\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 61, pp. 1--23.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/61\hfil Existence and concentration of positive solutions] {Existence and concentration of positive solutions for a quasilinear elliptic equation in $\mathbb{R}$} \author[E. Gloss\hfil EJDE-2010/61\hfilneg] {Elisandra Gloss} \address{Departamento de Matem\'atica\\ Universidade Federal da Para\'iba\\ 58000-000, Jo\~ao Pessoa--PB, Brazil} \email{elisandra@mat.ufpb.br} \thanks{Submitted January 16, 2010. Published May 5, 2010.} \thanks{Supported by a grant from CAPES/Brazil} \subjclass[2000]{35J20, 35J62} \keywords{Schr\"odinger equation; quasilinear equation; concentration; \hfill\break\indent variational methods} \begin{abstract} We study the existence and concentration of positive solutions for the quasilinear elliptic equation $$-\varepsilon^2u'' -\varepsilon^2(u^2)''u+V(x) u = h(u)$$ in $\mathbb{R}$ as $\varepsilon\to 0$, where the potential $V:\mathbb{R}\to \mathbb{R}$ has a positive infimum and $\inf_{\partial \Omega}V>\inf_{ \Omega}V$ for some bounded domain $\Omega$ in $\mathbb{R}$, and $h$ is a nonlinearity without having growth conditions such as Ambrosetti-Rabinowitz. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In this article, we consider the quasilinear elliptic equation $$\label{p1} -\varepsilon^2 u''-\varepsilon^2 (u^2)''u + V(x)u=h(u)\quad\text{in }\mathbb{R}$$ where $\varepsilon>0$ is a small real parameter. Here our goal is to prove, by a variational approach, the existence and concentration of positive weak solutions. We say that $u\in H^1(\mathbb{R})$ is a (weak) solution of \eqref{p1} if \begin{align*}%\label{wsol} &\varepsilon^2\int_{\mathbb{R}^N}(1+2u^2)u'\varphi'\,\mathrm{d} x + 2\varepsilon^2\int_{\mathbb{R}^N}|u'|^2 u\varphi\,\mathrm{d} x +\int_{\mathbb{R}^N}V(x)u\varphi\,\mathrm{d} x\\ &= \int_{\mathbb{R}^N}h(u)\varphi\,\mathrm{d} x\quad\text{for all } \varphi\in C_c^{\infty}(\mathbb{R}). \end{align*} Solutions of equations like \eqref{p1} are related with existence of standing wave solutions for quasilinear equations of the form $$\label{qse} i\frac{\partial \psi}{\partial t}=-\varepsilon^2 \psi''+W(x)\psi-\eta(|\psi|^2)\psi-\varepsilon^2 \kappa[\rho(|\psi|^2)]''\rho'(|\psi|^2)\psi$$ where $\psi:\mathbb{R}\times\mathbb{R}\to\mathbb {C}$, $\kappa$ is a positive constant, $W:\mathbb{R}\to \mathbb{R}$ is a given potential and $\eta, \rho:\mathbb{R}^+\to \mathbb{R}$ are suitable functions. Quasilinear equations of the form \eqref{qse} arise in several areas of physics in correspondence to different type of functions $\rho$. For physical motivations and developing of the physical aspects we refer to \cite{POSCWA} and references therein. Here we consider the case where $\rho(s)=s$. Looking for standing wave solutions of \eqref{qse} we set $\psi(t,x)=e^{-i\xi t}u(x)$, where $\xi\in\mathbb{R}$ and $u>0$ is a real function. So one obtains a corresponding equation of elliptic type which has the formal variational structure given by \eqref{p1}, where without loss of generality we set $\kappa=1$. Motivated by the physical aspects, equation \eqref{p1} has recently attracted a lot of attention and existence results have been obtained in the case of a bounded potential $V(x)$ or in the coercive case. Direct variational methods by using constrained minimization arguments were used in \cite{POSCWA} to provide existence of positive solutions up to an unknown Lagrange multiplier. The authors study the following problem $$\label{la} -u''+ V(x)u - (u^2)''u = \theta |u|^{p-1}u,\quad x\in \mathbb{R}.$$ Ambrosetti and Wang in \cite{AMWA}, by using variational methods, proved the existence of positive solutions for the following class of quasilinear elliptic equations $-u''+ (1+\varepsilon a(x))u - (1+\varepsilon b(x))(u^2)''u = (1+\varepsilon c(x))u^p,\quad u\in H^1(\mathbb{R})$ for $p>1$ and $\varepsilon>0$ sufficiently small, where $a(x),\, b(x)$ and $c(x)$ are real functions satisfying certain hypotheses. Subsequently a general existence result for \eqref{p1} was derived in \cite{LIWAWA}. In this paper, which deals also with higher dimensions, to overcome the undefiniteness of natural functional associated to the equation the idea is to introduce a change of variable and to rewrite the functional with this new variable which turns the problem into finding solutions of an auxiliary semilinear equation. Then critical points are search in an associated Orlicz space and existence results are given in the case of bounded, coercive or radial potentials. Following the strategy developed in \cite{C} on a related problem the authors in \cite{COJE2004} also make use of a change of unknown and define an associated equation that they call dual. A simple and shorter proof of the results in \cite{LIWAWA} is presented for bounded potentials, which does not use Orlicz spaces and permit to cover a different class of nonlinearities. We observe that this change of variables is not necessary in dimension one because in this case the functional associated is well defined. We mention some works that study problem \eqref{p1} without make this change of variables \cite{ALCAMI07}, \cite{ALMISO} and \cite{SE}. In \cite{ALCAMI07} and \cite{SE} the authors study \eqref{la} for p-laplacian or more general operator and $\theta=1$. In \cite{ALMISO} the authors study existence and concentration of positive solutions for equation \eqref{p1} with $h(t)=t^p$, $p\geq3$. There the potential $V:\mathbb{R}\to \mathbb{R}$ is a continuous function satisfying the following conditions: \begin{itemize} \item[(V1)] $V$ is bounded from below by a positive constant; that is, $\inf_{x\in\mathbb{R}}V(x)=V_0>0;$ \item[(V2)] there exists a bounded domain $\Omega$ in $\mathbb{R}$ such that $m\equiv\inf_{x\in\Omega}V(x)<\inf_{x\in\partial\Omega}V(x).$ \end{itemize} We should also mention that equation \eqref{p1} has been also considered in $\mathbb{R}^N$ for $N\geq2$, we refer the reader to the works of \cite{CAdoOMO2009, C, COJE2004, deMoraes2, LIWAWA} among others and references therein. Here we also assume that $V\in\mathcal{C}(\mathbb{R},\mathbb{R})$ satisfies the assumptions (V1)-(V2). Hereafter we use the following notation: $\mathcal{M}\equiv\{x\in\Omega : V(x)=m\}$ and without loss of generality we may assume that $0\in\mathcal M$. We emphasize that besides the local condition (V2), introduced in \cite{PIFE96} and so far well known for semilinear elliptic problems, we do not require any global condition other than (V1). We also suppose that $h:\mathbb{R}_+\to \mathbb{R}$ is a locally Lipschitz continuous function satisfying: \begin{itemize} \item[(H1)] $\lim_{t\to 0^+}{h(t)}/{t}=0;$ \item[(H2)] there exists $T>0$ such that $h(T)>mT,\quad H(T)=\frac{m}{2}T^2,\quad H(t)< \frac{m}{2}t^2\quad\text{for all } t\in(0,T)$ where $H(t)=\int_0^th(s)\,\mathrm{d} s$. \end{itemize} Similar hypothesis on the nonlinearity were used in \cite{BYJETA2008} for the semilinear case. Following the strategy developed there, using variational methods, we shall prove existence and concentration of positive solutions for \eqref{p1} without assuming Ambrosetti-Rabinowitz and monotonicity conditions on $h$. In particular we improve the results in \cite{ALMISO} where $h$ is a pure power. Next we state our main result. \begin{theorem}\label{teo1} Suppose that {\rm (V1)--(V), (H1)-(H2)} hold. Then there exists $\varepsilon_0>0$ such that \eqref{p1} has a positive solution $u_\varepsilon\in C^{1,\alpha}_{\rm loc}(\mathbb{R})$ for all $0<\varepsilon<\varepsilon_0$, satisfying the following: \begin{itemize} \item[(i)] $u_\varepsilon$ admits a maximum point $x_\varepsilon$ such that $\lim_{\varepsilon\to0}\mathop{\rm dist}(x_\varepsilon,\mathcal{M})=0$ and for any sequence $\varepsilon_n\to0$ there exist $x_0\in\mathcal M$ and a solution $u_0$ of $$\label{p3} - u'' -(u^2)''u+ mu=h(u),\quad u>0,\quad u\in H^1(\mathbb{R})$$ such that, up to subsequences, $x_{\varepsilon_n}\to x_0\quad\text{and}\quad u_{\varepsilon_n}(\varepsilon_n\cdot+x_{\varepsilon_n})\to u_0\quad \text{in } H^1(\mathbb{R}) \text{ as} \quad n\to\infty.$ \item[(ii)] There exist positive constants $C$ and $\zeta$ such that $u_\varepsilon(x)\leq C \exp \big(-\frac{\zeta}{\varepsilon}(|x-x_\varepsilon|)\big)\quad \text{for all } x\in\mathbb{R}.$ \end{itemize} \end{theorem} The proof of this theorem relies on the study of a semilinear equation obtained after making the chance of variables introduced in \cite{LIWAWA}. In order to prove existence of solutions for this equation we study some properties of the least energy solutions for a limit equation obtained from \eqref{p3} by the same change of variables. Using these properties, after some technical lemmata, we can find a bounded Palais-Smale sequence in a suitable space for the associated functional. Thus we obtain a solution for the semilinear equation which gives us a solution for the original problem \eqref{p1}. This paper is organized as follows: In Section $2$ we a change of variables and study some properties of the functional, $J_\varepsilon$, associated to the new semilinear equation obtained from \eqref{p1}, and of the space where it is defined. Section $3$ is devoted to prove that the mountain pass level of $J_\varepsilon$ is well defined and converges to the least energy level of the functional associated to the limit problem. In Section $4$ we prove the existence of a nontrivial critical point for $J_\varepsilon$ and finally Section $5$ brings the results that complete the proof of Theorem \ref{teo1}. \section{Preliminaries results} Since we are looking for positive solutions we define $h(t)=0$ for $t<0$. Observe that defining $v(x)=u(\varepsilon x)$ equation \eqref{p1} becomes equivalent to $$\label{p2} -v''-(v^2)''v+V(\varepsilon x)v=h(v),\quad v>0\text{ in } \mathbb{R}.$$ The natural energy functional associated with \eqref{p2}, namely $I_\varepsilon(v)=\frac{1}{2}\int_{\mathbb{R}} [(1+2v^2)|v^{\prime}|^2+V(\varepsilon x)v^2]\,\mathrm{d} x -\int_{\mathbb{R}}H(v)\,\mathrm{d} x,$ is well defined on $H_\varepsilon:=\big\{v\in H^1(\mathbb{R}):\int_{\mathbb{R}}V(\varepsilon x)v^2\,\mathrm{d} x<\infty\big\}$ due the imbedding $H^1(\mathbb{R})\hookrightarrow L^\infty(\mathbb{R})$ and (V1). Despite this, following the strategy developed in \cite{CAdoOMO2009}, \cite{COJE2004}, \cite{doOSE} and \cite{LIWAWA} on a related problem for higher dimensions, we introduce a change of variables $u=f^{-1}(v)$ where $f$ is a $C^\infty$ function defined by $f'(t)=\left(1+2f^2(t)\right)^{-1/2}\quad\text{if } t>0,\quad f(0)=0,\quad\text{and}\quad f(t)=-f(-t)\quad\text{if } t<0.$ This change of variables allows us to consider more general nonlinearities. To make easier the reference we list here some properties of $f(t)$ whose proofs can be found in \cite[Lemma 2.1]{doOSE} (see also \cite{COJE2004} and \cite{LIWAWA}). The proof of the last item is found in \cite{deMoraes2}. \begin{lemma} \label{lema f} The function $f(t)$ satisfies: \begin{itemize} \item[(1)] $f$ is $C^{\infty}$, invertible and uniquely defined; \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)] $f(t)/{\sqrt t}\to2^{1/4}$ as $t\to +\infty$; \item[(6)] $f(t)/2\leq tf'(t)\leq f(t)$ for all $t\geq 0$; \item[(7)] $|f(t)|\leq 2^{1/4}|t|^{1/2}$ for all $t\in \mathbb{R}$; \item[(8)] The function $f^2(t)$ is strictly convex; \item[(9)] There exists a positive constant $C$ such that $|f(t)| \geq \begin{cases} C|t|, & |t| \leq 1 \\ C|t|^{1/2}, & |t| \geq 1; \end{cases}$ \item[(10)] $|f(t)f'(t)|\leq 1/\sqrt{2}$ for all $t\in \mathbb{R}$; \item[(11)] For each $\lambda>1$ we have $f^2(\lambda t)\leq \lambda^2 f^2(t)$ for all $t\in\mathbb{R}$. \end{itemize} \end{lemma} After this change of variable from $I_\varepsilon$ we obtain a new functional $P_\varepsilon(u)=I_\varepsilon(f(u))=\frac{1}{2} \int_{\mathbb{R}}[|u^{\prime}|^2+V(\varepsilon x)f^2(u)]\,\mathrm{d} x-\int_{\mathbb{R}}H(f(u))\,\mathrm{d} x,$ which is well defined on $E_{\varepsilon}:=\big\{u\in H^1(\mathbb{R}):\int_{\mathbb{R}}V(\varepsilon x)f^2(u)\,\mathrm{d} x<\infty\big\}.$ Using the properties of $f(t)$ we can see that $E_\varepsilon$ is a normed space with norm $$\label{norma} \|u\|_\varepsilon:=\|u^{\prime}\|_2 +\inf_{\lambda>0}\lambda\big\{1+\int_{\mathbb{R}}V(\varepsilon x)f^2(\lambda^{-1}u)\,\mathrm{d} x\big\}:=\|u'\|_2+|\|u\||_\varepsilon.$$ The following proposition is crucial to prove convergence results. \begin{proposition}\label{despri} There exists $C>0$ independent of $\varepsilon>0$ such that $$\label{desigualdade primordial} {\int_{\mathbb{R}} V(\varepsilon x)f^2(u)\,\mathrm{d} x}\leq C|\|u\||_\varepsilon {\Big[1+\Big(\int_{\mathbb{R}} V(\varepsilon x)f^2(u)\,\mathrm{d} x\Big)^{1/2}\Big]}$$ for all $u\in E_\varepsilon$. \end{proposition} The proof of the above proposition is the same as in \cite[Proposition 2.1]{doOSE}, since the constant $C$ that appearing there depends only on $f$. From this result we obtain that $E_\varepsilon$ is a Banach space and the embedding $E_\varepsilon\hookrightarrow H^1(\mathbb{R})$ is continuous. Also can be proved that the space $\mathcal{C}_c^{\infty}(\mathbb{R})$ is dense in $E_\varepsilon$ (see \cite{CAdoOMO2009}, \cite{doOMOSE}, \cite{doOSE} and \cite{LIWAWA} for details). Moreover due to the imbedding $H^1(\mathbb{R})\hookrightarrow L^\infty(\mathbb{R})$ we can see that the functional $P_\varepsilon$ is of class $\mathcal{C}^1$ on $E_\varepsilon$. This does not occurs in general for higher dimensions. For $N\geq2$ some regularity results can be found in \cite{CAdoOMO2009, doOMOSE, doOSE} where the authors prove that $P_\varepsilon$ is continuous in $E_\varepsilon$ and G\^ateaux differentiable with derivative given by $\langle P_\varepsilon'(u),\varphi\rangle = \int_{\mathbb{R}^N}\nabla u\nabla\varphi \,\mathrm{d} x +\int_{\mathbb{R}^N}f'(u)\left[V(\varepsilon x)f(u)-h(f(u))\right]\varphi \,\mathrm{d} x.$ They also prove that $P_\varepsilon'$ is continuous from the norm topology of $E_\varepsilon$ to the weak-* topology of $E_\varepsilon'$; i.e., if $u_n\to u$ strongly in $E_\varepsilon$ then $$\langle P_\varepsilon'(u_n),\varphi\rangle \to \langle P_\varepsilon'(u),\varphi\rangle\quad\text{for each } \varphi\in E_\varepsilon.$$ In our case, for $N=1$, we have $P_\varepsilon$ of class $\mathcal{C}^1$ and for each $\varphi\in E_\varepsilon$ it holds $\langle P_\varepsilon'(u),\varphi\rangle = \int_{\mathbb{R}} u'\varphi' \;\mathrm{d} x +\int_{\mathbb{R}}f'(u)\left[V(\varepsilon x)f(u)-h(f(u))\right]\varphi \,\mathrm{d} x.$ We observe that nontrivial critical points for $P_\varepsilon$ are weak solutions for $$\label{p4} -u''=f'(u)\left[h(f(u))-V(\varepsilon x)f(u)\right]\quad \text{in } \mathbb{R}.$$ In Proposition \ref{relation} below we relate the solutions of \eqref{p4} to the solutions of \eqref{p2}. From now on, for any set $A\subset\mathbb{R}$ and $\varepsilon>0$, we define $A_\varepsilon\equiv\{x\in\mathbb{R}: \varepsilon x\in A\}$. We define $$\chi_\varepsilon(x)= \begin{cases} 0 & \text{if } x \in \Omega_\varepsilon\\ \varepsilon^{-1} & \text{if } x \notin \Omega_\varepsilon, \end{cases}$$ and $\label{eq5} Q_\varepsilon (u)=\Big(\int_{\mathbb{R}} \chi_\varepsilon(x)u^2\,\mathrm{d} x-1 \Big)^{2}_+.$ The functional $Q_\varepsilon:H^1(\mathbb{R})\to\mathbb{R}$ is of class $\mathcal{C}^1$ with Frechet derivative given by $\langle Q_\varepsilon'(u),\varphi\rangle= {4}\Big(\int_{\mathbb{R}}\chi_\varepsilon(x) u^2\,\mathrm{d} x-1\Big)_+ \int_{\mathbb{R}}\chi_\varepsilon(x) u\varphi\,\mathrm{d} x.$ It will act as a penalization to force the concentration phenomena to occur inside $\Omega$. This type of penalization was first introduced in \cite{BYWA2003} for the semilinear case in $\mathbb{R}^N$ with $N\geq2$. Finally let $J_\varepsilon: E_\varepsilon\to\mathbb{R}$ be given by $$J_\varepsilon (u)= P_\varepsilon (u)+Q_\varepsilon(u).$$ The next proposition relates solutions of \eqref{p2} and \eqref{p4}. \begin{proposition} \label{relation} \begin{itemize} \item[(i)] If $u \in E_\varepsilon$ is a critical point of $P_\varepsilon$ then $v=f(u) \in E_\varepsilon$ is a weak solution of \eqref{p2}; \item[(ii)] If $u$ is a classical solution of \eqref{p4} then $v=f(u)$ is a classical solution of \eqref{p2}. \end{itemize} \end{proposition} \begin{proof} The second claim was proved in \cite{COJE2004} and to prove (i) we follow the same idea. If $v=f(u)$ by Lemma \ref{lema f} we have $|v|\leq|u|$ and $|v'|=f'(u)|u'|\leq |u'|$ which imply $v\in E_\varepsilon$. Since $u$ is a critical point for $P_\varepsilon$, $u$ is a weak solution for \eqref{p4}. So $$\label{ws} \int_{\mathbb{R}}u'\varphi'\;\mathrm{d} x=\int_{\mathbb{R}}f'(u)\left[h(f(u)) -V(\varepsilon x)f(u)\right]\varphi\;\mathrm{d} x \quad\text{for all } \varphi\in E_\varepsilon.$$ Since $(f^{-1})'(t)=[f'(f^{-1}(t))]^{-1}$, it follows that $(f^{-1})'(t)=\left[1+2f^2(f^{-1}(t))\right]^{1/2}=(1+2t^2)^{1/2},\quad (f^{-1})''(t)=\frac{2t}{(1+2t^2)^{1/2}}$ which yields $$u'=(f^{-1})'(v) v'= (1+2v^2)^{1/2}v'.$$ For each $\psi\in \mathcal{C}^\infty_c(\mathbb{R})$ we have $\varphi:=(f'(u))^{-1}\psi=(f^{-1})'(v)\psi\in E_\varepsilon$ with $\varphi'=\frac{2v\psi}{(1+2v^2)^{1/2}} v'+(1+2v^2)^{1/2}\psi'.$ Hence by \eqref{ws} we obtain $\int_{\mathbb{R}}\left[2|v'|^2v\psi+(1+2v^2)^{1/2}v'\psi\right]\mathrm{d} x= \int_{\mathbb{R}}\left[h(v)-V(\varepsilon x)v\right]\psi\;\mathrm{d} x$ and concludes the proof of $(i)$. \end{proof} Following this result, to prove existence of solutions for \eqref{p1}, we shall look for critical points to $J_\varepsilon$ for which ones $Q_\varepsilon$ is zero. Initially we will study the limiting problem \eqref{p3}. \subsection{The limiting problem} In this subsection we shall study some properties of the solutions of \eqref{p3}, namely $-v''-(v^2)''v+mv=h(v),\quad v>0 \quad\text{in } \mathbb{R}.$ Using the same change of variables $f$, we will do it dealing with classical solutions for the problem $$\label{p3l} -u''=g(u),\quad\lim_{|x|\to\infty}u(x)=0,\quad u(x_0)>0\quad \text{for some }x_0\in\mathbb{R},$$ where $g(t)=f'(t)[h(f(t))-mf(t)]$ for $t\geq0$ and $g(t)=-g(-t)$ for $t<0$. Like in Proposition \ref{relation} we see that if $u\in H^1(\mathbb{R})$ is a classical solution of \eqref{p3l} then $v=f(u)$ is a classical solution for \eqref{p3}. From assumptions on $h$ and Lemma \ref{lema f} we can see that the function $g(t)$ is locally Lipschitz continuous and satisfies: \begin{itemize} \item[(G1)] $\lim_{t\to0}g(t)/t=-m<0$; \item[(G2)] for $\tilde T=f^{-1}(T)$ and $G(t)=\int_0^tg(s)\,\mathrm{d} s$ it holds $\tilde T>0$ and $$\label{ttil} G(\tilde T)=0,\quad g(\tilde T)>0,\quad G(t)<0\quad\text{for all } t\in(0,\tilde T).$$ \end{itemize} In \cite[Theorem 5]{BELI83}, the authors prove that \eqref{ttil} is a necessary and sufficient condition for the existence of a solution of \eqref{p3l}. They also show some properties of this solutions when they there exist. Thus from \cite[Theorem 5 and Remark 6.3]{BELI83} we have the following result. \begin{theorem}\label{beli} Assume {\rm (H1), (H2)}. Then \eqref{p3l} has a solution $U\in\mathcal{C}^2(\mathbb{R})$, which is unique up to translation, positive and satisfies: \begin{itemize} \item[(i)] $U(0)=\tilde T$, $U$ is radially symmetric and decreases with respect to $|x|$; \item[(ii)] $U$ together with its derivatives up to order $2$ have exponential decay at infinity $0\leq U(x)+|U'(x)|+|U''(x)|\leq C\exp\left(-c|x|\right)\quad\text{for all } x\in\mathbb{R};$ \item[(iii)] $-[U'(x)]^2=2G(U(x))$ for all $x\in\mathbb{R}$. \end{itemize} \end{theorem} Now we consider $L_m:H^1(\mathbb{R})\to\mathbb{R}$, the functional associated to equation \eqref{p3l}, $L_m(u)=\frac{1}{2}\int_{\mathbb{R}}\left(|\nabla u|^2+mf^2(u)\right)\;\mathrm{d} x-\int_{\mathbb{R}}H(f(u))\;\mathrm{d} x$ which is well defined and of class $\mathcal{C}^1$. Let $E_m := L_m(U).$ Since $U$ is unique up to translation we have $L_m(w)=E_m$ for each solution $w$ of \eqref{p3l}. By a result of Jeanjean and Tanaka \cite{JETA2003} we know that these solutions have a mountain pass characterization, that is $$\label{mpca} L_m(w)=c_m:=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}L_m(\gamma(t))$$ where $\Gamma=\{\gamma\in\mathcal{C}([0,1],H^1(\mathbb{R})): \gamma(0)=0\,\,\text{and}\,\,L_m(\gamma(1))<0\}$. Using the same arguments as in \cite[Proposition 2]{BYJETA2008} we prove the next result. \begin{proposition}\label{path} There exist $t_0>1$ and a continuous path $\theta:[0,t_0]\to H^1(\mathbb{R})$ satisfying: \begin{itemize} \item[(i)] $\theta(0)=0,\,L_m(\theta(t_0))<-1$ and $\max_{t\in[0,t_0]}L_m(\theta(t))=E_m$; \item[(ii)] $\theta(1)=U$ and $L_m(\theta(t))0$ such that for any $t\in[0,t_0]$ it holds $|\theta(t)(x)|+|[\theta(t)]'(x)|\leq C\exp(-c|x|)\quad x\in\mathbb{R}.$ \end{itemize} \end{proposition} \section{The mountain pass level} For the rest of this article, we fix $\beta=\mathop{\rm dist}(\mathcal{M}, \mathbb{R}^{N}\backslash\Omega)/10$ and choose a cut-off function $\varphi\in \mathcal{C}^\infty_c(\mathbb{R})$ such that $0\leq\varphi\leq1,\,\varphi(x)=1$ for $|x|\leq \beta$ and $\varphi(x)=0$ for $|x|\geq2\beta$. We define $\varphi_\varepsilon(x)=\varphi(\varepsilon x)$ and for $z\in\mathcal{M}^\beta$ $U^z_\varepsilon(x):=\varphi_\varepsilon(x-z/\varepsilon)U(x-z/\varepsilon),\quad x\in\mathbb{R}.$ For sufficiently small $\varepsilon$ we will find a solution near the set $X_\varepsilon:=\{U_\varepsilon^z:\ z\in\mathcal{M}^\beta\}.$ \begin{remark}\label{compactness} {\rm For $\varepsilon\in(0,10)$ we have $X_\varepsilon$ uniformly bounded and moreover for each $\varepsilon$ it is compact in $E_\varepsilon$. Indeed, let $U^z_\varepsilon\in X_\varepsilon$ for some $z\in\mathcal{M}^\beta$. So \begin{align*} \|U^z_\varepsilon\|_\varepsilon &\leq \Big[\int_{\mathbb{R}}|(\varphi_\varepsilon U)'|^2\,\mathrm{d} x\Big]^{1/2} + \Big[1+\int_{\mathbb{R}}V(\varepsilon x+z)f^2(\varphi_\varepsilon U) \,\mathrm{d} x\Big]\\ &\leq \Big[2\int_{\mathbb{R}}\left(\varepsilon^2|\varphi'(\varepsilon x)|^2 U^2 +\varphi_\varepsilon^2|U'|^2\Big]\,\mathrm{d} x\right]^{1/2}+ 1+\sup_{x\in\Omega}V(x)\int_{\mathbb{R}}(\varphi_\varepsilon U)^2\,\mathrm{d} x\\ &\leq c\|U\|+\tilde c\|U\|^2+1\leq C \end{align*} independently of $z\in\mathcal M^\beta$ and $\varepsilon\in(0,10)$. This proves the uniform boundedness of $X_\varepsilon$. Now let $\{U_\varepsilon^{z_n}\}$ be a sequence in $X_\varepsilon$. The compactness of $\mathcal{M}^\beta$ implies the existence of $z_0\in\mathcal M^\beta$ such that $z_n\to z_0$ in $\mathbb{R}$, up to subsequences. Hence $U_\varepsilon^{z_0}\in X_\varepsilon$ and due to the exponential decay of $U+|U'|$ and the boundedness of $\{z_n\}$ we get \begin{align*} \int_{\mathbb{R}}V(\varepsilon x)f^2\left(U_\varepsilon^{z_n}-U_\varepsilon^{z_0}\right)\,\mathrm{d} x \leq \sup_{\Omega}V(x)\int_{\mathbb{R}} \left|U_\varepsilon^{z_n}-U_\varepsilon^{z_0}\right|^2\,\mathrm{d} x \to 0, \\ \int_{\mathbb{R}}| \left(U_\varepsilon^{z_n}-U_\varepsilon^{z_0}\right)' |^2\,\mathrm{d} x \to 0 \quad\text{as } n\to\infty. \end{align*} Now for $\lambda\in(0,1)$ it follows from (ii) in Lemma \ref{lema f} that $\lambda\big\{1+\int_{\mathbb{R}}V(\varepsilon x)f^2(\lambda^{-1}(U_\varepsilon^{z_n}-U_\varepsilon^{z_0}))\,\mathrm{d} x\big\} \leq \lambda+\lambda^{-1}\int_{\mathbb{R}}V(\varepsilon x)f^2 \left({U_\varepsilon^{z_n}-U_\varepsilon^{z_0}}\right)\,\mathrm{d} x.$ Thus $|\|U_\varepsilon^{z_n}-U_\varepsilon^{z_0}\||_\varepsilon\leq 2\lambda$ for large $n$ which proves that $U_\varepsilon^{z_n}\to U_\varepsilon^{z_0}$ in $E_\varepsilon$ as $n\to\infty$. }\end{remark} \begin{lemma}\label{lema1} We have $\sup_{t\in[0,t_0]}|J_\varepsilon(\varphi_\varepsilon\theta(t)) -L_m(\theta(t))|\to0\quad\text{as } \varepsilon\to0.$ \end{lemma} \begin{proof} Since $\mathop{\rm supp}(\varphi_\varepsilon\theta(t))\subset\Omega_\varepsilon$ and $\mathop{\rm supp}(\chi_{\varepsilon})\subset\mathbb{R}\backslash\Omega_\varepsilon$ we have $Q_\varepsilon(\varphi_\varepsilon\theta(t))=0$ and $J_\varepsilon(\varphi_\varepsilon\theta(t)) =P_\varepsilon(\varphi_\varepsilon\theta(t))$. Then for $t\in(0,t_0]$ we get \begin{align*} &|P_\varepsilon(\varphi_\varepsilon\theta(t)) - L_m(\theta(t))|\\ &\leq \frac{1}{2}\Big|\int_{\mathbb{R}} \left[|\left(\varphi_\varepsilon\theta(t)\right)^{\prime}|^2 -|\theta({t})'|^2+ V(\varepsilon x)f^2(\varphi_\varepsilon\theta(t))-mf^2(\theta_{t})\right] \mathrm{d} x\Big|\\ &\quad +\int_{\mathbb{R}}\left|H(f(\varphi_\varepsilon\theta(t))) -H(f(\theta(t)))\right| \mathrm{d} x. \end{align*} At first, using a change of variables and the exponential decay of $\theta(t),\,\theta(t)'$, we get $\int_{\mathbb{R}}|\left(\varphi_\varepsilon\theta(t)\right)' -\theta(t)'|^2\,\mathrm{d} x \leq C\int_{\mathbb{R}}\left[\varepsilon^2 +(1-\varphi_\varepsilon)^2\right]\exp(-c|x|)\,\mathrm{d} x$ for all $t\in(0,t_0]$. Now since $f(t)f'(t)<2^{-1/2}$ for all $t\in[0,t_0]$ we obtain \begin{align*} &\int_{\mathbb{R}} \big|V(\varepsilon x) f^2(\varphi_\varepsilon\theta(t)) -mf^2(\theta({t}))\big|\,\mathrm{d} x\\ &\leq \int_{\mathbb{R}} \left|V(\varepsilon x)-m\right|f^2(\varphi_\varepsilon\theta(t)) \,\mathrm{d} x+m\int_{\mathbb{R}} \left|f^2(\varphi_\varepsilon\theta(t)) -f^2(\theta(t))\right|\,\mathrm{d} x\\ &\leq 2^{1/2}C\int_{\mathbb{R}} \left[|V(\varepsilon x)-m|\chi_{\{|x|\leq2\beta/\varepsilon\}} +m(1-\varphi_\varepsilon)\right]\exp (-c|x|)\,\mathrm{d} x. \end{align*} Recalling that $$\label{tfc} H(f(a+b))-H(f(a))=b\int_0^1f'(a+sb)h(f(a+sb))\,\mathrm{d} s$$ due to the imbedding $H^1(\mathbb{R})\hookrightarrow L^\infty(\mathbb{R})$ and the boundedness of $\{\theta(t)\}$ in $L^\infty(\mathbb{R})$ it follows from (H1) that \begin{align*} \int_{\mathbb{R}}\left|H(f(\varphi_\varepsilon\theta(t)))-H(f(\theta(t)))\right| \mathrm{d} x&\leq C\int_{\mathbb{R}}\left|\varphi_\varepsilon\theta(t)-\theta(t)\right| \left[\theta(t)+\varphi_\varepsilon\theta(t)\right]\mathrm{d} x\\ &\leq C\int_{\mathbb{R}}\left(1-\varphi_\varepsilon\right)\exp\left(-c|x|\right)\,\mathrm{d} x \end{align*} for $t\in(0,t_0]$. Therefore, $J_\varepsilon(\varphi_\varepsilon\theta(t))\to L_m(\theta(t))$ as $\varepsilon\to0$, uniformly in $t\in[0,t_0]$. This is the end of the proof. \end{proof} For Lemma \ref{lema1} there exists $\varepsilon_0$ sufficiently small such that $|J_\varepsilon(\varphi_\varepsilon\theta({t_0}))-L_m(\theta({t_0}))|\leq -L_m(\theta({t_0}))-1$ and so $J_\varepsilon(\varphi_\varepsilon\theta({t_0}))<-1$ for all $\varepsilon\in(0,\varepsilon_0)$. From now on we consider $\varepsilon\in(0,\varepsilon_0)$. We define the minimax level $C_\varepsilon=\inf_{\gamma\in\Gamma_\varepsilon}\max_{s\in[0,1]} J_\varepsilon(\gamma(s)),$ where $\Gamma_\varepsilon=\{\gamma\in \mathcal{C}([0,1],E_\varepsilon) :\gamma(0)=0,\ \gamma(1)=\varphi_\varepsilon\theta(t_0)\}.$ \begin{proposition}\label{prop2} $C_\varepsilon$ converges to $E_m$ as $\varepsilon$ goes to zero. \end{proposition} \begin{proof} At first we will prove that $\label{limsup} \limsup_{\varepsilon\to0}C_\varepsilon\leq E_m.$ Since $\theta:[0,t_0]\to H^1(\mathbb{R})$ is a continuous function using arguments as in Remark \ref{compactness} we prove that $\gamma_\varepsilon:[0,1]\to E_\varepsilon$ given by $$\label{gama} \gamma_\varepsilon(s):=\varphi_{\varepsilon}\theta(st_0)\quad \text{for } s\in[0,1]$$ is continuous. So $\gamma_\varepsilon\in\Gamma_\varepsilon$ and by Lemma \ref{lema1} and Proposition \ref{path} we obtain \begin{align*} \limsup_{\varepsilon\to0}C_\varepsilon &\leq \limsup_{\varepsilon\to0}\max_{s\in[0,1]} J_\varepsilon(\gamma_\varepsilon(s))\\ &=\limsup_{\varepsilon\to0}\max_{t\in[0,t_0]} J_\varepsilon(\varphi_{\varepsilon}\theta(t))\\ &\leq \max_{t\in[0,t_0]}L_m(\theta(t))=E_m \end{align*} which concludes the first part of the proof. Next we are going to prove that $$\label{liminf} \liminf_{\varepsilon\to0}C_\varepsilon\geq E_m.$$ Let us assume $\liminf_{\varepsilon\to0}C_\varepsilon< E_m$ instead. Then there exist $\alpha>0,\ \varepsilon_n\to0$ and $\gamma_n\in\Gamma_{\varepsilon_n}$ satisfying $\max_{s\in[0,1]}J_{\varepsilon_n}(\gamma_n(s))\max_{s\in[0,1]}J_\varepsilon(\gamma(s)) \geq\max_{s\in[0,s_0]}P_\varepsilon(\gamma(s)), \] by the estimates above we obtain $E_m-\alpha>E_m-\frac{m}{2}\varepsilon [1+(1+E_m)^{1/2}]>E_m-\alpha.$ This contradiction completes the proof. \end{proof} At this point, denoting $D_\varepsilon\equiv\max_{s\in[0,1]}J_\varepsilon(\gamma_\varepsilon(s))$ where$\gamma_\varepsilon$was defined in \eqref{gama}, we see that$C_\varepsilon \leq D_\varepsilon$and also$\lim_{\varepsilon\to0}D_\varepsilon=E_m$. \section{Existence of a critical point for$J_\varepsilon$} We define $J_\varepsilon^\alpha\equiv\{u\in E_\varepsilon:J_\varepsilon(u) \leq\alpha\}, \quad A^\alpha\equiv\{u\in E_\varepsilon:\inf_{v\in A}\|u-v\|_\varepsilon \leq\alpha\}$ for any$A\subset E_\varepsilon$and$\alpha>0$. Moreover in the next propositions, for any$\varepsilon>0$and$R>0$, we consider the functional$J_\varepsilon$restricted to the space$H_0^1((-R/\varepsilon,R/\varepsilon))$endowed with the norm $\|v\|_\varepsilon=\|v'\|_{L^2((-R/\varepsilon,R/\varepsilon))} +\inf_{\lambda>0}\lambda\Big\{1 +\int_{-R/\varepsilon}^{R/\varepsilon}V(\varepsilon x) f^2(\lambda^{-1}v)\,\mathrm{d} x\Big\}.$ We will denote this space by$E_\varepsilon^R$. We can see that$E_\varepsilon^R$is a Banach space and$J_\varepsilon$is of class$\mathcal{C}^1$on$E_{\varepsilon}^R$. \begin{proposition}\label{prop4} There exist$d>0$sufficiently small such that if$\varepsilon_n\to0$,$R_n\to\infty$and$u_n\in X_{\varepsilon_n}^d\cap E_{\varepsilon_n}^{R_n}$satisfy $\lim_{n\to\infty}J_{\varepsilon_n}(u_n)\leq E_m,\quad \lim_{n\to\infty}\|J_{\varepsilon_n}'(u_n)\|_{(E_{\varepsilon_n}^{R_n})'}=0$ then, up to subsequences, there exist$\{y_n\}\subset\mathbb{R}$and$z_0\in \mathcal{M}$satisfying $\lim_{n\to\infty}|\varepsilon_n y_n-z_0|=0,\quad \lim_{n\to\infty}\|u_n-\varphi_{\varepsilon_n}(\cdot-y_n) U(\cdot-y_n)\|_{\varepsilon_n}=0.$ \end{proposition} \begin{proof} From now on we suppose$d\in(0,10)$. Since$u_n\in X_{\varepsilon_n}^d$by definition of$X_{\varepsilon_n}^d$there exists$v_n\in X_{\varepsilon_n}$such that $$\label{18} \|u_n-v_n\|_{\varepsilon_n}\leq d.$$ We have$v_n(x)=\varphi_{\varepsilon_n}(x-z_n/\varepsilon_n)U(x-z_n/\varepsilon_n)$,$x\in \mathbb{R}$, for$\{z_n\}\subset \mathcal{M}^\beta$. From Remark \ref{compactness} we have $\|u_n\|_{\varepsilon_n}\leq C\quad \text{for all } n\in\mathbb N,\; d\in(0,10).$ By compactness of$\mathcal{M}^\beta$, up to subsequences, we may assume that$z_n\to z_0$in$\mathbb{R}$for some$z_0\in\mathcal{M}^\beta$. We divide the proof of this proposition in five steps. \subsection*{Step 1:} For small$d>0$, defining$A(y;r_1,r_2)=\{x\in\mathbb{R}: r_1\leq|y-x|\leq r_2\}$for$00. \] Indeed, suppose that there exist $R>0$ and a sequence $\{\tilde z_n\}$ satisfying $\tilde z_n\in A\Big(\frac{z_n}{\varepsilon_n};\frac{\beta}{2\varepsilon_n}, \frac{3\beta}{\varepsilon_n}\Big), \quad \lim_{n\to \infty}\int_{\tilde{z}_n-R}^{\tilde{z}_n+R}|u_n|^2\,\mathrm{d} x>0.$ Since Remark \ref{compactness} implies that $X_\varepsilon^d$ is uniformly bounded on $\varepsilon\in(0,\varepsilon_0)$ and $d\in(0,10)$, due to Proposition \ref{despri} and the imbedding $H^1(\mathbb{R})\hookrightarrow L^4(\mathbb{R})$ we get $\{u_n'\}_{n}$ bounded in $L^2(\mathbb{R})$ and \begin{align*} \int_{\mathbb{R}}|u_n|^2\,\mathrm{d} x & \leq C\int_{\mathbb{R}}\left[f^2(u_n)+f^4(u_n)\right]\,\mathrm{d} x\\ &\leq C\int_{\mathbb{R}}V(\varepsilon x)f^2(u_n)\,\mathrm{d} x +C\|f(u_n)\|_{H^1}^4\\ &\leq C\Big\{\|u_n\|_{\varepsilon_n}+ \Big[\int_{\mathbb{R}}\left(|u_n'|^2+V(\varepsilon x)f^2(u_n)\right) \,\mathrm{d} x\Big]^2\Big\}\\ &\leq C\left(\|u_n\|_{\varepsilon_n}+\|u_n\|_{\varepsilon_n}^2 +\|u_n\|_{\varepsilon_n}^4\right)\leq\tilde C. \end{align*} Consequently $\{u_n\}$ is bounded in $H^1(\mathbb{R})$. Hence we may assume that $\varepsilon_n\tilde z_n\to \tilde z_0$ and that $\tilde w_n:=u_n(\cdot+\tilde z_n)\rightharpoonup \tilde w$ in $H^1(\mathbb{R})$ for some $\tilde z_0\in A\left({z_0};{\beta}/{2},3\beta\right)$ and $\tilde w\in H^1(\mathbb{R})$. By the compactness of the imbedding $H^1((-R,R))\hookrightarrow \mathcal{C}([-R,R])$ we get $\int_{-R}^R|\tilde w|^{2}\,\mathrm{d} x=\lim_{n\to\infty}\int_{-R}^R|\tilde w_n|^{2} \,\mathrm{d} x= \lim_{n\to\infty}\int_{\tilde z_n-R}^{\tilde z_n+R}|u_n|^{2}\,\mathrm{d} x>0$ and so $\tilde w\neq0$. Now given $\phi\in\mathcal{C}^\infty_c(\mathbb{R})$ let $\phi_n(x)=\phi(x-\tilde z_n)$, $n\in\mathbb N$. We have $\varepsilon_n\tilde z_n\in\mathcal M^{4\beta}$ and so we obtain $\phi_n\in E_{\varepsilon_n}^{R_n}$ for large $n$. Since $\|J_{\varepsilon_n}'(u_n)\|_{(E_{\varepsilon_n}^{R_n})'}\to0$ and $\|\phi_n\|_{\varepsilon_n}\leq C$ we have $\lim_{n\to\infty}\langle J_{\varepsilon_n}'(u_n),\phi_n\rangle =0.$ Consequently the boundedness of $\mathop{\rm supp} (\phi)$ implies that $\int_{\mathbb{R}}\left[\tilde w'\phi'+V(\tilde z_0)f'(\tilde w)f(\tilde w)\phi\right])\,\mathrm{d} x= \int_{\mathbb{R}}f'(\tilde w)h(f(\tilde w))\phi\,\mathrm{d} x.$ Since $\phi$ is arbitrary it follows that $\tilde w$ satisfies $$\label{p3l0} -\tilde w''=f'(\tilde w)[h(f(\tilde w))-V(\tilde z_0)f(\tilde w)]=g_0(\tilde w), \quad\tilde w\geq0\quad\text{in } \mathbb{R}.$$ By assumptions on $h$ we get $g_0$ locally Lipschitz continuous, $g_0(0)=0$ and so due to (\cite{BELI83}, Theorem 5) we know that the function $g_0$ must satisfy \eqref{ttil} for some $T>0$. Thus Theorem \ref{beli} hods for problem \eqref{p3l0} and $\tilde w(x)=w_0(x+c)$ where $w_0$ is radial. Then for $L_{V(\tilde z_0)}$ defined as $L_m$ with $V(\tilde x_0)$ instead of $m$ we denote $E_{V(\tilde z_0)}=L_{V(\tilde z_0)}(\tilde w)$. By (\cite{BOMU92}, Theorem 2.1) we obtain $\tilde w_n'(x)\to\tilde w'(x)$ a.e. in $A$ for any set $A\subset\mathbb{R}$. So using the Fatou's Lemma for $R>0$ sufficiently large we get $\frac{1}{2}\int_{\mathbb{R}}|\tilde{w}'|^2\,\mathrm{d} x\leq\int_{-R}^R|\tilde{w}'|^2\,\mathrm{d} x \leq \liminf_{n\to\infty}\int_{-R}^R|\tilde w_n'|^2\,\mathrm{d} x= \liminf_{n\to\infty}\int_{\tilde{z}_n-R}^{\tilde{z}_n+R}|u_n'|^2\,\mathrm{d} x.$ Since $V(\tilde z_0)\geq m$ and the least energy levels for equations \eqref{p3l} and \eqref{p3l0} are equal to the mountain pass levels (see \cite{JETA2003}) we have $E_{V(\tilde z_0)}\geq E_m$. Using item (iii) in Theorem \ref{beli} we see that $$\int_{\mathbb{R}}|\tilde{w}'|^2\,\mathrm{d} x=L_{V(\tilde z_0)}(\tilde{w}).$$ Thus we obtain $\liminf_{n\to\infty}\int_{\tilde{z}_n-R}^{\tilde{z}_n+R}|u_n'|^2\,\mathrm{d} x \geq\frac{1}{2}L_{V(\tilde z_0)}(\tilde{w})\geq\frac{1}{2}E_m>0.$ On the other hand, from \eqref{18} we have $$\int_{\tilde{z}_n-R}^{\tilde{z}_n+R}|u_n'|^2\,\mathrm{d} x\leq4d^2$$ for large $n$ ($n\geq n_0(d)$). Then $\frac{1}{2}E_m\leq\liminf_{n\to\infty}\int_{\tilde{z}_n-R}^{\tilde{z}_n+R}|u_n'|^2 \,\mathrm{d} x\leq4d^2$ which is impossible for $d\in(0,\sqrt{E_m/8})$. This proves Step 1. \subsection*{Step 2:} Defining $u_{n,1}=\varphi_{\varepsilon_n}(\cdot-z_n/\varepsilon_n)u_n$ and $u_{n,2}=u_n-u_{n,1}$ we have $$\label{passo2} J_{\varepsilon_n}(u_n)\geq J_{\varepsilon_n}(u_{n,1})+J_{\varepsilon_n}(u_{n,2})+o(1)$$ where $o(1)$ indicates the quantity that vanishes as $n\to\infty$. Indeed, we can see that $Q_{\varepsilon_n}(u_{n,1})=0$ and $Q_{\varepsilon_n}(u_n)=Q_{\varepsilon_n}(u_{n,2})$. Then the boundedness of $\{u_n\}$ and the convexity of $f^2$ imply that \begin{align*} &J_{\varepsilon_n} (u_{n,1})+J_{\varepsilon_n}(u_{n,2})\\ &= J_{\varepsilon_n}(u_{n})+\frac{1}{2}\int_{\mathbb{R}}\left\{ \varphi_{\varepsilon_n}^2(x-z_n/\varepsilon_n) +\left[1-\varphi_{\varepsilon_n}(x-z_n/\varepsilon_n)\right]^2-1 \right\}|u_n'|^2\,\mathrm{d} x\\ &\quad +\frac{1}{2}\int_{\mathbb{R}}V(\varepsilon_n x) \left[ f^2(u_{n,1})+f^2(u_{n,2})-f^2(u_n) \right] \mathrm{d} x\\ &\quad +\int_{\mathbb{R}} \left[H(f(u_n))-H(f(u_{n,1}))-H(f(u_{n,2}))\right]\,\mathrm{d} x+o(1)\\ &\leq J_{\varepsilon_n}(u_{n})+\int_{\mathbb{R}} \left[H(f(u_{n}))-H(f(u_{n,1}))-H(f(u_{n,2}))\right]\,\mathrm{d} x+o(1). \end{align*} To conclude Step 2 we need to estimate this last integral. We have \begin{align*} &\int_{\mathbb{R}} [H(f(u_{n})) - H(f(u_{n,1}))-H(f(u_{n,2}))]\,\mathrm{d} x\\ &= \int_{A\left(\frac{z_n}{\varepsilon_n};\frac{\beta}{\varepsilon_n}, \frac{2\beta}{\varepsilon_n}\right)} \left[H(f(u_{n}))-H(f(u_{n,1}))-H(f(u_{n,2}))\right]\,\mathrm{d} x. \end{align*} Choose $\psi\in \mathcal{C}^\infty_c(\mathbb{R})$ such that $0\leq\psi\leq1$, $\psi\equiv1$ on $A(0;\beta,2\beta)$ and $\psi\equiv0$ on $\mathbb{R}\backslash A(0;\beta/2,3\beta)$. Setting $\psi_n(x)=\psi(\varepsilon_n x-z_n)u_n(x)$, for large $n$ we get \begin{align*} \sup_{y\in A\left(\frac{z_n}{\varepsilon_n};\frac{\beta}{2\varepsilon_n}, \frac{3\beta}{\varepsilon_n}\right)} \int_{y-R}^{y+R}|u_n|^2\,\mathrm{d} x &\geq \sup_{y\in A\left(\frac{z_n}{\varepsilon_n};\frac{\beta}{2\varepsilon_n}, \frac{3\beta}{\varepsilon_n}\right)} \int_{y-R}^{y+R}|\psi_n|^2\,\mathrm{d} x\\ &= \sup_{y\in\mathbb{R}}\int_{y-R}^{y+R}|\psi_n|^2\,\mathrm{d} x. \end{align*} Using Step 1 and a result of Lions \cite[Lemma 1.1]{LI84}, we see that $\psi_n\to0$ in $L^{p}(\mathbb{R})$ as $n\to\infty$ for all $p\in(2,\infty)$. Since $\psi_n=u_n$ in $A(z_n/\varepsilon_n;\beta/\varepsilon_n,2\beta/\varepsilon_n)$ we obtain $\lim_{n\to\infty}\int_ {A\left(\frac{z_n}{\varepsilon_n};\frac{\beta}{\varepsilon_n}, \frac{2\beta}{\varepsilon_n}\right)} |u_n|^{p}\,\mathrm{d} x=0.$ Thus for $p>2$ fixed using the fact that $|u_{n,1}|,\ |u_{n,2}|\leq|u_n|$ and (H1) we see that given $\sigma>0$ there exists $c=c(\sigma,p)>0$ such that \begin{align*} &\int_{A\left(\frac{z_n}{\varepsilon_n};\frac{\beta}{\varepsilon_n}, \frac{2\beta}{\varepsilon_n}\right)} \left|H(f(u_{n}))-H(f(u_{n,1}))-H(f(u_{n,2}))\right| \mathrm{d} x\\ &\leq \sigma\|u_n\|_{L^2}+c \int_{A\left(\frac{z_n}{\varepsilon_n};\frac{\beta}{\varepsilon_n}, \frac{2\beta}{\varepsilon_n}\right)} |u_n|^{p}\,\mathrm{d} x \leq C\sigma \end{align*} for large $n$. So \eqref{passo2} is proved. \subsection*{Step 3:} Given $d>0$ sufficiently small there exists $n_0=n_0(d)$ such that $J_{\varepsilon_n}(u_{n,2})\geq\frac{1}{8}\Big[ \int_{\mathbb{R}}\left(|u_{n,2}'|^2+V({\varepsilon_n}x)f^2(u_{n,2})\right)\mathrm{d} x\Big]\quad\text{for all } n\geq n_0.$ In fact, using \eqref{18} we can see that there exists $n_0=n_0(d)$ such that \begin{align*} \|u_{n,2}'\|_{L^2} &\leq \|[1-\varphi_{\varepsilon_n}(\cdot-z_n/\varepsilon_n)]' u_n\|_{L^2}+ \|u_{n}'-v_n'\|_{L^2}+\|(1-\varphi_{\varepsilon_n}) (\varphi_{\varepsilon_n} U)'\|_{L^2}\\ &\leq o(1)+d\leq 2d\quad\text{for all } n\geq n_0 \end{align*} where $v_n=\varphi_{\varepsilon_n}(\cdot-z_n/\varepsilon_n) U(\cdot-z_n/\varepsilon_n)$. Moreover by Proposition \ref{despri} we get $\int_{\mathbb{R}}V(\varepsilon_nx)f^2(u_{n,2})\,\mathrm{d} x\leq c_0d\quad\text{for all } n\geq n_0$ for large $n_0$. Since $\{u_{n,2}\}$ is bounded in $H^1(\mathbb{R})$ it is also bounded in $L^{\infty}(\mathbb{R})$. So by (H1) we get $H(f(u_{n,2}))\leq (V_0/4)f^2(u_{n,2})+Cf^4(u_{n,2}).$ Due to the imbedding $H^1(\mathbb{R})\hookrightarrow L^4(\mathbb{R})$ and (V1) we see that $\int_{\mathbb{R}}H(f(u_{n,2}))\leq\frac{1}{4} \int_{\mathbb{R}}V(\varepsilon x)f^2(u_{n,2})\,\mathrm{d} x+ C\Big[\int_{\mathbb{R}}\left(|u_{n,2}'|^2+V(\varepsilon x)f^2(u_{n,2})\right)\,\mathrm{d} x\Big]^2.$ Hence we obtain \begin{align*} J_{\varepsilon_n}(u_{n,2}) & \geq \frac{1}{2}\|u_{n,2}'\|_{L^2}^2+\frac{1}{4} \int_{\mathbb{R}}V(\varepsilon_n x)f^2(u_{n,2})\,\mathrm{d} x-C\|f(u_{n,2})\|_{H^1}^4\\ &\geq \big(\frac{1}{2}-C(2d)^2\big)\|u_{n,2}'\|_{L^2}^2 +\big(\frac{1}{4}-C(c_0d)\big)\int_{\mathbb{R}}V(\varepsilon_n x)f^2(u_{n,2})\,\mathrm{d} x \end{align*} for $n\geq n_0$. This proves Step 3 for small $d>0$. \subsection*{Step 4:} We have $\lim_{n\to\infty} J_{\varepsilon_n}(u_{n,1})= E_m$ and $z_0\in\mathcal{M}$. Indeed, let $w_n:=u_{n,1}(\cdot+z_n/\varepsilon_n)$. After extracting a subsequence, we may assume $w_n\rightharpoonup w$ in $H^1(\mathbb{R})$, $w_n(x)\to w(x)$ for almost every $x\in \mathbb{R}$ and $w_n\to w$ in $L^2((0,1))$. As we see in Step 3 using (8) and (11) of Lemma \ref{lema f} and \eqref{desigualdade primordial} it follows from \eqref{18} \begin{align*} &\frac{V_0}{2}\int_{0}^1f^2(\varphi_{\varepsilon_n}U)\,\mathrm{d} x - V_0\int_{0}^1f^2(w_n)\,\mathrm{d} x\\ &\leq V_0\int_{0}^1f^2(w_n-\varphi_{\varepsilon_n}U)\,\mathrm{d} x\\ &\leq \int_{\mathbb{R}}V(\varepsilon_n x)f^2(u_{n,1}-v_n)\,\mathrm{d} x\\ &\leq 2\int_{\mathbb{R}}V(\varepsilon_n x)\left[f^2(u_{n}-v_n)+f^2(u_{n,2})\right]\,\mathrm{d} x\leq c_0d \end{align*} for large $n$. Since $\varphi_{\varepsilon_n}U=U$ in $[0,1]$ for large $n$, we obtain $$\int_{0}^1f^2(w)\,\mathrm{d} x=\lim_{n\to\infty}\int_{0}^1f^2(w_n)\,\mathrm{d} x \geq c\int_{0}^1f^2(U)\,\mathrm{d} x-cd>0$$ for small $d$. Consequently $w\neq0$. Moreover for any $r>0$ it follows that $u_{n,1}(x+z_n/\varepsilon_n)=u_n(x+z_n/\varepsilon_n)\quad\text{in } (-r,r)$ for large $n$. Then as in Step 1, we can see that $w$ satisfies $-w''=f'(w)\left[h(f(w))-V(z_0)f(w)\right],\quad w>0\quad\text{in } \mathbb{R}.$ Now we shall consider two cases: Case 1: $\lim_{n\to\infty}\sup_{z\in\mathbb{R}} \int_{z-1}^{z+1}|w_n-w|^2\,\mathrm{d} x=0$. Case 2: $\lim_{n\to\infty}\sup_{z\in\mathbb{R}}\int_{z-1}^{z+1}|w_n-w|^2 \,\mathrm{d} x>0$. If Case 1 occurs we have that $w_n\to w$ in $L^{p}(\mathbb{R})$ for all $p\in(2,\infty)$. By (H1), \eqref{tfc} and the boundedness of $\|w_n\|_\infty$, given $\sigma>0$ there exists $C=C(\sigma)$ such that \begin{align*} &\int_{\mathbb{R}} |H(f(w_n))-H(f(w))|\mathrm{d} x\\ &\leq \int_{\mathbb{R}}|w_n-w|\left[\sigma\left(|w| + |w_n|\right) + C\left(|w|^3 + |w_n-w|^3\right)\right]\mathrm{d} x\\ &\leq c\sigma+C\left(\|w_n-w\|_{L^{4}}+\|w_n-w\|_{L^{4}}^{4}\right) \leq (c+1)\sigma \end{align*} for large $n$. Thus $$\label{FnF} \int_{\mathbb{R}}H(f(w_n))\,\mathrm{d} x\to\int_{\mathbb{R}}H(f(w))\,\mathrm{d} x\quad\text{as } n\to\infty.$$ Now if Case 2 occurs there exists $\{\hat z_n\}\subset\mathbb{R}$ such that $\lim_{n\to\infty}\int_{\hat z_n-1}^{\hat z_n+1}|w_n-w|^2\,\mathrm{d} x>0.$ Since $w_n\rightharpoonup w$ in $H^1(\mathbb{R})$ we have $$\label{zn} |\hat z_n|\to\infty.$$ Therefore, $\lim_{n\to\infty}\int_{\hat z_n-1}^{\hat z_n+1}|w|^2\,\mathrm{d} x=0 \quad\text{and so}\quad \lim_{n\to\infty}\int_{\hat z_n-1}^{\hat z_n+1}|w_n|^2\,\mathrm{d} x>0.$ Since $w_n(x)=\varphi_{\varepsilon_n}(x)u_n(x+z_n/\varepsilon_n)$, it is easily seen that $|\hat z_n|\leq3\beta/\varepsilon_n$ for large $n$. If $|\hat z_n|\geq\beta/2\varepsilon_n$ for a subsequence from Step 1, we would have $0<\lim_{n\to\infty}\int_{\hat z_n-1}^{\hat z_n+1}|w_n|^2\,\mathrm{d} x \leq\lim_{n\to\infty}\sup_{z\in A\left(\frac{z_n}{\varepsilon_n}; \frac{\beta}{2\varepsilon_n},\frac{3\beta}{\varepsilon_n}\right)} \int_{z-1}^{z+1}|u_n|^2\,\mathrm{d} x=0$ which is impossible. So $|\hat z_n|\leq\beta/2\varepsilon_n$ for large $n$. We may assume that $\varepsilon_n \hat z_n\to \hat z_0\quad\text{and}\quad u_{n,1}(\cdot+\hat z_n+z_n/\varepsilon_n)\rightharpoonup\hat{w},$ and we see that $|\hat z_0|\leq\beta/2$ and $\hat{w}\in H^1(\mathbb{R})\backslash\{0\}$. Then, given any $r>0$ we have $$u_{n,1}(\cdot+\hat z_n+z_n/\varepsilon_n) =u_n(\cdot+\hat z_n+z_n/\varepsilon_n)\quad\text{in } [-r,r]$$ for large $n$. Consequently as in Step 1 it follows that $\hat{w}$ satisfies $-\hat{w}''=f'(\hat w)\left[h(f(\hat w))-V(\hat z_0+z_0)f(\hat{w})\right],\quad \hat w>0\quad\text{in }\mathbb{R}.$ Analogous to Step 1, \eqref{zn} leads us to a contradiction with \eqref{18} if $d>0$ is sufficiently small. At this point we have proved that Case $2$ does not hold and so Case $1$ takes place. Now from (\cite{BOMU92}, Theorem 2.1) we see that $w_n'(x)\to w'(x)$ a.e. in $\mathbb{R}$. Then by \eqref{FnF} and Fatou's Lemma we have \begin{align*} &\liminf_{n\to\infty}J_{\varepsilon_n}(u_{n,1})\\ &=\liminf_{n\to\infty}\Big\{\frac{1}{2}\int_{\mathbb{R}}\left[|w_n'|^2+ V(\varepsilon_nx+z_n)f^2(w_n)\right]\,\mathrm{d} x- \int_{\mathbb{R}}H(f(w_n))\,\mathrm{d} x\Big\}\\ &\geq\frac{1}{2}\int_{\mathbb{R}}\left[|w'|^2+V(z_0)f^2(w)\right]\,\mathrm{d} x- \int_{\mathbb{R}}H(f(w))\,\mathrm{d} x\\ &\geq L_{V(z_0)}(w)\geq E_{V(z_0)}\geq E_m. \end{align*} On the other hand, since $\lim_{n\to\infty}J_{\varepsilon_n}(u_n)\leq E_m$ and $J_{\varepsilon_n}(u_{n,2})\geq0$ because of \eqref{passo2} we get $$\limsup_{n\to\infty}J_{\varepsilon_n}(u_{n,1})\leq E_m.$$ Hence $E_{V(z_0)}=E_m$ and $\lim_{n\to\infty}J_{\varepsilon_n}(u_{n,1})= E_m$. Moreover from the mountain pass characterization to the least energy solution and Proposition \ref{path} we can see that $a>b$ implies $E_a>E_b$. So $V(z_0)=m$ and this concludes the proof of Step 4. \subsection*{Step 5:} Conclusion. From Step 4, we have $\lim_{n\to\infty}\int_{\mathbb{R}} \left[|w_n'|^2+V(\varepsilon_nx+z_n)f^2(w_n)\right]\mathrm{d} x =\int_{\mathbb{R}}\left(|w'|^2+mf^2(w)\right)\,\mathrm{d} x.$ Since $w$ is a solution for \eqref{p3l} there exists $\zeta\in\mathbb{R}$ such that $w=U(\cdot-\zeta)$. We have $w_n(x)\to w(x)$ and $w_n' (x)\to w'(x)$ a.e. in $\mathbb{R}$ which imply the following convergence results \begin{gather*} \int_{A}|w_n'|^2\,\mathrm{d} x\to\int_{A}|w'|^2\,\mathrm{d} x,\quad \int_{A}V(\varepsilon_nx+z_n)f^2(w_n)\;\mathrm{d} x\to \int_{A}mf^2(w)\;\mathrm{d} x, \\ \int_{A}V(\varepsilon_nx+z_n)f^2(\varphi_{\varepsilon_n} (x-\zeta)w)\;\mathrm{d} x \to \int_{A}mf^2(w)\;\mathrm{d} x \end{gather*} for any $A\subset\mathbb{R}$. Then given $\sigma>0$ there exist $R>0$ and $n_0\in\mathbb N$ such that $\int_{\{|x|\geq R\}}V(\varepsilon_nx+z_n) \left[f^2(w_n)+f^2(\varphi_{\varepsilon_n}(x-\zeta)w)\right]\,\mathrm{d} x \leq\frac{\sigma}{4}$ for all $n\geq n_0$. On the other hand, due the convergence $w_n\to w$ in $L^2((-R,R))$ we obtain $\int_{-R}^RV(\varepsilon_nx+z_n)f^2(w_n-\varphi_{\varepsilon_n} (x-\zeta)w)\,\mathrm{d} x\leq \frac{\sigma}{2}\quad\text{for all } n\geq n_0$ for large $n_0$ . This implies $\int_{\mathbb{R}}V(\varepsilon_nx+z_n)f^2(w_n-\varphi_{\varepsilon_n}(x-\zeta)w) \,\mathrm{d} x \leq{\sigma}\quad\text{for all } n\geq n_0.$ By the definition of $|\|\cdot\||_{\varepsilon_n}$ (see also Remark \ref{compactness}), we obtain $|\|u_{n,1}-\varphi_{\varepsilon_n}(\cdot-\zeta-z_n/\varepsilon_n) w(\cdot-z_n/\varepsilon_n)\||_{\varepsilon_n}\to0.$ Now let $y_n:=z_n/\varepsilon_n+\zeta$. Since $w_n'(x)\to w'(x)$ a.e. in $\mathbb{R}$ and $\|w_n'\|_{L^2}\to\|w'\|_{L^2}$ from Brezis-Lieb Lemma (see \cite{BRELI}) it follows that $w_n'\to w'$ in $L^2(\mathbb{R})$. Consequently $[u_{n,1}-\varphi_{\varepsilon_n}(\cdot-y_n)U(\cdot-y_n)]'\to0$ in $L^2(\mathbb{R})$. Hence $\|u_{n,1}-\varphi_{\varepsilon_n}(\cdot-y_n) U_0(\cdot-y_n)\|_{\varepsilon_n}\to0\quad \text{as } n\to\infty.$ On the other hand, using Steps 2, 3, and 4, we obtain $E_m\geq\lim_{n\to\infty}J_{\varepsilon_n}(u_n)\geq E_m+\frac{1}{8}\limsup_{n\to\infty}\int_{\mathbb{R}}[|u_{n,2}'|^2 +V({\varepsilon_n}x)f^2(u_{n,2})]\,\mathrm{d} x,$ which implies that $\|u_{n,2}\|_{\varepsilon_n}\to0$. This completes the proof. \end{proof} We observe that the result of Proposition \ref{prop4} holds for $d\in(0,d_0)$, with $d_0>0$ sufficiently small, independently of the sequences satisfying the assumptions. \begin{corollary}\label{prop5} For any $d\in(0,d_0)$ there exist constants $\omega_d,\, R_d, \,\varepsilon_d>0$ such that $$\|J_{\varepsilon}'(u)\|_{(E_\varepsilon^R)'}\geq\omega_d$$ for any $u\in E_\varepsilon^R\cap J_\varepsilon^{D_\varepsilon}\cap (X_\varepsilon^{d_0}\backslash X_\varepsilon^d)$, $R\ge R_d$ and $\varepsilon\in(0,\varepsilon_d)$. \end{corollary} \begin{proof} By contradiction we suppose that for some $d\in(0,d_0)$ there exist sequences $\{\varepsilon_n\}$, $\{R_n\}$ and $\{u_n\}$ such that $R_n\geq n,\quad \varepsilon_n\leq 1/n,\quad u_n\in E_{\varepsilon_n}^{R_n}\cap J_{\varepsilon_n}^{D_{\varepsilon_n}} \cap(X_{\varepsilon_n}^{d_0}\backslash X_{\varepsilon_n}^d),\quad \|J_{\varepsilon_n}'(u_n)\|_{(E_{\varepsilon_n}^{R_n})'}<\frac{1}{n}.$ By Proposition \ref{prop4} there exist $\{y_n\}\subset\mathbb{R}$ and $z_0\in \mathcal{M}$ such that $\lim_{n\to\infty}|\varepsilon_n y_n-z_0|=0,\quad \lim_{n\to\infty}\|u_n-\varphi_{\varepsilon_n}(\cdot-y_n)U(\cdot-y_n) \|_{\varepsilon_n}=0.$ So for sufficiently large $n$, we have $\varepsilon_ny_n\in\mathcal{M}^\beta$ and then, by the definition of $X_{\varepsilon_n}$ and $X_{\varepsilon_n}^d$, we obtain $\varphi_{\varepsilon_n}(\cdot-y_n)U(\cdot-y_n) \in X_{\varepsilon_n}$ and $u_n\in X_{\varepsilon_n}^d$. This contradicts $u_n\in X_{\varepsilon_n}^{d_0}\backslash X_{\varepsilon_n}^d$ and completes the proof. \end{proof} The next lemmas are necessary to obtain a suitable bounded Palais-Smale sequence in $E_\varepsilon^R$. \begin{lemma}\label{obs1} Given $\lambda>0$ there exist $\varepsilon_0$ and $d_0>0$ small enough such that $J_\varepsilon(u)>E_m-\lambda\quad \text{for all } u\in X_\varepsilon^{d_0}\,\; \varepsilon\in(0,\varepsilon_0).$ \end{lemma} \begin{proof} For $u\in X_\varepsilon$ we have $u(x)=\varphi_\varepsilon(x-z/\varepsilon)U(x-z/\varepsilon),\, x\in\mathbb{R}$, for some $z\in\mathcal{M}^\beta$. Since $L_m(U)=E_m$ by (V2) we obtain \begin{align*} J_\varepsilon(u)-E_m &\geq \frac{1}{2}\int_{\mathbb{R}} \left[\left(|(\varphi_\varepsilon U)'|^2- |U'|^2\right) +m\left(f^2(\varphi_\varepsilon U)-f^2(U)\right)\right]\mathrm{d} x\\ &\quad -\int_{\mathbb{R}}\left| H(f(\varphi_\varepsilon U))-H(f(U))\right|\,\mathrm{d} x \end{align*} independently of $z\in\mathcal{M}^\beta$. It is easily seen that $\varphi_\varepsilon U\to U$ in $H^1(\mathbb{R})$ as $\varepsilon\to 0$. Hence using \eqref{tfc} we can see that there exists $\varepsilon_0>0$ such that $J_\varepsilon(u)-E_m>-\frac{\lambda}{2}\quad\text{for all } u\in X_\varepsilon,\; \varepsilon\in(0,\varepsilon_0).$ Now, if $v\in X_\varepsilon^d$ there exists $u\in X_\varepsilon$ such that $\|u-v\|_\varepsilon\leq d$. We have $v=u+w$ with $\|w\|_\varepsilon\leq d$. Since $Q_\varepsilon(u)=0$ we see that \begin{align*} J_\varepsilon(v)-J_\varepsilon(u) &\geq \frac{1}{2}\int_{\mathbb{R}}\left[|(u+w)'|^2-|u'|^2+ V(\varepsilon x)\left(f^2(u+w)-f^2(u)\right)\right]\,\mathrm{d} x\\ &\quad-\int_{\mathbb{R}}\left[ H(f(u+w))-H(f(u))\right]\,\mathrm{d} x. \end{align*} From \eqref{desigualdade primordial} and Lemma \ref{lema f} we obtain \begin{align*} &\int_{\mathbb{R}}V(\varepsilon x)\big|f^2(u+w)-f^2(u)\big|\,\mathrm{d} x\\ &\leq \int_{\{|w|\leq1\}}V(\varepsilon x)\left|f(u+w)-f(u)\right|\left|f(u+w) +f(u)\right|\,\mathrm{d} x\\ &\quad +\int_{\{|w|>1\}}V(\varepsilon x)\left|f^2(u+w)-f^2(u)\right|\,\mathrm{d} x\\ &\leq C(|\|w\||_\varepsilon^{1/2}+|\|w\||_\varepsilon)\\ &\leq Cd \leq \frac{\lambda}{6} \end{align*} provided $d$ is small enough. With the same arguments as used before we see that there exists small $d_0>0$ such that $J_\varepsilon(v)>J_\varepsilon(u)-\frac{\lambda}{2}>E_m-\lambda \quad\text{for all } v\in X_\varepsilon^{d_0},\; \varepsilon\in(0,\varepsilon_0).$ This completes the proof. \end{proof} Following Corollary \ref{prop5} and Lemma \ref{obs1}, we fix $d_0>0$, $d_1\in (0,d_0/3)$ and corresponding $\omega>0$, $R_0>0$ and $\varepsilon_0>0$ satisfying $$\label{bd1} \begin{gathered} \|J_{\varepsilon}'(u)\|_{(E_\varepsilon^R)'}\geq\omega \quad \text{for all } u\in E_\varepsilon^R\cap J_\varepsilon^{D_\varepsilon} \cap(X_\varepsilon^{d_0}\backslash X_\varepsilon^{d_1}),\\ J_\varepsilon(u)>{E_m}/{2}\quad \text{for all } u\in X_\varepsilon^{d_0} \end{gathered}$$ for any $R\geq R_0$ and $\varepsilon\in(0,\varepsilon_0)$. Thus we obtain the following result. \begin{lemma}\label{prop6} There exists $\alpha>0$ such that $|s-1/t_0|\leq\alpha$ implies $\gamma_\varepsilon(s)\in X_\varepsilon^{d_1}$ for all $\varepsilon\in(0,\varepsilon_0)$, where $\gamma_\varepsilon$ is given by \eqref{gama}. \end{lemma} \begin{proof} At first we observe that \begin{align*} \|\varphi_\varepsilon v\|_\varepsilon &\leq \|(\varphi_\varepsilon v)'\|_{L^2}+\|v\|_{L^2} \Big\{1+\int_{\mathbb{R}}V(\varepsilon x)f^2 \left(\|v\|_{L^2}^{-1}\varphi_\varepsilon v\right)\,\mathrm{d} x\Big\}\\ &\leq \|\varepsilon\varphi'(\varepsilon\cdot) v +\varphi_\varepsilon v'\|_{L^2}+\|v\|_{L^2} \big(1+\sup_\Omega V(x)\big)\\ &\leq C_0\|v\|_{H^1}\quad\text{for all } \varepsilon\in (0,\varepsilon_0),\; v\in H^1(\mathbb{R}). \end{align*} Since the function $\theta:[0,t_0]\to H^1(\mathbb{R})$ given by Proposition \ref{path} is continuous and $\theta(1)=U$ there exists $\sigma>0$ such that $|t-1|\leq\sigma\quad \Rightarrow\quad \|\theta(t)-U\|_{H^1}<\frac{d_1}{C_0}.$ So if $|st_0-1|\leq\sigma$, which means $|s-1/t_0|\leq\sigma/t_0=:\alpha$, this inequality yields $\|\gamma_\varepsilon(s)-\varphi_\varepsilon U\|_\varepsilon =\|\varphi_\varepsilon[\theta(st_0)- U]\|_\varepsilon \leq C_0\|\theta(st_0)- U\|<{d_1}\quad\text{for } \varepsilon\in(0,\varepsilon_0).$ Since $\varphi_\varepsilon U\in X_\varepsilon$ we have $\gamma_\varepsilon(s)\in X_\varepsilon^{d_1}$. \end{proof} \begin{lemma}\label{obs2} For $\alpha$ given in Lemma \ref{prop6} there exist $\rho>0$ and $\varepsilon_0>0$ such that $J_\varepsilon(\gamma_\varepsilon(s))0 satisfying \[ L_m(\theta(t))0 such that \[ \sup_{t\in[0,t_0]}|J_\varepsilon(\varphi_{\varepsilon} \theta(t))-L_m(\theta(t))|<{\rho}{}\quad \text{for } \varepsilon\in(0,\varepsilon_0).$ So for $|t-1|\geq t_0\alpha$ and $\varepsilon\in(0,\varepsilon_0)$ we obtain $J_\varepsilon(\varphi_{\varepsilon}\theta(t)) \leq L_m(\theta (t))+|J_\varepsilon(\varphi_{\varepsilon}\theta(t)) -L_m(\theta(t))| 0 and large R>0 there exists a sequence \{u_n^R\}\subset E^R_\varepsilon\cap X_\varepsilon^{d_0} \cap J_\varepsilon^{D_\varepsilon} such that J_\varepsilon'(u_n^R)\to0 in \left(E_{\varepsilon}^R\right)' as n\to\infty. \end{proposition} \begin{proof} We take R_0>0 such that \Omega\subset B(0,R_0). Then \gamma_\varepsilon([0,1])\subset E_{\varepsilon}^R for all R\geq R_0. Suppose that the statement of Proposition \ref{prop7} does not hold. Then for small \varepsilon>0 and large R>R_0 there exists a(\varepsilon,R)>0 such that  \|J_\varepsilon'(u)\|_{(E_\varepsilon^R)'}\geq a(\varepsilon,R) \quad\text{on } E_\varepsilon^R\cap X_\varepsilon^{d_0}\cap J_\varepsilon^{D_\varepsilon}.  From \eqref{bd1} that there exists \omega independent of \varepsilon\in(0,\varepsilon_0) and R>R_0 satisfying  \|J_\varepsilon'(u)\|_{(E_\varepsilon^R)'}\geq \omega\quad\text{on}\quad E_\varepsilon^R\cap(X_\varepsilon^{d_0}\backslash X_\varepsilon^{d_1}) \cap J_\varepsilon^{D_\varepsilon}.  So there exists a pseudo-gradient vector field, T_\varepsilon^R, for J_\varepsilon on a neighborhood Z_\varepsilon^R\subset E^R_\varepsilon of E_\varepsilon^R\cap X_\varepsilon^{d_0}\cap J_\varepsilon^{D\varepsilon}. We refer to \cite{ST90} for details. Let \tilde Z^R_\varepsilon\subset Z^R_\varepsilon for which one \|J_\varepsilon'(u)\|_{(E_\varepsilon^R)'}> a(\varepsilon,R)/2 and take a Lipschitz continuous function on E_\varepsilon^R, \eta_\varepsilon^R, such that \[ 0\leq\eta_\varepsilon^R\leq1,\quad \eta_\varepsilon^R\equiv 1\text{ on } E_\varepsilon^R\cap X_\varepsilon^{d_0}\cap J_\varepsilon^{D_\varepsilon},\quad \text{and}\quad\eta_\varepsilon^R\equiv 0\text{ on } E_\varepsilon^R\backslash \tilde Z_\varepsilon^R.$ Letting $\xi:\mathbb{R}\to\mathbb{R}^+$ be a Lipschitz continuous function such that $\xi\leq1,\quad\xi(a)=1\quad \text{if } |a-E_m|\leq E_m/2,\quad \text{and}\quad \xi(a)= 0\quad \text{if } |a-E_m|\geq E_m$ and defining $e_\varepsilon^R(u)=\begin{cases} -\eta_\varepsilon^R(u)\xi(J_\varepsilon(u))T_\varepsilon^R(u) &\text{if } u\in Z_\varepsilon^R\\ 0&\text{if } u\in E_\varepsilon^R\backslash Z_\varepsilon^R, \end{cases}$ there exists a global solution $\Psi_\varepsilon^R:E_\varepsilon^R\times\mathbb{R}\to E_\varepsilon^R$, which is unique, of the initial value problem $$\label{edo} \begin{gathered} \frac{d}{d t}\Psi_\varepsilon^R(u,t) = e_\varepsilon^R(\Psi_\varepsilon^R(u,t))\\ \Psi_\varepsilon^R(u,0) = u. \end{gathered}$$ Since $\lim_{\varepsilon\to0}D_\varepsilon=E_m$, we have $D_\varepsilon\leq E_m+(1/2)\min\left\{E_m,\omega^2d_1\right\}$ for small $\varepsilon>0$. Hence, by the choice of $d_0$ and $d_1$, $\Psi^R_\varepsilon$ has the following properties: \begin{itemize} \item[(i)] $\Psi^R_\varepsilon(u,t)=u$ if $t=0$ or $u\in E_\varepsilon^R\backslash Z_\varepsilon^R$ or $J_\varepsilon(u)\notin(0,2E_m)$. \item[(ii)] $\|\frac{d}{d t}\Psi^R_\varepsilon(u,t)\|\leq2$ for all $(u,t)$. \item[(iii)] $\frac{d}{d t}\left(J_\varepsilon \left(\Psi^R_\varepsilon(u,t)\right)\right)\leq0$ for all $(u,t)$. \item[(iv)] $\frac{d}{d t}\left(J_\varepsilon(\Psi^R_\varepsilon(u,t))\right) \leq -\omega^2$ if $\Psi^R_\varepsilon(u,t)\in E_\varepsilon^R\cap (X_\varepsilon^{d_0}\backslash X_\varepsilon^{d_1})\cap J_\varepsilon^{D_\varepsilon}$. \item[(v)] $\frac{d}{d t}(J_\varepsilon(\Psi^R_\varepsilon(u,t))) \leq-(a(\varepsilon,R))^2$ if $\Psi^R_\varepsilon(u,t)\in E_\varepsilon^R\cap X_\varepsilon^{d_1}\cap J_\varepsilon^{D_\varepsilon}$. \end{itemize} Due to Lemmas \ref{prop6} and \ref{obs2}, there exist $\alpha$ and $\rho>0$ such that $|s-1/t_0|\leq\alpha\Longrightarrow\gamma_\varepsilon(s)\in X_\varepsilon^{d_1}\quad \text{and}\quad |s-1/t_0|>\alpha\Longrightarrow J_\varepsilon(\gamma_\varepsilon(s))\alpha it follows that \[ J_\varepsilon(\Psi^R_\varepsilon(\gamma_\varepsilon(s),t)) \leq J_\varepsilon(\gamma_\varepsilon(s))0.$ So \eqref{mmx} holds for any $t_\varepsilon^R$. Now, if $s\in I:=[1/t_0-\alpha,1/t_0+\alpha]$, we get $\gamma_\varepsilon(s)\in X_\varepsilon^{d_1}$ and two distinct cases are considered: \begin{itemize} \item[(a)] $\Psi^R_\varepsilon(\gamma_\varepsilon(s),t) \in X_\varepsilon^{d_0}$ for all $t\in[0,\infty)$. \item[(b)] $\Psi^R_\varepsilon(\gamma_\varepsilon(s),t_s) \notin X_\varepsilon^{d_0}$ for some $t_s>0$. \end{itemize} If $s\in I$ satisfies (a), then (i), (iv) and (v) yield \begin{align*} J_\varepsilon(\Psi^R_\varepsilon(\gamma_\varepsilon(s),t)) &= J_\varepsilon(\gamma_\varepsilon(s)) +\int_0^t\frac{d}{d \tau}\left(J_\varepsilon(\Psi^R_\varepsilon(\gamma_\varepsilon(s), \tau))\right) \mathrm{d} \tau\\ &\leq D_\varepsilon-\min\left\{\omega^2,(a(\varepsilon,R))^2\right\}t \end{align*} and so $J_\varepsilon(\Psi^R_\varepsilon(\gamma_\varepsilon(s),t)) \to-\infty$ as $t\to\infty$ which is in contradiction with \eqref{bd1}. Thus any $s\in I$ satisfies $(b)$. We fix $s_0$ and a neighborhood $I^{s_0}=I^{s_0}(\varepsilon,R)\subset I$ such that $\Psi^R_\varepsilon(\gamma_\varepsilon(s),t_{s_0}) \notin X_\varepsilon^{d_0}$ for all $s\in I^{s_0}$. Since $\gamma_\varepsilon(s)\in X_\varepsilon^{d_1}$ for any $s\in I^{s_0}$, we can observe from $(i)-(v)$ that there exists an interval $[t_s^1,t_s^2]\subset[0,t_{s_0}]$ for which one $\Psi^R_\varepsilon(\gamma_\varepsilon(s),t)\in X_\varepsilon^{d_0} \backslash X_\varepsilon^{d_1}\quad \text{for}\quad t\in[t_s^1,t_s^2] \quad \text{and}\quad |t_s^1-t_s^2|\geq d_1.$ So (i), (iii) and (iv) lead to \begin{align*} J_\varepsilon\left(\Psi^R_\varepsilon(\gamma_\varepsilon(s),t_{s_0}) \right) &\leq J_\varepsilon\left(\gamma_\varepsilon(s)\right) +\int_{t_s^1}^{t_s^2}\frac{d}{d \tau}\left(J_\varepsilon(\Psi^R_\varepsilon (\gamma_\varepsilon(s),\tau))\right)\,\mathrm{d}\tau\\ &\leq D_\varepsilon-\omega^2\left(t_s^2-t_s^1\right)\\ &\leq E_m-\frac{1}{2}\omega^2d_1\quad\text{for all } s\in I^{s_0}. \end{align*} By compactness there exist $s_1,\cdots,s_l$, $l=l(\varepsilon,R)$, such that $I=\bigcup_{i=1}^{l} I^{s_i}$. Let $t_\varepsilon^R=\max_{1\leq i\leq l}t_{s_i}$. Then for any $s\in I$ we have $s\in I^{s_i}$ for some $i$ and so $J_\varepsilon(\Psi^R_\varepsilon(\gamma_\varepsilon(s), t_{\varepsilon}^R)) \leq J_\varepsilon(\Psi^R_\varepsilon(\gamma_\varepsilon(s),t_{s_i})) \leq E_m-\frac{1}{2}\omega^2d_1.$ Therefore, \eqref{mmx} holds. Since ${\gamma}^R_\varepsilon\in\Gamma_\varepsilon$ we obtain $C_\varepsilon\leq\max_{s\in[0,1]}J_\varepsilon({\gamma}^R_\varepsilon(s))\leq E_m-\min\big\{\rho,\frac{\omega^2 d_1}{2}\big\},$ which is in contradiction with Proposition \ref{prop2}. This completes the proof. \end{proof} \begin{proposition}\label{prop8} There exists a critical point $u_\varepsilon\in X_\varepsilon^{d_0} \cap J_\varepsilon^{D_\varepsilon}$ of $J_\varepsilon$ if $\varepsilon>0$ is sufficiently small. \end{proposition} \begin{proof} From Proposition \ref{prop7} there exist $\varepsilon_0>0$ and $R_0>0$ for which ones we can find $\{u_n\}_n \subset E^R_\varepsilon\cap X_\varepsilon^{d_0} \cap J_\varepsilon^{D_\varepsilon}$ such that $J_\varepsilon'(u_n)\to0$ in $\left(E^R_\varepsilon\right)'$ as $n\to\infty$, for each $R\geq R_0$ and $\varepsilon\in(0,\varepsilon_0)$. Since $\{u_n\}_n$ is bounded in $E_\varepsilon^R$ it is also bounded in $H^1_0((-R/\varepsilon,R/\varepsilon))$ with the usual norm. So we may assume that $u_n\rightharpoonup u$ in $H^1_0((-R/\varepsilon,R/\varepsilon))$, $u_n\to u$ in $L^r((-R/\varepsilon,R/\varepsilon))$ for $r=2$ and $4$ and $u_n(x)\to u(x)$ a.e. in $\mathbb{R}$ where $u=u_{\varepsilon,R}$. Because $\|J_\varepsilon'(u_n)\|_{(E_\varepsilon^R)'}\to0$ we see that $u$ is a nonnegative solution for $$\label{pr} -u''=f'(u)\left[h(f(u))-V(\varepsilon x)f(u)\right] -g_{\varepsilon,R}(u)\chi_\varepsilon u\quad \text{in } (-R/\varepsilon,R/\varepsilon)$$ where $g_{\varepsilon,R}(u)=4\Big(\int_{-R/\varepsilon}^{R/\varepsilon} \chi_\varepsilon|u|^2\,\mathrm{d} x-1\Big)_+.$ Then we can see that $u_n\to u$ in $H^1_0((-R/\varepsilon,R/\varepsilon))$ which implies $\int_{B(0,R/\varepsilon)}\left[|u_n'-u'|^2+V(\varepsilon x) f^2(u_n-u)\right]\,\mathrm{d} x \to 0\quad\text{as } n\to \infty$ and so $u_n\to u$ in $E_\varepsilon$. Thus $u\in X_\varepsilon^{d_0}\cap J_\varepsilon^{D_\varepsilon}$. Due to boundedness of $\{u_{\varepsilon,R}\}$ in $H^1(\mathbb{R})$ we get $\|u_{\varepsilon,R}\|_\infty\leq C_0$ for all $R\geq R_0$ and $\varepsilon\in(0,\varepsilon_0)$. So from (H1) and Lemma \ref{lema f} there exists $C>0$ depending on $C_0$ such that $%\label{deur} -u'' \leq Cf'(u)f(u)^2\leq Cu\quad \text{in } (-R/\varepsilon,R/\varepsilon).$ Hence by \cite[Theorem 9.26]{GITRU}, there exists $C_0=C_0(N, C)$ such that $$\label{tru} \sup_{B(y,1)}u\leq C_0\left\|u\right\|_{L^2(B(y,2))}\quad \text{for all } y\in\mathbb{R}.$$ Due to the boundedness of $\{\|u_{\varepsilon,R}\|_\varepsilon\}$ and $\{J_\varepsilon(u_{\varepsilon,R})\}$ we get $\{Q_\varepsilon(u_{\varepsilon,R})\}$ uniformly bounded on $R\geq R_0$ and $\varepsilon\in(0,\varepsilon_0)$. So there is $C_1>0$ such that $$\label{qrbb} \int_{\{|x|\geq R_0/\varepsilon\}}|u_{\varepsilon,R}|^2\,\mathrm{d} x\leq\varepsilon\int_{\mathbb{R}} \chi_\varepsilon|u_{\varepsilon,R}|^2\,\mathrm{d} x\leq \varepsilon C_1$$ for any $R\geq R_0$ and $\varepsilon \in(0,\varepsilon_0)$. Hence for sufficiently small $\varepsilon_0$ and $\varepsilon\in(0,\varepsilon_0)$ fixed, it follows from \eqref{tru}, \eqref{qrbb} and by (H1) $h(f(u_{\varepsilon,R}(x)))\leq \frac{V_0}{2}f(u_{\varepsilon,R}(x)) \quad\text{for any } |x|\geq\frac{R_0}{\varepsilon}+2,\; R\geq R_0.$ Then after some calculations we obtain $$\label{lur} \lim_{A\to\infty}\int_{\mathbb{R}^N\backslash B(0,A)}\left[|u'_{\varepsilon,R}|^2+V(\varepsilon x)f^2(u_{\varepsilon,R})\right]\,\mathrm{d} x=0$$ uniformly on $R\geq R_0$. We take $R_k\to\infty$ and denote $u_k=u_{\varepsilon,{R_k}}$. We may assume $u_k\rightharpoonup u_\varepsilon$ in $H^1(\mathbb{R})$ as $k\to\infty$. Since $u_k$ is a solution for \eqref{pr}, using \eqref{lur} and (\cite{BOMU92}, Theorem 2.1) we see that $\int_{\mathbb{R}}|u_k'|^2\,\mathrm{d} x\to\int_{\mathbb{R}}|u_\varepsilon'|^2\,\mathrm{d} x\quad\text{and}\quad \int_{\mathbb{R}}V(\varepsilon x)f^2(u_k-u_\varepsilon)\,\mathrm{d} x\to0$ as $k\to\infty$, up to subsequences. From this result we get $u_k\to u_\varepsilon$ in $E_\varepsilon$ which implies that $u_\varepsilon\in X_\varepsilon^{d_0}\cap J_\varepsilon^{D_\varepsilon}$ and $J_\varepsilon'(u_\varepsilon)=0$ in $E_\varepsilon'$. This completes the proof. \end{proof} \section{Proof of Theorem \ref{teo1}} Until now we have proved the existence of a critical point for $J_\varepsilon$, $u_\varepsilon\in X_\varepsilon^{d_0}\cap J_{\varepsilon}^{D_\varepsilon}$, for $\varepsilon\in(0,\varepsilon_0)$ with $\varepsilon_0>0$ and $d_0>0$ sufficiently small. We also have $u_\varepsilon\geq0$ and $J_\varepsilon(u_\varepsilon)\geq(E_m/2)$ which imply $u_\varepsilon\neq0$. The function $u_\varepsilon$ satisfies $$\label{ep sol} -u_\varepsilon''=f'(u_\varepsilon)\left[h(f(u_\varepsilon))-V(\varepsilon x)f(u_\varepsilon)\right]-4\Big(\int_{\mathbb{R}}\chi_\varepsilon|u|^2\, \mathrm{d} x-1\Big)_+ \chi_\varepsilon u_\varepsilon\quad \text{in }\mathbb{R}.$$ Since $u_\varepsilon\in \mathcal{C}^{1,\alpha}_{\rm loc}(\mathbb{R})$ by the Maximum Principle we get $u_\varepsilon>0$. Moreover from \eqref{ep sol} we can see that there exists $\rho>0$ such that $\|u_\varepsilon\|_{L^\infty}\geq\rho$ for small $\varepsilon>0$. We observe that by Proposition \ref{prop4} there exists $\{y_\varepsilon\}\subset\mathbb{R}$ such that $\varepsilon y_\varepsilon\in\mathcal M^{2\beta}$ and for any sequence $\varepsilon_n\to0$ there exists $z_0\in\mathcal M$ satisfying $\varepsilon_ny_{\varepsilon_n}\to z_0\quad\text{and}\quad \|u_{\varepsilon_n}-\varphi_{\varepsilon_n}(\cdot-y_{\varepsilon_n}) U(\cdot-y_{\varepsilon_n})\|_{\varepsilon_n} \to 0,$ and so $\|u_{\varepsilon_n}(\cdot+y_{\varepsilon_n})-U\|_{H^1}\to 0.$ Consequently given $\sigma>0$ there exist $A>0$ and $\varepsilon_0>0$ such that $$\label{wep0} \sup_{\varepsilon\in(0,\varepsilon_0)}\int_{\{|x|\geq A\}}u_\varepsilon^2 (x+y_\varepsilon)\,\mathrm{d} x\leq\sigma.$$ Denoting $w_\varepsilon=u_\varepsilon(\cdot+y_\varepsilon)$, the equation \eqref{ep sol} and the uniform boundedness of $\{u_\varepsilon\}$ in $L^\infty(\mathbb{R}^N)$ give us $-w''_\varepsilon\leq C w_{\varepsilon}\quad \text{in } \mathbb{R}.$ Hence from \cite[Theorem 8.17]{GITRU}, there exists $C_0=C_0(C)$ such that $\sup_{(y-1,y+1)}w_\varepsilon(x)\leq C_0\left\|w_\varepsilon\right\|_{L^2((y-2,y+2))}\quad \text{for all } y\in\mathbb{R}.$ From this inequality and by \eqref{wep0} we have $\lim_{|x|\to\infty}w_\varepsilon(x)=0$ uniformly on $\varepsilon$. So we can prove the exponential decay of $w_\varepsilon$ $w_\varepsilon(x)\leq C\exp(-c|x|)\quad\text{for all } x\in\mathbb{R},\; \varepsilon\in(0,\varepsilon_0)$ for some $C,c>0$. Now we consider $\zeta_\varepsilon\in\mathbb{R}$ a maximum point of $w_\varepsilon$. Since $w_\varepsilon(x)\to 0\quad\text{as}\quad |x|\to\infty\quad\text{and}\quad \|w_\varepsilon\|_\infty\geq\rho \quad\text{for all }\varepsilon\in(0,\varepsilon_0)$ we conclude that $\{\zeta_\varepsilon\}$ is bounded. Hence $x_\varepsilon:=\zeta_\varepsilon+y_\varepsilon$ is a maximum point for $u_\varepsilon$ and the following exponential decay holds $$\label{epdecay} u_\varepsilon(x)= w_\varepsilon(x-y_\varepsilon)\leq C\exp\left(-c|x-x_\varepsilon|\right)\quad\text{for all } x\in\mathbb{R}.$$ So $Q_\varepsilon(u_\varepsilon)=0$ for small $\varepsilon$ and $u_\varepsilon$ is a critical point for $P_\varepsilon$. From Proposition \ref{relation} we have $v_\varepsilon =f(u_\varepsilon)$ a positive solution for \eqref{p2}. Since $f$ is increasing, $x_\varepsilon$ is also a maximum point for $v_\varepsilon$. Moreover by the choice of $\{y_\varepsilon\}$ for any sequence $\varepsilon_n\to0$ there are $z_0\in\mathcal M$ and $\zeta_0\in\mathbb{R}$ such that $$\label{limites} \zeta_{\varepsilon_n}\to \zeta_0,\quad \varepsilon_n x_{\varepsilon_n}\to z_0\quad \text{and}\quad \|u_{\varepsilon_n}(\cdot+x_{\varepsilon_n}) -U(\cdot+\zeta_0)\|_{H^1}\to0,$$ up to subsequences. We observe that $U(\cdot+\zeta_0)$ is also a solution of \eqref{p3l} and so $v_0=f(U(\cdot+\zeta_0))$ is a solution of \eqref{p3}. We have \begin{align*} \|v_{\varepsilon_n}(\cdot+x_{\varepsilon_n})-v_0\|_{H^1}^2 &\leq 2\|u_{\varepsilon_n}(\cdot+x_{\varepsilon_n}) -U(\cdot+\zeta_0)\|_{H^1}^2\\ &\quad +2\int_{\mathbb{R}} |f'\left(u_{\varepsilon_n} (x+x_{\varepsilon_n})\right)-f'(U(x+\zeta_0))|^2|U'(x+\zeta_0)|^2 \,\mathrm{d} x \end{align*} and by \eqref{limites} and properties of $f$ we get $v_{\varepsilon_n}(\cdot+x_{\varepsilon_n})\to v_0\quad\text{in } H^1(\mathbb{R})\quad\text{as } n\to\infty.$ At this point we have proved that, for small $\varepsilon$, $\tilde u_{\varepsilon}(x):=v_\varepsilon(x/\varepsilon)$ is a solution for the quasilinear equation \eqref{p1} and satisfies (i)-(ii) in Theorem \ref{teo1} with maximum point $\tilde x_\varepsilon=\varepsilon x_\varepsilon$. \subsection*{Acknowledgments} The author would like to thank professor Jo\~ao Marcos do \'O for his valuable suggestions and comments. \begin{thebibliography}{00} \bibitem{AMWA} A.~Ambrosetti and Z.-Q.~Wang, \emph{Positive solutions to a class of quasilinear elliptic equation on $\mathbb{R}$}, Discrete Contin. 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