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\noindent {\tt Dynamic Systems and Applications 22 (2013) xx-xx}
\title[Combined effects for a stationary problem]{Combined effects for a stationary problem \\ with indefinite nonlinearities and lack of compactness}
\author{Vicen\c tiu D. R\u adulescu}
\email[V. R\u adulescu]{vicentiu.radulescu@imar.ro}

\begin{abstract}
\vspace{-.2cm}
\begin{center}
``Simion Stoilow" Mathematics Institute of the Romanian Academy, Bucharest, Romania\\
Department of Mathematics, University of Craiova, 200585 Craiova, Romania
\end{center}
\vspace{-.2cm}
\end{abstract}

\thanks{\hspace{-.5cm}\tt
	Received xxxx, 2012
	\hfill 1056-2176 \$15.00 \copyright Dynamic Publishers, Inc.}
\maketitle
\thispagestyle{empty}


{\footnotesize
\noindent{\bf ABSTRACT.} In this paper we study the existence  of entire solutions for a class of quasilinear elliptic equations on the whole space, provided that the nonlinear term has a subcritical growth. Our main results establish related nonexistence or multiplicity results.\\

\noindent{\bf 2000 Mathematics Subject Classification:}  35J60, 35J65, 35J70, 58E05. \\
\noindent{\bf Key words:}  quasilinear elliptic equation, entire solution, mountain pass geometry.}

\section{INTRODUCTION}
In \cite{AT2}, Alama and Tarantello studied the existence and multiplicity of solutions of the equation
\begin{equation}\label{alama}
\left\{\begin{array}{lll}
-\Delta u-\lambda u=k(x)u^q-h(x)u^p, &\mbox{if}& x\in\Omega\\
u>0, &\mbox{if}& x\in\Omega\\
u=0, &\mbox{if}& x\in\partial\Omega\,,
\end{array}\right.
\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 neighborhood of the first eigenvalue of the Laplace operator in $H_0^1(\Omega)$,
they obtained the solvability of problem \eq{alama}, as well as  corresponding multiplicity properties, 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 $k(x)^{p-1}/h(x)^{q-1}$.

The work of Alama and Tarantello was carried on by Chabrowski \cite{C} who obtained similar results for the
problem
\begin{equation}\label{chab}
\left\{\begin{array}{lll}
-\Delta u+u=\lambda|u|^{q-2}u-h(x)|u|^{p-2}u, &\mbox{if}& x\in\RR^N\\
u>0, &\mbox{if}& x\in\RR^N\,,
\end{array}\right.
\end{equation}
where $h>0$ is a positive continuous function on $\RR^N$ satisfying some integrability condition,
$\lambda>0$ is a positive parameter and $2<q<p<2N/(N-2)$, $N\geq 3$. More exactly, Chabrowski proved
that there exists $\lambda_0>0$ such that for any $\lambda\in(0,\lambda_0)$ equation \eq{chab} does not
have any solution while for any $\lambda\geq\lambda_0$ equation \eq{chab} has at least a nontrivial solution.

Related studies with those presented above can be found in \cite{AT1}, \cite{ARC}, \cite{BW}, \cite{rare},
\cite{T}. We also refer to the recent monographs \cite{agre1} and \cite{agre2} for related qualitative results.

In this paper, motivated by \cite{AT2} and \cite{C}, we study the existence and multiplicity of
solutions for the quasilinear problem
\begin{equation}\label{1}
\left\{\begin{array}{lll}
-{\rm div}(|\nabla u|^{m-2}\nabla u)+|u|^{m-2}u=\lambda|u|^{q-2}u-h(x)|u|^{p-2}u, &\mbox{if}& x\in\RR^N\\
u\geq 0, &\mbox{if}& x\in\RR^N\,,
\end{array}\right.
\end{equation}
where $h(x)$ is a positive continuous function on $\RR^N$ ($N\geq 3$) satisfying
the condition
\begin{equation}\label{H}
\int_{\RR^N}\frac{1}{h(x)^{q/(p-q)}}\;dx<\infty,
\end{equation}
$\lambda>0$ is a positive parameter and $2\leq m<q<p<m^\star=Nm/(N-m)$, $m<N$.

If $h=0$, problem \eq{1} is called the Lane-Emden-Fowler equation and it arises in the boundary-layer theory of viscous fluids
(see \cite{wong}). This equation goes back to the paper by Lane \cite{lane} in 1869 and is originally motivated by Lane's interest in
computing both the temperature and the density of mass on the surface of the sun. Problem \eq{1} describes the behavior of
the density of a gas sphere in hydrostatic equilibrium and the index $p$, which is called the polytropic index in astrophysics and
is related to the ratio of the specific heats of the gas.

The main results in the present paper point out the following perturbation effects: (i) if the perturbation in the right-hand side of \eq{1} is weak, then there is no solution; (ii) if the positive term in the right-hand side of \eq{1} is big (this corresponding to a strong perturbation) then there are at least two different entire solutions.
More precisely, we establish the non-existence of nontrivial solutions for problem \eq{1} if
$\lambda$ is small enough (see Section 3) and the existence of at
least two nontrivial solutions for problem \eq{1} if $\lambda$ is large enough
(see Section 4).
\bigskip

In this pa\-per we use stan\-dard no\-ta\-tions and ter\-mi\-no\-lo\-gy. We de\-note by
$W^{1,m}(\RR^N)$ the Sobolev space equipped with the norm
$$\|u\|_{W^{1,m}(\RR^N)}=\left(\int_{\RR^N}(|\nabla u|^m+|u|^m)\;dx\right)^{1/m}.$$
For simplicity we will often denote the above norm by $\|u\|$.

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

\section{MAIN RESULTS}
In this paper we seek weak solutions for problem \eq{1} in a subspace of
$W^{1,m}(\RR^N)$. Let $E$ be the weighted Sobolev space defined by
$$E=\left\{u\in W^{1,m}(\RR^N);\;\int_{\RR^N}h(x)|u|^p\;dx<\infty\right\},$$
equipped with the norm
$$\|u\|_E^m=\int_{\RR^N}(|\nabla u|^m+|u|^m)\;dx+
\left(\int_{\RR^N}h(x)|u|^p\;dx\right)^{m/p}.$$


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

The main results of this paper are the following.
\smallskip

\begin{teo}\label{t1}
There exists $\lambda^\star>0$ such that for any $\lambda\in(0,\lambda^\star)$
problem \eq{1} does not have a nontrivial weak solution.
\end{teo}

\begin{teo}\label{t2}
There exists $\lambda_0>0$ such that for $\lambda>\lambda_0$ problem \eq{1} admits
at least two nontrivial weak solutions.
\end{teo}


\noindent{\bf Remark.} In the linear case, when $m=2$, similar results as those
presented above were obtained in \cite{AT2}, \cite{C}, and \cite{T}, while in the general case we refer
to \cite{YZ}.

\section{PROOF OF THEOREM \ref{t1}}
Let $\Phi:E\rightarrow\RR$ be the energy functional defined by
$$\Phi(u)=\frac{1}{m}\int_{\RR^N}(|\nabla u|^m+|u|^m)\;dx-\frac{\lambda}{q}
\int_{\RR^N}|u|^q\;dx+\frac{1}{p}\int_{\RR^N}h(x)|u|^p\;dx.$$
Standard arguments assure that $\Phi\in C^1(E,\RR)$ with the derivative given by
$$\langle\Phi^{'}(u),v\rangle=\int_{\RR^N}(|\nabla u|^{m-2}\nabla u\nabla v+
|u|^{m-2}uv)\;dx-\lambda\int_{\RR^N}|u|^{q-2}uv\;dx+\int_{\RR^N}h(x)|u|^{p-2}uv\;dx$$
for any $u$, $v\in E$. Solutions of problem \eq{1} will be found as critical points
of functional $\Phi$.

We assume by contradiction that $u\in E$ is a weak solution of problem \eq{1}.
Then $u$ satisfies
\begin{equation}\label{e2}
\int_{\RR^N}(|\nabla u|^m+|u|^m)\;dx+\int_{\RR^N}h(x)|u|^p\;dx=\lambda
\int_{\RR^N}|u|^q\;dx.
\end{equation}
To proceed further, we need Young's inequality
$$ab\leq\frac{a^\alpha}{\alpha}+\frac{b^\beta}{\beta},\;\;\;\forall\;a,b>0$$
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 that
$$h(x)^{q/p}|u|^q\frac{\lambda}{h(x)^{q/p}}\leq\frac{q}{p}(h(x)^{q/p}|u|^q)^{p/q}+
\frac{p-q}{p}\left(\frac{\lambda}{h(x)^{q/p}}\right)^{p/(p-q)}.$$
Integrating over $\RR^N$ we have
$$\lambda\int_{\RR^N}|u|^q\;dx\leq\frac{q}{p}\int_{\RR^N}h(x)|u|^p\;dx+
\frac{p-q}{p}\lambda^{p/(p-q)}\int_{\RR^N}\frac{1}{h(x)^{q/(p-q)}}\;dx.$$
The above inequality and relation \eq{e2} imply
$$\int_{\RR^N}(|\nabla u|^m+|u|^m)\;dx\leq\frac{p-q}{p}\lambda^{p/(p-q)}
\int_{\RR^N}\frac{1}{h(x)^{q/(p-q)}}\;dx+\frac{q-p}{p}\int_{\RR^N}h(x)|u|^p\;dx.$$
Since $q<p$ it results that $\frac{q-p}{p}\int_{\RR^N}h(x)|u|^p\;dx<0$ and thus
\begin{equation}\label{e3}
\int_{\RR^N}(|\nabla u|^m+|u|^m)\;dx\leq\frac{p-q}{p}\lambda^{p/(p-q)}
\int_{\RR^N}\frac{1}{h(x)^{q/(p-q)}}\;dx.
\end{equation}
Since $m<q<m^\star$ the Sobolev embedding of $W^{1,m}(\RR^N)$ into
$L^q(\RR^N)$ implies the existence of a positive constant $C_q$ such
that
$$C_q\left(\int_{\RR^N}|u|^q\;dx\right)^{m/q}\leq\int_{\RR^N}(|\nabla u|^m+
|u|^m)\;dx.$$
We note that $\int_{\RR^N}h(x)|u|^p\;dx\geq 0$. It follows from
\eq{e2} that
$$\int_{\RR^N}(|\nabla u|^m+|u|^m)\;dx\leq\lambda\int_{\RR^N}|u|^q\;dx.$$
Combining the last two inequalities we obtain
\begin{equation}\label{e4}
C_q\left(\int_{\RR^N}|u|^q\;dx\right)^{m/q}\leq\int_{\RR^N}(|\nabla u|^m+
|u|^m)\;dx\leq\lambda\int_{\RR^N}|u|^q\;dx.
\end{equation}
Retaining the first  and the last terms of \eq{e4} we get
$$(C_q\lambda^{-1})^{q/(q-m)}\leq\int_{\RR^N}|u|^q\;dx.$$
That inequality combined with \eq{e4} leads to
$$C_q[(C_q\lambda^{-1})^{q/(q-m)}]^{m/q}\leq\int_{\RR^N}(|\nabla u|^m+
|u|^m)\;dx$$
By relation \eq{e3} and the above inequality we have
$$C_q(C_q\lambda^{-1})^{m/(q-m)}\leq\int_{\RR^N}(|\nabla u|^m+
|u|^m)\;dx\leq\frac{p-q}{p}\lambda^{p/(p-q)}\int_{\RR^N}\frac{dx}{h(x)^{q/(p-q)}}\;dx.$$
Retaining the first and the last term it follows that
$$\lambda>\left[C_q^{q/(q-m)}\frac{p}{p-q}\left(\int_{\RR^N}
\frac{dx}{h(x)^{q/(p-q)}}\;dx\right)^{-1}\right]^{(p-q)(q-m)/(q(p-m))}.$$
Denoting the term in the right-hand side of the above inequality by $\lambda^\star$,
we conclude that Theorem \ref{t1} holds true.  \qed

\section{PROOF OF THEOREM \ref{t2}}
We first establish some
auxiliary results.
\begin{lemma}\label{l1}
The functional $\Phi$ is coercive.
\end{lemma}
\proof
To proceed to the proof of Lemma \ref{l1} we need the following inequality:
\smallskip

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)},\;\;\;\forall\;t\in\RR,
\end{equation}
where $C_{rs}>0$ is a constant depending on $r$ and $s$.
\smallskip

If we take in inequality \eq{6} $k_1=\frac{\lambda}{q}$, $k_2=\frac{(m-1)h(x)}{mp}$, $s=q$ and
$r=p$ ($s<r$ is verified since $q<p$) we obtain
\begin{eqnarray*}
\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))}\\
&=&C_{pq}\lambda^{(p/(p-q))}\frac{1}{h(x)^{q/(p-q)}}\left(\frac{mp}{q(m-1)}\right)^{q/(p-q)}
\frac{1}{q},\\&&\forall\;x\in\RR^N,
\end{eqnarray*}
where $C_{pq}>0$ is a constant depending on $p$ and $q$. Relabeling $C_{pq}\left(\frac{mp}{q(m-1)}
\right)^{q/(p-q)}\frac{1}{q}$ by $C_{pq}$ and integrating the above inequality
over $\RR^N$ it follows that
$$\int_{\RR^N}\left(\frac{\lambda}{q}|u|^q-\frac{(m-1)h(x)}{mp}|u|^p\right)\;dx\leq C_{pq}
\lambda^{(p/(p-q))}\int_{\RR^N}\frac{dx}{h(x)^{q/(p-q)}}\;dx.$$
Using the hypotheses \eq{H} we deduce that there exists a constant $C_1>0$ such that
$$\int_{\RR^N}\left(\frac{\lambda}{q}|u|^q-\frac{(m-1)h(x)}{mp}|u|^p\right)\;dx\leq C_1.$$
Therefore
\begin{equation}\label{7}
\begin{array}{lll}
\Phi(u)&=&\di\frac{1}{m}\di\int_{\RR^N}(|\nabla u|^m+|u|^m)\;dx-\frac{\lambda}{q}
\di\int_{\RR^N}|u|^q\;dx+\di\frac{1}{p}\di\int_{\RR^N}h(x)|u|^p\;dx\\
&=&\di\frac{1}{m}\di\int_{\RR^N}(|\nabla u|^m+|u|^m)\;dx-\left[\di\int_{\RR^N}
\left(\di\frac{\lambda}{q}|u|^q-
\di\frac{(m-1)h(x)}{mp}|u|^p\right)\right]\;dx-\\
&&\di\int_{\RR^N}\di\frac{(m-1)h(x)}{mp}|u|^p+\di\frac{1}{p}\di\int_{\RR^N}h(x)|u|^p\;dx\\
&\geq&\di\frac{1}{m}\di\int_{\RR^N}(|\nabla u|^m+|u|^m)\;dx-C_1+\di\frac{1}{mp}
\di\int_{\RR^N}h(x)|u|^p\;dx,
\end{array}
\end{equation}
and thus $\Phi$ is coercive.
\qed
\begin{lemma}\label{l2}
Assume that $\{u_n\}$ is a sequence in $E$ such that $\Phi(u_n)$ is bounded. Then there exists a
subsequence of $\{u_n\}$, relabeled again $\{u_n\}$, which converges weakly  in $E$ to some $u_0\in E$ and
$$\Phi(u_0)\leq\liminf\limits_{n\rightarrow\infty}\Phi(u_n).$$
\end{lemma}
\proof
Using inequality \eq{7} we obtain
$$\Phi(u_n)\geq\frac{1}{m}\int_{\RR^N}(|\nabla u_n|^m+|u_n|^m)\;dx+\frac{1}{mp}
\int_{\RR^N}h(x)|u_n|^p\;dx-C_1.$$
Since $\Phi(u_n)$ is bounded the above inequality implies that
$\int_{\RR^N}(|\nabla u_n|^m+|u_n|^m)\;dx$ and $\int_{\RR^N}h(x)|u_n|^p\;dx$
are bounded. Therefore, $\{\|u_n\|_E\}$ is bounded. In fact, 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}\;\;
s\in[1,m^\star).
\end{eqnarray*}
We define
$$F(x,u)=\frac{\lambda}{q}|u|^q-h(x)\frac{|u|^p}{p}$$
and
$$f(x,u)=F_u(x,u)=\lambda|u|^{q-2}u-h(x)|u|^{p-2}u.$$
We see 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{eqnarray*}
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)},\;\;\;\forall\;x\in E,
\end{eqnarray*}
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 $\Phi$ and $F$ we obtain the following estimate for
$\Phi(u_0)-\Phi(u_n)$
\begin{equation}\label{9}
\begin{array}{lll}
\Phi(u_0)-\Phi(u_n)&=&\di\frac{1}{m}\di\int_{\RR^N}(|\nabla u_0|^m+|u_0|^m)\;dx-
\di\frac{1}{m}\di\int_{\RR^N}(|\nabla u_n|^m+|u_n|^m)\;dx+\\
&&\di\int_{\RR^N}[F(x,u_n)-F(x,u_0)]\;dx.
\end{array}
\end{equation}
It is clear that
\begin{eqnarray*}
\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{eqnarray*}
Integrating the above relation over $[0,1]$ we obtain
\begin{eqnarray*}
\int_0^1\left(\int_0^sf_u(x,u_0+t(u_n-u_0))\;dt\right)\;ds&=&\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\\
&=&\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{eqnarray*}
The above equality can be written in the following way
\begin{equation}\label{ee}
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+
(u_n-u_0)f(x,u_0).
\end{equation}
Introducing relation \eq{ee} in relation \eq{9} we get
\begin{equation}\label{91}
\begin{array}{lll}
\Phi(u_0)-\Phi(u_n)&=&\di\frac{1}{m}\di\int_{\RR^N}(|\nabla u_0|^m+|u_0|^m)\;dx-
\di\frac{1}{m}\di\int_{\RR^N}(|\nabla u_n|^m+|u_n|^m)\;dx+\\
&&\di\int_{\RR^N}(u_n-u_0)f(x,u_0)\;dx+\di\int_{\RR^N}(u_n-u_0)^2
\di\int_0^1\di\int_0^sf_u(x,u_0+\\&&+t(u_n-u_0))\;dt\;ds\;dx\\
&\leq&\di\frac{1}{m}\di\int_{\RR^N}(|\nabla u_0|^m+|u_0|^m)\;dx-
\di\frac{1}{m}\di\int_{\RR^N}(|\nabla u_n|^m+|u_n|^m)\;dx+\\
&&\di\int_{\RR^N}(u_n-u_0)f(x,u_0)\;dx+C_1\di\int_{\RR^N}(u_n-u_0)^2\di\frac{dx}{h(x)^{(q-2)/(p-q)}},
\end{array}
\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
$$J(v)=\int_{\RR^N}f(x,u_0)v\;dx.$$
Obviously, $J$ is linear. We prove that $J$ is also continuous. Indeed, we have
\begin{equation}\label{10}
\begin{array}{lll}
|J(v)|&\leq&\di\int_{\RR^N}|f(x,u_0)|\cdot|v|\;dx=\di\int_{\RR^N}|\lambda|\cdot||u_0|^{q-1}-
h(x)|u_0|^{p-1}|\cdot|v|\;dx\\
&\leq&\lambda\di\int_{\RR^N}|u_0|^{q-1}|v|\;dx+\di\int_{\RR^N}h(x)|u_0|^{p-1}|v|\;dx.
\end{array}
\end{equation}
On the other hand, using H\"{o}lder's inequality, it results
$$\int_{\RR^N}|u_0|^{q-1}|v|\;dx\leq\left(\int_{\RR^N}|u_0|^q\;dx\right)^{(q-1)/q}
\left(\int_{\RR^N}|v|^q\;dx\right)^{1/q}=\|u_0\|_{q}^{q-1}\|v\|_q.$$
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
$$\|v\|_q\leq C\|v\|_{W^{1,m}(\RR^N)},\;\;\;\forall\;v\in W^{1,m}(\RR^N).$$
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{array}{lll}
\di\int_{\RR^N}h(x)|u_0|^{p-1}|v|\;dx&\leq&\di\int_{\RR^N}(h(x)^{(p-1)/p}|u_0|^{p-1})(h(x)^{1/p}|v|)\;dx\\
&\leq&\left(\di\int_{\RR^N}h(x)|u_0|^p\;dx\right)^{(p-1)/p}\left(\di\int_{\RR^N}h(x)|v|^p\;dx\right)^{1/p}\\
&\leq& C_3\|v\|_{h,p}\leq C_3\|v\|_E,
\end{array}
\end{equation}
where $c_3$ is a positive constant.

By \eq{10}, \eq{11} and \eq{12} we conclude that there exists a positive constant $C_4$ such that
$$|J(v)|\leq C_4\|v\|_E,\;\;\;\forall\;v\in E,$$
and thus, $J$ is continuous. Since $\{u_n\}$ converges weakly to $u_0$ in $E$ and $J$ is linear and
continuous we deduce
$$J(u_n)\rightarrow J(u_0)$$
or
\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
$$\lim_{n\rightarrow\infty}\int_{\RR^N}\frac{(u_n-u_0)^2}{h(x)^{(q-2)/(p-q)}}\;dx=0$$
we take $R>0$ sufficiently large and we observe that
\begin{equation}\label{14}
\begin{array}{lll}
\di\int_{\RR^N}\di\frac{(u_n-u_0)^2}{h(x)^{(q-2)/(p-q)}}\;dx&=&\di\int_{|x|<R}
\di\frac{(u_n-u_0)^2}{h(x)^{(q-2)/(p-q)}}\;dx+\di\int_{|x|\geq R}\di\frac{(u_n-u_0)^2}{h(x)^{(q-2)/(p-q)}}\;dx\\
&\leq&\left(\di\int_{|x|<R}\di\frac{dx}{h(x)^{q/(p-q)}}\;dx\right)^{(q-2)/q}\cdot
\left(\di\int_{|x|<R}|u_n-u_0|^q\;dx\right)^{2/q}+\\
&&\left(\di\int_{|x|\geq R}\di\frac{dx}{h(x)^{q/(p-q)}}\right)^{(q-2)/q}\cdot
\left(\di\int_{|x|\geq R}|u_n-u_0|^q\;dx\right)^{2/q}.
\end{array}
\end{equation}
By hypothesis \eq{H} we have
$$\int_{|x|<R}\frac{dx}{h(x)^{q/(p-q)}}\;dx<\int_{\RR^N}\di\frac{dx}{h(x)^{q/(p-q)}}\;dx<
\infty,\;\;\;\forall\;R>0.$$
On the other hand, for all $\epsilon>0$ there exists $R_\epsilon>0$ such that
$$\int_{|x|\geq R_\epsilon}\di\frac{dx}{h(x)^{q/(p-q)}}\;dx<\epsilon.$$
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 $\{u_n-u_0\}$ is bounded in $E$ it follows that it is bounded in $L^q(\RR^N)$ and we find that
there exists a positive constant $M>0$ such that
$$\left(\int_{|x|\geq R_\epsilon}|u_n-u_0|^q\;dx\right)^{2/q}\leq
\left(\int_{\RR^N}|u_n-u_0|^q\;dx\right)^{2/q}<M.$$
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
$$\int_{\RR^N}\frac{(u_n-u_0)^2}{h(x)^{(q-2)/(p-q)}}\;dx\leq\epsilon+M\cdot\epsilon^{(q-2)/q}.$$
Therefore,
\begin{equation}\label{15}
\lim_{n\rightarrow\infty}\int_{\RR^N}\frac{(u_n-u_0)^2}{h(x)^{(q-2)/(p-q)}}\;dx=0.
\end{equation}
Since $\{u_n\}$ 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{15} hold true we obtain
$$\Phi(u_0)\leq\liminf_{n\rightarrow\infty}\Phi(u_n).$$
Thus, $\Phi$ is weakly lower semicontinuous.

The proof of Lemma \ref{l2} is now complete.  \qed
{\sc Proof of Theorem \ref{t2}.}
Using Lemmas \ref{l1}, \ref{l2} and Theorem 1.2 in \cite{S} we deduce that
there exists $u\in E$ a global minimizer of $\Phi$, i.e.
$$\Phi(u)=\inf_{v\in E}\Phi(v).$$
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 $\inf_E\Phi<0$ providing that the parameter $\lambda$ is sufficiently large.

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

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

Fix $\lambda\geq\lambda_0$. Set
$$
g(x,t)=\left\{\begin{array}{lll}
0, &\mbox{for}& t<0\\
\lambda t^{q-1}-h(x) t^{p-1}, &\mbox{for}& 0\leq t\leq u_1(x)\\
\lambda u_1(x)^{q-1}-h(x) u_1(x)^{p-1}, &\mbox{for}& t>u_1(x)
\end{array}\right.
$$
and
$$G(x,t)=\int_0^tg(x,s)\;ds.$$
Define the functional $\Psi:E\rightarrow\RR$ by
$$\Psi(u)=\frac{1}{m}\int_{\RR^N} (|\nabla u|^m+|u|^m)\;dx-\int_{\RR^N} G(x,u)\;dx.$$
The same arguments as those used for functional $I$ imply that
$J\in C^1(E,\RR)$ 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 is clear that if $u$ is a critical point of
$\Psi$ then $u\geq 0$ a.e. in $\RR^N$.

Next, we prove
\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{eqnarray*}
0&=&\langle\Psi^{'}(u)-\Phi^{'}(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+\\
&&\int_{\RR^N}(|u|^{m-2}u-|u_1|^{m-2}u_1)(u-u_1)^+\;dx-\\
&&\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+\\
&&\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+\\
&&\int_{[u>u_1]}(|u|^{m-1}-|u_1|^{m-1})(|u|-|u_1|)\;dx\geq 0.
\end{eqnarray*}
Thus, we obtain $u\leq u_1$ and the proof of Lemma \ref{l3} is complete.
\qed

In the following we determine a critical point $u_2\in E$ of $\Psi$
such that $\Psi(u_2)>0$ via the mountain pass theorem. By the above lemma
we will deduce that $0\leq u_2\leq u_1$ in $\Omega$. Therefore
$$g(x,u_2)=\lambda u_2^{q-1}-h(x) u_2^{p-1}\;\;\;{\rm and}\;\;\;
G(x,u_2)=\frac{\lambda}{q}u_2^q-\frac{h(x)}{p}u_2^p$$
and thus
$$\Psi(u_2)=\Phi(u_2)\;\;\;{\rm and}\;\;\;\Psi^{'}(u_2)=\Phi^{'}(u_2).$$
More exactly we find
$$\Phi(u_2)>0=\Phi(0)>\Phi(u_1)\qquad\mbox{and}\qquad \Phi^{'}(u_2)=0\,.$$
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
\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{eqnarray*}
\Psi(u)&=&\frac{1}{m}\|u\|^m-\int_{\RR^N}G(x,u)\;dx\\
&=&\frac{1}{m}\|u\|^m-\int_{[u>u_1]}G(x,u)\;dx-\int_{[u<u_1]}G(x,u)\;dx\\
&=&\frac{1}{m}\|u\|^m-\frac{\lambda}{q}\int_{[u>u_1]}u_1^q\;dx+\frac{1}{p}\int_{[u>u_1]}h(x)u_1^p\;dx
-\frac{\lambda}{q}\int_{[u>u_1]}u^q\;dx+\\
&&+\frac{1}{p}\int_{[u>u_1]}h(x)u^p\;dx\\
&\geq&\frac{1}{m}\|u\|^m-\frac{\lambda}{q}\int_{\RR^N}|u|^q\;dx.
\end{eqnarray*}
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
$$|u|_q\leq L\cdot\|u\|,\;\;\;\forall\;u\in E.$$
The above inequalities imply
$$\Psi(u)\geq\frac{1}{m}\|u\|^m-L_1\|u\|^q=\|u\|^m\left[\frac{1}{m}-L_1\|u\|^{q-m}\right]\,,$$
where $L_1$ is a positive constant. Since $q>m$ it is clear that Lemma \ref{l4} holds true.
\qed

\begin{lemma}\label{l5}
The functional $\Psi$ is coercive.
\end{lemma}
\proof
For each $u\in E$ we have
\begin{eqnarray*}
\Psi(u)&=&\frac{1}{m}\|u\|^m-\frac{\lambda}{q}\int_{[u>u_1]}u_1^q\;dx+\frac{1}{p}\int_{[u>u_1]}h(x)u_1^p\;dx
-\frac{\lambda}{q}\int_{[u>u_1]}u^q\;dx+\\
&&+\frac{1}{p}\int_{[u>u_1]}h(x)u^p\;dx\\
&\geq&\frac{1}{m}\|u\|^m-\frac{\lambda}{q}\int_{\RR^N}u_1^q\;dx\\
&=&\frac{1}{m}\|u\|^m-L_2\,,
\end{eqnarray*}
where $L_2$ is a positive constant. The above inequality implies that $\Psi(u)\rightarrow
\infty$ as $\|u\|\rightarrow\infty$, that is, $\Psi$ is coercive.
The proof of Lemma \ref{l5} is complete.
\qed

\medskip
{\sc Proof of Theorem \ref{t2} 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 $(u_n)\subset E$ such that
\begin{equation}\label{MPT}
\Psi(u_n)\rightarrow c>0\;\;\;{\rm and}\;\;\;\Psi^{'}(u_n)\rightarrow 0
\end{equation}
where
$$c=\inf\limits_{\gamma\in\Gamma}\max\limits_{t\in[0,1]}
\Psi(\gamma(t))$$
and
$$\Gamma=\{\gamma\in C([0,1],E);\;\gamma(0)=0,\;\gamma(1)=u_1\}.$$
By relation \eq{MPT} and Lemma \ref{l5} we obtain that $(u_n)$ is
bounded and thus passing eventually to a subsequence, still denoted
by $(u_n)$, we may assume that there exists $u_2\in E$ such that
$u_n$ converges weakly to $u_2$. Standard arguments based on the Sobolev embeddings will
show that
$$\lim_{n\rightarrow\infty}\langle\Psi^{'}(u_n),v\rangle=\langle\Psi^{'}(u_2),v\rangle\,,$$
for any $v\in C_0^\infty(\RR^N)$. 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} has two nontrivial weak solutions. The proof of Theorem \ref{t2} is
complete. \qed

\bigskip
We point out that the proof of Theorem \ref{t2} is similar with those of Theorems 2.1 and 2.2 in
\cite{AT2}. However, our method in finding the second solution is somewhat different since we use
the mountain pass theorem while in \cite{AT2} the authors appeals to sub and super-solutions
method. Our idea is frequently used when we deal with quasilinear problems see, e.g.,  Perera \cite{P}
or Mih\u ailescu and R\u adulescu \cite{MR}.

\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 will be coercive. In that
cases standard regularity results will lead to stronger results in what concerns the smoothness of
solutions since in that case $W^{1,m}$ is embedded in $C^1$.


\medskip

\textbf{Acknowledgements.} The author acknowledges the support through Grant CNCS PCE-47/2011.





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