\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 15, pp. 1--10.\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-2010/15\hfil Entire solutions] {Entire solutions for a class of $p$-Laplace equations in $\mathbb{R}^2$} \author[Z. Zhou\hfil EJDE-2010/15\hfilneg] {Zheng Zhou} \address{Zheng Zhou \newline College of Mathematics and Econometrics, Hunan University, Changsha, China} \email{zzzzhhhoou@yahoo.com.cn} \thanks{Submitted September 15, 2009. Published January 21, 2010.} \subjclass[2000]{35J60, 35B05, 35B40} \keywords{Entire solution; $p$-Laplace Allen-Cahn equation; \hfill\break\indent Variational methods} \begin{abstract} We study the entire solutions of the $p$-Laplace equation $-\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)+a(x,y)W'(u(x,y))=0, \quad (x,y)\in {\mathbb{R}}^2$ where $a(x,y)$ is a periodic in $x$ and $y$, positive function. Here $W:\mathbb{R}\to\mathbb{R}$ is a two well potential. Via variational methods, we show that there is layered solution which is heteroclinic in $x$ and periodic in $y$ direction. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In this paper we consider the $p$-Laplacian Allen-Cahn equation $$\label{eq1.1} \begin{gathered} -\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)+a(x,y)W'(u(x,y))=0, \quad (x,y)\in {\mathbb{R}}^2\\ \lim_{x\to \pm\infty}u(x,y)=\pm\sigma \quad \text{uniformly w.r.t. }y\in \mathbb{R}. \end{gathered}$$ where we assume $2\sigma$ such that $W(s)>W(R_0)$ for any $|s|>R_0$. \end{itemize} \end{itemize} For example, here we may take $W(t)=\frac{p-1}{p}|\sigma^2-t^2|^{p}$. This is similar with case $p=2$, where the typical examples of $W$ are given by $W(t)=\frac{1}{4}\prod_{i=1}^{k}(t-z_{i})^2$, where $z_{i}$, $i=1,2,\ldots k<\infty$ are zeros of $W(t)$. The case $p=2$ can be viewed as stationary Allen-Cahn equation introduced in 1979 by Allen and Cahn. We recall that the Allen-Cahn equation is a model for phase transitions in binary metallic alloys which corresponds to taking a constant function $a$ and the double well potential $W(t)$. The function $u$ in these models is considered as an order parameter describing pointwise the state of the material. The global minima of $W$ represent energetically favorite pure phases and different values of $u$ depict mixed configurations. In 1978, De Giorgi \cite{De} formulated the following question. Assume $N>1$ and consider a solution $u\in C^2({\mathbb{R}}^{N})$ of the scalar Ginzburg-Laudau equation: $$\label{eq1.2} \Delta u = u(u^2-1)$$ satisfying $|u(x)|\leq1$, $\frac{\partial u}{\partial x_N}>0$ for every $x=(x', x_N)\in {\mathbb{R}}^{N}$ and $\lim\limits_{x_N\to\pm\infty} u(x',x_N)=\pm1$. Then the level sets of $u(x)$ must be hyperplanes; i.e., there exists $g\in C^2(\mathbb{R})$ such that $u(x)=g(ax'-x_n)$ for some fixed $a\in{\mathbb{R}}^{N-1}$. This conjecture was first proved for $N=2$ by Ghoussoub and Gui in \cite{GG} and for $N=3$ by Ambrosio and Cabr\'{e} in \cite{AC}. For $4\leq N\leq8$ and assuming an additional limiting condition on $u$, the conjecture has been proved by Savin in \cite{Sa} . Alessio, Jeanjean and Montecchiari \cite{AJM} studied the equation $-\triangle u+a(x)W'(u)=0$ and obtained the existence of layered solutions based on the crucial condition that there is some discrete structure of the solutions to the corresponding ODE. In \cite{AJc}, when $a(x,y)>0$ is periodic in $x$ and $y$, the authors got the existence of infinite multibump type solutions, where $a(x,y)=a(x,-y)$ takes an important role \cite{AJc}(see also \cite{AJc,R,Ra,Rab,RS1,RS2}). Inherited from the above results, I wonder under what condition p-Laplace type equation \eqref{eq1.1} would have two dimensional layered solutions periodical in $y$. Adapting the renormalized variational introduced in \cite{AJM,AJc} (see also \cite{Ra,Rab}) to the p-Laplace case, we prove \begin{theorem}\label{th1.1} Assume {\rm (H1)--(H2)}. Then there exists entire solution for \eqref{eq1.1}, which behaves heteroclinic in $x$ and periodic in $y$ direction. \end{theorem} \section{The periodic problem} To prove Theorem \ref{th1.1}, we first consider the equation $$\label{eq2.1} \begin{gathered} -\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)+a(x,y)W'(u(x,y))=0, \quad (x,y)\in {\mathbb{R}}^2\\ u(x,y)=u(x,y+1)\\ \lim_{x\to\pm\infty}u(x,y)=\pm\sigma \quad \text{uniformly w.r.t. } y\in \mathbb{R}. \end{gathered}$$ The main feature of this problem is that it has mixed boundary conditions, requiring the solution to be periodic in the $y$ variable and of the heteroclinic type in the $x$ variable. Letting $S_0={\mathbb{R}}\times[0,1]$, we look for minima of the Euler-Lagrange functional $I(u)=\int_{S_0}\frac{1}{p}|\nabla u(x,y)|^p+a(x,y)W(u(x,y))\,dx\,dy$ on the class $$\Gamma=\{u\in W_{\rm loc}^{1,p}(S_0):\|u(x,\cdot)\mp\sigma\|_{L^p(0,1)}\to0.\; x\to\pm\infty\}$$ where $\|u(x_1,\cdot)-u(x_2,\cdot)\|_{L^p(0,1)}^p=\int_0^1 |u(x_1,y)-u(x_2,y)|^pdy$. Setting \begin{gather*} \Gamma_p=\{u\in\Gamma:u(x,0)=u(x,1)\text{for a.e. } x\in{\mathbb{R}}\} \\ c_p=\inf_{\Gamma_p} I\quad \text{and}\quad {\mathcal {K}}_p=\{u\in\Gamma_p : I(u)=c_p\} \end{gather*} Then we use the reversibility assumption (H1)-(ii) to show that the minima $c$ on $\Gamma$ equals minima $c_p$ on $\Gamma_p$, and so solutions of \eqref{eq2.1}. Note the assumptions on $a$ and $W$ are sufficient to prove that $I$ is lower semicontinuous with respect to the weak convergence in $W_{\rm loc}^{1,p}(S_0)$; i.e., if $u_n\to u$ weakly in $W_{\rm loc}^{1,p}(\Omega)$ for any $\Omega$ relatively compact in $S_0$, then $I(u)\leq\liminf_{n\to\infty}I(u_n)$. Moreover we have \begin{lemma} \label{lem2.1} If $(u_n)\subset W_{\rm loc}^{1,p}(S_0)$ is such that $u_n\to u$ weakly in $W_{\rm loc}^{1,p}(S_0)$ and $I(u_n)\to I(u)$, then $I(u)\leq\liminf_{n\to\infty}u_n$ and \begin{gather*} \int_{S_0}a(x,y)W(u_n)\,dx\,dy\to\int_{S_0}a(x,y)W(u)\,dx\,dy\\ \int_{S_0}|\nabla u_n|^p\,dx\,dy\to\int_{S_0}|\nabla u|^p\,dx\,dy \end{gather*} \end{lemma} \begin{proof} Since $u_n\to u$ weakly in $W_{\rm loc}^{1,p}(S_0)$, $\|\nabla u\|_{L^p(S_0)}\leq\liminf_{n\to\infty} \|\nabla u_n\|_{L^p(S_0)}$ by the lower semicontinuous of the norm. By compact embedding theorem, we have $u_n\to u$ in $L_{\rm loc}^p(S_0)$, using pointwise convergence and Fatou lemma, we have $\int_{S_0}a(x,y)W(u)\,dx\,dy\leq\liminf_{n\to\infty} \int_{S_0}a(x,y)W(u_n)\,dx\,dy$, then \begin{align*} \int_{S_0}a(x,y)W(u)\,dx\,dy&\leq \limsup_{n\to\infty} \int_{S_0}a(x,y)W(u_n)\,dx\,dy\\ &= \limsup_{n\to\infty}\Big[I(u_n)-\int_{S_0}\frac{1}{p} |\nabla u_n|^p\,dx\,dy\Big]\\ &= I(u)-\liminf_{n\to\infty}\int_{S_0}\frac{1}{p} |\nabla u_n|^p\,dx\,dy\\ &\leq \int_{S_0}a(x,y)W(u)\,dx\,dy. \end{align*} Thus, $\int_{S_0}a(x,y)W(u_n)\,dx\,dy\to\int_{S_0}a(x,y)W(u)\,dx\,dy$, and since $I(u_n)\to I(u)$, we have $\int_{S_0}|\nabla u_n|^p\,dx\,dy\to\int_{S_0}|\nabla u|^p\,dx\,dy$. \end{proof} By Fubini's Theorem, if $u\in W_{\rm loc}^{1,p}(S_0)$, then $u(x,\cdot)\in W^{1,p}(0,1)$, and for all $x_1,x_2\in{\mathbb{R}}$, we have \begin{align*} \int_0^1|u(x_1,y)-u(x_2,y)|^pdy &= \int_0^1|\int_{x_1}^{x_2} \partial_xu(x,y)dx|^pdy\\ &\leq |x_1-x_2|^{p-1}\int_0^1 \int_{x_1}^{x_2}|\partial_xu(x,y)dx|^p\,dx\,dy\\ &\leq pI(u)|x_1-x_2|^{p-1}. \end{align*} If $I(u)<+\infty$, the function $x\to u(x,\cdot)$ is H\"older continuous from a dense subset of $\mathbb{R}$ with values in $L^p(0,1)$ and so it can be extended to a continuous function on $\mathbb{R}$. Thus, any function $u\in W_{\rm loc}^{1,p}(S_0)\cap\{I<+\infty\}$ defines a continuous trajectory in $L^p(0,1)$ verifying \begin{aligned} \mathrm{d}(u(x_1,\cdot),u(x_2,\cdot))^p &= \int_0^1|u(x_1,y)-u(x_2,y)|^pdy \\ &\leq pI(u)|x_1-x_2|^{p-1}, \forall x_1,x_2\in \mathbb{R}. \end{aligned}\label{eq2.2} \begin{lemma}\label{lem2.2} For all $r>0$, there exists $\mu_r>0$, such that if $u\in W_{\rm loc}^{1,p}(S_0)$ satisfies $\min\|u(x,\cdot)\pm\sigma\|_{W^{1,p}(0,1)}\geq r$ for a.e. $x\in (x_1,x_2)$, then \label{eq2.3} \begin{aligned} &\int_{x_1}^{x_2}\Big[\int_0^1\frac{1}{p}|\nabla u|^p+a(x,y)W(u(x,y))dy\Big]dx\\ &\geq \frac{1}{p(x_2-x_1)^{p-1}}\mathrm{d}(u(x_1,\cdot),u(x_2,\cdot))^p +\frac{p-1}{p}\mu_r^{\frac{p}{p-1}}(x_2-x_1)\\ &\geq \mu_r\mathrm{d}(u(x_1,\cdot),u(x_2,\cdot)) \end{aligned} \end{lemma} \begin{proof} We define the functional $F(u(x,\cdot))=\int_0^1\frac{1}{p}|\partial_yu(x,y)|^p+ \underline{a}W(u(x,y))dy$ on $W^{1,p}(0,1)$, where $\underline{a}=\min_{{\mathbb{R}}^2}a(x,y)>0$. To prove the lemma, we first to claim that: \par For any $r>0$, there exists $\mu_r>0$, such that if $q(y)\in W^{1,p}(0,1)$ is such that $\min\|q(y)\pm\sigma\|_{W^{1,p}(0,1)}\geq r$, then$F(q(y))\geq\frac{p-1}{p}\mu_r^{\frac{p}{p-1}}$. Namely, if $q_n(\cdot)\in W^{1,p}(0,1)$ and $F(q_n)\to0$, then $\min\|q_n\pm\sigma\|_{W^{1,p}(0,1)}\to0$. Assume by contradiction that if $F(q_n)\to0$ and $\min\|q_n\pm\sigma\|_{L^\infty(0,1)}\geq\varepsilon_0>0$. Then there exists a sequence $(y_n^1)\subset[0,1]$ such that $\min|q_n(y_n^1)\pm\sigma|\geq\varepsilon_0$. Since $\int_0^1\underline{a}W(q_n)dy\to0$ there exists a sequence $(y_n^2)\subset[0,1]$ such that $|q_n(y_n^2)\pm\sigma|<\frac{\varepsilon_0}{2}$. Then \begin{align*} \frac{\varepsilon_0}{2} &\leq |q_n(y_n^2)-q_n(y_n^1)|\\ &\leq |\int_{y_n^1}^{y_n^2}|\dot{q}_n(t)|dt~|\\ &\leq |y_n^2-y_n^1|^{1-\frac{1}{p}}\Big[\int_0^1|\dot{q}_n(t)|^pdt\Big] ^{1/p}\\ &\leq p^{\frac{1}{p}}(F(q_n))^{1/p}\to0. \end{align*} It is a contradiction. Since $\min\|q_n\pm\sigma\|_{L^\infty(0,1)}\to0$ as $F(q_n)\to0$, then $\int_0^1|\dot{q}_n(y)|^pdy\to0$, and it follows that $\|q_n-\sigma\|_{W^{1,p}(0,1)}\to0$ as $F(q_n)\to0$. Observe that if $(x_1,x_2)\subset\mathbb{R}$ and $u\in W_{\rm loc}^{1,p}(S_0)$ are such that $F(u(x,\cdot))\geq\frac{p-1}{p}\mu_r^{\frac{p}{p-1}}$ for a.e. $x\in(x_1,x_2)$, by H\"older's and Yung's inequalities we have \begin{align*} & \int_{x_1}^{x_2}\Big[\int_0^1\frac{1}{p}|\nabla u|^p+a(x,y)W(u(x,y))dy\Big]dx\\ &\geq \int_{x_1}^{x_2}\int_0^1\frac{1}{p}|\partial_xu|^p\,dy\,dx+ \int_{x_1}^{x_2}\int_0^1\frac{1}{p}|\partial_yu|^p+\underline{a}W(u)\,dy\,dx\\ &= \frac{1}{p}\int_0^1\int_{x_1}^{x_2}|\partial_xu|^p\,dx\,dy+ \int_{x_1}^{x_2}F(u(x,\cdot))dx\\ &\geq \frac{1}{p(x_2-x_1)^{p-1}}\mathrm{d}(u(x_1,\cdot),u(x_2,\cdot))^p +\frac{p-1}{p}\mu_r^{\frac{p}{p-1}}(x_2-x_1)\\ &\geq \mu_r\mathrm{d}(u(x_1,\cdot),u(x_2,\cdot)). \end{align*} The proof is complete. \end{proof} As a direct consequence of Lemma \ref{lem2.2}, we have the following result. \begin{lemma}\label{lem2.3} If $u\in W_{\rm loc}^{1,p}(S_0)\cap \{I<+\infty\}$, then $\mathrm{d}\big(u(x,\cdot),\pm\sigma\big)\to0$ as $x\to\pm\infty$. \end{lemma} \begin{proof} Note that since $I(u)=\int_{S_0}\frac{1}{p}|\nabla u|^p+a(x,y)W(u(x,y))\,dx\,dy<+\infty,$ $W(u(x,y))\to0$ as $|x|\to+\infty$. Then by Lemma \ref{lem2.2}, $\liminf_{x\to+\infty}\mathrm{d}\big(u(x,\cdot),\sigma\big)=0$. Next we show that $\limsup_{x\to+\infty} \mathrm{d}\big(u(x,\cdot), \sigma\big)=0$ by contradiction. We assume that there exists $r\in(0,\sigma/4)$ such that $\limsup_{x\to+\infty}\mathrm{d}(u(x,\cdot),\sigma)>2r$, by \eqref{eq2.2} there exists infinite intervals $(p_i,s_i),i\in \mathbb{N}$ such that $\mathrm{d}\big(u(p_i,\cdot),\sigma\big)=r$, $\mathrm{d}\big(u(s_i,\cdot),\sigma\big)=2r$ and $r\leq \mathrm{d}\big(u(x,\cdot),\sigma\big)\leq2r$ for $x\in\cup_i(p_i,s_i)$, $i\in \mathbb{N}$ by Lemma \ref{lem2.2}, this implies $I(u)=+\infty$, it's a contradiction. Similarly, we can prove that $\lim_{x\to-\infty}\mathrm{d}\big(u(x,\cdot),-\sigma\big)=0$. \end{proof} Now we consider the functional on the class $$\Gamma=\{u\in W_{\rm loc}^{1,p}(S_0):I(u)<+\infty, ~{\rm d}\big(u(x,\cdot),\pm\sigma\big)\to0{\rm ~as~x\to\pm\infty}\}$$ Let $$\label{eq2.4} c=\inf_\Gamma I\quad\text{and}\quad {\mathcal {K}}= \{u\in\Gamma:I(u)=c\}$$ We will show that $\mathcal {K}$ is not empty, and we start noting that the trajectory in $\Gamma$ with action close to the minima has some concentration properties. For any $\delta>0$, we set $$\label{eq2.5} \lambda_\delta=\frac{1}{p}\delta^p+\max_{\mathbb{R}^2}a(x,y)\cdot \max_{|s\pm\sigma|\leq p^{1/p}\delta}W(s).$$ \begin{lemma}\label{lem2.4} There exists $\bar{\delta}_0\in(0,\sigma/2)$ such that for any $\delta\in(0,\bar{\delta}_0)$ there exists $\rho_\delta>0$ and $l_\delta>0$, for which, if $u\in\Gamma$ and $I(u)\leq c+\lambda_\delta$, then \begin{itemize} \item[(i)] $\min\|u(x,\cdot)\pm\sigma\|_{W^{1,p}(0,1)}\geq\delta$ for a.e. $x\in(s,p)$ then $p-s\leq l_\delta$. \item[(ii)] if $\|u(x_-,\cdot)+\sigma\|_{W^{1,p}(0,1)}\leq\delta$, then $\mathrm{d}(u(x_-,\cdot),-\sigma)\leq\rho_\delta$ for any $x\leq x_-$, and if $\|u(x_+,\cdot)-\sigma\|_{W^{1,p}(0,1)}\leq\delta$, then $\mathrm{d}(u(,\cdot),\sigma)\leq\rho_\delta$ for any $x\geq x_+$. \end{itemize} \end{lemma} \begin{proof} By Lemma \ref{lem2.2}, as in this case, there exists $\mu_\delta>0$ such that $\int_s^p\int_0^1\frac{1}{p}|\nabla u|^p+a(x,y)W(u)\,dx\,dy\geq \mu_\delta(p-s).$ Since $I(u)\leq c+\lambda_\delta$ there exists $l_\delta<+\infty$ such that $p-s2\sigma$, there exists $\bar{x}\in\mathbb{R}$ such that $u(\bar{x},\cdot)\notin B_R$ for $u\in\Gamma\cap\{I(u)\leq c+\lambda\},\lambda>0$, such that $\|u(\bar{x},\cdot)\|_{L^p(0,1)}\geq R$, then $\mathrm{d}(u(\bar{x},\cdot),\sigma)\geq\|u(\bar{x},\cdot)\|_{L^p(0,1)} -\|\sigma\|_{L^p(0,1)}\geq R-\sigma$. Since $\mathrm{d}(u(x,\cdot),\pm\sigma)\to0$ as $x\to\pm\infty$, by continuity there exists $x_1>\bar{x}$ such that $\mathrm{d}(u(x_1,\cdot),\sigma)\leq\sigma/2$ and $\mathrm{d}(u(x,\cdot),\sigma)\geq\sigma/2$ for $x\in(\bar{x},x_1)$. Using Lemma 2.2, we get $$c+\lambda\geq I(u)\geq \mu_{\sigma/2}\mathrm{d}(u(x_1,\cdot), u(\bar{x},\cdot)) \geq \mu_{\sigma/2}(R-3\sigma/2).$$ which is a contradiction for $R$ large enough. We conclude that $(u_n)$ is bounded in $W_{\rm loc}^{1,p}(S_0)$, thus there exists $u_0\in W_{\rm loc}^{1,p}(S_0)$ such that up to a sequence, $u_n\to u_0$ weakly in $W_{\rm loc}^{1,p}(S_0)$. We shall prove that $u_0\in\Gamma$; i.e., $\mathrm{d}(u_0(x,\cdot),\pm\sigma)\to0$ as $x\to\pm\infty$. First we claim that: \begin{quote} For any small $\varepsilon>0$, there exists $\lambda(\varepsilon)\in(0,\lambda_{\bar{\delta}})$ and $l(\varepsilon)>l_{\bar{\delta}}$ such that if $u\in\Gamma\cap\{I(u)\leq c+\lambda(\varepsilon)\}$ then $$\label{eq2.8} \int_{|x-X(u)|\geq l(\varepsilon)}\int_0^1 W(u(x,y))\,dy\,dx\leq\varepsilon.$$ \end{quote} Indeed, let $\delta<\bar{\delta}$ be such that $3\lambda_\delta\leq\underline{a}w_1\varepsilon$ where $\underline{a}=\min_{{\mathbb{R}}^2}a(x,y)$. Given any $u\in\Gamma\cap\{I(u)\leq c+\lambda_\delta\}$, by Lemma \ref{lem2.4}, there exists $x_-\in(X(u)-l_\delta,X(u))$ and $x_+\in(X(u),X(u)+l_\delta)$ such that $\|u(x_-,\cdot)+\sigma)\|_{W^{1,p}(0,1)}\leq\delta$ and $\|u(x_+,\cdot)-\sigma\|_{W^{1,p}(0,1)}\leq\delta$. We define the function $\tilde{u}(x,y)=\begin{cases} -\sigma& \text{if } xx_++1 \end{cases}$ which belongs to $\Gamma$, and $I(\tilde{u})\geq c$, \begin{align*} &\int_{|x-X(u)|\geq l_\delta}\int_0^1\frac{1}{p}|\nabla u|^p+a(x,y)W(u)\,dy\,dx\\ &\leq I_{-\infty}^{x_-}(u)+I_{x_+}^{+\infty}(u)\\ &=I(u)-I(\tilde{u})+I_{x_--1}^{x_-}(\tilde{u}) +I_{x_+}^{x_++1}(\tilde{u}) \\ &\leq 3\lambda_\delta \end{align*} then \eqref{eq2.8} follows setting $l(\varepsilon)=l_{\bar{\delta}}$ and $\lambda(\varepsilon)=\lambda_\delta$. From \eqref{eq2.8} it is easy to see that $u(x,y)\to \sigma$ as $x\to +\infty$. Combining \eqref{eq2.8} and \eqref{eq2.7} we obtain $\int_{|x-X(u)|\geq l(\varepsilon)}\int_0^1 w_1|u(x,y)-\sigma|^p\,dx\,dy \leq \int_{|x-X(u)|\geq l(\varepsilon)}\int_0^1W(u(x,y))\,dy\,dx \leq\varepsilon;$ i.e., $\mathrm{d}\big(u(x,\cdot),\sigma\big)\to0$ as $x\to+\infty$. Analogously, we can get that $\mathrm{d}\big(u(x,\cdot),-\sigma\big)\to0$ as $x\to-\infty$, it follows that $u_0\in\Gamma$. \end{proof} As a consequence, we get the following existence result. \begin{proposition}\label{p2.1} $\mathcal {K}\neq\emptyset$ and any $u\in\mathcal {K}$ satisfies $u\in C^{1,\alpha}({\mathbb{R}}^2)$ is a solution of $-\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)+a(x,y)W'(u(x,y))=0$ on $S_0$ with $\partial_yu(x,0)=\partial_yu(x,1)=0$ for all $x\in\mathbb{R}$, and $\|u\|_{L^\infty}(S_0)\leq R_0$. Finally, $u(x,y)\to\pm\sigma$ as $x\to\pm\infty$ uniformly in $y\in[0,1]$. \end{proposition} \begin{proof} By Lemma \ref{lem2.5}, the set $\mathcal {K}$ is not empty. By $(H_2)$, $\|u\|_{L^\infty(S_0)}\leq R_0$. Indeed, $\tilde{u}=\max\{-R_0,\min\{R_0,u\}\}$ is a fortiori minimizer. Let $\eta\in C_0^\infty(S_0)$ and $\tau\in\mathbb{R}$, then $u+\tau\eta\in\Gamma$ and since $u\in\mathcal {K}$, $I(u+\tau\eta)$ is a $C^1$ function of $\tau$ with a local minima at $\tau=0$. Therefore, $I'(u)\eta=\int_{S_0}|\nabla u|^{p-2}\nabla u\nabla\eta+aW'(u)\eta \,dx\,dy=0$ for all such $\eta$, namely $u$ is a weak solution of the equation $-\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)+a(x,y)W'(u(x,y))=0$ on $S_0$. Standard regularity arguments show that $u\in C^{1,\alpha}(S_0)$ for some $\alpha\in(0,1)$ and satisfies the Neumann boundary condition (see \cite{GT}\cite{L2}\cite{To}). Since $\|u\|_{L^\infty(S_0)}\leq R_0$, there exists $C>0$ such that $\|u\|_{C^{1,\alpha}(S_0)}\leq C$, which guarantees that $u$ satisfies the boundary conditions. Indeed, assume by contradiction that $u$ does not verify $u(x,y)\to-\sigma$ as $x\to-\infty$ uniformly with respect to $y\in[0,1]$. Then there exists $\delta>0$ and a sequence $(x_n,y_n)\in S_0$ with $x_n\to-\infty$ and $|u(x_n,y_n)+\sigma|\geq2\delta$ for all $n\in\mathbb{N}$. The $C^{1,\alpha}$ estimate of $u$ implies that there exists $\rho>0$ such that $|u(x,y)+\sigma|\geq\delta$ for $\forall\,(x,y)\in B_\rho(x_n,y_n),n\in\mathbb{N}$. Along a subsequence $x_n\to-\infty,~y_n\to y_0\in[0,1],~|u(x,y)+\sigma|\geq\delta$ for $(x,y)\in B_{\rho/2}(x_n,y_0)$, which contradicts with the fact that $\mathrm{d}(u(x,\cdot),-\sigma)\to0$ as $x\to-\infty$ since $u\in\Gamma$. The other case is similar. \end{proof} We shall explore the reversibility condition of (H1)-(ii), and we will prove that the minimizer on $\Gamma$ is in fact a solution of \eqref{eq2.1}. \begin{lemma} \label{lem2.6} $c_p=c$. \end{lemma} \begin{proof} Since $\Gamma_p\subset\Gamma$, $c_p\geq c$. Assume by contradiction that $c_p>c$, then there exists $u\in\Gamma$ such that \$I(u)