\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 56, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/56\hfil Existence of weak solutions] {Existence of weak solutions for quasilinear elliptic equations involving the $p$-Laplacian} \author[U. Severo\hfil EJDE-2008/56\hfilneg] {Uberlandio Severo} \address{Uberlandio Severo \newline Departamento de Matem\'atica \\ Universidade Federal da Para\'\i ba \\ 58051-900, Jo\~ao Pessoa, PB , Brazil} \email{uberlandio@mat.ufpb.br Tel.\ 55 83 3216 7434 Fax 55 83 3216 7277} \thanks{Submitted October 27, 2007. Published April 17, 2008.} \thanks{Supported by CAPES/MEC/Brazil, CNPq and Millennium Institute for the Global \hfill\break\indent Advancement of Brazilian Mathematics-IM-AGIMB} \subjclass[2000]{35J20, 35J60, 35Q55} \keywords{Quasilinear Schr\"{o}dinger equation; solitary waves; $p$-Laplacian; \hfill\break\indent variational method; mountain-pass theorem} \begin{abstract} This paper shows the existence of nontrivial weak solutions for the quasilinear elliptic equation $$ -\big(\Delta_p u +\Delta_p (u^2)\big) +V(x)|u|^{p-2}u= h(u) $$ in $\mathbb{R}^N$. Here $V$ is a positive continuous potential bounded away from zero and $h(u)$ is a nonlinear term of subcritical type. Using minimax methods, we show the existence of a nontrivial solution in $C^{1,\alpha}_{\rm loc}(\mathbb{R}^N)$ and then show that it decays to zero at infinity when $1
0$ and putting \[ \psi(t,x)=\exp(-iFt)u(x),\quad F\in \mathbb{R}, \] we obtain a corresponding equation \begin{equation}\label{simp} -\Delta u -\Delta (u^2)u +V(x)u= h(u)\quad\mbox{in }\mathbb{R}^N\\ \end{equation} where we have renamed $V(x)-F$ to be $V(x)$, $h(u)=\widetilde{h}(u^2)u$ and we assume, without loss of generality, that $\kappa =1$. Our paper was motivated by the quasilinear Schr\"{o}dinger equation (\ref{simp}), to which much attention has been paid in the past several years. This problem was studied in \cite{Jeanjean-Colin,OMS,Liu-Wang I, Liu-Wang II,Liu-wang-wang, Poppenberg-Schmitt-Wang} and references therein. Many important results on the existence of nontrivial solutions of (\ref{simp}) were obtained in these papers and give us very good insight into this quasilinear Schr\"{o}dinger equation. The existence of a positive ground state solution has been proved in \cite{Poppenberg-Schmitt-Wang} by using a constrained minimization argument, which gives a solution of (\ref{simp}) with an unknown Lagrange multiplier $\lambda$ in front of the nonlinear term. In \cite{Liu-Wang II}, by a change of variables the quasilinear problem was transformed to a semilinear one and an Orlicz space framework was used as the working space, and they were able to prove the existence of positive solutions of (\ref{simp}) by the mountain-pass theorem. The same method of change of variables was used recently also in \cite{Jeanjean-Colin}, but the usual Sobolev space $H^1(\mathbb{R}^N)$ framework was used as the working space and they studied different class of nonlinearities. In \cite{OMS}, for $N=2$ the authors treated the case where the nonlinearity $h:\mathbb{R} \to \mathbb{R}$ has critical exponential growth, that is, $h$ behaves like $\exp(4\pi s^4)-1$ as $ |s| \to \infty$. They establish an existence result for the problem by combining Ambrosetti-Rabinowitz mountain-pass theorem with a version of the Trudinger-Moser inequality in $\mathbb{R}^2$. In \cite{Liu-wang-wang}, it was established the existence of both one-sign and nodal ground states of soliton type solutions by the Nehari method. Here, our goal is to prove by variational approach the existence of nontrivial weak solutions of \eqref{principal}. A function $u:\mathbb{R}^N\to \mathbb{R}$ is called a weak solution of \eqref{principal} if $u\in W^{1,p}(\mathbb{R}^N)\cap L^{\infty}_{\rm loc}(\mathbb{R}^N)$ and for all $\varphi\in C_0^{\infty}(\mathbb{R}^N)$ it holds \begin{equation} \label{solucao fraca} \begin{aligned} &\int_{\mathbb{R}^N}(1+2^{p-1}|u|^p)|\nabla u|^{p-2}\nabla u\nabla\varphi\,\mathrm{d} x+2^{p-1}\int_{\mathbb{R}^N}|\nabla u|^p|u|^{p-2} u\varphi\,\mathrm{d} x\\ &=\int_{\mathbb{R}^N}g(x,u)\varphi\,\mathrm{d} x. \end{aligned} \end{equation} where $g(x,u):= h(u)-V(x)|u|^{p-2}u$. We notice that we can not apply directly such methods because the natural functional associated to \eqref{principal} given by \[ J(u) = \frac{1}{p}\int_{\mathbb{R}^N} (1+2^{p-1}|u|^p)|\nabla u |^p\,\mathrm{d} x + \frac{1}{p}\int_{\mathbb{R}^N} V(x)|u|^p\,\mathrm{d} x - \int_{\mathbb{R}^N} H(u)\,\mathrm{d} x, \] where $H(s)=\int_0^s h(t)\; \mathrm{d}t$, is not well defined in general, for instance, in $W^{1,p}(\mathbb{R}^N)$. For example, if $1
0$ such that $V(x)\geq V_0$
for all $x\in\mathbb{R}^N$;
\item[(V2)] $\lim_{|x|\to\infty}V(x)=V_{\infty}$ and
$V(x)\leq V_{\infty}$ for all $x\in\mathbb{R}^N$;
\item[(H0)] $h$ is odd and $h(s)=o(|s|^{p-2}s)$ at the origin;
\item[(H1)] There exists a constant $C>0$ such that for all $s\in \mathbb{R}$
\[
|h(s)|\leq C(1+|s|^r),
\]
where $2p-1 2p$;
\item[(b)] {\rm (H2)} holds with $\theta=2p$ and $p-1< r< p^{*}-1$ if
$1 p-1$ if $p=N$ in {\rm (H1)}.
\end{itemize}
Moreover, if $1 0$.
\noindent$\bullet$ $C_0^{\infty}(\mathbb{R}^N)$ denotes functions
infinitely differentiable with compact support in $\mathbb{R}^N$.
\noindent$\bullet$ For $1 \leq p \leq \infty$, $L^p(\mathbb{R}^N)$
denotes the usual Lebesgue space with the norms
\begin{gather*}
\| u \|_p:=\Big(\int_{\mathbb{R}^N}|u|^p\,\mathrm{d} x\Big)^{1/p},\quad
1 \leq p < \infty;\\
\|u\|_{\infty}:=\inf\{C>0:|u(x)|\leq C \mbox{ almost
everywhere in }\mathbb{R}^N\}.
\end{gather*}
\noindent$\bullet$ $W^{1,p}(\mathbb{R}^N)$ denotes the Sobolev spaces modelled
on
$L^p(\mathbb{R}^N)$ with its usual norm
\[
\|u\|:=\left(\|\nabla u\|_p^p+ \|u\|_p^p\right)^{1/p}.
\]
\noindent$\bullet$ $\langle\cdot,\cdot\rangle$ denotes the duality pairing
between $X$ and its dual $X^*$.
\noindent$\bullet$ The weak convergence in $X$ is denoted by
$\rightharpoonup$, and the strong convergence by $\to$.\\
The outline of the paper is as follows. In Section 2, we
give the properties of the change of variables $f(t)$ and some
preliminary results. In Section 3, we present an auxiliary problem
and some related results and Section 4 is devoted to the proof of Theorem
\ref{Main2}.
\section{Preliminary results}
We begin with some preliminary results. Let us collect some
properties of the change of variables
$f:\mathbb{R}\to \mathbb{R}$ defined in (\ref{mudanda de variavel}),
which will be usual in the sequel of the paper.
\begin{lemma}\label{Lema f}
The function $f(t)$ and its derivative satisfy the following
properties:
\begin{itemize}
\item[(1)] $f$ is uniquely defined, $C^2$ and invertible;
\item[(2)] $|f'(t)|\leq 1$ for all $t\in \mathbb{R}$;
\item[(3)] $|f(t)|\leq |t|$ for all $t\in \mathbb{R}$;
\item[(4)] $f(t)/t\to 1$ as $t\to 0$;
\item[(5)] $|f(t)|\leq 2^{1/2p}|t|^{1/2}$ for all $t\in
\mathbb{R}$;
\item[(6)] $f(t)/2\leq tf'(t)\leq f(t)$ for all $t\geq 0$;
\item[(7)] $f(t)/{\sqrt t}\to a>0$ as $t\to +\infty$.
\item[(8)] there exists a positive constant
$C$ such that
\[
|f(t)| \geq
\begin{cases}
C|t|, & |t| \leq 1 \\
C|t|^{1/2}, & |t| \geq 1.
\end{cases}
\]
\end{itemize}
\end{lemma}
\begin{proof}
To prove (1), it is sufficient to remark that the function
\[
y(s):=\frac{1}{(1+2^{p-1}|s|^p)^{1/p}}
\]
has bounded derivative. The point (2) is immediate by the
definition of $f$. Inequality (3) is a consequence of (2) and the
fact that $f(t)$ is odd and concave function for $t>0$. Next, we
prove (4). As a consequence of the mean value theorem for
integrals, we see that
\[
f(t)=\int_0^t \frac{1}{(1+2^{p-1}|f(s)|^p)^{1/p}}\,\mathrm{d} s =
\frac{t}{(1+2^{p-1}|f(\xi)|^p)^{1/p}}
\]
where $\xi\in (0,t)$. Since $f(0)=0$, we get
\[
\lim_{t\to 0}\frac{f(t)}{t}=\lim_{\xi\to
0}\frac{1}{(1+2^{p-1}|f(\xi)|^p)^{1/p}}=1.
\]
To show the item (5), we integrate $f'(t)(1+2^{p-1}|f(t)|^p)^{1/p} = 1$
and we obtain
\[
\int_0^t f'(s)(1+2^{p-1}|f(s)|^p)^{1/p}\,\mathrm{d} s = t
\]
for $t>0$. Using the change of variables $y=f(s)$, we get
\[
t=\int_0^{f(t)}(1+2^{p-1}y^p)^{1/p}\,\mathrm{d} y\geq
2^{(p-1)/p}\frac{(f(t))^2}{2}=2^{-1/p}(f(t))^2
\]
and thus $(5)$ is proved for $t\geq 0$. For $t<0$, we use the fact
that $f$ is odd. The first inequality in (6) is equivalent to
$2t\geq (1+2^{p-1}(f(t))^p)^{1/p}f(t)$. To show this inequality, we
study the function $G:\mathbb{R}^{+}\to \mathbb{R}$, defined
by $G(t)=2t-(1+2^{p-1}(f(t))^p)^{1/p}f(t)$. Since $G(0)=0$ and using
the definition of $f$ we obtain for all $t\geq 0$
\[
G'(t)=1-\frac{2^{p-1}(f(t))^p}{1+2^{p-1}(f(t))^p}
=\frac{1}{1+2^{p-1}(f(t))^p}=(f'(t))^p> 0,
\]
and the first inequality is proved. The second one is obtained in a
similar way. Now, by point (4) it follows that
$\lim_{t\to 0^+}f(t)/\sqrt t=0$ and inequality (6) implies that
for all $t>0$
\[
\frac{d}{dt}\Big(\frac{f(t)}{\sqrt
t}\Big)=\frac{2f'(t)t-f(t)}{2t\sqrt{t}}\geq 0.
\]
Thus, the function $f(t)/\sqrt t$ is nondecreasing for $t>0$ and
this together with estimate $(5)$ shows the item $(7)$. Point $(8)$
is a immediate consequence of the limits $(4)$ and $(7)$.
\end{proof}
We readily deduce that the functional
$I:W^{1,p}(\mathbb{R}^N)\to \mathbb{R}$ is of class $C^1$
under the conditions {\rm (V1)--(V2)} and {\rm (H1)--(H2)}. Moreover,
\[
\langle I'(v),w\rangle=\int_{\mathbb{R}^N}|\nabla v|^{p-2}\nabla
v\nabla w\,\mathrm{d} x-\int_{\mathbb{R}^N}g(x,f(v))f'(v)w\,\mathrm{d} x
\]
for $v,w\in W^{1,p}(\mathbb{R}^N)$. Thus, the critical points of $I$
correspond exactly to the weak solutions of (\ref{equacao dual}). We
have the following result that relates the solutions of
(\ref{equacao dual}) to the solutions of \eqref{principal}.
\begin{proposition} \label{prop1}
\begin{itemize}
\item[(1)] If $v \in W^{1,p}(\mathbb{R}^N)\cap L^{\infty}_{\rm loc}(\mathbb{R}^N)$ is a critical point
of the functional $I$, then $u=f(v)$ is a weak solution of
\eqref{principal};
\item[(2)] If $v$ is a classical solution of (\ref{equacao dual}) then
$u=f(v)$ is a classical solution of \eqref{principal}.
\end{itemize}
\end{proposition}
\begin{proof}
First, we prove (1). We have that $|u|^p=|f(v)|^p\leq |v|^p$ and
$|\nabla u|^p=|f'(v)|^p|\nabla v|^p\leq |\nabla v|^p$.
Consequently,
$u \in W^{1,p}(\mathbb{R}^N)\cap L^{\infty}_{\rm loc}(\mathbb{R}^N)$.
As $v$ is a critical
point of $I$, we have for all $w\in W^{1,p}(\mathbb{R}^N)$
\begin{equation}\label{solucao fraca mod}
\int_{\mathbb{R}^N}|\nabla v|^{p-2}\nabla v\nabla w\,\mathrm{d} x
=\int_{\mathbb{R}^N}g(x,f(v))f'(v)w\,\mathrm{d} x.
\end{equation}
Since $(f^{-1})'(t)=\frac{1}{f'(f^{-1}(t))}$, it follows that
\begin{equation}\label{derivada}
(f^{-1})'(t)=(1+2^{p-1}|f(f^{-1}(t))|^p)^{1/p}=(1+2^{p-1}|t|^p)^{1/p}
\end{equation}
which implies that
\begin{equation}\label{perna longa}
\nabla v=(f^{-1})'(u)\nabla u=(1+2^{p-1}|u|^p)^{1/p}\nabla u.
\end{equation}
For all $\varphi\in C_0^{\infty}(\mathbb{R}^N)$, we have
\[
f'(v)^{-1}\varphi=(1+2^{p-1}|u|^p)^{1/p}\varphi\in
W^{1,p}(\mathbb{R}^N)
\]
and
\begin{equation}\label{ligeirinho}
\nabla(f'(v)^{-1}\varphi)=2^{p-1}(1+2^{p-1}|u|^p)^{(1-p)/p}|u|^{p-2}
u\varphi\nabla u +(1+2^{p-1}|u|^p)^{1/p}\nabla \varphi
\end{equation}
Taking $w=f'(v)^{-1}\varphi$ in (\ref{solucao fraca mod}) and
using \eqref{perna longa}--\eqref{ligeirinho}, we obtain
(\ref{solucao fraca}) which shows that $u=f(v)$ is a weak solution
of \eqref{principal}.
Next, we prove (2). We have
\[
\Delta_p v= \sum_{i=1}^{N}\frac{\partial}{\partial x_i}
\Big(|\nabla v|^{p-2}\frac{\partial v}{\partial x_i}\Big)
= \sum_{i=1}^{N}\frac{\partial}{\partial x_i}
\Big(|(f^{-1})'(u)\nabla u|^{p-2}(f^{-1})'(u)\frac{\partial u}{\partial
x_i}\Big)
\]
and deriving
\begin{align*}
\Delta_p v&=\sum_{i=1}^{N}\frac{\partial}{\partial
x_i}\Big(|\nabla u|^{p-2}\frac{\partial u}{\partial
x_i}\Big)|(f^{-1})'(u)|^{p-2}(f^{-1})'(u)\\
&\quad + \sum_{i=1}^{N}\frac{\partial}{\partial x_i}
\Big(|(f^{-1})'(u)|^{p-2}(f^{-1})'(u)\big)|\nabla u|^{p-2}
\frac{\partial u}{\partial x_i}\,.
\end{align*}
Using (\ref{derivada}), we get
\[
\Delta_p v= (1+2^{p-1}|u|^p)^{(p-1)/p}\Delta_p u +
(p-1)2^{p-1}|u|^{p-2}u\left((1+2^{p-1}|u|^p\right)^{-1/p}|\nabla
u|^p.
\]
Thus,
\begin{align*}
&(1+2^{p-1}|u|^p)^{(p-1)/p}\Delta_p u +
(p-1)2^{p-1}|u|^{p-2}u\left((1+2^{p-1}|u|^p\right)^{-1/p}|\nabla
u|^p\\
&=- \frac{1}{(1+2^{p-1}|u|^p)^{1/p}}g(x,u);
\end{align*}
consequently
\[
\Delta_p u +2^{p-1}|u|^{p}\Delta_p u+(p-1)2^{p-1}|u|^{p-2}u|\nabla
u|^p =-g(x,u)
\]
Finally, observing that
\[
2^{p-1}|u|^{p}\Delta_p u+(p-1)2^{p-1}|u|^{p-2}u|\nabla u|^p=\Delta_p
(u^2)u
\]
we conclude that
$-\Delta _pu -\Delta _p(u^2)u = g(x,u)$.
\end{proof}
At this moment, it is clear that to obtain a weak solution of
\eqref{principal}, it is sufficient to obtain a critical point of
the functional $I$ in $L^{\infty}_{\rm loc}(\mathbb{R}^N)$.
\section{Auxiliary problem}
To prove our main result, we shall use results due to do
\'O - Medeiros \cite{JMBO e Veve} for the equation
\begin{equation}
\label{Medeiros} -\Delta_p v=k(v)\quad \mbox{in } \mathbb{R}^N.
\end{equation}
The energy functional corresponding to
(\ref{Medeiros}) is
\[
\mathcal{F}(v)=\frac{1}{p}\int_{\mathbb{R}^N}|\nabla v|^p\,\mathrm{d} x
-\int_{\mathbb{R}^N}K(v)\,\mathrm{d} x,
\]
where $K(s):=\int_0^s k(t)\textrm{d}t$. This functional is of
class $C^1$
on $W^{1,p}(\mathbb{R}^N)$ under the assumptions
on $k(s)$ below. The authors consider the following conditions on
the nonlinearity $k(s)$:
\begin{enumerate}
\item [(K0)] $k\in C(\mathbb{R},\mathbb{R})$ and is odd;
\item [(K1)] When $1 0$ and $\alpha_0>0$, that
\[
|k(s)|\leq C[\exp(\alpha_0
|s|^{N/(N-1)})-S_{N-2}(\alpha_0,s)],
\]
for all $|s|\geq R>0$, where
\[
S_{N-2}(\alpha_0,s)=\sum_{k=0}^{N-2}\frac{\alpha_0^k}{k!}|s
|^{kN/N-1};
\]
\item[(K2)] When $1 0$ such that $K(\zeta)>0$.
\end{enumerate}
Let
\begin{equation}
\label{m} m:=\inf\{\mathcal{F}(v): v\in
W^{1,p}(\mathbb{R}^N)\setminus \{0\}\mbox{ is a solution of
(\ref{Medeiros})}\}.
\end{equation}
By a \emph{least energy solution (or ground state)} of
(\ref{Medeiros}) we mean a minimizer of $m$. Therefore, if $w$ is
a minimizer of (\ref{m}) and $v$ is any nontrivial solution of
(\ref{Medeiros}) then $\mathcal{F}(w)\leq \mathcal{F}(v)$.\\
The following results are proved in \cite[Theorems
1.4, 1.6 and 1.8]{JMBO e Veve}.
\begin{theorem}\label{Medeiros2}
Let $1 0,\ \delta>0$ such that
$\mathcal{F}(v)\geq \alpha \|v\|^p$ if $\|v\|\leq \delta$.
\end{remark}
\section{Proof of Theorem \ref{Main2}}
To prove Theorem \ref{Main2} we first show that the functional $I$
possesses the mountain-pass geometry. To do this, we shall use
some results related to an auxiliary problem.
\subsection{Mountain-pass geometry}
\begin{lemma} \label{Geometria}
Under the hypotheses {\rm (V1)--(V2)} and
{\rm (H0)--(H1)}, the functional $I$ has a mountain-pass geometry.
\end{lemma}
\begin{proof}
Let the energy functionals associated with the equations
$-\Delta_pv=g_0(v)$ and $-\Delta_pv=g_{\infty}(v)$, respectively, be
\begin{gather*}
J_0(v)=\frac{1}{p}\int_{\mathbb{R}^N}|\nabla v|^p\,\mathrm{d} x +
\frac{1}{p}\int_{\mathbb{R}^N}V_0|f(v)|^p\,\mathrm{d} x
-\int_{\mathbb{R}^N}H(f(v))\,\mathrm{d} x
\\
J_{\infty}(v)=\frac{1}{p}\int_{\mathbb{R}^N}|\nabla v|^p\,\mathrm{d} x +
\frac{1}{p}\int_{\mathbb{R}^N}V_{\infty}|f(v)|^p\,\mathrm{d} x
-\int_{\mathbb{R}^N}H(f(v))\,\mathrm{d} x,
\end{gather*}
where
\begin{gather*}
g_0(v):= f'(v)[h(f(v))-V_0|f(v)|^{p-2}f(v)], \\
g_{\infty}(v):= f'(v)[h(f(v))-V_{\infty}|f(v)|^{p-2}f(v)].
\end{gather*}
Note that $J_0(v)\leq I(v)\leq J_{\infty}(v)$ for all $v\in
W^{1,p}(\mathbb{R}^N)$. It is not difficult to see that the
nonlinearity $g_0$ satisfies the hypotheses {\rm (K0)--(K2)}.
Thus, from Remark \ref {coercivo}, we deduce that there exist
$\beta_0
>0$ and $\delta_0 >0$ such that
\begin{equation}
\label{coercividade de I} I(v)\geq J_0(v)\geq \beta_0 \| v
\|^{p}\quad \mbox{if}\quad \| v\| \leq \delta_0.
\end{equation}
Namely the origin is a strict local minimum for $I$. Moreover,
since $g_{\infty}$ also satisfies {\rm (K0)--(K2)}, applying
Theorem \ref{Medeiros2} to the functional $J_{\infty}$, we see
that there exists $e\in W^{1,p}(\mathbb{R}^N)$ with $\|e\| >
\delta_0$ such that $J_{\infty}(e)< 0$ which implies that $I(e)<
0$. Thus $\Gamma \neq \emptyset$, where
\[
\Gamma = \{\gamma \in C([0,1],W^{1,p}(\mathbb{R}^N)): \gamma(0)=0,\;
I(\gamma (1))<0\}\,.
\]
The lemma is proved.
\end{proof}
\begin{remark} \label{crescimento} \rm
By the condition (H2), there exists $C>0$
such that $H(s)\geq Cs^{\theta}$ for $s\geq 1$. In particular, we
get $\lim_{s\to +\infty}H(s)/s^p=+\infty$. Thus, there
exists $\zeta>0$ such that $G_0(\zeta)>0$ and $G_{\infty}(\zeta)>0$
where
\begin{gather*}
G_{\infty}(s)=\int_0^sg_{\infty}(t)\
\textrm{d}t=H(f(s))-\frac{V_{\infty}}{p}|f(s)|^p;\\
G_0(s)=\int_0^sg_0(t)\ \textrm{d}t=H(f(s))-\frac{V_0}{p}|f(s)|^p.
\end{gather*}
Therefore $g_0$ and $g_{\infty}$ also satisfy (K3). As a
consequence of Theorem \ref{Medeiros1}, the problems
\[
-\Delta_pv=g_0(v)\quad\mbox{and}\quad
-\Delta_pv=g_{\infty}(v)\quad\mbox{in}\quad \mathbb{R}^N
\]
have least energy solutions in $W^{1,p}(\mathbb{R}^N)$ which are
positive.
\end{remark}
\subsection{Cerami sequences}
We recall that a sequence $(v_n)$ in $W^{1,p}(\mathbb{R}^N)$ is
called {\it Cerami sequence} for $I$ at the level $c$ if
\[
I(v_n)\to c\quad\mbox{and}\quad
\|I'(v_n)\|(1+\|v_n\|)\to 0\quad \mbox{as } n\to \infty.
\]
We have the following lemma:
\begin{lemma}\label{limitacao}
Suppose that {\rm (V1)--(V2)} and {\rm (H0)--(H2)} hold. Then each
Cerami sequence for $I$ at the level $c>0$ is bounded in
$W^{1,p}(\mathbb{R}^N)$.
\end{lemma}
\begin{proof}
First, we will show that if a sequence $(v_n)$ in
$W^{1,p}(\mathbb{R}^N)$ satisfies
\begin{equation}\label{essencial}
\int_{\mathbb{R}^N}|\nabla v_n|^p\,\mathrm{d} x
+\int_{\mathbb{R}^N}V(x)|f(v_n)|^p\,\mathrm{d} x\leq C
\end{equation}
for some constant $C>0$, then it is bounded in
$W^{1,p}(\mathbb{R}^N)$. Indeed, we need just to prove that
$\int_{\mathbb{R}^N}|v_n|^p\,\mathrm{d} x$ is bounded. We write
\[
\int_{\mathbb{R}^N}|v_n|^p\,\mathrm{d} x
=\int_{\{|v_n|\leq 1\}}|v_n|^p\,\mathrm{d} x+\int_{\{|v_n|> 1\}}|v_n|^p\,\mathrm{d} x.
\]
By (8) and Remark \ref{crescimento} there exists $C>0$ such that
$H(f(s))\geq Cs^p$ for all $s\geq 1$. This implies that
\[
\int_{\{|v_n|> 1\}}|v_n|^p\,\mathrm{d} x\leq\frac{1}{C}\int_{\{|v_n|> 1\}}
H(f(v_n))\,\mathrm{d} x\leq\frac{1}{C}\int_{\mathbb{R}^N} H(f(v_n))\,\mathrm{d} x.
\]
Again using (8) in Lemma \ref{Lema f}, it follows that
\[
\int_{\{|v_n|\leq1\}}|v_n|^p\,\mathrm{d} x\leq\frac{1}{C^p}\int_{\{|v_n|\leq1\}} |f(v_n)|^p\,\mathrm{d} x\leq\frac{1}{C^pV_0}\int_{\mathbb{R}^N} V(x)|f(v_n)|^p\,\mathrm{d} x.
\]
These estimates prove that $(v_n)$ is bounded in
$W^{1,p}(\mathbb{R}^N)$.
Now let $(v_n)$ be in $W^{1,p}(\mathbb{R}^N)$ an arbitrary Cerami
sequence for $I$ at the level $c>0$. We have that
\begin{equation}\label{morango}
\frac{1}{p}\int_{\mathbb{R}^N}|\nabla v_n|^p\,\mathrm{d} x +
\frac{1}{p}\int_{\mathbb{R}^N}V(x)|f(v_n)|^p\,\mathrm{d} x
-\int_{\mathbb{R}^N}H(f(v_n))\,\mathrm{d} x=c+o_n(1)
\end{equation}
and for all $\varphi \in W^{1,p}(\mathbb{R}^N)$
\begin{equation} \label{derivada de Gateaux}
\begin{aligned}
\langle I'(v_n),\varphi\rangle
&=\int_{\mathbb{R}^N}|\nabla
v_n|^{p-2}\nabla v_n\nabla \varphi\,\mathrm{d} x +
\int_{\mathbb{R}^N}V(x)\frac{|f(v_n)|^{p-2}f(v_n)\varphi}{(1+2^{p-1}|f(v_n)|^p)^{1/p}}\,\mathrm{d} x\\
&\quad -\int_{\mathbb{R}^N}\frac{h(f(v_n))\varphi}{(1+2^{p-1}|f(v_n)|^p)^{1/p}}\,\mathrm{d} x
\end{aligned}
\end{equation}
Considering the function
$\varphi_n(x):=(1+2^{p-1}|f(v_n(x))|^p)^{1/p}f(v_n(x))$ and
using points $(3)$ and (6) in Lemma \ref{Lema f}, we obtain that
$|\varphi_n|\leq |v_n|$ and
\[
|\nabla\varphi_n
|=\left(1+\frac{2^{p-1}|f(v_n)|^p}{1+2^{p-1}|f(v_n)|^p}\right)|\nabla
v_n|\leq 2|\nabla v_n|.
\]
Thus $\|\varphi_n\|\leq 2\|v_n\|$. Taking $\varphi=\varphi_n$ in
(\ref{derivada de Gateaux}) and since $(v_n)$ is a Cerami
sequence, we conclude that
\begin{equation}
\begin{aligned}\label{baba}
&\int_{\mathbb{R}^N}\left(1+\frac{2^{p-1}|f(v_n)|^p}{1+2^{p-1}
|f(v_n)|^p}\right)|\nabla v_n|^p\,\mathrm{d} x \\
&+ \int_{\mathbb{R}^N}V(x)|f(v_n)|^p\,\mathrm{d} x
-\int_{\mathbb{R}^N}h(f(v_n))f(v_n)\,\mathrm{d} x \\
&= \langle I'(v_n),\varphi_n \rangle=o_n(1).
\end{aligned}
\end{equation}
From (\ref{morango}), (\ref{baba}) and (H2) it follows that
\begin{equation}
\begin{aligned}
\label{cocada}
&\int_{\mathbb{R}^N}\Big[\frac{1}{p}-\frac{1}{\theta}
\Big(1+\frac{2^{p-1}|f(v_n)|^p}{1+2^{p-1}|f(v_n)|^p}\Big)\Big]|
\nabla v_n|^p\,\mathrm{d} x
+ \frac{1}{2p}\int_{\mathbb{R}^N}V(x)|f(v_n)|^p\,\mathrm{d} x\\
&\leq c + o_n(1).
\end{aligned}
\end{equation}
If $\theta > 2p$, we get
\[
\Big(\frac{\theta - 2p}{p\theta}\Big)\int_{\mathbb{R}^N}|\nabla
v_n|^p\,\mathrm{d} x + \frac{1}{2p}\int_{\mathbb{R}^N}V(x)|f(v_n)|^p\,\mathrm{d} x
\leq c + o_n(1)
\]
which shows that (\ref{essencial}) holds and thus $(v_n)$ is
bounded. Now if $\theta =2p$ we deduced from (\ref{cocada})
\begin{equation}\label{boneca}
\frac{1}{2p}\int_{\mathbb{R}^N}\frac{|\nabla v_n|^p}{1+2^{p-1}|f(v_n)|^p}\,\mathrm{d} x
+ \frac{1}{2p}\int_{\mathbb{R}^N}V(x)|f(v_n)|^p\,\mathrm{d} x\leq c + o_n(1).
\end{equation}
Denoting $u_n=f(v_n)$, we have that $|\nabla v_n|^p
=(1+2^{p-1}|f(v_n)|^p)|\nabla u_n|^p$ and (\ref{boneca}) implies
that
\begin{equation}\label{caja}
\frac{1}{2p}\int_{\mathbb{R}^N}|\nabla u_n|^p\,\mathrm{d} x +
\frac{1}{2p}\int_{\mathbb{R}^N}V(x)|u_n|^p\,\mathrm{d} x\leq c + o_n(1).
\end{equation}
From (\ref{caja}) we achieved that $(u_n)$ is bounded in
$W^{1,p}(\mathbb{R}^N)$. Using the hypotheses {\rm (H0)--(H1)}, we
get
\begin{equation}\label{manga}
H(s)\leq |s|^p+C|s|^{r+1}
\end{equation}
and by Sobolev embedding $\int_{\mathbb{R}^N}H(f(v_n))\,\mathrm{d}
x=\int_{\mathbb{R}^N}H(u_n)\,\mathrm{d} x$ is bounded, where we
are supposing that the condition $(b)$ in Theorem \ref{Main2}
holds. Hence, using (\ref{morango}) we obtain (\ref{essencial}).
Thus $(v_n)$ is bounded in $W^{1,p}(\mathbb{R}^N)$ and this
concludes the proof.
\end{proof}
\subsection{Existence of nontrivial critical points for $I$}
Since $I$ has the mountain-pass geometry, we know (see, for
example, \cite{Djairo} and \cite{Ekeland}) that $I$ possesses a
Cerami sequence $(v_n)$ at the level
\[
c=\inf_{\gamma \in \Gamma} \max_{0\leq t\leq 1} I(\gamma (t))>0.
\]
By Lemma \ref{limitacao}, $(v_n)$ is bounded. Thus, we can assume
that, up to a subsequence, $v_n\rightharpoonup v$ in
$W^{1,p}(\mathbb{R}^N)$. We claim that $I'(v)=0$. Indeed, since
$C_0^{\infty}(\mathbb{R}^N)$ is dense in $W^{1,p}(\mathbb{R}^N)$,
we only need to show that $\langle I'(v),\psi\rangle=0$ for all
$\psi \in C_0^{\infty}(\mathbb{R}^N)$. Observe that
\begin{align*}
&\langle I'(v_n),\psi\rangle - \langle I'(v),\psi\rangle\\
&=\int_{\mathbb{R}^N}\left(|\nabla v_n|^{p-2}\nabla v_n - |\nabla
v|^{p-2}\nabla v\right)\nabla\psi\,\mathrm{d} x
\\
&\quad
+\int_{\mathbb{R}^N}\Big(\frac{|f(v_n)|^{p-2}f(v_n)}{(1+2^{p-1}|f(v_n)|^p)^{1/p}}-
\frac{|f(v)|^{p-2}f(v)}{(1+2^{p-1}|f(v)|^p)^{1/p}}\Big)V(x)\psi\,\mathrm{d} x
\\
&\quad
+\int_{\mathbb{R}^N}\Big(\frac{h(f(v))}{(1+2^{p-1}|f(v)|^p)^{1/p}}
-\frac{h(f(v_n))}{(1+2^{p-1}|f(v_n)|^p)^{1/p}}\Big)\psi\,\mathrm{d} x.
\end{align*}
Using the fact that $v_n \to v$ in $L_{\rm loc}^q(\mathbb{R}^N)$
for $q\in [1,p^*)$ if $1 0$,
$R>0$ and $(y_n)$ in $\mathbb{R}^N$ such that
\[
\lim_{n\to \infty}\int_{B_R(y_n)}|v_n|^p\,\mathrm{d} x\geq \alpha.
\]
\noindent {\bf Verification.} We suppose that the claim is not
true. Therefore, it holds that
\[
\lim_{n\to \infty}\sup_{y\in
\mathbb{R}^N}\int_{B_R(y)}|v_n|^p\,\mathrm{d} x=0,\quad\forall\; R>0.
\]
By \cite[Lemma I.1]{Lions}, we have $v_n \to 0$ in
$L^q(\mathbb{R}^N)$ for any $q \in (p,p^{*})$ if $1 p$
if $p=N$. From (H0)--(H1), for each $\epsilon >0$ there exists
$C_{\epsilon}>0$ such that for all $s\in \mathbb{R}$
\[
h(f(s))f(s)\leq \epsilon |f(s)|^p + C_{\epsilon}|f(s)|^{r+1}.
\]
From this estimate, using (3) and (5) in Lemma \ref{Lema f}, for
$v\in W^{1,p}(\mathbb{R}^N)$ we get
\begin{gather}
\label{e1}\int_{\mathbb{R}^N}h(f(v))f(v)\,\mathrm{d} x
\leq \epsilon \int_{\mathbb{R}^N}|v|^p\,\mathrm{d} x +
C_{\epsilon}\int_{\mathbb{R}^N}|v|^{r+1}\,\mathrm{d} x\\
\label{e2}\int_{\mathbb{R}^N}h(f(v))f(v)\,\mathrm{d} x
\leq \epsilon \int_{\mathbb{R}^N}|v|^p\,\mathrm{d} x +
C_{\epsilon}\int_{\mathbb{R}^N}|v|^{(r+1)/2}\,\mathrm{d} x.
\end{gather}
We use inequality (\ref{e1}) when $\theta=2p$ and (\ref{e2}) when
$\theta>2p$. We are going consider only the case $\theta>2p$ because
the other one is similar. By (6) in Lemma \ref{Lema f} and
(\ref{e2}) we see that for all $\epsilon >0$
\begin{align*}
\lim_{n\to \infty}\int_{\mathbb{R}^N}h(f(v_n))f'(v_n)v_n\,\mathrm{d} x
&\leq \lim_{n\to
\infty}\int_{\mathbb{R}^N}h(f(v_n))f(v_n)\,\mathrm{d} x\\
& \leq \lim_{n\to \infty}\Big(\epsilon
\int_{\mathbb{R}^N}|v_n|^p\,\mathrm{d} x +
C_{\epsilon}\int_{\mathbb{R}^N}|v_n|^{(r+1)/2}\,\mathrm{d} x\Big)\\
&\leq \epsilon \lim_{n\to \infty}
\int_{\mathbb{R}^N}|v_n|^p\,\mathrm{d} x
\end{align*}
because $(r+1)/2\in (p,p^*)$ if $1 p$ if $p=N$.
We then obtain
\begin{equation}\label{limites}
\lim_{n\to \infty}\int_{\mathbb{R}^N}h(f(v_n))f(v_n)\,\mathrm{d} x=0,\quad
\lim_{n\to \infty}\int_{\mathbb{R}^N}h(f(v_n))f'(v_n)v_n\,\mathrm{d} x=0.
\end{equation}
Since $\langle I'(v_n),v_n\rangle\to 0$, it follows that
\[
\int_{\mathbb{R}^N}|\nabla v_n|^p\,\mathrm{d} x +
\int_{\mathbb{R}^N}V(x)|f(v_n)|^{p-2}f(v_n)f'(v_n)v_n\,\mathrm{d} x\to 0.
\]
Using again (6) in Lemma \ref{Lema f} we get
\[
\int_{\mathbb{R}^N}|\nabla v_n|^p\,\mathrm{d} x +
\int_{\mathbb{R}^N}V(x)|f(v_n)|^{p}\,\mathrm{d} x\to 0.
\]
By the first limit in (\ref{limites}) and (H2), we conclude that
\[
\lim_{n\to \infty}\int_{\mathbb{R}^N}H(f(v_n))\,\mathrm{d} x=0.
\]
This implies that $I(v_n)\to 0$ in contradiction with
the fact that $I(v_n)\to c>0$ and the Claim is proved.
\smallskip
Now, define $\widetilde{v}_n(x)=v_n(x+y_n)$. As $(v_n)$ is a Cerami sequence
for $J_{\infty}$, it is not difficult to see that
$\widetilde{v}_n$ is also a Cerami sequence for $J_{\infty}$.
Proceeding as in the case of $(v_n)$, up to a subsequence, we
obtain $\widetilde{v}_n\rightharpoonup \widetilde{v}$ with
$J_{\infty}'(\widetilde{v})=0$. As $\widetilde{v}_n\to
\widetilde{v}$ in $L^p(B_R)$, by Claim 1 we conclude that
\[
\int_{B_R}|\widetilde{v}|^p\,\mathrm{d} x=\lim_{n\to
\infty}\int_{B_R}|\widetilde{v}_n|^p\,\mathrm{d} x=\lim_{n\to
\infty}\int_{B_R(y_n)}|v_n|^p\,\mathrm{d} x\geq \alpha.
\]
what implies that $\widetilde{v}\neq 0$.\\
By (6) in Lemma \ref{Lema f}, for all $n$ we obtain
\[
f^2(\widetilde{v}_n)-f(\widetilde{v}_n)f'(\widetilde{v}_n)\widetilde{v}_n\geq0
\]
which implies
\[
|f(\widetilde{v}_n)|^p-|f(\widetilde{v}_n)|^{p-2}f(\widetilde{v}_n)f'(\widetilde{v}_n)\widetilde{v}_n\geq
0.
\]
Furthermore, from the condition (H2) we conclude for all $n$
that
\[
\frac{1}{p}h(f(\widetilde{v}_n))f'(\widetilde{v}_n)\widetilde{v}_n-H(f(\widetilde{v}_n))\geq
\frac{1}{2p}h(f(\widetilde{v}_n))f(\widetilde{v}_n)-H(f(\widetilde{v}_n))\geq
0.
\]
Thus, from Fatou's lemma and since $\widetilde{v}_n$ is a Cerami
sequence for $J_{\infty}$, we obtain
\begin{align*}
c&=\lim_{n\to \infty}\left[J_{\infty}(\widetilde{v}_n)-\frac{1}{p}\langle J_{\infty}'(\widetilde{v}_n),\widetilde{v}_n\rangle\right]\\
&=\limsup_{n\to
\infty}\frac{1}{p}\int_{\mathbb{R}^N}V_{\infty}\left[|f(\widetilde{v}_n)|^p-|f(\widetilde{v}_n)|^{p-2}f(\widetilde{v}_n)f'(\widetilde{v}_n)\widetilde{v}_n\right]\,\mathrm{d} x\\
&\quad +\limsup_{n\to
\infty}\int_{\mathbb{R}^N}\left[\frac{1}{p}h(f(\widetilde{v}_n))f'(\widetilde{v}_n)\widetilde{v}_n-H(f(\widetilde{v}_n))\right]\,\mathrm{d} x\\
&\geq\frac{1}{p}\int_{\mathbb{R}^N}V_{\infty}\left[|f(\widetilde{v})|^p-|f(\widetilde{v})|^{p-2}f(\widetilde{v})f'(\widetilde{v})\widetilde{v}\right]\,\mathrm{d} x\\
&\quad +\int_{\mathbb{R}^N}\left[\frac{1}{p}h(f(\widetilde{v}))f'(\widetilde{v})\widetilde{v}-H(f(\widetilde{v}))\right]\,\mathrm{d} x\\
&=
J_{\infty}(\widetilde{v})-\frac{1}{p}\langle J_{\infty}'(\widetilde{v}),\widetilde{v}\rangle=J_{\infty}(\widetilde{v}).
\end{align*}
Therefore, $\widetilde{v}\neq 0$ is a critical point of
$J_{\infty}$ satisfying $J_{\infty}(\widetilde{v})\leq c$. We
deduce that the least energy level $m_{\infty}$ for $J_{\infty}$
satisfies $m_{\infty}\leq c$. We denote by $\widetilde{w}$ a least
energy solution of the equation $-\Delta_pv=g_{\infty}(v)$ (see
Remark \ref{crescimento}). Now applying Theorem \ref{Medeiros2} to
the functional $J_{\infty}$ we can find a path $\gamma \in
C([0,1],W^{1,p}(\mathbb{R}^N))$ such that $\gamma(0)=0$,
$J_{\infty}(\gamma (1))<0$, $\widetilde{w}\in \gamma([0,1])$ and
\[
\max_{t\in [0,1]}J_{\infty}(\gamma(t))=J_{\infty}(\widetilde{w}).
\]
We can assume that $V\not\equiv V_{\infty}$ in (V2), otherwise
there is nothing to prove. Thus
\[
I(\gamma(t))