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\title{\sc Combined effects\\in quasilinear elliptic problems\\with lack of compactness}

%\author{\sc  Patrizia Pucci and Vicen\c{t}iu R\u{a}dulescu}

\author{\sc Patrizia Pucci\\
{\small Dipartimento di Matematica e Informatica,}\\
{\small Universit\`a degli Studi di Perugia,  06123 Perugia, Italy}\\
 {\small \texttt{pucci@dmi.unipg.it} }\\
\and \sc Vicen\c{t}iu R\u{a}dulescu\\
{\small ``Simion Stoilow" Mathematics Institute of the Romanian Academy,}\\
{\small P.O. Box 1-764, 014700 Bucharest, Romania}\\
{\small \texttt{vicentiu.radulescu@imar.ro}}
}


\date{}

\begin{document}



\maketitle

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\begin{abstract} We are concerned with the study of weak solutions to a class of subcritical quasilinear elliptic equations
with weight, variable parameter, and lack of compactness.
By means of variational methods, we establish that this problem does not have any solution if the positive parameter is small enough, while at least two distinct solutions exist, provided that the parameter is sufficiently large. \\
{\bf KEY WORDS}: Quasilinear Elliptic Equations; Weighted Sobolev Spaces; Weak Solutions; Mountain Pass Geometry.\\
{\bf 2010 Mathematics Subject Classification}: 35J60, 35J65, 35J70, 58E05.
\end{abstract}


\begin{center}{\it This paper is dedicated to the memory of Professor Giovanni Prodi}\end{center}

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\section{Introduction and the main result}

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Nonlinear elliptic equations with convex--concave nonlinearities in bounded domains have been studied starting with the seminal paper by Ambrosetti, Brezis and Cerami \cite{ARC}. They considered the Dirichlet problem
\begin{equation}\label{abc}
\begin{cases}
\di -\Delta u=\lambda u^{q-1}+u^{p-1}   &\mbox{ in }\ \Omega\\
\di u>0 &\mbox{ in } \Omega\\
\di u=0 &\mbox{ on }  \partial\Omega\,,
\end{cases}
\end{equation}
where $\lambda$ is a positive parameter, $\Omega\subset\RR^N$ is a bounded domain with smooth boundary, and $1<q<2<p<2^\star$ ($2^\star=2N/(N-2)$ if $N\geq 3$, $2^\star=\infty$ if $N=1,\, 2$). Ambrosetti, Brezis and Cerami proved that there exists $\lambda_0>0$ such that problem \eq{abc} admits at least two solutions for all $\lambda\in (0,\lambda_0)$, has one solution for $\lambda=\lambda_0$, and no solution exists provided that $\lambda>\lambda_0$.

In \cite{AT2}, Alama and Tarantello studied the related Dirichlet problem with indefinite weights
\begin{equation}\label{alama}
\begin{cases}
-\Delta u-\lambda u=k(x)u^q-h(x)u^p&\mbox{ in }\ \Omega\\
u>0&\mbox{ in }\ \Omega\\
u=0&\mbox{ on }\ \partial\Omega\,,
\end{cases}
\end{equation}
where $\lambda\in\RR$, $\Omega\subset\RR^N$, $N\geq 3$, is a bounded open set with smooth boundary,
the functions $h$, $k\in L^1(\Omega)$ are nonnegative, and $1<p<q$.
For $\lambda\in\RR$ in a neighbourhood of $\lambda_1$ (the first eigenvalue of the Laplace operator in $H_0^1(\Omega)$),
they obtained the solvability of \eq{alama} (and corresponding multiplicities) under
various assumptions on $h$ and $k$. More exactly, they proved existence, nonexistence
and multiplicity results depending on $\lambda$ and according to the integrability
properties of the ratio $k^{p-1}/h^{q-1}$.

For more general results of \eq{alama} involving variable weights functions in unbounded domains, we refer to Chabrowski \cite{C}, Chabrowski and do \'O \cite{chao}, Liu and Wang \cite{liuwang}, and Wu \cite{wu}.

Related studies can be also found in \cite{BW}, \cite{chao}, \cite{filpucrad}, \cite{tang}, \cite{T} and \cite{YZ}.

Motivated by these results, we are concerned in this paper with the existence and the multiplicity of
solutions in the quasilinear case. More precisely, we consider the problem
\begin{equation}\label{1}
\begin{cases}
-{\rm div}(|\nabla u|^{m-2}\nabla u)+|u|^{m-2}u=\lambda|u|^{q-2}u-h(x)|u|^{p-2}u &\mbox{in }\ \RR^N\\
u\geq 0, &\mbox{in }\ \RR^N\,,
\end{cases}
\end{equation}
where $h:\RR^N\ri\RR$ {\it is a positive continuous function, whose growth is described by
the condition
\begin{equation}\label{H}
\int_{\RR^N}h(x)^{q/(q-p)}dx=H\in\RR^+,
\end{equation}
$\lambda$ is a positive parameter and $2\leq m<q<p<m^\star$}, with $m^\star=Nm/(N-m)$ if $N> m$ and $m^\star=\infty$ if $N\leq m$.

\medskip
Without altering the proof arguments below, the coefficient 1 of the dominating
term $|u|^{m-2}u$ can be replaced by any function $f\in L^\infty (\RR^N)$ with $\mbox{inf}\, \mbox{ess}\,\|f\|_{L^\infty} > 0$. Hence
equation \eq{1} is the renormalized form.
Problem \eq{1} may be viewed as a prototype of pattern formation in biology
and is related to the steady-state problem for a chemotactic aggregation model
introduced by Keller and Segel \cite{kelseg}. Problem \eq{1} also plays an important role in
the study of activator-inhibitor systems modeling biological pattern formation, as
proposed by Gierer and Meihardt \cite{giem}.

In this paper we use standard notations and terminology. We denote by
$W^{1,m}(\RR^N)$ the Sobolev space equipped with the norm
$$\|u\|_{W^{1,m}}=\left(\int_{\RR^N}(|\nabla u|^m+|u|^m)\;dx\right)^{1/m}.$$
For simplicity we  often denote the above norm by $\|u\|$.

By $L^p_r(\RR^N)$, $1\leq p<\infty$, we denote the weighted Lebesgue space
$$L^p_r(\RR^N)=\left\{u\in L^1_{\text{\scriptsize{\rm loc}}}(\RR^N)\,:\,\int_{\RR^N}r(x)|u|^{p}dx<\infty\right\},$$
where $r(x)$ is a positive continuous function on $\RR^N$, equipped with the norm
$$\|u\|_{r,p}=\left(\int_{\RR^N}r(x)|u|^pdx\right)^{1/p}.$$
If $r(x)\equiv 1$ on $\RR^N$, the norm is denoted by $\|\cdot\|_p$.
\medskip

In this paper we seek weak solutions of problem \eq{1} in a subspace $E$ of
$W^{1,m}(\RR^N)$, which is defined by
$$E=\left\{u\in W^{1,m}(\RR^N)\,:\,\int_{\RR^N}h(x)|u|^pdx<\infty\right\}.$$
Then $E$ is a Banach space if equipped with the norm
$$\|u\|_E=\left(\|u\|_{W^{1,m}}^m+\|u\|_{r,p}^m\right)^{1/m}.$$

We define a {\it weak solution} of problem \eq{1} as a function $u\in E$ with $u(x)\geq 0$ a.e. in $\RR^N$, satisfying
$$\begin{aligned}
\int_{\RR^N}|\nabla u|^{m-2}\nabla u\nabla vdx+\int_{\RR^N}|u|^{m-2}uvdx&-\lambda
\int_{\RR^N}|u|^{q-2}uvdx\\ &  +\int_{\RR^N}h(x)|u|^{p-2}uvdx=0,\end{aligned}$$
for all $v\in E$.

The main result in the present paper establishes the following properties: the non-existence of nontrivial solutions to problem \eq{1} if
$\lambda$ is small enough; the existence of at
least two nontrivial solutions for problem \eq{1} if $\lambda$ is large enough.

\begin{theorem}\label{t1} Under the above hypotheses there exists $\lambda^\star>0$ such that

\begin{enumerate}
\item[$\phantom{ii}(i)$] if $0<\lambda <\lambda^\star$, then
problem \eq{1} does not possess any nontrivial weak solution;
\item[$\phantom{i}(ii)$] if $\lambda >\lambda^\star$, then problem \eq{1} admits
at least two nontrivial weak solutions;
\item[$(iii)$] if $\lambda =\lambda^\star$, then problem \eq{1} has at least one nontrivial weak solution.
\end{enumerate}
\end{theorem}

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\section{Proof of Theorem \ref{t1}}

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We point out in what follows the main ideas in the proof:

(a) There exists $\lambda^\star>0$ such that problem \eq{1} does not have any solution for any $\lambda<\lambda^\star$. This means that if a solution exists then $\lambda$ must be sufficiently large. One
of the key arguments in this proof is based on the assumption $p>q$. In particular,
this proof yields an energy lower bound of solutions in term of $\lambda$ that will be useful
to conclude that problem \eq{1} has a non-trivial solution if $\lambda=\lambda^\star$.

(b) There exists $\lambda^{\star\star}>0$ such that problem \eq{1} admits at least two solutions for all $\lambda>\lambda^{\star\star}$. Next, by the properties of $\lambda^{\star}$ and $\lambda^{\star\star}$ we deduce that $\lambda^{\star}=\lambda^{\star\star}$.

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\subsection{Nonexistence if $\lambda$ is small}

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Let $\Phi_\lambda:E\rightarrow\RR$ be the energy functional given by
$$\Phi_\lambda(u)=\frac{1}{m}\|u\|^m-\frac{\lambda}{q}
\|u\|_q^q+\frac{1}{p}\|u\|_{h,p}^p\,.$$
Then $\Phi_\lambda\in C^1(E,\RR)$ and for all $u$, $v\in E$
$$\begin{aligned} \langle\Phi'_\lambda(u),v\rangle=\int_{\RR^N}(|\nabla u|^{m-2}\nabla u\nabla v+
|u|^{m-2}uv)dx&-\lambda\int_{\RR^N}|u|^{q-2}uvdx\\ &+\int_{\RR^N}h(x)|u|^{p-2}uvdx.\end{aligned}$$
Weak solutions of problem \eq{1} are found as the critical points
of the functional $\Phi_\lambda$ in $E$.

\medskip
Let us now assume that $u\in E$ is a weak solution of problem \eq{1}.
Then
\begin{equation}\label{e2}
\|u\|^m+\|u\|_{h,p}^p=\lambda \|u\|_q^q.
\end{equation}
To proceed further, we need Young's inequality
$$ab\leq\frac{a^\alpha}{\alpha}+\frac{b^\beta}{\beta}\quad\mbox{for all }a,b>0,$$
where $\alpha$, $\beta>1$ satisfy $1/\alpha+1/\beta=1$.

Taking $a=h(x)^{q/p}|u|^q$, $b=\lambda/[h(x)]^{q/p}$, $\alpha=p/q$ and $\beta=p/(p-q)$,
we obtain
$$h(x)^{q/p}|u|^q\frac{\lambda}{h(x)^{q/p}}\leq\frac{q}{p}(h(x)^{q/p}|u|^q)^{p/q}+
\frac{p-q}{p}\left(\frac{\lambda}{h(x)^{q/p}}\right)^{p/(p-q)}.$$
Integrating over $\RR^N$ we have
$$\lambda\|u\|_q^q\leq\frac{q}{p}\|u\|_{h,p}^p+
\frac{p-q}{p}\lambda^{p/(p-q)}\int_{\RR^N}h(x)^{q/(q-p)}dx.$$
The above inequality and relation \eq{e2} imply
\begin{equation}\label{e3}\begin{aligned}\|u\|^m&\leq\frac{p-q}{p}\lambda^{p/(p-q)}
\int_{\RR^N}h(x)^{q/(q-p)}dx+\frac{q-p}{p}\|u\|_{h,p}^p\\
&\le \frac{p-q}{p}\lambda^{p/(p-q)}H
,\end{aligned}\end{equation}
being $q<p$.

Since $m<q<m^\star$, the Sobolev embedding $W^{1,m}(\RR^N)\subset L^q(\RR^N)$ is continuous, so that there exists a positive constant $C_q$ such
that
$$C_q\|v\|_q^m\leq\|v\|^m\quad\mbox{for all }v\in W^{1,m}(\RR^N).$$
On the other hand, being  $\|u\|_{h,p}\geq 0$, it follows from \eq{e2} that
$$\|u\|^m\leq\lambda\|u\|_q^q.$$
Combining the last two inequalities we obtain
\begin{equation}\label{e4}
C_q\|u\|_q^m\leq\|u\|^m\leq\lambda\|u\|_q^q.\end{equation}
Retaining the first  and the last terms of \eq{e4} we get
$$(C_q\lambda^{-1})^{q/(q-m)}\leq\|u\|_q^q.$$
That inequality combined with \eq{e4} leads to
$$C_q[(C_q\lambda^{-1})^{q/(q-m)}]^{m/q}\le\|u\|^m$$
By relation \eq{e3} and the above inequality we have
$$C_q(C_q\lambda^{-1})^{m/(q-m)}\leq\|u\|^m\leq\frac{p-q}{p}\lambda^{p/(p-q)}H.$$
Retaining the first and the last term it follows that
$$\lambda>\left(C_q^{q/(m-q)}\frac{p-q}{p} H\right)^{(q-p)(q-m)/q(p-m)},$$
being $H>0$ by \eqref{H}.
Denoting the term in the right-hand side of the above inequality by $\mu$,
we conclude that Theorem \ref{t1}--$(i)$ holds true, by putting
\begin{equation}\label{star}\lambda^\star:=\sup\{\lambda>0\,:\, \mbox{\eq{1} does not admit any nontrivial weak solution}\}.\end{equation}
Clearly $\lambda^\star\ge\mu>0$.

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\subsection{Existence if $\lambda$ is large}

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We start with several auxiliary results.
\begin{lemma}\label{l1}
The functional $\Phi_\lambda$ is coercive.
\end{lemma}
\proof
We need the following elementary inequality:
for every $k_1>0$, $k_2>0$ and $0<s<r$ we have
\begin{equation}\label{6}
k_1|t|^s-k_2|t|^r\leq C_{rs}k_1\left(\frac{k_1}{k_2}\right)^{s/(r-s)}\quad\mbox{for all }t\in\RR,
\end{equation}
where $C_{rs}>0$ is a constant depending on $r$ and $s$.


Taking  $k_1={\lambda}/{q}$, $k_2={(m-1)h(x)}/{mp}$, $s=q$ and
$r=p$ (so that $s<r$ is verified, being $q<p$), in \eq{6} for all $x\in\RR^N$ we obtain
$$\begin{aligned}
\frac{\lambda}{q}|u(x)|^q-\frac{(m-1)h(x)}{mp}|u(x)|^p&\leq C_{pq}\frac{\lambda}{q}
\left(\frac{\lambda/q}{(m-1)h(x)/mp}\right)^{q/(p-q)}\\
&=\frac{C_{pq}}{q}\left(\frac{mp}{q(m-1)}\right)^{q/(p-q)}
\lambda^{p/(p-q)}h(x)^{q/(q-p)},\end{aligned}
$$
where $C_{pq}>0$ is a constant depending on $p$ and $q$. Relabeling the constant $\frac{C_{pq}}q\left(\frac{mp}{q(m-1)}
\right)^{q/(p-q)}$ by $K$ and integrating the above inequality
over $\RR^N$, we get
$$\int_{\RR^N}\left(\frac{\lambda}{q}|u|^q-\frac{(m-1)h(x)}{mp}|u|^p\right)dx\leq K
\lambda^{p/(p-q)}\int_{\RR^N}h(x)^{q/(q-p)}dx.$$
By assumption \eq{H} there exists a constant $C_\lambda>0$ such that
$$\frac{\lambda}{q}\|u\|_q^q-\frac{m-1}{mp}\|u\|_{h,p}^p\leq C_\lambda.$$
Therefore
\begin{equation}\label{7}
\begin{aligned}
\Phi_\lambda (u)&=\frac{1}{m}\|u\|^m-\left[\frac{\lambda}{q}\|u\|_q^q-\frac{m-1}{mp}\|u\|_{h,p}^p\right]-\frac{m-1}{mp}\|u\|_{h,p}^p+\frac{1}{p}\|u\|_{h,p}^p\\
&\geq\frac{1}{m}\|u\|^m+\frac{1}{mp}\|u\|_{h,p}^p-C_\lambda,\end{aligned}
\end{equation}
and so $\Phi_\lambda$ is coercive in $E$.
\endproof

\begin{lemma}\label{l2}
If $(u_n)_n$ is a sequence in $E$ such that $(\Phi_\lambda (u_n))_n$ is bounded in $\RR$, then there exists a
subsequence of $(u_n)_n$, still relabeled  $(u_n)_n$, which converges weakly  in $E$ to some $u_0\in E$ and
$$\Phi_\lambda (u_0)\leq\liminf\limits_{n\rightarrow\infty}\Phi_\lambda (u_n).$$
\end{lemma}
\proof
By \eq{7} and the fact that  $(\Phi_\lambda(u_n))_n$ is bounded, it follows that  both
sequences $(\|u_n\|)_n$ and $(\|u_n\|_{h,p})_n$ are bounded.
Therefore, $(\|u_n\|_E)_n$ is bounded and there exists $u_0\in E$ such that
\begin{eqnarray*}
u_n&\rightharpoonup& u_0\;\;{\rm in}\;\;W^{1,m}(\RR^N)\\
u_n&\rightarrow& u_0\;\;{\rm in}\;\;L^p_h(\RR^N)\\
u_n&\rightarrow& u_0\;\;{\rm in}\;\;L^s_{\rm loc}(\RR^N)\;\;{\rm for all}\;\;
s\in[1,m^\star).
\end{eqnarray*}
Let us define
$$F(x,u)=\frac{\lambda}{q}|u|^q-h(x)\frac{|u|^p}{p}$$
and
$$f(x,u)=F_u(x,u)=\lambda|u|^{q-2}u-h(x)|u|^{p-2}u,$$
so that
$$f_u(x,u)=\lambda(q-1)|u|^{q-2}-h(x)(p-1)|u|^{p-2}.$$
Using again inequality \eq{6} for $k_1=\lambda(q-1)$, $k_2=h(x)(p-1)$,
$s=q-2$, $r=p-2$, we obtain
$$\begin{aligned}
f_u(x,u)&=\lambda(q-1)|u|^{q-2}-h(x)(p-1)|u|^{p-2}\\
&\leq C\cdot\lambda\cdot(q-1)\cdot\left(\frac{\lambda(q-1)}{h(x)(p-1)}
\right)^{(q-2)/(p-q)},
\end{aligned}$$
where $C$ is a positive constant depending only of $p$ and $q$.

This yields,
\begin{equation}\label{8}
f_u(x,u)\leq C_{pq}\cdot\lambda\cdot\left(\frac{\lambda}{h(x)}\right)^{(q-2)/(p-q)},
\end{equation}
where $C_{pq}$ is a positive constant depending only of $p$ and $q$.
According to the definition of $\Phi_\lambda$ and $F$ we obtain the following estimate for
$\Phi_\lambda(u_0)-\Phi_\lambda(u_n)$
\begin{equation}\label{9}
\begin{aligned}
\Phi_\lambda(u_0)-\Phi_\lambda(u_n)= \frac{1}{m}&\left(\|u_0\|^m-\|u_n\|^m\right)\\&\hskip 1.2truecm+\int_{\RR^N}[F(x,u_n)-F(x,u_0)]dx.
\end{aligned}
\end{equation}
By position
$$\begin{aligned}
\int_0^sf_u(x,u_0+t(u_n-u_0))dt&=\frac{1}{u_n-u_0}[f(x,u_0+s(u_n-u_0))-f(x,u_0)]\\
&=\frac{1}{u_n-u_0}[F_u(x,u_0+s(u_n-u_0))-F_u(x,u_0)].
\end{aligned}$$
Integrating the above relation over $[0,1]$, we obtain
$$\begin{aligned}
\int_0^1&\left(\int_0^sf_u(x,u_0+t(u_n-u_0))dt\right)ds\\ &\quad=\frac{1}{u_n-u_0}
\int_0^1[F_u(x,u_0+s(u_n-u_0))-F_u(x,u_0)]ds\\
&\quad=\frac{1}{(u_n-u_0)^2}[F(x,u_n)-F(x,u_0)]-\frac{f(x,u_0)}{u_n-u_0}.\end{aligned}
$$
The above equality can be rewritten in the following way
\begin{equation}\label{ee}\begin{aligned}
F(x,u_n)-F(x,u_0)=(u_n-u_0)^2\int_0^1&\left(\int_0^sf_u(x,u_0+t(u_n-u_0))dt\right)ds\\
\\
&\hskip 1.5 truecm+(u_n-u_0)f(x,u_0).\end{aligned}
\end{equation}
Introducing relation \eq{ee} in \eq{9} we get
\begin{equation}\label{91}
\begin{aligned}
\Phi_\lambda(u_0) &-\Phi_\lambda(u_n)=\frac{1}{m}\left(\|u_0\|^m-\|u_n\|^m\right)+
\int_{\RR^N}(u_n-u_0)f(x,u_0)dx\\
&\quad+\int_{\RR^N}(u_n-u_0)^2\int_0^1\di\int_0^sf_u(x,u_0+t(u_n-u_0))dtdsdx\\
&\leq\frac{1}{m}\left(\|u_0\|^m-\|u_n\|^m\right)
+\int_{\RR^N}(u_n-u_0)f(x,u_0)dx\\
&\quad+C_1\di\int_{\RR^N}(u_n-u_0)^2h(x)^{(q-2)/(q-p)}dx,
\end{aligned}
\end{equation}
where the last inequality follows from \eq{8} and $C_1=C_{pq}\lambda^{(p-2)/(p-q)}$.
It remains to show that the last two integrals converge to $0$ as $n\rightarrow\infty$.

We define $J:E\rightarrow\RR$ by
$$J(v)=\int_{\RR^N}f(x,u_0)\,vdx.$$
Obviously, $J$ is linear. We shall show that $J$ is also continuous. Indeed,
\begin{equation}\label{10}
\begin{aligned}
|J(v)|&\leq\int_{\RR^N}|f(x,u_0)|\cdot|v|\,dx\\
&\leq\lambda\int_{\RR^N}|u_0|^{q-1}|v|\,dx+\di\int_{\RR^N}h(x)|u_0|^{p-1}|v|\,dx.
\end{aligned}
\end{equation}
On the other hand, using H\"{o}lder's inequality, it follows that
$$\int_{\RR^N}|u_0|^{q-1}|v|dx\leq \|u_0\|_{q}^{q-1}\|v\|_q.$$
Since $W^{1,m}(\RR^N)$ is continuously embedded in $L^q(\RR^N)$ we deduce that there exists
a constant $C>0$ such that
$$\|v\|_q\leq C\|v\|_{W^{1,m}(\RR^N)}\quad\mbox{for all }v\in W^{1,m}(\RR^N).$$
Combining the last two inequalities with the fact that
$$\|v\|_{W^{1,m}(\RR^N)}\leq\|v\|_E\,,$$
we deduce that there exists a positive constant $c_q>0$ such that
\begin{equation}\label{11}
\int_{\RR^N}|u_0|^{q-1}|v|\;dx\leq c_q\|v\|_E\,.
\end{equation}
Applying again H\"{o}lder's inequality we obtain
\begin{equation}\label{12}
\begin{aligned}
\int_{\RR^N}h(x)|u_0|^{p-1}|v|dx&=\int_{\RR^N}(h(x)^{(p-1)/p}|u_0|^{p-1})(h(x)^{1/p}|v|)dx\\
&\leq\|u_0\|_{h,p}^{p-1}\|v\|_{h,p}\leq C_0\|v\|_{h,p}\\
&\leq C_0\|v\|_E\,,
\end{aligned}
\end{equation}
where $C_0$ is a positive constant.

By \eq{10}, \eq{11} and \eq{12} we conclude that there exists a positive constant $\kappa$ such that
$$|J(v)|\leq \kappa\|v\|_E\quad\mbox{for all }v\in E,$$
and so $J$ is continuous.

Since $(u_n)_n$ converges weakly to $u_0$ in $E$ and $J$ is linear and
continuous we deduce that
$$J(u_n)\rightarrow J(u_0),$$
in other words
\begin{equation}\label{13}
\lim_{n\rightarrow\infty}\int_{\RR^N}f(x,u_0)(u_n-u_0)\;dx=0.
\end{equation}
In order to show that
\begin{equation}\label{io}\lim_{n\rightarrow\infty}\int_{\RR^N}(u_n-u_0)^2 h(x)^{(q-2)/(q-p)}dx=0,\end{equation}
we first note that for all $R>0$
\begin{equation}\label{14}
\begin{aligned}
\int_{\RR^N}&(u_n-u_0)^2 h(x)^{(q-2)/(q-p)}dx=\int_{\{|x|<R\}}(u_n-u_0)^2 h(x)^{(q-2)/(q-p)}dx\\
&\quad+\int_{\{|x|\geq R\}}(u_n-u_0)^2 h(x)^{(q-2)/(q-p)}dx\\
&\leq\left(\di\int_{\{|x|<R\}}h(x)^{q/(q-p)}dx\right)^{(q-2)/q}\cdot
\left(\int_{\{|x|<R\}}|u_n-u_0|^qdx\right)^{2/q}\\
&\quad+\left(\int_{\{|x|\geq R\}}h(x)^{q/(q-p)}\right)^{(q-2)/q}\cdot
\left(\di\int_{\{|x|\geq R\}}|u_n-u_0|^qdx\right)^{2/q}.
\end{aligned}
\end{equation}
By hypothesis \eq{H} we have
$$\int_{\{|x|<R\}}h(x)^{q/(q-p)}dx\le\int_{\RR^N}h(x)^{q/(q-p)}dx=H<\infty\quad\mbox{for all }R>0.$$
On the other hand, for all $\epsilon>0$ there exists $R_\epsilon>0$ such that
$$\int_{\{|x|\geq R_\epsilon\}}h(x)^{q/(q-p)}dx<\epsilon.$$
Using the  fact that $m<q<m^\star$ we deduce that $W^{1,m}(B_{R_\epsilon}(0))$ is compactly embedded in
$L^q(B_{R_\epsilon}(0))$ and thus
$$\lim_{n\rightarrow\infty}\left(\int_{|x|<R_\epsilon}|u_n-u_0|^q\;dx\right)^{2/q}=0.$$
Since $(u_n-u_0)_n$ is bounded in $E$, it is also bounded  in $L^q(\RR^N)$ and so
there exists a positive constant $M>0$ such that
$$\left(\int_{|x|\geq R_\epsilon}|u_n-u_0|^qdx\right)^{2/q}\leq
\|u_n-u_0\|_q^{2}\le M.$$
Combining the above information with relation \eq{14}, we conclude that for any $\epsilon>0$ there
exists $N_\epsilon>0$ such that for all $n\geq N_\epsilon$ we have
$$\int_{\RR^N}(u_n-u_0)^2h(x)^{(q-2)/(q-p)}dx\leq H\epsilon+M \epsilon^{(q-2)/q}.$$
Therefore, \eqref{io} holds true.

Since $(u_n)_n$ converges weakly to $u_0$ in $W^{1,m}(\RR^N)$ Proposition III.5 in \cite{B} implies
$$\liminf_{n\rightarrow\infty}\|u_n\|_{W^{1,m}(\RR^N)}^m\geq\|u_0\|_{W^{1,m}(\RR^N)}^m.$$
Passing to the limit in \eq{91} and taking into account that \eq{13} and \eq{io} hold true, we obtain
$$\Phi_\lambda(u_0)\leq\liminf_{n\rightarrow\infty}\Phi_\lambda(u_n).$$
Thus, $\Phi_\lambda$ is weakly lower semi-continuous.

The proof of Lemma \ref{l2} is now complete.  \endproof

\medskip
{\sc Proof of Theorem \ref{t1} continued.}
Using Lemmas \ref{l1}, \ref{l2} and Theorem 1.2 in \cite{S} we deduce that
there exists  a global minimizer $u\in E$ of $\Phi_\lambda$, that is,
$$\Phi_\lambda(u)=\inf_{v\in E}\Phi_\lambda(v).$$
It is obvious that $u$ is a weak solution of problem \eq{1}. We prove that  $u\not\equiv 0$ in $E$.
To do that we show that $\inf_E\Phi_\lambda<0$, provided that the parameter $\lambda$ is sufficiently large.

Let us set
$$\overline\lambda=\inf_{u\in E}\left\{\frac{q}{m}\|u\|^m+\frac{q}{p}
\|u\|_{h,p}^p\,:\,\|u\|_q=1\right\}.$$
We point out that $\overline\lambda>0$. Indeed, for any $u\in E$ with $\|u\|_q=1$ by
H\"{o}lder's inequality and by \eqref{H} we have
$$1=\|u\|_q^q\leq\left(\int_{\RR^N}h(x)^{q/(q-p)}dx\right)^{(p-q)/p}\cdot
\left(\int_{\RR^N}h(x)|u|^p\;dx\right)^{q/p}=H^{(p-q)/p}\|u\|_{h,p}^q,$$
so  that
$$\overline\lambda\geq\frac{q}{p}H^{(q-p)/q}>0,$$
being $H>0$ by assumption. Let $\lambda>\overline\lambda$. Then there exists a function $u_1\in E$, with $\|u_1\|_q=1$, such that
$$\lambda\,\|u_1\|_q^q=\lambda>\frac{q}{m}\|u_1\|^m+\frac{q}{p}\|u_1\|_{h,p}^p.$$
This can be rewritten as
$$\Phi_\lambda(u_1)=\frac{1}{m}\|u_1\|^m-\frac{\lambda}{q}\|u_1\|_q^q+\frac{1}{p}\|u_1\|_{h,p}^p<0$$
and consequently $\inf_{u\in E}\Phi_\lambda(u)<0$. Therefore, there exists $\lambda_0=\overline\lambda>0$ such that
problem \eq{1} has a nontrivial weak solution $u_1\in E$ for any $\lambda>\lambda_0$, and
$\Phi_\lambda(u_1)<0$. Since $\Phi_\lambda(u_1)=\Phi_\lambda(|u_1|)$ and $|u_1|\in E$, we may assume that $u_1\geq 0$ a.e. in $\RR^N$.
\endproof

In the following we are looking for a second nontrivial weak solution for problem \eq{1}.

Fix $\lambda\geq\lambda_0$. Set
$$
g(x,t)=\begin{cases}
0, &\mbox{if }t<0\\
\lambda t^{q-1}-h(x) t^{p-1}, &\mbox{if } 0\leq t\leq u_1(x)\\
\lambda u_1(x)^{q-1}-h(x) u_1(x)^{p-1}, &\mbox{if }t>u_1(x)
\end{cases}
$$
and
$$G(x,t)=\int_0^tg(x,s)ds.$$
Define the functional $\Psi:E\rightarrow\RR$ by
$$\Psi(u)=\frac{1}{m}\|u\|^m-\int_{\RR^N} G(x,u)dx.$$
Clearly, $\Psi\in C^1(E,\RR)$ and
$$\langle \Psi'(u),v\rangle=\int_{\RR^N}(|\nabla u|^{m-2}\nabla u\nabla v+
|u|^{m-2}uv)\,dx-\int_{\RR^N} g(x,u)\,v\,dx,$$
for all $u$, $v\in E$. Moreover, {\it if $u$ is a critical point of
$\Psi$, then $u\geq 0$ a.e. in $\RR^N$}.

Next, we are concerned with the location of critical points of the energy functional $\Psi$.

\begin{lemma}\label{l3}
If $u$ is a critical point of $\Psi$, then $u\leq u_1$.
\end{lemma}
\proof
For a function $v$ we define the positive part $v^+(x)=\max\{v(x),0\}$. By Theorem 7.6 in \cite{GT}
we deduce that if $v\in E$ then $v^+\in E$. We have
$$
\begin{aligned}
0&=\langle\Psi'(u)-\Phi'_\lambda(u_1),(u-u_1)^+\rangle\\
&=\int_{\RR^N}(|\nabla u|^{m-2}\nabla u-|\nabla u_1|^{m-2}\nabla u_1)\nabla(u-u_1)^+dx\\ &
\quad+\int_{\RR^N}(|u|^{m-2}u-|u_1|^{m-2}u_1)(u-u_1)^+dx\\
&\quad-\int_{\RR^N}[g(x,u)-\lambda u_1^{q-1}+h(x)u_1^{p-1}](u-u_1)^+dx\\
&=\int_{\{u>u_1\}}(|\nabla u|^{m-2}\nabla u-|\nabla u_1|^{m-2}\nabla u_1)(\nabla u-\nabla u_1)\,dx\\
&\quad+\int_{\{u>u_1\}}(|u|^{m-2}u-|u_1|^{m-2}u_1)(u-u_1)\,dx\\
&\geq\int_{\{u>u_1\}}(|\nabla u|^{m-1}-|\nabla u_1|^{m-1})(|\nabla u|-|\nabla u_1|)\,dx\\
&\quad+\int_{\{u>u_1\}}(|u|^{m-1}-|u_1|^{m-1})(|u|-|u_1|)\,dx\geq 0.
\end{aligned}
$$
Thus, we obtain $u\leq u_1$ and the proof of Lemma \ref{l3} is complete.
\endproof

In the following, via the mountain pass theorem, we determine a critical point $u_2\in E$ of $\Psi$
such that $\Psi(u_2)>0$. By the above lemma
we shall deduce that $0\leq u_2\leq u_1$ in $\Omega$. Therefore,
$$g(x,u_2)=\lambda u_2^{q-1}-h(x) u_2^{p-1}\;\;\;{\rm and}\;\;\;
G(x,u_2)=\frac{\lambda}{q}u_2^q-\frac{h(x)}{p}u_2^p,$$
so that
$$\Psi(u_2)=\Phi_\lambda(u_2)\;\;\;{\rm and}\;\;\;\Psi'(u_2)=\Phi'_\lambda(u_2).$$
More precisely, we find
$$\Phi_\lambda(u_2)>0=\Phi_\lambda(0)>\Phi_\lambda(u_1)\qquad\mbox{and}\qquad \Phi'_\lambda(u_2)=0\,.$$
This shows that $u_2$ is a weak solution of problem \eq{1} such
that $0\leq u_2\leq u_1$, $u_2\neq 0$ and $u_2\neq u_1$.

In order to find $u_2$ described above we prove the following result.

\begin{lemma}\label{l4}
There exists $\rho\in(0,\|u_1\|)$ and $a>0$ such that
$\Psi(u)\geq a$ for all $u\in E$, with $\|u\|=\rho.$
\end{lemma}

\proof
We have
$$\begin{aligned}
\Psi(u)&=\frac{1}{m}\|u\|^m-\int_{\{u>u_1\}}G(x,u)\,dx-\int_{\{u\le u_1\}}G(x,u)\,dx\\
&=\frac{1}{m}\|u\|^m-\frac{\lambda}{q}\int_{\{u>u_1\}}u_1^qdx+\frac{1}{p}\int_{\{u>u_1\}}h(x)u_1^pdx\\
&\hskip 1.7truecm-\frac{\lambda}{q}\int_{\{0\le u\le u_1\}}u^qdx+\frac{1}{p}\int_{\{0\le u\le u_1\}}h(x)u^pdx\\
&\geq\frac{1}{m}\|u\|^m-\frac{\lambda}{q}\|u\|_q^q.
\end{aligned}$$
On the other hand, the continuous Sobolev embedding of $E$ into $L^q(\RR^N)$ implies
that there exists a positive constant $L>0$ such that
$$\|v\|_q\leq L\,\|v\|\quad\mbox{for all }v\in E.$$
The above inequalities yield
$$\Psi(u)\geq\frac{1}{m}\|u\|^m-L_1\|u\|^q=\|u\|^m\left(\frac{1}{m}-L_1\|u\|^{q-m}\right),$$
where $L_1=\lambda\,L^q/q$ is a positive constant. Since $q>m$ it is clear that Lemma~\ref{l4} holds true.
\endproof

\begin{lemma}\label{l5}
The functional $\Psi$ is coercive.
\end{lemma}

\proof
For each $u\in E$ we have
$$\begin{aligned}
\Psi(u)&=\frac{1}{m}\|u\|^m-\frac{\lambda}{q}\int_{\{u>u_1\}}u_1^qdx+\frac{1}{p}\int_{\{u>u_1\}}h(x)u_1^pdx\\
&\hskip 1.7truecm-\frac{\lambda}{q}\int_{\{0\le u\le u_1\}}u^qdx+\frac{1}{p}\int_{\{0\le u\le u_1\}}h(x)u^pdx\\
&\geq\frac{1}{m}\|u\|^m-\frac{\lambda}{q}\int_{\RR^N}u_1^qdx\\
&=\frac{1}{m}\|u\|^m-L_2,
 \end{aligned}$$
where $L_2$ is a positive constant, being $u_1\not=0$. The above inequality implies that $\Psi(u)\rightarrow
\infty$ as $\|u\|\rightarrow\infty$, that is, $\Psi$ is coercive, as required.
\endproof

\medskip
{\sc Proof of Theorem \ref{t1} completed.}
Using Lemma \ref{l4} and the mountain pass theorem (see \cite{AR}
with the variant given by Theorem 1.15 in \cite{W}) we deduce that
there exists a sequence $(v_n)_n\subset E$ such that
\begin{equation}\label{MPT}
\Psi(v_n)\rightarrow c>0\;\;\;{\rm and}\;\;\;\Psi'(v_n)\rightarrow 0,
\end{equation}
where
$$c=\inf\limits_{\gamma\in\Gamma}\max\limits_{t\in[0,1]}
\Psi(\gamma(t))$$
and
$$\Gamma=\{\gamma\in C([0,1],E)\,:\,\gamma(0)=0,\;\gamma(1)=u_1\}.$$
We also refer to Ambrosetti and Prodi \cite{ambpro}, Pucci and Serrin \cite{pucciserrin} for various extensions of the mountain pass theorem, cf. also the recent survey \cite{pucrad}.

By relation \eq{MPT} and Lemma \ref{l5} we obtain that $(v_n)_n$ is
bounded and thus passing eventually to a subsequence, still denoted
by $(v_n)_n$, we may assume that there exists $u_2\in E$ such that
$v_n$ converges weakly to $u_2$. Standard arguments based on the Sobolev embeddings will
show that
$$\lim_{n\rightarrow\infty}\langle\Psi'(v_n),\varphi\rangle=\langle\Psi'(u_2),\varphi\rangle$$
for any $\varphi\in C_0^\infty(\RR^N)$. Taking into account that $E\subset W^{1,m}(\RR^N)$ and
$C_0^\infty(\RR^N)$ is dense in $W^{1,m}(\RR^N)$, the above information implies that $u_2$ is a
weak solution of problem \eq{1}.

We conclude that problem \eq{1} admits at least two nontrivial weak solutions for all $\lambda>\lambda_0$.

Put
$$\lambda^{\star\star}:=\inf\{\lambda>0\,:\, \mbox{problem \eq{1} admits a nontrivial weak solution}\}.$$
Then $\lambda^{\star\star}\geq \lambda^{\star}>0$, where $\lambda^{\star}$ is the parameter defined in \eqref{star}.

Let us consider the constrained minimization problem
\begin{equation}\label{constr}\Lambda:= \inf_{v\in E}\left\{\frac{1}{m}\|v\|^m+\frac{1}{p}
\|v\|_{h,p}^p\,:\,\|v\|_q^q=q\right\}.\end{equation}
Let $(v_n)_n\subset E$  be a minimizing sequence of \eq{constr}. Then $(v_n)_n$ is bounded in $E$, hence we can assume, without loss of generality, that it converges weekly to some $v\in E$ with $\|v\|^q_q=q$. Moreover, by lower semi-continuity arguments we have
$$\Lambda=\frac{1}{m}\|v\|^m+\frac{1}{p}\|v\|_{h,p}^p.$$
Thus, $\Phi_\lambda (v)=\Lambda-\lambda$ for all $\lambda>\Lambda$.\smallskip

To complete the proof of Theorem \ref{t1} it is enough to show the following crucial facts:
\begin{enumerate}
\item[(a)]
 problem \eq{1} has at least two distinct solutions for any $\lambda >\lambda^{\star\star}$;
\item[(b)]  $\lambda^{\star\star}=\lambda^{\star}$ and problem \eq{1} admits a nontrivial weak solution if $\lambda=\lambda^{\star}$.\end{enumerate}

Claim (a) follows by standard monotonicity techniques. Claim (b) uses the same proof as in Filippucci, Pucci and R\u adulescu \cite[p.~712]{filpucrad}.
The proof of Theorem \ref{t1} is now complete.\endproof
\medskip

We point out that the proof of Theorem \ref{t1}--$(ii)$ uses some ideas found in the proofs of Theorems 2.1 and 2.2 in
\cite{AT2}. However, our method in finding the second solution is different, since we use
the mountain pass theorem, while in \cite{AT2} the authors appeal to sub- and super-solutions
method. Our idea is frequently used when we deal with quasilinear problems, see e.g., Filippucci, Pucci and R\u adulescu \cite{filpucrad}, Mih\u ailescu and R\u adulescu \cite{MR}, Perera \cite{P}, Pucci and Servadei \cite{patraf1,patraf2}.
\smallskip

On the other hand, we point out that equation \eq{1} can be studied also in the case when $p$ is
supercritical using similar arguments, since the $|u|^p$ term in the energy continues to be coercive. In these
cases standard regularity results will lead to stronger results in what concerns the smoothness of
solutions, since $W^{1,m}$ is embedded into $C^1$.
\medskip

\noindent {\bf Acknowledgments.}  P.~Pucci was supported by the
Italian MIUR project titled {\it ``Metodi Va\-ria\-zio\-na\-li ed
Equazioni Differenziali alle Derivate Parziali non Lineari''}. V.~R\u adulescu acknowledges the support through  Grant CNCSIS PCCE--8/2010
{\it ``Sisteme di\-fe\-ren\-\c{t}iale \^{\i}n analiza neliniar\u{a} \c{s}i aplica\c{t}ii"}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\bigskip\small

%\noindent{\sc Dipartimento di Matematica e Informatica, Universit\`a degli Studi
%di Perugia,  06123 Perugia, Italy}\\
%{\it E-mail address}:  {\tt pucci@dmi.unipg.it}

%\smallskip

%\noindent{\sc Institute of Mathematics ``Simion Stoilow" of the
%Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania}\\
%{\it E-mail address}: {\tt vicentiu.radulescu@imar.ro}

\end{document}

We are concerned with the study of weak solutions to the quasilinear elliptic equation $-{\rm div}(|\nabla
u|^{m-2}\nabla u)+|u|^{m-2}u=\lambda|u|^{q-2}u-h(x)|u|^{p-2}u$ in $\RR^N$,
where $\lambda $ is a positive parameter, $p,\, q$ are subcritical exponents, and $h$ is a positive continuous function.
By means of variational methods, we establish that this problem does not have any solution if $\lambda$ is small enough, while at least two distinct solutions exist, provided that $\lambda$ is sufficiently large. 