\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 $10$ and putting $\psi(t,x)=\exp(-iFt)u(x),\quad F\in \mathbb{R},$ we obtain a corresponding equation $$\label{simp} -\Delta u -\Delta (u^2)u +V(x)u= h(u)\quad\mbox{in }\mathbb{R}^N\/extract_tex] 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 \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} 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 $10$ 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 $2p-12p-1$ if $p=N$; \item[(H2)] There exists $\theta \geq 2p$ such that $0<\theta H(s)\leq sh(s)$ for all $s>0$ where $H(s)=\int_0^s h(t)\textrm{d}t$. \end{enumerate} The following theorem contains our main result. \begin{theorem}\label{Main2} Let $12p$; \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 $10$. \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 $w\in W^{1,p}(\mathbb{R}^N)$ $$\label{solucao fraca mod} \int_{\mathbb{R}^N}|\nabla v|^{p-2}\nabla v\nabla w\,\mathrm{d} x =\int_{\mathbb{R}^N}g(x,f(v))f'(v)w\,\mathrm{d} x.$$ Since $(f^{-1})'(t)=\frac{1}{f'(f^{-1}(t))}$, it follows that $$\label{derivada} (f^{-1})'(t)=(1+2^{p-1}|f(f^{-1}(t))|^p)^{1/p}=(1+2^{p-1}|t|^p)^{1/p}$$ which implies that $$\label{perna longa} \nabla v=(f^{-1})'(u)\nabla u=(1+2^{p-1}|u|^p)^{1/p}\nabla u.$$ 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 $$\label{ligeirinho} \nabla(f'(v)^{-1}\varphi)=2^{p-1}(1+2^{p-1}|u|^p)^{(1-p)/p}|u|^{p-2} u\varphi\nabla u +(1+2^{p-1}|u|^p)^{1/p}\nabla \varphi$$ 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 $$\label{Medeiros} -\Delta_p v=k(v)\quad \mbox{in } \mathbb{R}^N.$$ 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 $10$ 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 $10$ such that $K(\zeta)>0$. \end{enumerate} Let $$\label{m} m:=\inf\{\mathcal{F}(v): v\in W^{1,p}(\mathbb{R}^N)\setminus \{0\}\mbox{ is a solution of (\ref{Medeiros})}\}.$$ 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 $10,\ \delta>0$ 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 $$\label{coercividade de I} I(v)\geq J_0(v)\geq \beta_0 \| v \|^{p}\quad \mbox{if}\quad \| v\| \leq \delta_0.$$ 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 $$\label{essencial} \int_{\mathbb{R}^N}|\nabla v_n|^p\,\mathrm{d} x +\int_{\mathbb{R}^N}V(x)|f(v_n)|^p\,\mathrm{d} x\leq C$$ 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 $$\label{morango} \frac{1}{p}\int_{\mathbb{R}^N}|\nabla v_n|^p\,\mathrm{d} x + \frac{1}{p}\int_{\mathbb{R}^N}V(x)|f(v_n)|^p\,\mathrm{d} x -\int_{\mathbb{R}^N}H(f(v_n))\,\mathrm{d} x=c+o_n(1)$$ and for all $\varphi \in W^{1,p}(\mathbb{R}^N)$ \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} 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{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} From (\ref{morango}), (\ref{baba}) and (H2) it follows that \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} 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}) $$\label{boneca} \frac{1}{2p}\int_{\mathbb{R}^N}\frac{|\nabla v_n|^p}{1+2^{p-1}|f(v_n)|^p}\,\mathrm{d} x + \frac{1}{2p}\int_{\mathbb{R}^N}V(x)|f(v_n)|^p\,\mathrm{d} x\leq c + o_n(1).$$ 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 $$\label{caja} \frac{1}{2p}\int_{\mathbb{R}^N}|\nabla u_n|^p\,\mathrm{d} x + \frac{1}{2p}\int_{\mathbb{R}^N}V(x)|u_n|^p\,\mathrm{d} x\leq c + o_n(1).$$ 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 $$\label{manga} H(s)\leq |s|^p+C|s|^{r+1}$$ 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 $10$, $R>0$ 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 $1p$ 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 $1p$ if $p=N$. We then obtain $$\label{limites} \lim_{n\to \infty}\int_{\mathbb{R}^N}h(f(v_n))f(v_n)\,\mathrm{d} x=0,\quad \lim_{n\to \infty}\int_{\mathbb{R}^N}h(f(v_n))f'(v_n)v_n\,\mathrm{d} x=0.$$ 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))0 we define \begin{gather*} v_k= \begin{cases} v & \hbox{if } v\leq k \\ k & \hbox{if } v\geq k, \end{cases} \\ \vartheta_k=v_k^{p(\beta-1)}v,\quad w_k=vv_k^{\beta-1} \end{gather*} with \beta>1 to be determined later. Taking \vartheta_k as a test function in (\ref{fraco}), using that \[ h(f(v))\leq \frac{V_0}{2}f(v)+Cf(v)^{r},$ and condition (V1) we obtain \begin{align*} &\int_{\mathbb{R}^N}v_k^{p(\beta-1)}|\nabla v|^p\,\mathrm{d} x +p(\beta-1)\int_{\mathbb{R}^N}v_k^{p(\beta-1)-1}v \nabla v_k\nabla v\,\mathrm{d} x\\ &\leq C\int_{\mathbb{R}^N}f(v)^rf'(v)v v_k^{p(\beta-1)}\,\mathrm{d} x. \end{align*} Because the second summand in the left side of the inequality above is not negative and using (5) and (6) in Lemma \ref{Lema f} we see that $$\label{ouro} \int_{\mathbb{R}^N}v_k^{p(\beta-1)}|\nabla v|^p\,\mathrm{d} x\leq C\int_{\mathbb{R}^N}v^{(r+1)/2} v_k^{p(\beta-1)}\,\mathrm{d} x= C\int_{\mathbb{R}^N}v^{\widetilde{r}-p}w_k^p\,\mathrm{d} x$$ where $\widetilde{r}:= (r+1)/2$. By the Gagliardo-Nirenberg inequality and (\ref{ouro}), we obtain \begin{align*} \Big(\int_{\mathbb{R}^N}w_k^{p^*}\,\mathrm{d} x\Big)^{p/p^*} &\leq C_1\int_{\mathbb{R}^N}|\nabla w_k|^p\,\mathrm{d} x\\ &\leq C_2\int_{\mathbb{R}^N}v_k^{p(\beta-1)}|\nabla v|^p\,\mathrm{d} x+C_3(\beta-1)^p\int_{\mathbb{R}^N}v^pv_k^{p(\beta-2)}|\nabla v_k|^p\,\mathrm{d} x\\ &\leq C_4\beta^p\int_{\mathbb{R}^N}v_k^{p(\beta-1)}|\nabla v|^p\,\mathrm{d} x\\ &\leq C_5\beta^p\int_{\mathbb{R}^N}v^{\widetilde{r}-p}w_k^p\,\mathrm{d} x, \end{align*} where we have used that $v_k\leq v$, $1\leq \beta^p$ and $(\beta-1)^p\leq\beta^p$. Using the H\"{o}lder inequality, $\Big(\int_{\mathbb{R}^N}w_k^{p^*}\,\mathrm{d} x\Big)^{p/p^*} \leq\beta^pC_5\Big(\int_{\mathbb{R}^N}v^{p^*}\,\mathrm{d} x\Big)^{(\widetilde{r}-p)/p^*} \Big(\int_{\mathbb{R}^N}w_k^{pp^*/(p^*-\widetilde{r}+p)}\,\mathrm{d} x\Big)^{(p^*-\widetilde{r}+p)/p^*}.$ Since that $|w_k|\leq|u|^\beta$, by the continuity of the embedding $W^{1,p}(\mathbb{R}^N)\hookrightarrow L^{p^*}(\mathbb{R}^N)$ we get $$\Big(\int_{\mathbb{R}^N}|vv_k^{\beta-1}|^{p^*}\,\mathrm{d} x\Big)^{p/p^*} \leq\beta^pC_6\|v\|^{\widetilde{r}-p} \Big(\int_{\mathbb{R}^N}v^{\beta pp^*/(p^*-\widetilde{r}+p)}\,\mathrm{d} x\Big)^{(p^*-\widetilde{r}+p)/p^*}.$$ Choosing $\beta=1+(p^*-\widetilde{r})/p$ we have $\beta pp^*/(p^*-\widetilde{r}+p)=p^*$. Thus, $$\Big(\int_{\mathbb{R}^N}|vv_k^{\beta-1}|^{p^*}\,\mathrm{d} x\Big)^{p/p^*} \leq\beta^pC_6\|v\|^{\widetilde{r}-p}\|v\|_{\beta\alpha^*}^{p\beta},$$ where $\alpha^*=pp^*/(p^*-\widetilde{r}+p)$. By the Fatou's lemma, $$\label{inf} \|v\|_{\beta p^*}\leq(\beta^pC_6\|v\|^{\widetilde{r}-p})^{1/p\beta}\|v\|_{\beta\alpha^*}.$$ For each $m=0,1,2,\dots$. let us define $\beta_{m+1}\alpha^*:= p^*\beta_m$ with $\beta_0:=\beta$. Using the previous argument for $\beta_1$, by (\ref{inf}) we have \begin{align*} \|u\|_{\beta_1p^*} &\leq(\beta_1^pC_6\|u\|^{\widetilde{r}-p})^{1/p\beta_1}\|u\|_{\beta_1\alpha^*}\\ &\leq(\beta_1^pC_6\|u\|^{\widetilde{r}-p})^{1/p\beta_1}(\beta^pC_6\|u\|^{r-p})^{1/p\beta} \|u\|_{\beta\alpha^*}\\ &\leq(C_6\|u\|^{\widetilde{r}-p})^{1/p\beta+1/p\beta_1}(\beta)^{1/\beta}(\beta_1)^{1/\beta_1} \|u\|_{p*}. \end{align*} Observing that $\beta_m=\chi^m\beta$ where $\chi =p^*/\alpha^*$, by iteration we obtain $\|u\|_{\beta_mp^*}\leq(C_6\|u\|^{\widetilde{r}-p})^{1/p \beta\sum_{i=0}^{m}\chi^{-i}}\beta^{1/\beta \sum_{i=0}^{m}\chi^{-i}}\chi^{1/\beta\sum_{i=0}^{m}i\chi^{-i}} \|u\|_{p*}.$ Since $\chi>1$ and $\lim_{m\to\infty}{1/(p\beta)\sum_{i=0}^{m}\chi^{-i}}=1/(p^*-\widetilde{r})$, we can take the limit as $m\to\infty$ to conclude that $v\in L^{\infty}(\mathbb{R}^N)$ and $\|v\|_\infty\leq C_7\|v\|^{(p^*-p)/(p^*-\widetilde{r})}.$ In the case $p=N$, by using Theorem 1 in \cite{Serrin}, we can conclude that $v$ is locally bounded in $\mathbb{R}^N$. Resuming, in both cases, as a consequence of a result due to Tolksdorf \cite{tolksdorf}, we obtain that $v\in C_{\rm loc}^{1,\alpha}(\mathbb{R}^N)$, $\alpha\in (0,1)$. Next, for \$1