\ref{L}) and arguing as in \cite{crespo},
for each $\lambda>0$ 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 
\begin{equation}
\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}},
\end{equation}
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 $f_\lambda(w)<f_\lambda(0)$ and
$$
c_{\lambda}=\inf_{\gamma\in\varTheta}
\max_{t\in[0,1]}f_{\lambda}(\gamma(t)),
$$
by combining the non-smooth mountain pass Lemma \cite{cd}
with Proposition~\ref{mountain} and Theorem~\ref{palsma},
we get a weak
solution $u_1\in W^{1,n}_0(\varOmega)$ of (\ref{P_lambda})
with $f_\lambda(u_1)=c_\lambda>f_\lambda(0)$.
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 $\varrho<R_1$,
then there exists $c_0>0$ 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
$f^\tau_\lambda(\eta(u_h,t))<c$ for sufficiently
large values of $h$. Thus $(u_h)$ is a $(CPS)_c-$sequence.\
Since $f^\tau_\lambda$ satisfies $(CPS)_c$,
there exists $u_2\in W^{1,n}_0(\varOmega)$ with
$u_2\not\equiv 0$ such that $u_h\to u_2$ in $W^{1,n}_0(\varOmega)$.
This yields
$$
f^\tau_\lambda(u_2)=\min_{B_{W^{1,n}_0}(0,R_1)}
f^\tau_\lambda<-\lambda\int_{\varOmega}
{\mathcal K}(x)\,dx.\
$$
Since $f_\lambda^\tau\equiv f_\lambda$ in a
neighbourhood of $u_2$, of course
$u_2$ solves problem (\ref{P_lambda}) and
$f_\lambda(u_2)<f_\lambda(0)$.
Moreover, being $f_\lambda(u_1)>f_\lambda(0)$, 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
\begin{equation}
\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} \end{equation}
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. Aguilar Crespo, I. Peral},
On an elliptic equation with exponential growth,
{\sl Rend. Sem. Mat. Univ. Padova} {\bf 96} (1996), 143--175.

\bibitem{ab}
{\sc D. Arcoya, L. Boccardo},
Critical points for multiple
integrals of the calculus of variations,
{\sl Arch. Rat. Mech. Anal.} {\bf 134} (1996), 249--274.

\bibitem{bpv}
{\sc L. Boccardo, I. Peral, J.L. Vazquez}, 
The $n-$Laplacian elliptic equation: variational versus entropy solutions, 
{\sl J. Math. Anal. Appl.} {\bf 201} (1996), 671--688.

\bibitem{bb}  
{\sc H. Br\'ezis, F.E. Browder}, 
Sur une propri\'{e}t\'{e} des espaces de Sobolev, 
{\sl C. R. Acad. Sc. Paris} {\bf 287} (1978), 113--115.

\bibitem{bm}  
{\sc H. Br\'ezis, F. Merle}, 
Uniform estimates and
blow up behaviour for solutions of
$-\varDelta u=V(x)\,e^u$ in two dimensions,
{\sl Comm. Partial Differential Equations} {\bf 16} (1992), 1223--1253.

\bibitem{cd}
{\sc A. Canino, M. Degiovanni},
Nonsmooth critical point theory and quasilinear elliptic equations,
{\sl Topological Methods in Differential Equations and Inclusions},
A. Granas, M. Frigon, G. Sabidussi Eds. -- Montreal (1994),
NATO ASI Series -- Kluwer A. P. (1995), 1--50.

\bibitem{crespo3}
{\sc J. Garcia Azorero, I. Peral},
On an Emden--Fowler type equation,
{\sl Nonlinear Anal.} {\bf 18} (1992), 1085--1097.

\bibitem{gilbarg}
{\sc D. Gilbarg, N.S. Trudinger},
Elliptic partial differential equations of second order,
{\sl Springer}, New York (1983).

\bibitem{kazdan}
{\sc J.L. Kazdan, J.L. Warner},
Curvature functions for compact $2-$manifolds,
{\sl Ann. Math.} {\bf 99} (1974), 14--47.

\bibitem{pell}
{\sc B. Pellacci},
Critical points for non differentiable functionals,
{\sl Boll. Un. Mat. Ital.} B {\bf 11} (1997), 733--749.

\bibitem{pucci}
{\sc P. Pucci, J. Serrin},
A general variational identity,
{\sl Indiana Univ. Math. J.} {\bf 35} (1986), 681--703.

\bibitem{toulouse}
{\sc M. Squassina},
Existence of weak solutions to general Euler's equations
via nonsmooth critical point theory,
{\sl Ann. Fac. Sci. Toulouse Math. (6)} {\bf 9} (2000), 113--131.

\bibitem{tolk}
{\sc P. Tolksdorf}, 
Regularity for a more general class of quasilinear elliptic equations, 
{\sl J. Differential Equations} {\bf 51} (1984), 126--150.

\end{thebibliography}

\end{document}