\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 270, pp. 1--21.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/270\hfil Existence of bounded solutions of Neumann problem] {Existence of bounded solutions of Neumann problem for a nonlinear degenerate elliptic equation} \author[S. Bonafede \hfil EJDE-2017/270\hfilneg] {Salvatore Bonafede} \address{Salvatore Bonafede \newline Dipartimento di Agraria, Universit\`a degli Studi Mediterranea di Reggio Calabria, Localit\'a Feo di Vito - 89122 Reggio Calabria, Italy} \email{salvatore.bonafede@unirc.it} \dedicatory{Communicated by Pavel Drabek} \thanks{Submitted April 7, 2017. Published Ocotber 31, 2017.} \subjclass[2010]{35J60, 35J70, 35B53, 35B40} \keywords{Degenerate elliptic equations; Neumann problem; \hfill\break\indent Phragm\'en-Lindel\"of theorem; asymptotic behavior} \begin{abstract} We prove the existence of bounded solutions of Neumann problem for nonlinear degenerate elliptic equations of second order in divergence form. We also study some properties as the Phragm\'en-Lindel\"of property and the asymptotic behavior of the solutions of Dirichlet problem associated to our equation in an unbounded domain. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} We consider the equation \begin{equation} \label{e1.1} \sum_{i =1}^{m} \frac{\partial}{\partial x_i} a_i (x, u, \nabla u) - c_0 |u|^{p-2} u = f(x, u, \nabla u) \quad \text{in } \Omega, \end{equation} where $\Omega$ is a bounded open set of $\mathbb{R}^{m}$, $m\geq 2$, $c_0$ is a positive constant, $\nabla u$ is the gradient of unknown function $u$ and $f$ is a nonlinear function which has the growth of rate $p$, $1 < p < m$, respect to gradient $\nabla u$. We assume that the following degenerate ellipticity condition is satisfied, \begin{equation} \label{e1.2} \lambda (|u|) \sum_{i=1}^m a_i (x,u, \eta) \eta_i \geq \nu(x) |\eta|^p, \end{equation} where $ \eta= (\eta_1 , \eta_2 , \dots ,\eta_m )$, $|\eta|$ denotes the modulus of $\eta$, $\nu$ and $\lambda$ are positive functions with properties to be specified later on. We study the nonlinear Neumann boundary problem for \eqref{e1.1} with the boundary condition \begin{equation}\label{e1.3} \sum_{i=1}^m a_i (x,u, \nabla u) \cos (\overrightarrow {n},x_i ) + c_2 |u |^{p-2}u + F(x,u) =0 \quad (c_2 > 0), \; x\in \partial \Omega, \end{equation} where $\partial \Omega$ is locally Lipschitz boundary (see \cite{Adam}) and $\overrightarrow {n} = \overrightarrow {n}(x)$ is the outwardly directed (relative to $\Omega$) unit vector normal to $\partial \Omega$ at every point $x \in \partial \Omega$. Many results for linear and quasilinear elliptic equations of second order have been established starting with pionering papers \cite{LadUra,Miranda}, and arriving to the most recent \cite{BDY,DMR,TPP, ZLP, XK}. For example, in the very recent paper \cite{ZLP} the existence of positive solutions for p-Laplacian, with nonlinear Neumann boundary conditions, is proved by a priori estimates and topological methods. The Dirichlet problem for the equation of the type \eqref{e1.1} in nondegenerate case on bounded domains was studied by Boccardo, Murat and Puel in \cite{Bocc1, Bocc2}, using the method of sub and supersolutions. Afterwards, Drabek and Nicolosi in \cite{DrabNic}, assuming condition \eqref{e1.2}, studied the weak solvability of general boundary value problem for equation \eqref{e1.1}, obtaining more general results than \cite{Bocc1, Bocc2}. Let us also mention, on the related topic and in degenerate-case, \cite{Bon1, Bon2} and \cite{GuglNicol, GuglNic}. In this article the basic idea of \cite{DrabNic} is used: the question of the existence of solutions is handled by priori estimates, in the energy space corresponding to the given problem and in $L^{\infty}$, together with the theory of equations with pseudomonotone operators. This article is organized as follows. In Section 2 we formulate the hypotheses, we state our problem and the main existence theorem. Section 3 consists of preliminary assertions which are sufficient in the proof of our main results. In Section 4 we prove the existence theorem and we give an example where all our assumptions are satisfied. In Section 5 we study asymptotic behavior of the solution of the Dirichlet problem associated to equation \eqref{e1.1} in an unbounded domain. Finally, in Section 6 we shall show that a theorem, like the Phragm\'{e}n-Lindel\"{o}f one, holds for Dirichlet problem, in the case of $p$-Laplacian, in a cylindrical unbounded domain of $\mathbb{R}^{m}$; the analogous question for higher-order linear equations was first investigated by P.D. Lax in \cite{Lax}. \section{Hypotheses and formulation of the main results} We shall suppose that $\mathbb{R}^{m}$ ($m \geq 2$) is the $m$-dimensional Euclidean space with elements $x=(x_{1}, x_{2}, \ldots, x_{m})$. Let $\Omega$ be an open bounded nonempty subset of $\mathbb{R}^{m}$, $\partial \Omega$ be locally lipschitzian. The symbols $\operatorname{meas}_m(\cdot)$ and $\operatorname{meas}(\cdot)$ will denote the $m$-dimensional Lebesguel measure and the $(m-1)$-dimensional Hausdorff measure, respectively. We denote by $L^q (\partial \Omega)$, $(1\leq q < \infty)$ the Lebesgue space of $q$-summable functions on $\partial \Omega$ with respect to the $(m-1)$-dimensional Hausdorff measure, with obvious modifications if $q=\infty$. Let $p$ be a real number such that $1< p 0$ it is $u|u|^{\gamma} \in W$. Details concerning this assertion can be found in Guglielmino and Nicolosi \cite{GuglNicol}. \end{remark} We have alsothe following hypotheses \begin{itemize} %\label{2.2} \item[(H2)] There exists $t> \frac {m}{p - 1}$ such that $$ \frac{1}{\nu (x)} \in L^{t}(\Omega). $$ \end{itemize} From (H1) and (H2) there is a continuous inclusion $\xi$ of $W^{1,p} (\nu, \Omega)$ in $W^{1, p\tau} (\Omega)$, where $\tau = (1 + \frac{1}{t} ) ^{-1}$. So, from Sobolev embedding, if we set $$ p^\star = \frac{mp}{m-p + m/t}, $$ then, we have ${W}^{1,p}(\nu ,\Omega) \subset L^{p^\star} (\Omega)$ and there exists $\hat{c} >0$ depending only on $m, p , t, \Omega$ and $\|1/\nu \|_{L^t (\Omega)}$ such that for every $u \in {W}^{1,p}(\nu ,\Omega)$ $$ \Big( \int_{\Omega}|u|^{p^\star}\,dx \Big)^{1/{p^\star}} \leq \hat{c} \|u\|_{1,p}. $$ In this connection see, for instance, \cite{GuglNic}, \cite{kuf} and \cite[Theorem 3.1]{MurtStamp}. Next, by the theorem of trace for Sobolev spaces (see for instance \cite[Cap. 2, pag.77]{N} or \cite{LadUra}), we know that for any $u \in W^{1, p\tau} (\Omega)$, there exists a unique element $\gamma_0 u \in L^{\tilde p} (\partial \Omega)$ where $$ \tilde p = p \tau (m-1) (m - p \tau)^{-1} = \frac{(m-1)p}{m-p+ m/t} $$ and, the mapping $\gamma_0$ is continuous linear from $W^{1, p\tau} (\Omega)$ to $ L^{\tilde p} (\partial \Omega)$. Obviously, $\gamma_0 \circ \xi$ is a continuous linear map of $W^{1,p} (\nu, \Omega)$ to $L^{\tilde p} (\partial \Omega)$ and for $u|_{\partial \Omega} = (\gamma_0 \circ \xi)(u)$, the trace of $u$ on $\partial \Omega$, the following inequality holds: $$ \Big( \int_{\partial \Omega}|u|_{\partial \Omega}|^{\tilde{p}}\,ds \Big)^{1/{\tilde{p}}} \leq c' \|u\|_{1,p}, \quad \text{for all } u \in {W}^{1,p}(\nu ,\Omega), $$ where $c'$ is a positive constant depending only on $m, p , t, \Omega$ and $\|1/\nu \|_{L^t (\Omega)}$ . When clear from the context, for $u \in W^{1,p}(\nu, \Omega)$, we shall write $u$ instead of $u|_{\partial \Omega}$. \begin{remark} \label{rmk2.3} \rm Hypotheses (H1) and (H2) imply that ${W}^{1,p}(\nu ,\Omega)$ is compactly embedded in $L^p (\Omega)$. The proof of this assertion is the same as that for $p=2$ (see \cite{GuglNic}). Furthermore, as the linear and continuous map $\gamma_0$ from $W^{1, p\tau} (\Omega)$ in $ L^q (\partial \Omega)$ is compact for every $q$: $1\leq q < \tilde p$ (see \cite[Cap. 2, pag.103]{N}), then, it is also compact the embedding $\gamma_0 \circ \xi$ of ${W}^{1,p}(\nu ,\Omega)$ in $L^q (\partial \Omega)$. It will be useful to note that ${W}^{1,p}(\nu ,\Omega)$ is reflexive. For the proof of this fact it is possible to use the same procedure as in \cite[pag.46]{Adam}. \end{remark} We need the following structural hypotheses: \begin{itemize} \item[(H3)] The functions $f(x,u,\eta)$, $a_i(x,u,\eta)$ ($i=1,2,\ldots, m$) are Caratheodory functions in $\Omega \times \mathbb{R} \times \mathbb{R}^{m}$, i.e. measurable with respect to $x$ for every $(u, \eta) \in \mathbb{R} \times \mathbb{R}^{m}$ and continuous with respect to $(u,\eta)$ for almost all $x \in\Omega$. \item[(H4)] The function $F(x,u)$ is a Caratheodory function in $\partial \Omega \times \mathbb{R}$, i.e. measurable with respect to $x$ for every $u \in \mathbb{R} $ and continuous with respect to $u$ for almost all $x \in \partial \Omega$. \item[(H5)] There exist a number $\sigma$ and a function $f^{*}(x)$ such that \begin{gather} \max \big( 0, \frac {2-p}{2} \big) < \sigma < 1 , \quad f^{*}\in L^{1}(\Omega), \nonumber \\ |f(x,u,\eta)| \leq \lambda(|u|) \Big[ f^{*}(x) + |u|^{p-1+ \sigma} + \big( \nu^{1/p}(x) |\eta| \big)^{p-1+ \sigma} + \nu(x) |\eta|^{p} \Big] \label{e2.1} \end{gather} holds for almost all $x \in \Omega$ and for all real numbers $u,\eta_{1}, \eta_{2}, \ldots, \eta_m$ \item[(H6)] %\label{2.6} There exists a function $F^{*} \in L^{\infty}(\partial \Omega)$ such that \begin{equation} \label{e2.2} |F(x,u)| \leq \lambda(|u|) + F^{*}(x) \end{equation} holds for almost all $x \in \partial \Omega$ and for every $u \in \mathbb{R}$. \item[(H7)] %\label{2.7} There exists a function $F_0 \in L^{\infty}(\partial \Omega)$ such that \begin{equation} \label{e2.3} uF(x,u) + F_0(x) \geq 0 \end{equation} holds for almost all $x \in \partial \Omega$ and for every $u \in \mathbb{R}$. \item[(H8)] %\label{2.8} There exist a nonnegative number $c_{1}< c_0$ and a function $f_0 \in L^{\infty}(\Omega)$ such that for almost all $x \in \Omega$ and for all real numbers $u,\eta_{1}, \eta_{2}, \ldots, \eta_{m}$, \begin{equation} \label{e2.4} u f(x,u, \eta) + c_{1} |u|^{p} + \lambda(|u|) \nu(x) |\eta|^{p} + f_0(x) \geq 0\,. \end{equation} \item[(H9)] %\label{2.9} There exists a function $a^{*} \in L^{p/(p-1)}(\Omega)$ such that for almost all $x \in \Omega$ and for real numbers $ u,\eta_{1}, \eta_{2}, \ldots, \eta_{m}$, \begin{equation} \label{e2.5} \frac{|a_i(x,u,\eta)|}{\nu^{1/p}(x)} \leq \lambda(|u|) \big[ a^{*}(x) + |u|^{p-1} +\nu^{(p-1)/p}(x) |\eta|^{p-1} \big]\,. \end{equation} \item[(H10)] % \label{2.10} Condition \eqref{e1.2} is satisfied for almost all $x \in \Omega$ and for all real numbers $u,\eta_{1}, \eta_{2}, \ldots, \eta_{m}$; the function $\lambda:[0,+\infty) \to [1,+\infty)$ is monotone and nondecreasing. \item[(H11)] %\label{2.11} For almost all $x \in \Omega$ and all real numbers $u$, $\eta_{1}$, $\eta_{2}$, \ldots, $\eta_{m}$, $\tau_{1}, \tau_{2}$, \ldots , $\tau_{m}$, the inequality \begin{equation} \label{e2.6} \sum_{i=1}^{m} \big[ a_i(x,u,\eta)-a_i(x,u,\tau) \big] (\eta_i- \tau_i) \geq 0 \end{equation} holds while the inequality holds if and only if $\eta \neq \tau$. \end{itemize} In this article we study the problem of finding a function $u \in W$ such that \begin{equation} \label{e2.7} \begin{aligned} &\int_{\Omega} \Big\{ \sum_{i=1}^{m} a_i(x,u, \nabla u) \frac{\partial w}{\partial x_i} +c_0 |u|^{p-2} u w + f(x,u, \nabla u) w \Big\} \,dx \\ &+ \int_{\partial \Omega} \{ c_{2} |u|^{p-2}u w + F(x,u) w\}\,ds =0 \end{aligned} \end{equation} holds for every $w \in W$. Hypotheses (H1)--(H6)and (H10) provide the correctness for this problem. We shall prove the following result: \begin{theorem}\label{thm2.1} Let {\rm (H1)--(H11)} be satisfied. Then \eqref{e2.7} has at least one solution. \end{theorem} \section{Auxiliary results} The first result of this section is an a priori estimate in $L^{\infty} (\Omega) \cap L^{\infty} (\partial \Omega)$ for every solution of \eqref{e2.7}. \begin{lemma}\label{lem3.1} Let {\rm (H1)--(H10)} be satisfied and let $u$ be a solution of \eqref{e2.7}. Then \begin{equation} \label{e3.1} \|u\|_{L^{\infty}(\Omega)} + \|u\|_{L^{\infty}(\partial \Omega)} \leq K \end{equation} where \[ K=2 \big\{ \frac{2}{c_{3}} [ \|f_0\|_{L^{\infty}(\Omega)} + \|F_0\|_{L^{\infty}(\partial \Omega)} ] \big\}^{1/p}, \quad c_{3} = \min (c_{2}, c_0-c_{1}). \] \end{lemma} \begin{proof} Let us take $w=u|u|^{\gamma}$ as a test function in \eqref{e2.7} (see Remark \ref{rmk2.2}) where $\gamma$ is a positive number. We deduce that \begin{align*} &\int_{\Omega} |u|^{\gamma} \Big\{ (\gamma +1) \sum_{i=1}^{m} a_i(x,u, \nabla u) \frac{\partial u}{\partial x_i} +c_0 |u|^{p} + \ f(x,u, \nabla u) u \Big\} \,dx \\ &+ \int_{\partial \Omega} \Big\{ c_{2} |u|^{\gamma + p} + F(x,u)u |u|^{\gamma} \Big\}\,ds =0. \end{align*} By using (H7), (H8) and (H10) we obtain \begin{align*} &\int_{\Omega} |u|^{\gamma} \Big\{ \big[\frac{\gamma +1}{\lambda(\|u\|_{L^{\infty}(\Omega)})} -\lambda(\|u\|_{L^{\infty}(\Omega)}) \big] \nu |\nabla u|^{p} +(c_0-c_{1})|u|^{p} -f_0 \Big\}\,dx \\ & + \int_{\partial \Omega}\{ c_{2} |u|^{\gamma + p} - F_0 |u|^{\gamma} \}\,ds \leq 0. \end{align*} Set $\gamma$ such that $\gamma >[\lambda(\|u\|_{L^{\infty}(\Omega)})]^2 - 1$, from the above inequality it follows that $$ c_3\Big[ \int_{\Omega} |u|^{\gamma+p} \,dx +\int_{\partial \Omega} |u|^{\gamma+p}\,ds \Big] \leq \int_{\Omega}|f_0||u|^{\gamma}\,dx + \int_{\partial \Omega}|F_0||u|^{\gamma}\,ds. $$ Then, by H\"older's inequality \begin{align*} &c_3\Big[ \int_{\Omega} |u|^{\gamma+p} \,dx +\int_{\partial \Omega} |u|^{\gamma+p}\,ds \Big]\\ &\leq \Big[ \Big( \int_{\Omega} |u|^{\gamma+p} \,dx \Big)^{\frac{\gamma}{\gamma+p}} + \Big( \int_{\partial \Omega} |u|^{\gamma+p}\,ds \Big)^{\frac{\gamma}{\gamma+p}} \Big] \\ &\quad\times \Big[ \Big( \int_{\Omega} |f_0|^{(\gamma+p)/p} \,dx \Big)^{\frac{p}{\gamma+p}} + \Big( \int_{\partial \Omega} |F_0|^{(\gamma+p)/p}\,ds \Big)^{\frac{p}{\gamma+p}} \Big]. \end{align*} The above inequality implies \begin{align*} &\Big( \int_{\Omega} |u|^{\gamma+p} \,dx \Big)^{\frac{p}{\gamma+p}} + \Big( \int_{\partial \Omega} |u|^{\gamma+p}\,ds \Big)^{\frac{p}{\gamma+p}} \\ &\leq \frac{2^{\frac{p}{p+\gamma}+1}}{c_3} \big\{\|f_0\|_{L^{\infty}(\Omega)}(\operatorname{meas}_m \Omega)^{\frac{p}{\gamma+p}} + \|F_0\|_{L^{\infty}(\partial \Omega)} (\operatorname{meas} \partial \Omega)^{\frac{p}{\gamma+p}}\big\} \end{align*} Letting $\gamma \to +\infty$ we obtain \eqref{e3.1}. The proof is complete. \end{proof} The second result of this Section is an a priori estimate for every solution $u$ of \eqref{e2.7}, in the norm of ${W}^{1,p}(\nu ,\Omega)$. \begin{lemma}\label{lem3.2} Let {\rm (H1)--(H10)} be satisfied and let $u$ be a solution of \eqref{e2.7}. Then there exists a constant $M>0$ such that $$ \|u\|_{1,p} \leq M , $$ where $M$ depends only on $c_0$, $c_{1}$, $c_{2}$, $\sigma$, $p$, $\|f_0\|_{L^{\infty}(\Omega)}$, $\|f^{*}\|_{L^1(\Omega)}$, $\lambda(s)$, $\operatorname{meas}_m\Omega$, $\operatorname{meas} \partial \Omega$ and $ \|F_0\|_{L^{\infty}(\partial \Omega)}$. \end{lemma} \begin{proof} We have (see the proof of the Lemma \ref{lem3.1}): \begin{align*} &\int_{\Omega} |u|^{\gamma} \Big\{ \big[\frac{\gamma +1}{\lambda(\|u\|_{L^{\infty}(\Omega)})} -\lambda(\|u\|_{L^{\infty}(\Omega)}) \big] \nu |\nabla u|^{p} +(c_0-c_{1})|u|^{p} \Big\}\,dx \\ & + \int_{\partial \Omega} c_{2} |u|^{\gamma + p}\,ds \\ &\leq \int_{\partial \Omega} |F_0 ||u|^{\gamma} \,ds + \int_{\Omega} |f_0 ||u|^{\gamma} \,dx. \end{align*} Set $\gamma$ such that $\gamma > \lambda(K) [1+\lambda(K)] -1$, where $K$ is the constant defined in previous Lemma. Then, from the last inequality we obtain \begin{equation} \label{e3.2} \int_{\Omega} |u|^{\gamma} [ \nu |\nabla u|^p + (c_0 - c_1)|u|^p ] \,dx \leq K^{\gamma}\Big( \int_{\Omega} |f_0 | \,dx + \int_{\partial \Omega} |F_0 | \,ds\Big). \end{equation} On the other hand if we take $w(x) = u(x)$ as a test function in relation \eqref{e2.7}, we have $$ \int_{\Omega} \Big\{ \sum_{i=1}^{m} a_i(x,u, \nabla u) \frac{\partial u}{\partial x_i} + c_0 | u |^p + \ f(x,u, \nabla u) u \Big\} \,dx + \int_{\partial \Omega} F(x,u) u\,ds \leq0. $$ Applying inequalities \eqref{e1.2}, \eqref{e2.1}, \eqref{e2.3} and Lemma \ref{lem3.1} we obtain \begin{align*} &\min \Big( \frac{1}{\lambda(K)} , c_0 \Big) \|u \|_{1,p}^p \\ &\leq \lambda(K) \int_\Omega \big[ f^* |u| + |u|^{p+\sigma} + |u| (\nu^{1/p} | \nabla u | )^{p-1+\sigma} + |u| \nu |\nabla u|^p \big] \,dx + \int_{\partial \Omega} |F_0| \,ds. \end{align*} Then, there exists a constant $K_1$, depending only on $c_0$, $c_1$, $c_2$, $\sigma$, $\lambda (s)$, $\|f_0\|_{L^{\infty}(\Omega)}$ and $ \|F_0\|_{L^{\infty}(\partial \Omega)}$, such that \begin{equation}\label{e3.3} \|u \|_{1,p}^p \leq K_1 \int_\Omega [ f^* |u| + |u|^{p+\sigma} +|u|^{\tau'} \nu | \nabla u |^{p} ] \,dx + \|F_0\|_{L^{\infty}(\partial \Omega)} \operatorname{meas}\ \partial \Omega, \end{equation} where $\tau' = \frac{\sigma}{2} \frac{p}{p-1+\sigma}$ (see also \cite[(3.4)]{DrabNic}). We use \eqref{e3.1}, \eqref{e3.2} to estimate the first term on the right-hand side of previous inequality: \begin{gather*} \int_\Omega f^* | u | \,dx \leq \|u\|_{L^\infty (\Omega)} \|f^* \| _{L^1 (\Omega)} \leq K \|f^* \|_{L^1(\Omega)} \\ \int_\Omega | u | ^{p+\sigma} \,dx \leq \|u\|_{L^\infty (\Omega)} ^{p+ \sigma}\operatorname{meas}_m \Omega \leq K^{p+\sigma} \operatorname{meas}_m \Omega, \\ \int_\Omega |u|^{\tau'} \nu | \nabla u | ^{p} \,dx \leq K^{\tau'} \Big( \int_\Omega | f_0 | \,dx + \int_{\partial \Omega} | F_0 | \,ds \Big) \quad \text{if } \tau' > \lambda(K)[1 + \lambda(K)] -1. \end{gather*} In the case $ \tau' \leq \lambda(K)[1 + \lambda(K)] -1$, we first apply Young's inequality to obtain $$ | u |^{\tau'} \leq \epsilon + C(\epsilon, \tau' , \gamma) | u |^\gamma, \quad \gamma > \lambda(K)[1 + \lambda(K)] -1; $$ hence, $$ \int_\Omega |u|^{\tau'} \nu | \nabla u | ^{p} \,dx \leq \epsilon \| u \|_{1,p}^p + C(\epsilon,\tau' , \gamma) K^\gamma \Big( \int_\Omega | f_0 | \,dx + \int_{\partial \Omega} | F_0 | \,ds \Big). $$ The above inequalities and \eqref{e3.3} give $\|u \|_{1,p} \leq M$, where $M$ depends only on $c_0$, $c_1$, $c_2$, $p$, $\sigma$, $\|f_0\|_{L^{\infty}(\Omega)}$ , $ \|F_0\|_{L^{\infty}(\partial \Omega)}$, $\operatorname{meas}_m\Omega$, $\operatorname{meas}\ \partial \Omega$, $ \|f^{*}\|_{L^1(\Omega)}$, $\lambda (s)$. The proof is complete. \end{proof} We want to emphasize that the constants $K$ and $M$ in previous Lemmas do not depend on $u$. Moreover, Hypothesis (H2) in such Lemmas is only used for defining the trace of $u$ on $\partial \Omega$. The following lemma will be useful in verifying the assumptions of the Leray-Lions Theorem in the proof of Lemma \ref{lem3.4}. \begin{lemma}\label{lem3.3} Let {\rm (H1), (H3), (H9)--(H11)} be satisfied. Let $u \in {W}^{1,p}(\nu ,\Omega)$ and $\{u_n\}$ be a sequence in ${W}^{1,p}(\nu ,\Omega)$ such that there exists a constant $\Lambda >0$ for which $\|u_n\|_{1,p} \leq \Lambda$ and $\lambda(|u_n(x)|) \leq \Lambda$ for almost all $x \in \Omega$ and for every $n=1,2, \ldots$. Moreover, let us suppose $\lim_{n\to + \infty}\| u_n-u \|_{L^{p}(\Omega)}=0$ and \begin{equation} \label{e3.4} \lim_{n \to +\infty} \int_{\Omega} \sum_{i=1}^{m} [ a_i(x,u_n , \nabla u_n) - a_i(x, u_n, \nabla u)] \frac{\partial (u_n-u)}{\partial x_i} \,dx =0. \end{equation} Then \[ \lim_{n \to +\infty} \int_{\Omega} \nu |\nabla u_n - \nabla u|^{p} \,dx =0 . \] \end{lemma} The proof of the above lemma is an easy modification of the proof of \cite[Lemma 3.3]{DrabNic}. The following Lemma is a direct application of the Leray-Lions Theorem. \begin{lemma}\label{lem3.4} Assume that $\lambda(s) \equiv \lambda$, with $\lambda$ a positive constant. Let us suppose that {\rm (H1)--(H4), (H9)--(H11)} are satisfied. Let us suppose moreover that for every $u \in \mathbb{R}$, $(\eta_{1},\ldots, \eta_{m}) \in \mathbb{R}^{m}$ and for almost all $x \in \Omega$, it holds \[ |f(x,u, \eta)| \leq \lambda, \] and for almost all $x \in \partial \Omega$ and for all $u \in \mathbb{R}$, \[ |F(x,u)| \leq \lambda. \] Then \eqref{e2.7} has at least one solution. \end{lemma} \begin{proof} Let us consider the operator $$ A(u,v) : W^{1,p}(\nu ,\Omega) \times W^{1,p}(\nu ,\Omega) \to (W^{1,p}(\nu ,\Omega))^\star , $$ defined by \begin{align*} \bigl\langle A(u,v),w \bigr\rangle &= \int_{\Omega} \Big\{ \sum_{i=1}^{m} a_i(x,u,\nabla v) \frac{\partial w}{\partial x_i} + c_0|u|^{p-2} uw + f(x,u,\nabla u) w \Big\} \,dx \\ &\quad + \int_{\partial \Omega} [ c_{2}|u|^{p-2} uw + F(x,u)w] \,ds \end{align*} for every $w \in W^{1,p}(\nu ,\Omega)$, and the operator $T :W^{1,p}(\nu ,\Omega) \to (W^{1,p}(\nu ,\Omega))^\star$ defined by \[ T(u) = A(u,u), \quad u\in W^{1,p}(\nu ,\Omega). \] Using (H9), it is easy to check that the operator $A(u ,v)$ is a bounded operator. Moreover, $$ \bigl\langle A(v,v),v\bigr\rangle \geq \min (\frac{1}{\lambda}, c_0) \|v\|_{1,p}^{p} - \lambda \|v\|_{1,p} \big[ (\operatorname{meas}_m\Omega)^{(p-1)/p} + c' (\operatorname{meas} \partial \Omega)^{(\tilde{p}-1)/\tilde{p}}\big]. $$ Hence \[ \lim_{\|v\|_{1,p} \to +\infty} \frac{\bigl\langle T(v),v\bigr\rangle}{\|v\|_{1,p}} = + \infty. \] Now, we shall verify that the operator $A(u,v)$ satisfies the other assumptions of the Leray-Lions Theorem (see \cite[Theorem 1]{LL}; see, also, \cite{Fucik}): (i) Continuity and monotony in $v$: from (H11), $$ \langle A(u,u) - A(u,v), u-v\rangle \geq 0. $$ Moreover, we observe that \[ \lim_{n \to + \infty} \bigl \langle A(u_n ,v_n ),w \bigr\rangle =\bigl \langle A(u,v),w \bigr\rangle\quad \text{for every } w \in W^{1,p}(\nu ,\Omega), \] if \[ (u_n,v_n) \to (u,v) \quad \text{in } W^{1,p}(\nu ,\Omega) \times W^{1,p}(\nu ,\Omega). \] For example, we prove that \begin{equation} \label{e3.5} \lim_{n \to +\infty} \int_{\partial \Omega} |v_n |^{p-2} v_n w \,ds = \int_{\partial \Omega} |v|^{p-2} v w \,ds. \end{equation} Now, Hypothesis (H2) implies $$ \| v_n - v \|_{L^p (\partial \Omega)} \leq c' (\operatorname{meas} \partial \Omega )^{(\tilde{p}-p)/p\tilde{p}} \|v_n - v \|_{1,p} $$ then $v_n \to v$ in $L^p(\partial \Omega)$. Let $E$ be an arbitrary measurable subset of $\partial \Omega$. It results $$ \int_{E} |v_n |^{p-1} |w| \,ds \leq \int_{E} |v_n |^{p}\,ds + \int_{E} |w |^{p}\,ds. $$ The strong convergence of $v_n$ to $v$ in $L^p(\partial \Omega)$ implies that $\{ | v_n |^p \}$ are equiintegrable. Then the above inequality together with Hypothesis (H2) imply that $\{ |v_n |^{p-1} |w|\}$ is also an equiintegrable sequence of functions. Hence \eqref{e3.5} follows from Vitali's theorem. (ii) Continuity of $A(u,v)$ with respect to $v$: let $u_n \rightharpoonup u$ in $W^{1,p} (\nu, \Omega)$ and $ \lim_{n \to \infty} \langle A(u_n ,u_n ) - A(u_n ,u), u_n -u\rangle =0$, then, by Lemma \ref{lem3.3}, $ u_n \to u $ in $W^{1,p} (\nu, \Omega)$; hence, by previous observation, we have that $A(u_n , v) \rightharpoonup A(u,v)$ in $(W^{1,p}(\nu ,\Omega))^\star$, for every $v \in W^{1,p} (\nu, \Omega)$. (iii) Continuity of $\bigl\langle A(u , v),u \bigr\rangle$ in $u$: we observe that if $v \in W^{1,p} (\nu, \Omega)$, $u_n \rightharpoonup u$ in $W^{1,p} (\nu, \Omega)$ and $A(u_n , v) \rightharpoonup v'$ in $(W^{1,p}(\nu ,\Omega))^\star$, then $u_n \to u$ in $L^p (\Omega)$, $u_n \to u$ in $L^p (\partial\Omega)$, hence $$ \lim_{n\to \infty}\bigl\langle A(u_n , v), u_n - u \bigr\rangle =0 $$ and, therefore, $\langle A(u_n , v), u_n\rangle \to \bigl \langle v' , u \bigr \rangle$ (see, also, \cite[note (15)]{GuglNic}, where the special case $p=2$ is treated, but for Dirichlet problem, and, Remark \ref{rmk2.3}). Thus, all the assumptions of the Leray-Lions theorem (Hypothesis II) are satisfied. Hence the equation $Tu=0$ has at least one solution $u \in W^{1,p}(\nu ,\Omega)$. We shall prove that $u \in L^{\infty} (\Omega) \cap L^{\infty}(\partial \Omega)$. We set: $$ \Omega_k = \{ x \in \Omega : u>k\} , \quad \partial \Omega_k = \{ x \in \partial \Omega : u>k \}. $$ From \eqref{e2.7}, choosing $w=u - \min (u,k)$, $k> K_0$ (for $K_0$ see Remark \ref{rmk2.1}), we have \begin{align*} &\int_{\Omega_k} \Big\{ \sum_{i=1}^m a_i (x,w+k,\nabla w)\frac{\partial w}{\partial x_i} + c_0 |w+k|^{p-1} w + f(x,w+k,\nabla w) w \Big\} \,dx \\ & +\int_{\partial \Omega_k} \{c_2 |w+k|^{p-1}w + F(x,w+k)w \}\,ds=0. \end{align*} Applying condition \eqref{e1.2} we obtain $$ \min \big(\frac{1}{\lambda} , c_0 \big) \|w \|_{1,p}^p \leq \lambda \int_{\Omega_k} w \,dx + \lambda \int_{\partial \Omega_k} w \,ds. $$ The above inequality and (H4) imply $$ \|w \|_{1,p}^{p-1} \leq \frac{ \lambda [\hat{c} (\operatorname{meas}_m\Omega) ^{(p^\star - \tilde{p})/p^\star \tilde{p}} + c']} {\min (\frac{1}{\lambda} , c_0 ) } [ (\operatorname{meas}_m \Omega_k )^{(\tilde{p}-1)/\tilde{p}} + (\operatorname{meas} \partial \Omega_k )^{(\tilde{p}-1)/\tilde{p}} ]. $$ For $h>k$ we have \begin{align*} &\Big( \int_{\Omega} |w|^{\tilde p} \,dx \Big)^{\frac{p-1}{\tilde p}} + \Big( \int_{\partial \Omega} |w|^{\tilde p} \,ds \Big) ^{\frac{p-1}{\tilde p}} \\ &\geq (h-k)^{p-1} \big\{ (\operatorname{meas}_m\Omega_h ) ^{(p-1)/\tilde{p}} + ( \operatorname{meas} \partial \Omega_h ) ^{(p-1)/\tilde{p}}\big\}. \end{align*} For $h>0$, denote $$ \varphi (h) = \{\operatorname{meas}_m\Omega_h + \operatorname{meas} \partial \Omega_h\}. $$ We have $$ \varphi (h) \leq \frac{\alpha}{(h-k)^{\tilde{p}}} [ \varphi (k) ]^{\frac {\tilde{p} - 1}{p-1}}, \quad \text{if } h> k > K_0 $$ where the positive constant $\alpha$ depends only on $\hat{c}$, $c'$, $c_0$, $\lambda$, $m$, $p$, $t$, $\Omega$. Note that $\frac {\tilde{p}-1}{p-1} > 1$, then it follows from a lemma of Stampacchia \cite[Lemma 3.11]{MurtStamp} that $\operatorname{ess\,sup}_{\Omega} u + \operatorname{ess\,sup}_{\partial \Omega} u < +\infty$. By this way also $\operatorname{ess\,sup}_{\Omega} (-u) + \operatorname{ess\,sup}_{\partial \Omega} (-u) < +\infty$. Hence $u \in L^{\infty} (\Omega) \cap L^{\infty} (\partial \Omega)$. \end{proof} \section{Proof of Theorem \ref{thm2.1}} \begin{proof} Let $K$ be the constant defined in Lemma \ref{lem3.1}. We define \[ A_i(x,u,\eta) =\begin{cases} a_i(x,-K,\eta) & \text{if $u <-K$} \\ a_i(x,u,\eta) & \text{if $|u| \leq K$}\\ a_i(x,K,\eta) & \text{if $u >K$}, \end{cases} \] in $\Omega \times \mathbb{R} \times \mathbb{R}^{m}$. For every positive integer $n$ we define: \[ f_n(x,u,\eta) = \begin{cases} f(x,u,\eta) & \text{if $|f| \leq n$} \\[4pt] n \frac{f(x,u,\eta)}{|f(x,u,\eta)|} &\text{if $|f| >n$} \end{cases} \] in $\Omega \times \mathbb{R} \times \mathbb{R}^{m}$, \[ F_n(x,u) = \begin{cases} F(x,u) & \text{if $|F| \leq n$} \\[4pt] n \frac{F(x,u)}{|F(x,u)|} & \text{if $|F| >n$} \end{cases} \] in $\partial \Omega \times \mathbb{R}$. The functions $A_i (x,u,\eta)$, $f_n (x,u,\eta)$, $F_n (x,u)$, satisfy (H3)--(H11). It is sufficient to note, for example, that in $\Omega \times \mathbb{R} \times \mathbb{R}^{m}$, $$ |f_n (x,u,\eta)| \leq |f(x,u,\eta)|, $$ and, that (H8) holds with $|f_0 (x)|$ instead of $f_0(x)$. Analogous considerations verify the others assumptions. On the other hand, for every $u \in \mathbb{R}$, $(\eta_{1},\ldots,\eta_{m}) \in \mathbb{R}^{m}$ and for almost all $x \in \Omega$ it holds that \[ |f_n (x,u, \eta)| \leq n\,, \] and for almost all $x \in \partial \Omega$ and for all $u \in \mathbb{R}$, \[ |F_n (x,u)| \leq n. \] Then, it follows from Lemma \ref{lem3.4} that, for every $n\in \mathbb{N}$, there exists $u_n \in W$ such that \begin{equation} \label{e4.1} \begin{aligned} & \int_{\Omega} \Big[ \sum_{i=1}^m A_i(x,u_n,\nabla u_n) \frac{\partial w}{\partial x_i} + c_0 |u_n |^{p-2} u_n w + f_n(x,u_n, \nabla u_n) w \Big] \,dx \\ &+ \int_{\partial \Omega} [ c_2 |u_n |^{p-2} u_n w + F_n(x,u_n)w ] \,ds =0 \end{aligned} \end{equation} for every $w \in W $. An a priori estimate of Lemma \ref{lem3.1} yields \begin{equation}\label{e4.2} \|u_n \|_{L^{\infty}(\Omega)} + \|u_n \|_{L^{\infty}(\partial \Omega)}\leq K, \quad \text{for every } n \in \mathbb{N}, \end{equation} and hence \eqref{e4.1} can be written in the equivalent form \begin{equation} \label{e4.3} \begin{aligned} &\int_{\Omega} \Big[ \sum_{i=1}^m a_i(x,u_n,\nabla u_n) \frac{\partial w}{\partial x_i} + c_0 |u_n |^{p-2} u_n w + f_n(x,u_n, \nabla u_n) w \Big] \,dx \\ &+ \int_{\partial \Omega} [ c_2 |u_n |^{p-2} u_n w + F_n(x,u_n)w ] \,ds =0. \end{aligned} \end{equation} It follows from Lemma \ref{lem3.2} that for every $n \in \mathbb{N}$, \begin{equation} \label{e4.4} \|u_n \|_{1,p} \leq M. \end{equation} On the basis of \eqref{e4.2} and \eqref{e4.4} there exists a subsequence of $\{u_n \}$ (denoted again by $\{u_n \}$) such that $\{u_n \}$ converges weakly to $u$ in $W^{1,p}(\nu ,\Omega)$ and $\{u_n\}$ converges weakly* in $L^{\infty}(\Omega)$ and in $L^{\infty}(\partial \Omega)$ where $ u \in W$ and $\|u\|_{L^{\infty}(\Omega)} + \|u\|_{L^{\infty}(\partial \Omega)}\leq K$. We shall prove that $u \in W$ is the solution of \eqref{e2.7}. To pass to the limit in \eqref{e4.3} for $n \to +\infty$ we have to prove that \begin{equation} \label{e4.5} \lim_{n \to + \infty} \int_{\Omega} \nu | \nabla u_n - \nabla u |^p \,dx =0. \end{equation} Now, the compact embedding of $W^{1,p}(\nu ,\Omega)$ in $L^p(\Omega)$ implies the strong convergence of ${u_n}$ to $u$ in $L^p(\Omega)$ and hence also almost everywhere in $\partial \Omega$ (see Remark \ref{rmk2.3}). Then, taking into account Lemma \ref{lem3.3}, to get \eqref{e4.5} it will be sufficient to prove that \eqref{e3.4} it holds. Let us take $w=|u_n -u|^{\gamma}(u_n-u)$ as a test function in \eqref{e4.3} where $\gamma$ is a positive number. We deduce \begin{align*} &\int_{\Omega} \Big\{ \sum_{i=1}^m a_i(x,u_n ,\nabla u_n) (\gamma +1) |u_n - u|^{\gamma} \frac{\partial(u_n -u)}{\partial x_i} \\ &+c_0|u_n |^{p-2} u_n |u_n -u|^{\gamma} (u_n -u) + f_n (x,u_n,\nabla u_n)|u_n -u|^{\gamma} (u_n -u) \Big\}\,dx \\ &+ \int_{\partial \Omega} \{ c_{2} |u_n |^{p-2} u_n|u_n -u|^{\gamma}(u_n-u) + F_n(x,u_n )|u_n -u|^{\gamma}(u_n -u) \} \,ds\\ & =0. \end{align*} From the above inequality, taking into account \eqref{e1.2}, \eqref{e2.1}, \eqref{e2.2}, \eqref{e4.2}, we obtain \begin{align*} &\int_{\Omega} |u_n - u|^{\gamma} |\nabla u_n |^p \nu \,dx \\ &\leq \int_{\Omega} \sum_{i=1}^m a_i(x,u_n ,\nabla u_n) (\gamma +1) |u_n - u|^{\gamma} \frac{\partial u}{\partial x_i} \\ &\quad + c_0 K^{p-1} \int_{\Omega} |u_n - u|^{\gamma + 1} \,dx +2K\lambda(K) \int_{\Omega} \ [ |f^{*}| + K^{p-1+ \sigma} +1 ] |u_n - u|^{\gamma} \,dx \\ &\quad + c_2 \int_{\partial\Omega} |u_n |^{p-1} |u_n - u|^{\gamma + 1} \,ds + \int_{\partial\Omega} [ |F^{*} | + \lambda(K)] |u_n - u|^{\gamma + 1} \,ds, \end{align*} where $\gamma$ is such that $\frac{\gamma +1}{\lambda(K)} - 4 K\lambda(K) >1$. By Lebesgue theorem, the first three addends in the right hand side of previous inequality go to $0$ as $n \to +\infty$ (see, \cite[Lemma 3.4, pp. 229-230]{DrabNic}). We prove, for example, that $$ \lim_{n\to \infty} \int_{\partial\Omega} [ |F^{*}| + \lambda(K)] |u_n - u|^{\gamma + 1} \,ds =0, $$ this integral is absent in \cite{DrabNic}. It results that a.e. $x \in \partial \Omega$, $$ [ |F^{*}| + \lambda(K) ] |u_n - u|^{\gamma + 1} \leq (2K)^{\gamma + 1} [ |F^{*}| + \lambda(K) ] \in L^1(\partial \Omega). $$ As $u_n \to u$ a.e. in $\partial\Omega$, it will be enough to apply Lebesgue theorem again. Then, it follows \[ \lim_{n \to +\infty} \int_{\Omega} |u_n - u|^\gamma |\nabla u_n |^p \nu \,dx =0, \] and, so, applying H\"older inequality \begin{equation} \label{e4.6} \lim_{n \to +\infty} \int_{\Omega} |u_n - u| |\nabla u_n |^p \nu \,dx =0. \end{equation} By \eqref{e4.3} we obtain \begin{align*} &\int_{\Omega} \sum_{i=1}^m [ a_i(x,u_n, \nabla u_n) - a_i(x,u_n, \nabla u) ] \frac{\partial(u_n -u)}{\partial x_i}\,dx \\ &= - \int_{\Omega} c_0 |u_n |^{p-2} u_n (u_n -u) \,dx- \int_{\Omega} f_n(x,u_n , \nabla u_n) (u_n -u) \,dx \\ &\quad - \int_{\Omega} \sum_{i=1}^m a_i(x,u, \nabla u) \frac{\partial(u_n -u)}{\partial x_i} \,dx \\ &\quad +\int_{\Omega} \sum_{i=1}^m [ a_i(x,u, \nabla u) - a_i(x, u_n , \nabla u) ] \frac{\partial(u_n -u)}{\partial x_i} \,dx \\ &\quad - \int_{\partial\Omega} c_2 |u_n |^{p-2} u_n (u_n -u)\,ds - \int_{\partial \Omega}F_n (x,u_n )(u_n -u) \,ds. \end{align*} Now, all addends in the right-hand side of previous inequality go to $0$ as $n \to +\infty$. For example, we shall estimate the second and the last addend. We have \begin{align*} &\int_{\Omega} | f_n (x, u_n , \nabla u_n ) | |u_n -u| \,dx \\ &\leq \lambda(K) \int_{\Omega} [ K^{p-1+\sigma}+ 1 + |f^* |]|u_n - u| \,dx + 2\lambda(K)\int_{\Omega}|u_n - u| |\nabla u_n |^p \nu \,dx. \end{align*} From the Lebesgue theorem and \eqref{e4.6}, the above inequality implies $$ \lim_{n \to +\infty} \int_{\Omega} f_n(x,u_n , \nabla u_n) (u_n -u)\,dx=0. $$ Next $$ \int_{\partial \Omega} |F_n (x,u_n )||u_n -u| \,ds \leq [ \lambda(K) + \|F^* \|_{L^\infty (\partial \Omega )} ] \int_{\partial \Omega} |u_n -u| \,ds. $$ Taking into account that the imbedding of $W^{1,p} (\Omega)$ in $L^1 (\partial \Omega)$ is compact (see Remark \ref{rmk2.3}), the above inequality implies $$ \lim_{n \to +\infty} \int_{\partial\Omega} F_n(x,u_n) (u_n -u) \,dx=0. $$ For details concerning others passage to the limit see \cite[pag. 228]{DrabNic}. Consequently \[ \int_{\Omega} \sum_{i=1}^m [ a_i(x,u_n, \nabla u_n) - a_i(x,u_n, \nabla u) ] \frac{\partial(u_n -u)}{\partial x_i} \,dx \] tends to zero as $n \to + \infty$. So, $u_n \to u \ \text{in} \ W^{1,p}(\nu ,\Omega)$. Now, to prove that the function $u \in W$ is the solution of \eqref{e2.7} it is sufficient to pass to the limit as $n\to \infty$. For example, we prove that \begin{equation} \label{e4.7} \lim_{n \to +\infty} \int_{\partial \Omega} F_n (x,u_n ) w \,ds = \int_{\partial \Omega} F (x,u ) w \,ds \end{equation} for every $w \in W$. We fix $\epsilon >0$ and a point $x_0 \in \partial \Omega$ such that $u_n (x_0 ) \to u(x_0 )$ as $n \to +\infty$ and the function $F(x_0 ,u)$ is continuous with respect $u$. Then there is a number $n_\epsilon \in \mathbb {N}$ such that for any $n> n_\epsilon$, $$ -n < F(x_0 , u(x_0 )) - \epsilon < F(x_0 , u_n (x_0 )) < \epsilon + F(x_0 , u(x_0 )) < n. $$ These inequalities and the definition of the function $F_n (x,u)$ imply that for any $n> n_\epsilon$, $F_n (x_0 , u_n (x_0 )) = F(x_0 , u_n (x_0 ))$ and $$ | F_n (x_0 , u_n (x_0 )) - F(x_0 , u(x_0 )) |< \epsilon. $$ In this way $ F_n (x, u_n (x) )\to F(x,u(x))$ a.e. on $\partial \Omega$. Next, from definition of $F_n (x,u)$ and \eqref{e2.2} we have $$ |F_n (x,u_n (x))w(x)|\leq [\lambda(K) + \| F^{*} \|_{L^{\infty}(\partial \Omega)} ] |w(x)| $$ a.e. $x \in \partial \Omega$. Now, a new application of the Lebesgue theorem gives \eqref{e4.7}. The proof is complete. \end{proof} Now, we show an example where all assumptions are satisfied. Let $\Omega$ be a bounded open set of $\mathbb{R}^{m}$ such that $0 \in \partial \Omega$. Put \[ \nu (x) = | x | ^{\gamma}\quad\text{for }0< \gamma < p-1. \] Then the function $\nu$ satisfies Hypotheses (H1) and (H2) with $t$ such that $$ \frac{m}{p-1} < t < \frac{m}{\gamma}. $$ Consider the boundary-value problem \begin{gather} \label{e4.8} - \operatorname{div} \Big( \frac{| x | ^\gamma }{1 + | u | ^p} | \nabla u | ^{p-2} \nabla u \Big) + e^u - | u | ^p + | x |^\gamma | \nabla u | ^p = g(x) \quad \text{in }\Omega, \\ \label{e4.9} \frac{| x | ^\gamma}{1 + | u | ^p} | \nabla u | ^{p-2} \sum_{i=1}^{m} \frac{\partial u}{\partial x_i} \cos (\overrightarrow {n},x_i ) + \frac{1}{e}u| u | ^{p-2} + \frac {e^{u-1}}{2} =0 \quad \text{on } \partial \Omega, \end{gather} where $g(x) \in L^\infty (\Omega)$. In this case we have: \begin{gather*} a_i (x,u, \nabla u) = \frac{| x | ^\gamma }{1 + | u | ^p} | \nabla u | ^{p-2} \frac {\partial u}{\partial x_i}, \quad i=1,2,\dots ,m; \\ f(x,u, \nabla u) = e^u - | u | ^p - u | u | ^{p-2} + | x | ^\gamma | \nabla u | ^p - g(x) , \quad c_0 =1;\\ F(x,u) =\frac{1}{2e}u| u | ^{p-2} + \frac {e^{u-1}}{2} ; \quad c_2 = \frac {1}{2e}\,. \end{gather*} If we put $\lambda(| u |) = e^{| u | ^p}$, it is possible to verify all the Hypotheses (H3)--(H11). To verify (H3), for example, it will be sufficient to note that the function $(| u |^p + ue^u)$ has minimum ($\leq 0$) in $(- \infty, + \infty)$. Hence, BVP \eqref{e4.8}, \eqref{e4.9} has at least one weak solution in the sense \eqref{e2.7}, i.e. there exists at least one $u \in W$ such that \begin{align*} &\int_{\Omega} \frac{| x | ^\gamma }{1 + | u | ^p} | \nabla u | ^{p-2} \nabla u \nabla w \,dx + \int_{\Omega} [e^u - | u | ^p + | x | ^\gamma | \nabla u | ^p]w \,dx\\ &+\int_{\partial \Omega} \big\{ \frac{1}{e}u| u | ^{p-2} + \frac {e^{u-1}}{2}\big\} w \,ds \\ &= \int_{\Omega} g w \,dx \end{align*} holds for every $w \in W$. Examples concerning the Dirichlet problem related to \eqref{e1.1} can be found in \cite[Section 6]{DrabNic}. \section{Asymptotic behavior near infinity of solutions to the Dirichlet problem for \eqref{e1.1}} Let $\Omega= \{x \in \mathbb{R}^m : |x|> r\}$, $r$ be a positive constant. For $n \in \mathbb{N}$, we denote $$ \Omega_n = \Omega \cap \{ x \in \mathbb{R}^m : |x| < n \}. $$ We introduce the hypothesis \begin{itemize} \item[(H12)] The function $\nu = \nu(x): \Omega \to (0, + \infty)$ is a measurable function such that $\nu \in L^{\infty} (\Omega)$. For every $n \in \mathbb{N}$, there exists a real number $ \delta_n > \max ( \frac{m}{p},\frac {1}{p-1})$ such that $1/\nu \in L^{\delta_n}(\Omega_n )$. \end{itemize} We set \[ L^{1}(\Omega) + L^{p/(p-1)}(\Omega) =\{ f_1 (x) + f_2 (x) : f_1 \in L^{1}(\Omega) , f_2 \in L^{p/(p-1)}(\Omega) \}. \] Let (H3), (H5), (H8)--(H12) be satisfied with $f_0 \in L^{1}(\Omega) \cap L^{\infty}(\Omega)$, $f^* \in L^{1}(\Omega) + L^{p/(p-1)}(\Omega)$ and let $u\in \mathaccent"7017 {W}^{1,p}(\nu ,\Omega) \cap L^{\infty}(\Omega)$ such that \begin{equation} \label{e5.1} \int_{\Omega} \Big\{ \sum_{i=1}^{m} a_i(x,u, \nabla u) \frac{\partial w}{\partial x_i} +c_0 |u|^{p-2} u w + \ f(x,u, \nabla u) w \Big\} \,dx=0 \end{equation} for every $w \in \mathaccent"7017 {W}^{1,p}(\nu ,\Omega) \cap L^{\infty}(\Omega)$. The function $u$ exists because of \cite[Theorem 2.2]{DrabNic}. \begin{theorem}\label{thm5.1} Let {\rm (H3), (H5), (H8)--(H12)} be satisfied, with the function $f^*$ in $L^{1}(\Omega) + L^{p/(p-1)}(\Omega)$, and \begin{equation} \label{e5.2} |f_0(x)| + |a^*(x)| \leq \tilde {c} e^{-\delta_1|x|}, \quad x \in \Omega, \end{equation} with $\tilde{c}$ and $\delta_1$ positive constants. Let us consider $u \in \mathaccent"7017 {W}^{1,p}(\nu ,\Omega) \cap L^{\infty}(\Omega)$ that satisfies \eqref{e5.1} for every $w \in \mathaccent"7017{W}^{1,p}(\nu ,\Omega) \cap L^{\infty}(\Omega)$. Then \begin{equation} \label{e5.3} \int_{|x|> \lambda} |u|^p \,dx \leq C e^{-\delta_3 \lambda} \end{equation} for every $\lambda \geq r$, where $\delta_3$ and $C$ are positive constants depending on known parameters. \end{theorem} \begin{proof} Let us define in $\mathbb{R}^m$ a Lipschitzian function $\theta(x)$, $0 \leq \theta(x) \leq 1$, such that $\theta(x)=0$ if $0<|x|< r+1$, $\theta(x)=1$ if $|x|>r+2$. Define in $\mathbb{R}^m$ the function $\theta_R(x)$, $0 \leq \theta_R(x) \leq 1$, such that $\theta_R(x) =1$ if $|x| R+1$, and let $\theta_R(x) $ be a Lipschitzian function. Take in \eqref{e5.1} as a test function $w = u |u|^{\gamma}e^{\gamma \tau(x)}\theta \theta_R$ where $\tau(x) = \beta|x|$ if $|x| L$ and the positive constants $\gamma, \beta$ will be stated later on. Moreover, let us suppose that real numbers $L$, $R$ are such that $r+2 < L < R$. After easy computations, by \eqref{e1.2} and \eqref{e2.4}, we obtain \begin{equation} \label{e5.4} \begin{aligned} &\int_{\mathbb{R}^m} e^{\gamma \tau(x)} |u|^{\gamma} \theta \theta_R \Big\{ \Big[ \frac{\gamma +1}{\lambda(\|u\|_{L^{\infty}(\Omega)})} -\lambda(\|u\|_{L^{\infty}(\Omega)})\Big] \nu |\nabla u |^p \\ &+ (c_0-c_1) |u|^p \Big \} \,dx \\ &\leq \gamma \int_{\mathbb{R}^m} \sum_{i=1}^m |a_i(x,u, \nabla u) | \Big|\frac{\partial \tau (x)}{\partial x_i} \Big| |u|^{\gamma+1} e^{\gamma \tau (x)} \theta \theta_R \,dx \\ &\quad + \int_{\mathbb{R}^m} \sum_{i=1}^m |a_i(x,u, \nabla u) | |u|^{\gamma+1} e^{\gamma \tau(x)} |\nabla \theta| \theta_R\,dx \\ &\quad + \int_{\mathbb{R}^m} \sum_{i=1}^m |a_i(x,u, \nabla u) | |u|^{\gamma+1} e^{\gamma \tau(x)} |\nabla \theta_R| \theta \,dx \\ &\quad + \int_{\mathbb{R}^m} e^{\gamma \tau (x)} |f_0||u|^{\gamma} \theta \theta_R \,dx. \end{aligned} \end{equation} Now, we choose $\gamma$ in such that \[ \frac{\gamma +1}{\lambda(\|u\|_{L^{\infty}(\Omega)})} -\lambda(\|u\|_{L^{\infty}(\Omega)}) =2. \] Then, we can estimate from below the left-hand side of \eqref{e5.4} by \begin{equation}\label{e5.5} 2 \int_{r+2<|x| r+2$, $\beta>0$; where $\gamma$ is a fixed real number, $\gamma >2$. Fix $\beta$ such that \[ 0< \beta < \min \Big( \frac{\delta_1}{\gamma}, \frac{c_0-c_1}{2m \gamma \lambda(\|u\|_{L^{\infty}(\Omega)}) \| \nu \|_{L^{\infty}(\Omega)} ^{1/p}}, \frac{2}{m \gamma \lambda(\|u\|_{L^{\infty}(\Omega)}) \| \nu \|_{L^{\infty}(\Omega)} ^{1/p}} \Big). \] Then, for every $L > r+2$, we obtain $$ \int_{r+2<|x|r} e^{\delta_2 |x|} |u|^{\gamma+p} \,dx \leq M_1 \end{equation} where $\delta_2 = \gamma \beta$ and $M_1 = e^{\gamma \beta (r+2)} \|u\|_{L^{\infty}(\Omega)}^{\gamma + p} \operatorname{meas}_m ( r < |x| < r+2) + M$. Hence \eqref{e5.3} follows from \eqref{e5.10}. The proof is complete. \end{proof} We give an example where Hypothesis (H12) is satisfied. Let $\Omega= \{x \in \mathbb{R}^m : |x| > 1\}$. We consider the function $\nu : \Omega \to (0, +\infty)$ defined by $$ \nu (x) = \big[ (|x| - 1) e^{-(|x| -1)} \big]^{\gamma}, \quad \gamma \in (0, (p-1)/m ). $$ Then $$ \nu (x) \leq \big( \frac{1}{e} \big)^\gamma , \quad x \in \Omega. $$ For every integer $n \geq 2$, we set $\Omega_n = \{x\in \mathbb{R}^m : 1< |x| \max (\frac{m}{p}, \frac{1}{p-1})$; \item[(H14)] %(6.2) Let $f(x,u,\eta )$ be a Caratheodory function in $\pi_0 \times \mathbb{R} \times \mathbb{R}^{m}$ such that for almost all $x= (x' , x_m) \in \pi_0$ and for all $(u,\eta) \in \mathbb{R} \times \mathbb{R}^m$, \begin{gather*} | f(x,u,\eta) | \leq \lambda (| u | ) [ f^{*} (x) + \hat \nu(x') | \eta |^p ], \quad f^{*} \in L^1 (\pi_0 ) + L^{p/(p-1)} (\pi_0 ), \\ c_1 | u | ^ p + uf(x,u,\eta) \geq -f_0 (x) , \ f_0 \in L^1 (\pi_0 ) \cap L^{\infty} (\pi_0 ), \end{gather*} where $\lambda : [0, +\infty) \to [1,+\infty)$ is a monotone nondecreasing function and $c_1$ is a positive constant. \end{itemize} \begin{theorem}\label{thm6.1} Let {\rm (H13), (H14)} be satisfied. Let $\tilde{\lambda} : [0, +\infty) \to [1,+\infty)$ be a nondecreasing function such that $\tilde{\lambda} (s) \leq \lambda (s)$ for all $s\geq 0$. Let $c_0$ be a positive constant such that $c_0 >c_1$. Let $u \in \mathaccent"7017{W}^{1,p}(\hat \nu ,\pi_0) \cap L^{\infty}(\pi_0 )$ satisfy \begin{equation} \label{e6.1} \int_{\pi_0}\Big\{ \frac{\hat \nu}{\tilde{\lambda}(|u|)}\sum_{i=1}^m \Big| \frac{\partial u}{\partial x_i} \Big|^{p-2} \frac{\partial u}{\partial x_i} \frac{\partial w}{\partial x_i} + c_0 |u|^{p-2} u w + f(x,u,\nabla u)w \Big\} \,dx=0 \end{equation} for an arbitrary function $w \in \mathaccent"7017{W}^{1,p}(\hat \nu ,\pi_0) \cap L^{\infty}(\pi_0 )$ (the function $u$ exists by \cite[Theorem 2.2]{DrabNic}). Let us assume that for some $a\geq 0$, \[ c_1 | u | ^p + uf(x,u,\eta) \geq 0 \] for almost all $x \in \pi_a$ and for all $(u,\eta) \in \mathbb{R} \times \mathbb{R}^m$. \\ Then there exists a positive constant $\alpha$, depending on $m$, $p$, $t$, $\Omega'$, $\| u \|_{L^{\infty}(\pi_0)}$, $\|u\|_{1,p}$, $\lambda (s)$, $\|\hat \nu\|_{L^{\infty}(\Omega' )}$ and $\| 1 /\hat \nu\|_{L^{t}(\Omega')}$, such that $$ \int_{\pi_0} e^{\alpha x_m} \hat \nu \sum_{i=1}^{m} \Big| \frac {\partial u}{\partial x_i} \Big| ^{p} \,dx \leq D\,, $$ where $D$ is a positive number depending only on known parameters. \end{theorem} \begin{proof} For the sake of simplicity, we will assume throughout that \begin{equation} \label{e6.2} c_1 | u | ^p + uf(x, u , \eta) \geq 0 \end{equation} for almost all $x \in \pi_0$ and for all $(u,\eta) \in \mathbb{R} \times \mathbb{R}^m$. Let $\theta (x)\in C^1 (\mathbb{R})$ be a function such that $\theta (x) = 1$ if $x<\frac{1}{2}$, $\theta (x) = 0$ if $x>1$, $0\leq \theta (x) \leq 1$, $| \theta' (x) | \leq \beta$. For every $b\geq 0$, we consider $\theta_b (x_m) = \theta (x_m -b) $. It results $0\leq \theta_b (x_m) \leq 1$ and $| \theta_b' (x_m) | \leq \beta$ for all $b\geq 0$. Let $b$ be a real number, $b> 0$. Let us prove that \begin{equation} \label{e6.3} \begin{aligned} &\int_{\pi_0} \frac{\hat \nu}{\tilde{\lambda}(|u|)} \sum_{i=1}^m \Big| \frac{\partial u}{\partial x_i} \Big|^{p} \,dx + \int_{\pi_0} \{ c_0 |u|^{p} + f(x,u,\nabla u)u \}\,dx \\ &=\int_{\pi_0} \frac{\hat \nu}{\tilde{\lambda}(|u|)} \sum_{i=1}^m \Big| \frac{\partial u}{\partial x_i} \Big|^{p-2} \frac{\partial u}{\partial x_i} \frac{\partial }{\partial x_i}(\theta_b u) \,dx \\ &\quad + \int_{\pi_0} \{ c_0 |u|^{p} + f(x,u,\nabla u)u \} \theta_b\,dx. \end{aligned} \end{equation} The function $w = (\theta_c (x_m ) - \theta_b (x_m )) u \in \mathaccent"7017{W}^{1,p} (\hat \nu ,\pi_0) \cap L^{\infty}(\pi_0 )$, $c>b>0$, so by \eqref{e6.1}, we have \begin{align*} &\int_{\pi_0}\frac{\hat \nu}{\tilde{\lambda}(|u|)} \sum_{i=1}^m \Big| \frac{\partial u}{\partial x_i} \Big|^{p-2} \frac{\partial u}{\partial x_i} \frac{\partial}{\partial x_i} [ (\theta_c - \theta_b) u ] + c_0 |u|^{p} (\theta_c - \theta_b) \\ &+ f(x,u,\nabla u)(\theta_c - \theta_b)u \,dx =0, \end{align*} hence, in \eqref{e6.3} the right hand side does not depend on $b$. It results \begin{align} &\int_{\pi_0} \frac{\hat \nu}{\tilde{\lambda}(|u|)} \sum_{i=1}^m \Big|\frac{\partial u}{\partial x_i} \Big|^{p-2} \frac{\partial u}{\partial x_i} \frac{\partial }{\partial x_i}(\theta_b u) \,dx + \int_{\pi_0} c_0 |u|^{p}\theta_b \,dx + \int_{\pi_0} f(x,u,\nabla u)u \theta_b\,dx \nonumber \\ &= \int_{\pi_0}\frac{\hat \nu}{\tilde{\lambda}(|u|)}\sum_{i=1}^m \Big| \frac{\partial u}{\partial x_i} \Big|^{p} \theta_b \,dx + \int_{\pi_0} \frac{\hat \nu}{\tilde{\lambda}(|u|)} \Big| \frac{\partial u}{\partial x_m} \Big|^{p-2} \frac{\partial u}{\partial x_m} u \theta'_b \,dx \label{e6.4} \\ &\quad + \int_{\pi_0} c_0 |u|^{p}\theta_b \,dx + \int_{\pi_0} f(x,u,\nabla u)u \theta_b\,dx. \nonumber \end{align} By (H13) and \eqref{e6.2}, H\"older's inequality and the definition of function $\theta_b$ it follows that \begin{equation} \label{e6.5} \begin{aligned} &\Big| \int_{\pi_0} \frac{\hat \nu}{\tilde{\lambda}(|u|)} \Big|\frac{\partial u}{\partial x_m} \Big|^{p-2} \frac{\partial u}{\partial x_m} u \theta'_b \,dx \Big| \\ &\leq \beta (\sup_{\Omega'} \hat \nu)^{1/p} \Big( \int_{\pi_{b+\frac{1}{2}, b+1}} \hat \nu \sum_{i=1}^{m} \Big|\frac{\partial u}{\partial x_i} \Big|^{p} \,dx \Big)^{(p-1)/p} \Big( \int_{\pi_{b+\frac{1}{2}, b+1}} |u|^p \,dx \Big)^{1/p}. \end{aligned} \end{equation} Next, from the weighted Friedrichs inequality (see, \cite[Corollary 3.3]{MurtStamp}), we have \begin{equation} \label{e6.6} \int_{\Omega'} | u | ^{p} \,dx' \leq \alpha_1 \int_{\Omega'} \hat \nu (x') \sum_{i=1}^{m-1} \Big| \frac {\partial u}{\partial x_i} \Big| ^{p} \,dx', \end{equation} where the positive constant $\alpha_1$ depends only on $m$, $p$, $\Omega'$ and $\| 1/ \hat \nu \|_{L^{t}(\Omega')}$. From \eqref{e6.5} and \eqref{e6.6} we obtain \begin{equation}\label{e6.7} \begin{aligned} \Big| \int_{\pi_0} \frac{\hat \nu}{\tilde{\lambda}(|u|)} \Big| \frac{\partial u}{\partial x_m} \Big|^{p-2} \frac{\partial u}{\partial x_m} u \theta'_b \,dx \Big| &\leq \int_{\pi_0} \hat \nu \Big| \frac{\partial u}{\partial x_m} \Big|^{p-1} |u|\, |\theta'_b |\,dx \\ &\leq \alpha_2 (\sup_{\Omega'} \hat \nu)^{1/p} \int_{\pi_{b+\frac{1}{2}, b+1}} \hat \nu \sum_{i=1}^{m} \Big| \frac {\partial u}{\partial x_i} \Big| ^{p} \,dx, \end{aligned} \end{equation} where the positive constant $\alpha_2$ depends only on $m$, $p$, $\beta$, $\Omega'$ and $\| 1/ \hat \nu \|_{L^{t}(\Omega')}$. Hence \begin{equation} \label{e6.8} \lim_{b \to +\infty} \int_{\pi_0} \frac{\hat \nu}{\tilde{\lambda}(|u|)} \Big|\frac{\partial u}{\partial x_m} \Big|^{p-2}\frac{\partial u}{\partial x_m} u \theta'_b \,dx =0. \end{equation} From \eqref{e6.4}, letting $b\to +\infty$, taking into account that the left hand side does not depend on $b$, by Lebesgue theorem and \eqref{e6.8} we obtain \eqref{e6.3}. Next, by \eqref{e6.2}, \eqref{e6.3}, $c_0 > c_1$, an easy computation gives \begin{equation} \label{e6.9} \int_{\pi_0} \frac{\hat \nu}{\tilde{\lambda}(|u|)} \sum_{i=1}^m \Big| \frac{\partial u}{\partial x_i} \Big|^{p} \,dx \leq \int_{\pi_0} \frac{\hat \nu}{\tilde{\lambda}(|u|)} \sum_{i=1}^m \Big|\frac{\partial u}{\partial x_i} \Big|^{p-2} \frac{\partial u}{\partial x_i}\frac{\partial }{\partial x_i} (\theta_b u ) \,dx, \end{equation} for every $b> 0$. From \eqref{e6.9} and \eqref{e6.7} we obtain \begin{align*} \int_{\pi_{b+\frac{1}{2}}} \hat \nu \sum_{i=1}^m \Big|\frac{\partial u}{\partial x_i} \Big|^{p} \,dx &\leq \lambda(\|u\|_{L^{\infty} (\pi_0)}) [\alpha_2 (\sup_{\Omega'} \hat \nu)^{1/p} + 1] \int_{\pi_{b+\frac{1}{2}, b+1}} \hat \nu \sum_{i=1}^m \Big|\frac{\partial u}{\partial x_i} \Big|^{p} \,dx \\ &= (\alpha_3 + 1)\int_{\pi_{b+\frac{1}{2}, b+1}} \hat \nu \sum_{i=1}^m \Big|\frac{\partial u}{\partial x_i} \Big|^{p}\,dx, \end{align*} for every $b> 0$, where the positive constant $\alpha_3$ depends on $m$, $p$, $\beta$, $\Omega'$, $\|\hat \nu\|_{L^{\infty}(\Omega' )}$, $\|u\|_{L^{\infty}(\pi_0 )}$, $\lambda (s)$ and $\| 1/ \hat \nu\|_{L^{t}(\Omega')}$ Consequently, $$ I_{b+1}(u) \leq \frac{\alpha_3}{\alpha_3 +1} I_b (u) , \quad \forall b> 0 , $$ where, for every $a \geq 0$, $$ I_{a} (u)= \int_{\pi_{a}} \hat \nu \sum_{i=1}^{m} \Big| \frac {\partial u}{\partial x_i} \Big| ^{p} \,dx, \quad A=I_0(u) < \infty. $$ This formula, by induction, gives $$ I_{b+n} (u) \leq s^n I_b (u) \leq A s^n, $$ for $n \in \mathbb{N}$, $b> 0$ and $s=\frac{\alpha_3}{\alpha_3 +1}$. We can write last relation in this way $$ I_{b+n} (u) \leq A e^{n\log s},\quad \text{for every } b > 0, \; n \in \mathbb{N} \cup \{0\}. $$ It is simple to verify that above inequality gives $$ I_{\lambda} (u) \leq \alpha_4 e^{-\lambda \tilde \alpha}, \quad \text{for all } \lambda >0, $$ where $\alpha_4= A e^{\tilde \alpha}$ and $\tilde \alpha = -\log s >0$. Now, fixing $\alpha$ such that $0<\alpha < \tilde \alpha$, we have \begin{align*} \int_{\pi_0} e^{\alpha x_m} \hat \nu \sum_{i=1}^{m} \Big| \frac {\partial u}{\partial x_i} \Big| ^{p} \,dx & = \sum_{j=0}^{+\infty} \int_{\pi_{j,j+1}} e^{\alpha x_m} \hat \nu \sum_{i=1}^{m} \Big| \frac {\partial u}{\partial x_i} \Big| ^{p} \,dx \\ &\leq \sum_{j=0}^{+\infty} e^{\alpha (j+1)} \int_{\pi_{j,j+1}} \hat \nu \sum_{i=1}^{m} \Big| \frac {\partial u}{\partial x_i} \Big| ^{p} \,dx \\ &\leq \sum_{j=0}^{+\infty} e^{\alpha (j+1)} I_j (u) \\ &\leq \alpha_4 \sum_{j=0}^{+\infty} e^{\alpha (j+1)} e^{-j \tilde \alpha } < + \infty. \end{align*} The proof is complete. \end{proof} As in Section 4, we will show an example where all assumptions are fulfilled. Let $\Omega ' = \{x'=(x_1 , x_2 , \dots ,x_{m-1}) \in \mathbb{R}^{m-1} : | x' | < 1 \}$. Put $$ \hat \nu(x') = [ d(x', \partial \Omega')]^\rho = (1 - | x' |) ^\rho $$ for $\rho: 0<\rho < \min \big( \frac{p}{m} , (p-1) \big)$. Then the function $\hat \nu$ satisfies (H13) with $t$ arbitrarily taken as follows: $$ \max \Big( \frac{m}{p} , \frac{1}{p-1} \Big) < t < \frac{1}{\rho}. $$ Let us define in $\pi_0 \times \mathbb{R} \times \mathbb{R}^m \to \mathbb{R}$ the function $f(x,u,\eta)$ by $$ f(x,u, \eta) = ue^u (1 - | x' |) ^\rho | \eta | ^p- g_1 (x), $$ where $g_1 (x) \in L^\infty (\pi_0)$ has compact support. It is possible to verify (H14) by setting $\lambda(| u |) = e^{2| u | }$, and, taking into account that \begin{equation} \label{e6.10} \frac{1}{2} | u | ^p + uf(x,u, \eta) \geq -2^{\frac{1}{p-1}} | g_1 (x) | ^{\frac{p}{p-1}} \end{equation} for almost all $x \in \pi_0$ and for all $(u,\eta) \in \mathbb{R} \times \mathbb{R}^m$. Then, from \cite[Theorem 2.2]{DrabNic}, there exists a function $u \in \mathaccent"7017{W}^{1,p}(\hat \nu ,\pi_0) \cap L^{\infty}(\pi_0 )$ such that \begin{align*} &\int_{\pi_0}\Big \{ \frac {(1 - | x' |) ^\rho }{e^{2| u |}} \sum_{i=1}^m \Big| \frac{\partial u}{\partial x_i} \Big|^{p-2} \frac{\partial u}{\partial x_i} \frac{\partial w}{\partial x_i} + |u|^{p-2} u w + ue^u (1 - | x' | )^\rho | \nabla u | ^p w \Big \} \,dx \\ &= \int_{\pi_0} g_1 w \,dx \end{align*} for every arbitrary function $w \in \mathaccent"7017{W}^{1,p}(\hat \nu ,\pi_0) \cap L^{\infty}(\pi_0 )$. In this case $c_0 = 1$. From \eqref{e6.10} because of the support of $g_1$, there exists a positive number $a$ such that $$ \frac{1}{2} | u | ^p + uf(x,u, \eta) \geq 0 $$ for almost all $x \in \pi_a$ and for all $(u,\eta) \in \mathbb{R} \times \mathbb{R}^m$. So, it is possible to apply Theorem \ref{thm6.1} to the function $u$. \subsection*{Acknowledgments} I would like to thank the referee for carefully reading my manuscript and for giving such constructive comments which substantially helped improving the quality of this article. \begin{thebibliography}{00} \bibitem{Adam} R. A. Adams; \emph{Sobolev Spaces}, Academic Press, London 1975. \bibitem{BDY} P. A. Binding, P. Drabek, Y. Xi Huang; On Neumann boundary value problems for some quasilinear elliptic equations, \emph{Electronic Journal of Differential Equations}, 1997 No. 5 (1997), 1--11. \bibitem{Bocc1} L. Boccardo, F. Murat, J. P. Puel; Existence de solutions faibles pour des equations elliptiques quasilineares \'a croissance quadratique, \emph{Nonlinear Partial Differential Equations and Their Applications}, College de France Seminar, Vol. IV, edited by M. Brezis and J. Lions Research Notes in Mathematics, \textbf{84}, Pitman, London (1983), 19--73. \bibitem{Bocc2} L. Boccardo, F. Murat, J. P. Puel; R\'esultat d'existence pour certains probl\'emes elliptiques quasi-lin\'eaires, \emph{ Annali Scuola Normale Superiore di Pisa}, \textbf{4}, No. 11 (1984), 213--235. \bibitem{Bon1} S. Bonafede; Quasilinear degenerate elliptic variational inequalities with discontinuous coefficients, \emph{Commentationes Mathematicae Universitatis Carolinae}, \textbf{34}, No.1 (1993), 55--61. \bibitem{Bon2} S. Bonafede; Existence and regularity of solutions to a system of degenerate nonlinear elliptic equations, \emph{ British Journal of Mathematics and Computer Science}, \textbf{18}, No.5 (2016), 1--18. \bibitem{DMR} J. D. B. de Godoi, O. H. Miyagaki, R. S. Rodrigues; On Steklov-Neumann boundary value problems for some quasilinear elliptic equations, \emph{Applied Mathematics Letters}, \textbf{45} (2015), 47--51. \bibitem{DrabNic} P. Drabek, F. Nicolosi; Existence of bounded solutions for some degenerated quasilinear elliptic equations, \emph{Ann. Mat. Pura Appl.}, \textbf{165} (1993), 217--238. \bibitem{Fucik} S. Fucik, A. Kufner; \emph{Nonlinear differential equations}, Studies in Applied Mechanics vol. 2, Elsevier Scientific Publishing Company, Amsterdam - Oxford - New York 1980. \bibitem{GuglNicol} F. Guglielmino, F. Nicolosi; $W$-solutions of boundary value problems for degenerate elliptic operators, \emph{Ricerche Mat., Suppl.} \textbf{36} (1987), 59--72. \bibitem{GuglNic} F. Guglielmino, F. Nicolosi; Existence theorems for boundary value problems associated with quasilinear elliptic equations, \emph{Ricerche Mat.},\textbf{37} (1988), 157--176. \bibitem{kuf} A. Kufner; \emph{Weighted Sobolev Spaces}, Translated from the Czech. A Wiley-Interscience Publication. John Wiley and Sons, Inc., New York 1985. \bibitem{LadUra} O. A. Ladyzhenskaya, N. N. Ural'tseva; \emph{Linear and quasilinear elliptic equations}, Academic Press, New York and London 1968. \bibitem{Lax} P. D. Lax; A Phragm\'{e}n-Lindel\"{o}f theorem in harmonic analysis and its application to some questions in the theory of elliptic equations, \emph{ Comm. on Pure and Appl. Math.}, \textbf{10}, (1957), 361--389. \bibitem{LL} J. Leray, J. L. Lions; Quelques r\'esultats de Vi\v{s}ik sur les problemes elliptiques non lin\'eaires par les m\'ethodes de Minty-Browder, \emph{Bulletin de la Soci\'et\'e Math\'ematique de France}, \textbf{Vol. 93}, (1965), 97--107. \bibitem{Miranda} C. Miranda; Sul Teorema del Massimo Modulo per una Classe di Sistemi Ellittici di Equazioni del Secondo Ordine e per le Equazioni a Coefficienti Complessi, \emph{ Ist. Lombardo Accad. Sci. Lett. Rend.}, A \textbf{104}, (1970), 736--745. \bibitem{MurtStamp} M. K. V. Murthy, G. Stampacchia; Boundary value problems for some degenarate elliptic operators, \emph{ Ann. Mat. Pura Appl.}, \textbf{80}, No. 4, (1968), 1--122. \bibitem{N} J. Necas; \emph{Direct Methods in the Theory of Elliptic Equations}, Springer 1967. \bibitem{Niko1} F. Nicolosi; Sulla limitatezza delle soluzioni deboli dei problemi al contorno per operatori parabolici degeneri, \emph{Rend. Circ. Mat. Palermo}, \textbf{2}, 31 (1982), 23--40. \bibitem{TPP} K. Taira, D. K. Palagachev, P. R. Popivanov; A degenerate Neumann problem for quasilinear elliptic equations, \emph{Tokyo J. Math}, \textbf{23}, No.1 (2000), 227--234. \bibitem{ZLP} Z. Hu, L. Wang, P. Zhao; A priori estimates and existence for quasilinear elliptic equations with nonlinear Neumann boundary conditions, \emph{ Electronic Journal of Differential Equations}, No.187 (2016), 1--8. \bibitem{XK} X. Wu, Kok-Keong Tan; On existence and multiplicity of solutions of Neumann boundary value problems for quasilinear elliptic equations, \emph{ Nonlinear Analysis}, \textbf{65} (2006), 1334--1347. \end{thebibliography} \end{document}