\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 44, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/44\hfil Existence of positive solutions] {Existence of positive solutions to nonlinear elliptic problem in the half space} \author[I. Bachar, H. M\^{a}agli, M. Zribi\hfil EJDE-2005/44\hfilneg] {Imed Bachar, Habib M\^{a}agli, Malek Zribi} % in alphabetical order \address{Imed Bachar\hfill\break D\'{e}partement de math\'{e}matiques, Facult\'{e} des Sciences de Tunis, Campus universitaire, 1060 Tunis, Tunisia} \email{Imed.Bachar@ipeit.rnu.tn} \address{Habib M\^{a}agli \hfill\break D\'{e}partement de math\'{e}matiques, Facult\'{e} des Sciences de Tunis, Campus universitaire, 1060 Tunis, Tunisia} \email{habib.maagli@fst.rnu.tn} \address{Malek Zribi \hfill\break D\'{e}partement de math\'{e}matiques, Facult\'{e} des Sciences de Tunis, Campus universitaire, 1060 Tunis, Tunisia} \email{Malek.Zribi@insat.rnu.tn} \date{} \thanks{Submitted December 20, 2004. Published April 14, 2005.} \subjclass[2000]{34B27, 34J65} \keywords{Green function; elliptic equation; positive solution} \begin{abstract} This paper concerns nonlinear elliptic equations in the half space $\mathbb{R}_{+}^{n}:=\{ x=(x',x_{n})\in \mathbb{R}^{n}:x_{n}>0\} $, $n\geq 2$, with a nonlinear term satisfying some conditions related to a certain Kato class of functions. We prove some existence results and asymptotic behaviour for positive solutions using a potential theory approach. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In the present paper, we study the nonlinear elliptic equation \begin{equation} \Delta u+f(.,u)=0,\text{ in }\mathbb{R}_{+}^{n} \label{e1.1} \end{equation} in the sense of distributions, with some boundary values determined below (see problems \eqref{e1.7}, \eqref{e1.12} and \eqref{e1.13}). Here $\mathbb{R}_{+}^{n}:=\{ x=(x',x_{n})\in \mathbb{R} ^{n}:x_{n}>0\} $, $(n\geq 2)$. Several results have been obtained for \eqref{e1.1}, in both bounded and unbounded domain $D\subset \mathbb{R}^{n}$ with different boundary conditions; see for example \cite{bm,bmm,b3,bk,c2,d1,e1,k1,lai,laz,mm,m3} and the references therein. Our goal of this paper is to undertake a study of \eqref{e1.1} when the nonlinear term $f(x,t)$ satisfies some conditions related to a certain Kato class $K^{\infty}( \mathbb{R}_{+}^{n}) $, and to answer the questions of existence and asymptotic behaviour of positive solutions. Our tools are based essentially on some inequalities satisfied by the Green function $G(x,y)$ of $( -\Delta ) $ in $\mathbb{R}_{+}^{n}$. This allows us to state some properties of functions in the class $K^{\infty }( \mathbb{R}_{+}^{n}) $ which was introduced in \cite{bm} for $n\geq 3$, and in \cite{bmm} for $n=2$. \begin{definition} \label{def1.1} \rm A Borel measurable function $q$ in $\mathbb{R}_{+}^{n}$ belongs to the class $K^{\infty }(\mathbb{R}_{+}^{n})$ if $q$ satisfies the following two conditions \begin{gather} \label{e1.2} \lim_{\alpha \to 0}( \sup_{x\in \mathbb{R}_{+}^{n}} \int_{\mathbb{R}_{+}^{n}\cap B(x,\alpha )}\frac{y_{n}}{x_{n}}% G(x,y)|q(y)|dy) =0\,, \\ \lim_{M\to \infty }( \sup_{x\in \mathbb{R}_{+}^{n}} \int_{\mathbb{R}_{+}^{n}\cap (| y| \geq M)} \frac{y_{n}}{x_{n}}G(x,y)|q(y)|dy) =0. \label{e1.3} \end{gather} \end{definition} The class $K^{\infty }(\mathbb{R}_{+}^{n})$ is sufficiently rich. It contains properly the classical Kato class $K_{n}^{\infty }(\mathbb{R}_{+}^{n})$, defined by Zhao \cite{z1}, for $n\geq 3$ in unbounded domains $D$ as follows: \begin{definition} \rm A Borel measurable function $q$ on $D$ belongs to the Kato class $K_{n}^{\infty }(D)$ if $q$ satisfies the following two conditions \begin{gather*} \lim_{\alpha \to 0} \sup_{x\in D} \int_{D\cap (| x-y| \leq \alpha )}\frac{| \psi (y)| }{| x-y| ^{d-2}}dy)=0\,,\\ \lim_{M\to \infty } \sup_{x\in D} \int_{D\cap (| y| \geq M)}\frac{| \psi (y)| }{| x-y| ^{d-2}}dy)=0. \end{gather*} \end{definition} Typical examples of functions $q$ in the class $K^{\infty }(\mathbb{R}_{+}^{n})$ are: $q\in L^{p}( \mathbb{R}_{+}^{n}) \cap L^{1}(\mathbb{R}_{+}^{n})$, where $p>\frac{n}{2}$ and $n\geq 3$; and $$ q(x)=\frac{1}{(| x| +1)^{\mu -\lambda }x_{n}^{\lambda }}, $$ where $\lambda <2<\mu $ and $n\geq 2$ (see \cite{bm,bmm}). We shall refer in this paper to the bounded continuous solution $Hg$ of the Dirichlet problem \begin{equation} \label{e1.4} \begin{gathered} \Delta u=0,\quad \text{in }\mathbb{R}_{+}^{n} \\ \lim_{x_{n}\to 0}u(x)=g(x'), \end{gathered} \end{equation} where $g$ is a nonnegative bounded continuous function in $\mathbb{R}^{n-1}$ (see \cite[p. 418]{a1}). We also refer to the potential of a measurable nonnegative function $f$, defined on $\mathbb{R}_{+}^{n}$ by \begin{equation*} % 1.5 Vf(x)=\int_{\mathbb{R}_{+}^{n}}G(x,y)f(y)dy. \end{equation*} Our paper is organized as follows. Existence results are proved in sections 3, 4, and 5. In section 2, we collect some preliminary results about the Green function $G$ and the class $K^{\infty }(\mathbb{R}_{+}^{n})$. We prove further that if $p>n/2$ and $a\in L^{p}( \mathbb{R}_{+}^{n}) $, then for $\lambda <2-\frac{n}{p}<\mu $, the function \begin{equation*} x\mapsto \frac{a(x)}{(| x| +1)^{\mu -\lambda }x_{n}^{\lambda }}, \end{equation*} is in $K^{\infty }(\mathbb{R}_{+}^{n})$. In section 3, we establish an existence result for equation \eqref{e1.1} where a singular term and a sublinear term are combined in the nonlinearity $f(x,t)$. The pure singular elliptic equation \begin{equation*} %\label{e1.5} \Delta u+p(x)u^{-\gamma }=0,\quad \gamma >0,\; x\in D\subseteq \mathbb{R}^{n} \end{equation*} has been extensively studied for both bounded and unbounded domains. We refer to \cite{c2,d1,e1,k1,lai,laz} and references therein, for various existence and uniqueness results related to solutions for above equation. For more general situations M\^{a}agli and Zribi showed in \cite{m3} that the problem \begin{equation} \label{e1.6} \begin{gathered} \Delta u+\varphi ( .,u) =0,\quad x\in D \\ u\big|_{\partial D}=0 \\ \lim_{| x| \to \infty }u(x)=0,\quad \text{if $D$ is unbounded} \end{gathered} \end{equation} admits a unique positive solution if $\varphi $ is a nonnegative measurable function on $( 0,\infty ) $, which is non-increasing and continuous with respect to the second variable and satisfies \begin{itemize} \item[(H0)] For all $c>0$, $\varphi ( .,c) \in K_{n}^{\infty}(D)$. \end{itemize} If $D=\mathbb{R}_{+}^{n}$, the result of M\^{a}agli and Zribi \cite{m3} has been improved later by Bachar and M\^{a}agli in \cite{bm}, where they gave an existence and an uniqueness result for \eqref{e1.6}, with the more restrictive condition \begin{itemize} \item[(H0')] For all $c>0$, $\varphi ( .,c) \in K^{\infty }(\mathbb{R}_{+}^{n})$. \end{itemize} On the other hand, \eqref{e1.1} with a sublinear term $f(.,u) $ have been studied in $\mathbb{R}^{n}$ by Brezis and Kamin in \cite{bk}. Indeed, the authors proved the existence and the uniqueness of a positive solution for the problem \begin{gather*} \Delta u+\rho (x)u^{\alpha }=0\quad \text{in }\mathbb{R}^{n}, \\ \liminf_{| x| \to \infty }u(x)=0, \end{gather*} with $0<\alpha <1$ and $\rho $ is a nonnegative measurable function satisfying some appropriate conditions. In this section, we combine a singular term and a sublinear term in the nonlinearity. Indeed, we consider the boundary value problem \begin{equation} \label{e1.7} \begin{gathered} \Delta u+\varphi (.,u)+\psi (.,u)=0,\quad \text{in }\mathbb{R}_{+}^{n} \\ u>0,\quad \text{in }\mathbb{R}_{+}^{n} \\ \lim_{x_{n}\to 0}u(x)=0, \\ \lim_{| x| \to +\infty }u(x)=0, \end{gathered} \end{equation} in the sense of distributions, where $\varphi $ and $\psi $ are required to satisfy the following hypotheses: \begin{itemize} \item[(H1)] $\varphi $ is a nonnegative Borel measurable function on $\mathbb{R}_{+}^{n}\times ( 0,\infty ) $, continuous and non-increasing with respect to the second variable. \item[(H2)] For all $c>0$, $x\mapsto \varphi( x,c\theta ( x) )$ belongs to $K^{\infty }(\mathbb{R}_{+}^{n})$, where \begin{equation*} \theta ( x) =\frac{x_{n}}{( 1+| x| ) ^{n}}. \end{equation*} \item[(H3)] $\psi $ is a nonnegative Borel measurable function on $\mathbb{R}_{+}^{n}\times ( 0,\infty ) $, continuous with respect to the second variable such that there exist a nontrivial nonnegative function $p$ and a nonnegative function $q\in K^{\infty }(\mathbb{R}_{+}^{n})$ satisfying for $x\in \mathbb{R}_{+}^{n}$ and $t>0$, \begin{equation} \label{e1.8} p( x) h( t) \leq \psi ( x,t) \leq q( x) f( t) , \end{equation} where $h$ is a measurable nondecreasing function on $[0,\infty )$ satisfying \begin{equation} \label{e1.9} \lim_{t\to 0^{+}}\frac{h( t) }{t}=+\infty \end{equation} and $f$ is a nonnegative measurable function locally bounded on $[0,\infty )$ satisfying \begin{equation} \label{e1.10} \limsup_{t\to \infty }\frac{f( t) }{t}<\| Vq\| _{\infty }. \end{equation} \end{itemize} Using a fixed point argument, we shall state the following existence result. \begin{theorem} \label{thm1.3} Assume (H1)--(H3). Then the problem \eqref{e1.7} has a positive solution $u\in C_{0}(\mathbb{R}_{+}^{n})$ satisfying for each $x\in \mathbb{R}_{+}^{n}$ \begin{equation*} a\theta ( x) \leq u(x)\leq V( \varphi (.,a\theta ) )( x) +bVq(x), \end{equation*} where $a,b$ are positive constants. \end{theorem} Note that M\^{a}agli and Masmoudi studied in \cite{mm,m4} the case $\varphi \equiv 0$, under similar conditions to those in (H3). Indeed the authors gave an existence result for \begin{equation} \label{e1.11} \Delta u+\psi ( .,u) =0\text{ in }D, \end{equation} with some boundary conditions, where $D$ is an unbounded domain in $\mathbb{R}^{n}$ $( n\geq 2) $ with compact nonempty boundary. Typical examples of nonlinearities satisfying (H1)--(H3) are: $$ \varphi ( x,t) =p(x)( \theta ( x) ) ^{\gamma }t^{-\gamma }, $$ for $\gamma \geq 0$, and $$ \psi ( x,t) =p(x)t^{\alpha }\log ( 1+t^{\beta }) , $$ for $\alpha,\beta \geq 0$ such that $\alpha +\beta <1$, where $p$ is a nonnegative function in $K^{\infty }(\mathbb{R}_{+}^{n})$. In section 4, we consider the nonlinearity $f(x,t)=-t\varphi (x,t)$ and we use a potential theory approach to investigate an existence result for \eqref{e1.1}. Let $\alpha \in [ 0,1] $ and $\omega $ be the function defined on $\mathbb{R}_{+}^{n}$ by $\omega ( x) =\alpha x_{n}+( 1-\alpha ) $. We shall prove in this section the existence of positive continuous solutions for the following nonlinear problem \begin{equation} \label{e1.12} \begin{gathered} \Delta u-u\varphi (.,u)=0,\quad \text{in }\mathbb{R}_{+}^{n} \\ u>0,\quad \text{in }\mathbb{R}_{+}^{n} \\ \lim_{x_{n}\to 0}u(x)=( 1-\alpha ) g(x'), \\ \lim_{x_{n}\to +\infty }\frac{u(x)}{x_{n}}=\alpha \lambda , \end{gathered} \end{equation} in the sense of distributions, where $\lambda $ is a positive constant, $g$ is a nontrivial nonnegative bounded continuous function in $\mathbb{R}^{n-1}$ and $\varphi $ satisfies the following hypotheses: \begin{itemize} \item[(H4)] $\varphi $ is a nonnegative measurable function on $\mathbb{R}_{+}^{n}\times [ 0,\infty )$. \item[(H5)] For all $c>0$, there exists a positive function $q_{c}\in K^{\infty }(\mathbb{R}_{+}^{n})$ such that the map $t\mapsto t( q_{c}( x) -\varphi ( x,t\omega (x)) ) $ is continuous and nondecreasing on $[ 0,c] $ for every $x\in \mathbb{R}_{+}^{n}$. \end{itemize} \begin{theorem} \label{thm1.4} Under assumptions (H4) and (H5), problem \eqref{e1.12} has a positive continuous solution $u$ such that for each $x\in \mathbb{R}_{+}^{n}$, \begin{equation*} c( \alpha \lambda x_{n}+( 1-\alpha ) Hg(x)) \leq u( x) \leq \alpha \lambda x_{n}+( 1-\alpha ) Hg(x), \end{equation*} where $c\in ( 0,1) $. \end{theorem} Note that if $\alpha =0$, then the solution $u$ satisfies $cHg(x)\leq u( x) \leq Hg(x)$, $c\in ( 0,1) $. In particular, $u$ is bounded on $\mathbb{R}_{+}^{n}$. Our techniques are similar to those used by M\^{a}agli and Masmoudi in \cite{mm,m4}. Section 5 deals with the question of existence of continuous bounded solutions for the problem \begin{equation} \label{e1.13} \begin{gathered} \Delta u-\varphi (.,u)=0,\quad \text{in }\mathbb{R}_{+}^{n}\\ u>0,\quad \text{in }\mathbb{R}_{+}^{n} \\ \lim_{x_{n}\to 0}u(x)=g(x'), \end{gathered} \end{equation} where $g$ is a nontrivial nonnegative bounded continuous function in $\mathbb{R}^{n-1}$. We also establish an uniqueness result for such solutions. Here the nonlinearity $\varphi $ satisfies the following conditions: \begin{itemize} \item[(H6)] $\varphi $ is a nonnegative measurable function on $\mathbb{R}_{+}^{n}\times [0,\infty )$, continuous and nondecreasing with respect to the second variable. \item[(H7)] $\varphi (.,0)=0$. \item[(H8)] For all $c>0$, $\varphi ( .,c) \in K^{\infty}(\mathbb{R}_{+}^{n})$. \end{itemize} \begin{theorem} \label{thm1.5} Under assumptions (H6)--(H8), problem \eqref{e1.13} has a unique positive solution $u$ such that for each $x\in \mathbb{R}_{+}^{n}$, \begin{equation*} 00,\quad \text{in }\mathbb{R}_{+}^{n} \\ \lim_{x_{n}\to 0}\frac{u(x)}{x_{n}^{m-1}}=g(x') \end{gathered} \end{equation} (in the sense of distributions). Here $\varphi $ is a nonnegative measurable function on $\mathbb{R}_{+}^{n}\times ( 0,\infty ) $, continuous and non-increasing with respect to the second variable and satisfies some conditions related to a certain Kato class appropriate to the $m$-polyharmonic case. In fact in \cite{b3}, we proved that for a fixed positive harmonic function $h_{0}$ in $\mathbb{R}_{+}^{n}$, if $g\geq (1+c)h_{0}$, for $c>0$, then the problem \eqref{e1.14} has a positive continuous solution $u$ satisfying $u(x)\geq x_{n}^{m-1}h_{0}(x)$ for every $x\in \mathbb{R}_{+}^{n}$. Thus a natural question to ask is for $t\to \varphi ( x,t) $ nondecreasing, whether or not \eqref{e1.14} has a solution, which we aim to study in the case $m=1$. \subsection*{Notation} To simplify our statements, we define the following symbols. \\ $\mathbb{R}_{+}^{n}:=\{ x=(x_{1},...,x_{n})=( x',x_{n}) \in \mathbb{R}^{n}:x_{n}>0\} $, $n\geq 2$. \\ $\overline{x}=(x',-x_{n})$, for $x\in \mathbb{R}_{+}^{n}$. \\ Let $\mathcal{B}( \mathbb{R}_{+}^{n}) $ denote the set of Borel measurable functions in $\mathbb{R}_{+}^{n}$ and $\mathcal{B}^{+}( \mathbb{R}_{+}^{n}) $ the set of nonnegative functions in this space. \\ $C_{b}(\mathbb{R}_{+}^{n})=\{ w\in C( \mathbb{R}_{+}^{n}) : w\text{ is bounded in }\mathbb{R}_{+}^{n}\} $ \\ $C_{0}(\mathbb{R}_{+}^{n})=\{ w\in C( \mathbb{R}% _{+}^{n}) :\lim_{x_{n}\to 0}w(x)=0\text{ \ and \ }% \lim_{| x| \to \infty }w(x)=0\} $ \\ $C_{0}(\overline{\mathbb{R}_{+}^{n}})=\{ w\in C( \overline{% \mathbb{R}_{+}^{n}}) :\text{\ }\lim_{|x|\to \infty }w(x)=0\} $. Note that $C_{b}(\mathbb{R}_{+}^{n}), $ $C_{0}(\mathbb{R}_{+}^{n}) $ and $C_{0}(\overline{\mathbb{R}_{+}^{n}})$ are three Banach spaces with the uniform norm $\| w\| _{\infty }=\sup_{x\in \mathbb{R}_{+}^{n}}| w(x)| $. For any $q\in \mathcal{B}( \mathbb{R}_{+}^{n}) $, we put \begin{equation*} \| q\| :=\sup_{x\in \mathbb{R}_{+}^{n}}\int_{\mathbb{R}_{+}^{n}} \frac{y_{n}}{x_{n}}G(x,y)| q(y)| dy. \end{equation*} Recall that the potential $Vf$ of a function $f\in \mathcal{B}^{+}( \mathbb{R}_{+}^{n}) $, is lower semi-continuous in $\mathbb{R}_{+}^{n}$. Furthermore, for each function $q\in \mathcal{B}^{+}( \mathbb{R}_{+}^{n}) $ such that $Vq<\infty $, we denote by $V_{q}$ the unique kernel which satisfies the following resolvent equation (see \cite{m1,n1}): \begin{equation} \label{e1.15} V=V_{q}+V_{q}(qV)=V_{q}+V(qV_{q}). %\label{1.6} \end{equation} For each $u\in \mathcal{B}( \mathbb{R}_{+}^{n}) $ such that $V(q| u| )<\infty $, we have \begin{equation} \label{e1.16} ( I-V_{q}(q.)) ( I+V(q.)) u=( I+V(q.)) ( I-V_{q}(q.)) u=u. %1.7 \end{equation} Let $f$ and $g$ be two positive functions on a set $S $. We call $f\sim g$, if there is $c>0$ such that \begin{equation*} \frac{1}{c}g( x) \leq f( x) \leq cg( x) \text{ \ \ \ for all }x\in S. \end{equation*} We call $f\preceq g$, if there is $c>0$ such that \begin{equation*} f( x) \leq cg( x) \text{ \ \ \ for all }x\in S. \end{equation*} The following properties will be used in this article: For $x,y\in \mathbb{R}_{+}^{n}$, note that $| x-\overline{y}|^2-| x-y|^2=4x_{n}y_{n}$. So we have \begin{gather} | x-\overline{y}| ^{2} \sim |x-y| ^{2}+x_{n}y_{n} \label{e1.17}\\ x_{n}+y_{n} \leq | x-\overline{y}| . \label{e1.18} \end{gather} Let $\lambda ,\mu >0$ and $0<\gamma \leq 1$, then for $t\geq 0$ we have \begin{gather} \log(1+\lambda t) \sim \log(1+\mu t), \label{e1.19}\\ \log(1+t) \preceq t^{\gamma }. \label{e1.20} \end{gather} \section{Properties of the Green function and the Kato class $K^{\infty }(\mathbb{R}_{+}^{n})$} In this section, we briefly recall some estimates on the Green function $G$ and we collect some properties of functions belonging to the Kato class $K^{\infty }(\mathbb{R}_{+}^{n})$, which are useful at stating our existence results. For $x,y\in \mathbb{R}_{+}^{n}$, we set \begin{equation*} G(x,y)=\begin{cases} \frac{\Gamma ( \frac{n}{2}-1) }{4\pi ^{n/2}} \big[\frac{1}{| x-y| ^{n-2}}-\frac{1}{| x-\overline{y}% | ^{n-2}}\big] ,&\text{if }n\geq 3 \\[3pt] \frac{1}{4\pi }\log\big( 1+\frac{4x_{2}y_{2}}{| x-y|^{2}}\big) , & \text{if }n=2, \end{cases} \end{equation*} the Green function of $( -\Delta ) $ in $\mathbb{R}_{+}^{n}$ (see \cite[p. 92]{a1}). Then we have the following estimates and inequalities whose proofs can be found in \cite{bm} for $n\geq 3$ and in \cite{bmm} for $n=2$. \begin{proposition}\label{prop2.1} For $x,y\in \mathbb{R}_{+}^{n}$, we have \begin{equation} \label{e2.1} G(x,y)\sim \begin{cases} \frac{x_{n}y_{n}}{{| x-y| }^{n-2}| x-\overline{y}|^{2}} &\text{if }n\geq 3, \\[3pt] \frac{x_{2}y_{2}}{| x-\overline{y}| ^{2}} {\log (}1+\frac{| x-\overline{y}| ^{2}}{| x-y| ^{2}}) &\text{if }n=2. \end{cases} \end{equation} \end{proposition} \begin{corollary} \label{coro2.2} For $x,y\in \mathbb{R}_{+}^{n}$, we have \begin{equation} \label{e2.2} \frac{x_{n}y_{n}}{( | x| +1) ^{n}( |y| +1) ^{n}}\preceq G(x,y). \end{equation} \end{corollary} \begin{theorem}[3G-Theorem] \label{thm2.3} There exists $C_{0}>0$ such that for each $x,y,z\in \mathbb{R}_{+}^{n}$, we have \begin{equation} \label{e2.3} \frac{G(x,z)G(z,y)}{G(x,y)}\leq C_{0}\big[ \frac{z_{n}}{x_{n}}G(x,z)+% \frac{z_{n}}{y_{n}}G(y,z)\big] . \end{equation} \end{theorem} Let us recall in the following properties of functions in the class $K^{\infty }(\mathbb{R}_{+}^{n})$. The proofs of these propositions can be found in \cite{bm,bmm}. \begin{proposition} \label{prop2.4} Let $q$ be a nonnegative function in $K^{\infty }(\mathbb{R}_{+}^{n})$. Then we have: (i) $\| q\| <\infty $; (ii) The function $x\mapsto \frac{x_{n}}{(|x| +1) ^{n}}q( x) $ is in $L^{1}( \mathbb{R}_{+}^{n})$. and (iii) \begin{equation} \label{e2.4} \frac{x_{n}}{( | x| +1) ^{n}}\preceq Vq(x). \end{equation} \end{proposition} For a fixed nonnegative function $q$ in $K^{\infty }(\mathbb{R}_{+}^{n})$, we put \begin{equation*} \mathcal{M}_{q}:=\{ \varphi \in B(\mathbb{R}_{+}^{n}),\text{ \ }% | \varphi | \preceq q\} . \end{equation*} \begin{proposition} \label{prop2.5} Let $q$ be a nonnegative function in $K^{\infty }(\mathbb{R}_{+}^{n})$, then the family of functions \begin{equation*} V( \mathcal{M}_{q}) =\left\{ V\varphi :\text{ }\varphi \in \mathcal{M}_{q}\right\} \end{equation*} is relatively compact in $C_{0}(\mathbb{R}_{+}^{n})$. \end{proposition} \begin{proposition} \label{prop2.6} Let $q$ be a nonnegative function in $K^{\infty }(\mathbb{R}_{+}^{n})$, then the family of functions \begin{equation*} \mathcal{N}_{q}=\big\{ \int_{\mathbb{R}_{+}^{n}}\frac{y_{n}}{x_{n}}G(x,y)| \varphi (y)| dy: \varphi \in \mathcal{M}_{q}\big\} \end{equation*} is relatively compact in $C_{0}(\overline{\mathbb{R}_{+}^{n}})$. \end{proposition} In the sequel, we use the following notation \begin{equation*} \alpha _{q}:=\sup_{x,y\in \mathbb{R}_{+}^{n}}\int_{\mathbb{R} _{+}^{n}}\frac{G(x,z)G(z,y)}{G(x,y)}| q(z)| dz. \end{equation*} \begin{lemma} \label{lem2.7} Let $q$ be a function in $K^{\infty }(\mathbb{R}_{+}^{n})$. Then we have \begin{equation*} \| q\| \leq \alpha _{q}\leq 2C_{0}\| q\| , \end{equation*} where $C_{0}$ is the constant given in \eqref{e2.3}. \end{lemma} \begin{proof} By \eqref{e2.3}, we obtain easily that $\alpha _{q}\leq 2C_{0}\|q\| $. On the other hand, we have by Fatou lemma that for each $x\in \mathbb{R}_{+}^{n}$ \begin{equation*} \int_{\mathbb{R}_{+}^{n}}\frac{z_{n}}{x_{n}}G(x,z)| q(z)| dz\leq \liminf_{| \zeta | \to \infty }\int_{\mathbb{R}_{+}^{n}}\frac{z_{n}}{x_{n}} \frac{| x-\zeta | ^{n}}{| z-\zeta | ^{n}} G(x,z)| q(z)| dz. \end{equation*} Now since for each $x,z\in \mathbb{R}_{+}^{n}$ and $\zeta \in \partial \mathbb{R}_{+}^{n}$, we have \begin{equation*} \lim_{y\to \zeta }\frac{G(z,y)}{G(x,y)}=\frac{z_{n}}{x_{n}}% \frac{| x-\zeta | ^{n}}{| z-\zeta |^{n}}. \end{equation*} Then by Fatou lemma we deduce that \[ \int_{\mathbb{R}_{+}^{n}}\frac{z_{n}}{x_{n}}\frac{| x-\zeta | ^{n}}{| z-\zeta | ^{n}}G(x,z)| q(z)| dz \leq \liminf_{y\to \zeta }\int_{\mathbb{R}_{+}^{n}}G(x,z) \frac{G(z,y)}{G(x,y)}| q(z)| dz \\ \leq \alpha _{q}. \] We derive obviously that $\| q\| \leq \alpha _{q}$. \end{proof} \begin{proposition} \label{prop2.8} Let $q$ be a function in $K^{\infty }(\mathbb{R}_{+}^{n})$ and $v$ be a nonnegative superharmonic function in $\mathbb{R}_{+}^{n}$. Then for each $x\in \mathbb{R}_{+}^{n}$, we have \begin{equation}\label{e2.5} \int_{\mathbb{R}_{+}^{n}}G(x,y)v(y)| q(y)| dy\leq \alpha _{q}v(x). \end{equation} \end{proposition} \begin{proof} Let $v$ be a nonnegative superharmonic function in $\mathbb{R}_{+}^{n}$, then there exists (see \cite[Theorem 2.1]{ps}) a sequence $(f_{k})_{k}$ of nonnegative measurable functions in $\mathbb{R}_{+}^{n}$ such that the sequence $(v_k)_K$ defined on $\mathbb{R}_{+}^{n}$ by $v_{k}:=Vf_{k}$ increases to $v$. Since for each $x,z\in \mathbb{R}_{+}^{n}$, we have \[ \int_{\mathbb{R}_{+}^{n}}G(x,y)G(y,z)|q(y)|\,dy\leq \alpha _{q}G(x,z), \] it follows that \[ \int_{\mathbb{R}_{+}^{n}}G(x,y)v_{k}(y)|q(y)|\,dy\leq \alpha _{q}v_{k}(x). \] Hence, the result holds from the monotone convergence theorem. \end{proof} \begin{corollary} \label{coro2.9} Let $q$ be a nonnegative function in $K^{\infty }(\mathbb{R}_{+}^{n})$ and $v$ be a nonnegative superharmonic function in $\mathbb{R}_{+}^{n}$, then for each $x\in \mathbb{R}_{+}^{n}$ such that $0n/2$ and $a$ be a function in $L^{p}( \mathbb{R}_{+}^{n}) $. Then for $\lambda <2-\frac{n}{p}<\mu $, the function $\varphi ( x) =\frac{a(x)}{(| x| +1)^{\mu-\lambda }x_{n}^{\lambda }}$ is in $K^{\infty}(\mathbb{R}_{+}^{n})$. \end{proposition} \begin{proof} Let $p>n/2$ and $q\geq 1$ such that $\frac{1}{p}+\frac{1}{q}=1$. Let $a$ be a function in $L^{p}( \mathbb{R}_{+}^{n}) $ and $\lambda <2-\frac{n}{p}<\mu $. First, we claim that the function $\varphi $ satisfies $( 1.2) $. Let $0<\alpha <1$. Since $x_{n}\leq ( 1+| x| ) $ and $y_{n}\leq ( 1+| y| ) $, then we remark that if $| x-y| \leq \alpha $, then $(| x| +1)\sim (| y| +1)$ and consequently \begin{equation} \label{e2.6} | x-\overline{y}| \preceq (| y| +1),\quad \text{for }y\in B(x,\alpha ). \end{equation} \noindent Put $\lambda ^{+}=\max ( \lambda ,0) $. So to show the claim, we use the H\"{o}lder inequality and we distinguish the following two cases: \noindent\textit{Case 1. }$n\geq 3 $. Using \eqref{e2.1}, \eqref{e1.18} and the fact that $|x-y| \leq | x-\overline{y}| $ and $y_{n}\leq ( 1+| y| ) $, we deduce that \begin{align*} &\int_{B(x,\alpha )\cap \mathbb{R}_{+}^{n}}\frac{y_{n}}{x_{n}} G(x,y)\varphi ( y) dy\\ &\leq \| a\| _{p}(\int_{B(x,\alpha )\cap \mathbb{R}_{+}^{n}}\frac{y_{n}^{2q}}{| x-y| ^{( n-2) q}| x-\overline{y}|^{2q}y_{n}^{\lambda q}(| y| +1)^{( \mu -\lambda ) q}}dy)^{\frac{1}{q}}\\ &\leq \| a\| _{p}(\int_{B(x,\alpha )\cap \mathbb{R}_{+}^{n}}\frac{dy}{| x-y| ^{(n-2+\lambda ^{+}) q}})^{\frac{1}{q}}\\ &\preceq \alpha ^{2-\frac{n}{p}-\lambda ^{+},} \end{align*} which tends to zero as $\alpha \to 0$.\smallskip \noindent\textit{Case 2. }$n=2$. Using \eqref{e2.1}, \eqref{e2.6}, \eqref{e1.18} and taking $\gamma \in (\frac{\lambda^{+}}{2},\frac{1}{q})$ in (1.19), we obtain that \begin{align*} &\int_{B(x,\alpha )\cap \mathbb{R}_{+}^{2}}\frac{y_{2}}{x_{2}}% G(x,y)\varphi ( y) dy\\ &\leq \| a\| _{p}(\int_{B(x,\alpha )\cap \mathbb{R}_{+}^{2}}\frac{y_{2}^{( 2-\lambda ) q}}{| x- \overline{y}| ^{2q}(| y| +1)^{( \mu -\lambda ) q}}(\log (1+\frac{| x-\overline{y}| ^{2}}{ | x-y| ^{2}}))^{q}dy)^{\frac{1}{q}}\\ &\leq \| a\| _{p}(\int_{B(x,\alpha )\cap \mathbb{R}_{+}^{2}}\frac{| x-\overline{y}| ^{( 2\gamma -\lambda ^{+}) q}}{(| y| +1)^{( \mu -\lambda ^{+}) q}| x-y| ^{2\gamma q}}dy)^{\frac{1}{q}}\\ &\leq \| a\| _{p}(\int_{B(x,\alpha )\cap \mathbb{R}_{+}^{2}}\frac{1}{| x-y| ^{2\gamma q}}dy)^{\frac{1}{q}}\\ & \preceq \alpha ^{2-2\gamma q} \end{align*} which tends to zero as $\alpha \to 0$. \noindent Now, we claim that the function $\varphi $ satisfies \eqref{e1.3}. Let $M>1$ and put $\Omega :=\{ y\in \mathbb{R}% _{+}^{n}:( | y| \geq M) \cap ( | x-y| \geq \alpha ) \} $ and \begin{equation*} I(x,M):=\int_{\Omega }\frac{y_{n}}{x_{n}}G(x,y)\varphi ( y) dy. \end{equation*} By the above argument, to show the claim we need only to prove that $% I(x,M)\longrightarrow 0$, as $M\longrightarrow \infty $, uniformly on $x\in \mathbb{R}_{+}^{n}$. So we use the H\"{o}lder inequality and we distinguish the following two cases: \noindent\textit{Case 1.} $y\in D_{1}$. From \eqref{e2.1}, it is clear that $G(x,y)\preceq \frac{x_{n}y_{n}}{| x-y| ^{n}}$. Then we have \begin{equation*} \int_{\Omega \cap D_{1}}\frac{y_{n}}{x_{n}}G(x,y)\varphi ( y) dy\preceq \| a\| _{p}(\int_{\Omega \cap D_{1}} \frac{y_{n}^{( 2-\lambda ) q}}{| y| ^{( \mu -\lambda) q}| x-y| ^{nq}}dy) ^{1/q}. \end{equation*} Now we write that $2-\lambda =(2-\lambda -\frac{n}{p})+\frac{n}{p}$ and we put $\gamma =\mu -2+\frac{n}{p}$. Hence, using the fact that $y_{n}\leq \max (| y| ,| x-y| )$, we deduce that \begin{equation*} \int_{\Omega \cap D_{1}}\frac{y_{n}}{x_{n}}G(x,y)\varphi ( y) dy\preceq \| a\| _{p}(\int_{\Omega \cap D_{1}}\frac{dy}{% | x-y| ^{n}| y| ^{\gamma q}})^{1/q}. \end{equation*} On the other hand \begin{align*} &\int_{\Omega \cap D_{1}}| x-y| ^{-n}|y| ^{-\gamma q}dy \\ \preceq& \sup_{| x| \leq \frac{M}{2}} \int_{\Omega \cap \mathbb{R}_{+}^{n}}| x-y|^{-n}| y| ^{-\gamma q}dy \\ &\quad +\underset{| x| \geq \frac{M}{2}}{\sup } \int_{(\max (M,\frac{| x| }{2})\leq | y| \leq 2| x| )\cap \mathbb{R}_{+}^{n}\cap (| x-y| \geq \alpha )}| x-y| ^{-n}| y| ^{-\gamma q}dy \\ &\quad +\sup_{| x| \geq \frac{M}{2}}\int_{(| y| \geq 2| x| )\cap \mathbb{R}_{+}^{n}\cap (| x-y| \geq \alpha )}| x-y| ^{-n}| y| ^{-\gamma q}dy \\ &\quad +\sup_{| x| \geq 2M}\int_{(M\leq | y| \leq \frac{| x| }{2})\cap \mathbb{R}_{+}^{n}\cap (| x-y| \geq \alpha )}| x-y| ^{-n}| y|^{-\gamma q}dy \\ &\preceq \int_{(| y| \geq M)}\frac{1}{| y| ^{n+\gamma q}}\,dy +\sup_{| z|\geq \frac{M}{2}}\frac{\log (\frac{3| z| }{\alpha })} {| z| ^{\gamma q}} \\ &\preceq \frac{1}{M^{\gamma q}}+\sup_{| z| \geq \frac{M}{2}} \frac{\log (\frac{3| z| }{\alpha })}{| z| ^{\gamma q}}. \end{align*} \textit{Case 2.} $y\in D_{2}$. From Lemma \ref{lem2.10}, we have that $| y| \sim | x| $, $y_{n}\sim x_{n}\sim | x-\overline{y}| $. This implies:\\ If $n\geq 3$, then by \eqref{e2.1}, we deduce that \begin{align*} \int_{\Omega \cap D_{2}}\frac{y_{n}}{x_{n}}G(x,y)\varphi ( y) dy &\preceq \| a\| _{p}\frac{1}{x_{n}^{\lambda }| x| ^{\mu -\lambda }}(\int_{\Omega \cap B(x,cx_{n})}\frac{dy}{% | x-y| ^{( n-2) q}})^{\frac{1}{q}}\\ &\preceq \| a\| _{p}\frac{x_{n}^{2-\lambda -\frac{n}{p}}}{| x| ^{\mu -\lambda }}\\ &\preceq \| a\| _{p}\frac{1}{M^{\mu -2+\frac{n}{p}}}. \end{align*} If $n=2$, then from \eqref{e2.1} and \eqref{e1.19} it follows that \begin{align*} \int_{\Omega \cap D_{2}}\frac{y_{2}}{x_{2}}G(x,y)\varphi ( y)dy &\preceq \| a\| _{p}\frac{1}{x_{2}^{\lambda }| x| ^{\mu -\lambda }}(\int_{\Omega \cap B(x,cx_{n})} (\log (1+\frac{x_{2}^{2}}{| x-y| ^{2}}))^{q}dy)^{\frac{1}{q}}\\ &\preceq \| a\| _{p}\frac{x_{n}^{\frac{2}{q}-\lambda }} {| x| ^{\mu -\lambda }}\\ &\preceq \| a\| _{p}\frac{1}{M^{\mu -2+\frac{2}{p}}}. \end{align*} Hence we conclude that $I(x,M)$ converges to zero as $M\to \infty $ uniformly on $x\in \mathbb{R}_{+}^{n}$. This completes the proof. \end{proof} \section{Proof of Theorem \ref{thm1.3}} Recall that $\theta ( x) =\frac{x_{n}}{( | x| +1) ^{n}}$ on $\mathbb{R}_{+}^{n}$. \begin{proof}[Proof of Theorem \ref{thm1.3}] Assuming (H1)--(H3), we shall use the Schauder fixed point theorem. Let $K $ be a compact of $\mathbb{R}_{+}^{n}$ such that, using (H3), we have \begin{equation*} 0<\alpha :=\int_{K}\theta ( y) p(y)dy<\infty . \end{equation*} We put $\beta :=\min \{\theta (x):x\in K\}$. We note that by \eqref{e2.2} there exists a constant $\alpha _{1}>0$ such that for each $x,y\in \mathbb{R}_{+}^{n}$ \begin{equation} \label{e3.1} \alpha _{1}\theta ( x) \theta ( y) \leq G(x,y). \end{equation} Then from \eqref{e1.9}, we deduce that there exists $a>0$ such that \begin{equation} \label{e3.2} \alpha _{1}\alpha h(a\beta )\geq a. \end{equation} On the other hand, since $q\in K^{\infty}(\mathbb{R}_{+}^{n})$, then by Proposition \ref{prop2.5} we have that $\|Vq\| _{\infty }<\infty $. So taking $0<\delta <\frac{1}{\| Vq\|_{\infty }}$ we deduce by \eqref{e1.10} that there exists $\rho >0$ such that for $t\geq \rho $ we have $f(t)\leq \delta t$. Put $\gamma =\sup_{0\leq t\leq \rho}f(t)$. So we have that \begin{equation} \label{e3.3} 0\leq f(t)\leq \delta t+\gamma ,\quad t\geq 0. \end{equation} Furthermore by \eqref{e2.4}, we note that there exists a constant $\alpha_{2}>0$ such that \begin{equation} \label{e3.4} \alpha _{2}\theta ( x) \leq Vq(x),\quad \forall x\in \mathbb{R}_{+}^{n}, \end{equation} and from (H2) and Proposition \ref{prop2.5}, we have that $\| V\varphi ( .,a\theta ) \| _{\infty }<\infty $. Let \[ b=\max \big\{ \frac{a}{\alpha _{2}},\frac{\delta \| V\varphi ( .,a\theta ) \| _{\infty }+\gamma }{1-\delta \| Vq\| _{\infty }}\big\} \] and consider the closed convex set \begin{equation*} \Lambda =\{u\in C_{0}(\mathbb{R}_{+}^{n}):a\theta ( x) \leq u(x)\leq V\varphi ( .,a\theta ) ( x) +bVq(x),\forall x\in \mathbb{R}_{+}^{n}\}. \end{equation*} Obviously, by \eqref{e3.4} we have that the set $\Lambda $ is nonempty. Define the integral operator $T$ on $\Lambda $ by \begin{equation*} Tu(x)=\int_{\mathbb{R}_{+}^{n}}G(x,y)[ \varphi (y,u(y))+\psi (y,u(y))] dy,\quad \forall x\in \mathbb{R}_{+}^{n}. \end{equation*} Let us prove that $T\Lambda \subset \Lambda $. Let $u\in \Lambda $ and $x\in \mathbb{R}_{+}^{n}$, then by \eqref{e3.3} we have \begin{align*} Tu(x) & \leq V\varphi ( .,a\theta ) ( x) +\int_{\mathbb{R}_{+}^{n}}G(x,y)q(y)f(u(y))dy \\ & \leq V\varphi ( .,a\theta ) ( x) +\int_{ \mathbb{R}_{+}^{n}}G(x,y)q(y)[ \delta u(y)+\gamma ] dy \\ &\leq V\varphi ( .,a\theta ) ( x) +\int_{\mathbb{R}_{+}^{n}}G(x,y)q(y)[ \delta ( \| V\varphi ( .,a\theta ) \| _{\infty }+b\| Vq\| _{\infty }) +\gamma ] dy \\ & \leq V\varphi ( .,a\theta ) ( x) +bVq(x). \end{align*} Moreover from the monotonicity of $h$, \eqref{e3.1} and \eqref{e3.2}, we have \begin{align*} Tu(x) & \geq \int_{\mathbb{R}_{+}^{n}}G(x,y)\psi (y,u(y))dy \\ & \geq \alpha _{1}\theta ( x) \int_{\mathbb{R} _{+}^{n}}\theta ( y) p(y)h(a\theta ( y) )dy \\ & \geq \alpha _{1}\theta ( x) h(a\beta )\int_{K}\theta ( y) p(y)dy \\ & \geq \alpha _{1}\alpha h(a\beta )\theta ( x) \medskip \\ & \geq a\theta ( x) . \end{align*} On the other hand, we have that for each $u\in \Lambda $, \begin{equation} \label{e3.5} \varphi ( .,u) \leq \varphi ( .,a\theta ) \quad\text{and}\quad \psi ( .,u) \leq [ \delta ( \| V\varphi ( .,a\theta ) \| +b\| Vq\| _{\infty }) +\gamma ] q. \end{equation} This implies by Proposition \eqref{e2.5} that $T\Lambda $ is relatively compact in $C_{0}(\mathbb{R}_{+}^{n})$. In particular, we deduce that $T\Lambda \subset \Lambda $. Next, we prove the continuity of $T$ in $\Lambda $. Let $(u_{k})_{k}$ be a sequence in $\Lambda $ which converges uniformly to a function $u$ in $\Lambda $. Then since $\varphi $ and $\psi $ are continuous with respect to the second variable, we deduce by the dominated convergence theorem that \begin{equation*} \forall x\in \mathbb{R}_{+}^{n}, \; Tu_{k}(x)\to Tu(x)\quad \text{as }k\to \infty . \end{equation*} Now, since $T\Lambda $ is relatively compact in $C_{0}(\mathbb{R}_{+}^{n})$, then we have the uniform convergence. Hence $T$ is a compact operator mapping from $\Lambda $ to itself. So the Schauder fixed point theorem leads to the existence of a function $u\in \Lambda $ such that \begin{equation} \label{e3.6} u(x)=\int_{\mathbb{R}_{+}^{n}}G(x,y)[ \varphi (y,u(y))+\psi (y,u(y))] dy,\quad \forall x\in \mathbb{R}_{+}^{n}. \end{equation} Finally, since $q$ and $\varphi ( .,a\theta ) $ are in $K^{\infty }(\mathbb{R}_{+}^{n})$, we deduce by \eqref{e3.5} and Proposition \eqref{e2.4}, that $y\mapsto \varphi (y,u(y))+\psi (y,u(y))\in L_{loc}^{1}( \mathbb{R}_{+}^{n}) $. Moreover, since $u\in C_{0}(\mathbb{R}_{+}^{n})$, we deduce from \eqref{e3.6}, that $V(\varphi (.,u)+\psi (.,u))\in L_{loc}^{1}( \mathbb{R}_{+}^{n}) $. Hence $u$ satisfies in the sense of distributions the elliptic equation \begin{equation*} \Delta u+\varphi (.,u)+\psi (.,u)=0,\text{\ in }\mathbb{R}_{+}^{n} \end{equation*} and so it is a solution of the problem \eqref{e1.7}. \end{proof} \begin{example}\label{ex3.1} \rm Let $\alpha ,\beta \geq 0$ such that $0\leq \alpha +\beta <1$ and $p\in K^{\infty }(\mathbb{R}_{+}^{n})$. Then the problem \begin{equation} \label{e3.7} \begin{gathered} \Delta u+p(x)[ ( u(x)) ^{-\gamma }( \theta ( x) ) ^{\gamma }+( u(x)) ^{\alpha }\log (1+( u(x)) ^{\beta })] =0,\quad \text{in }\mathbb{R}_{+}^{n} \\ u>0,\quad \text{in }\mathbb{R}_{+}^{n} \end{gathered} \end{equation} has a solution $u\in C_{0}(\mathbb{R}_{+}^{n})$ satisfying $\ a\theta ( x) \leq u(x)\leq bVp(x)$, where $a,b>0$. \end{example} \begin{remark} \rm Taking in Example \ref{ex3.1} the function $p(x)=\frac{1}{x_{n}^{\lambda }( 1+| x| ) ^{\mu -\lambda }}$, for $\lambda <2<\mu $, we deduce from \cite{bm,bmm} that the solution of \eqref{e3.7} has the following behaviour \begin{itemize} \item[(i)] $u(x)\preceq \frac{x_{n}^{2-\lambda }}{( 1+| x| ) ^{n+2-2\lambda }}$, if $1<\lambda <2$ and $\mu \geq n+2-\lambda $ \item[(ii)] $u(x)\preceq \theta ( x) \log (\frac{2( 1+| x| ) ^{2}}{x_{n}})$, if $\lambda =1\text{ and }\mu \geq n+1$ or $\lambda <1\text{ and }\mu =n+1$. \item[(iii)] $u(x)\preceq \theta ( x) $, if $\lambda <1$ and $\mu >n+1$ \item[(iv)] $u(x)\preceq \frac{x_{n}^{\mu -n}}{( 1+|x| ) ^{2\mu -n-2}}$, if $n<\mu <\min (n+1,n+2-\lambda)$. \end{itemize} \end{remark} \section{Proof of Theorem \ref{thm1.4}} In this section, we are interested in the existence of continuous solutions for the problem \eqref{e1.12}. We recall that $\omega ( x) =\alpha x_{n}+( 1-\alpha ) $, $x\in \mathbb{R}_{+}^{n}$, where $\alpha \in [ 0,1] $. We aim to prove Theorem \ref{thm1.4}. So we need the following lemma \begin{lemma} \label{lem4.1} Let $q$ be a nonnegative function in $K^{\infty}(\mathbb{R}_{+}^{n})$, then the family of functions \begin{equation*} \big\{ \int_{\mathbb{R}_{+}^{n}}\frac{\omega (y)}{\omega (x)} G(x,y)| \varphi (y)| dy\text{ }:\text{ }\varphi \in \mathcal{M}_{q}\big\} \end{equation*} is relatively compact in $C_{0}(\overline{\mathbb{R}_{+}^{n}})$. \end{lemma} \begin{proof} We remark that \begin{equation*} \frac{\omega (y)}{\omega (x)}=\frac{\alpha y_{n}+( 1-\alpha ) }{ \alpha x_{n}+( 1-\alpha ) }\leq \max (1,\frac{y_{n}}{x_{n}})\leq 1+\frac{y_{n}}{x_{n}}. \end{equation*} So the result holds from Propositions \ref{prop2.5} and \ref{prop2.6}. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.4}] Let $\lambda >0$ and $c:=\sup \{ \lambda ,\| g\| _{\infty }\}$. Then by (H5), there exists a nonnegative function $q:=q_{c}\in K^{\infty }(\mathbb{R}_{+}^{n})$, such that the map \begin{equation} \label{e4.1} t\mapsto t( q(x)-\varphi ( x,t\omega ( x) ) ) \end{equation} is continuous and nondecreasing on $[ 0,c]$. We denote by $h(x)=\alpha \lambda x_{n}+( 1-\alpha ) Hg(x)$. Let \begin{equation*} \Lambda :=\left\{ u\in \mathcal{B}^{+}( \mathbb{R}_{+}^{n}) :\exp (-\alpha _{q})h\leq u\leq h\right\} . \end{equation*} Note that since for $u\in \Lambda $, we have $u\leq h\leq c$ $\omega $, then \eqref{e4.1} implies in particular that for $u\in \Lambda $ \begin{equation} \label{e4.2} 0\leq \varphi (.,u)\leq q. \end{equation} We define the operator $T$ on $\Lambda $ by \begin{equation*} Tu( x) :=h( x) -V_{q}( qh) ( x) +V_{q}[ ( q-\varphi ( .,u) ) u] (x) . \end{equation*} First, we claim that $\Lambda $ is invariant under $T$. Indeed, for each $u\in \Lambda $ we have \[ Tu( x) \leq h( x) -V_{q}( qh) (x) +V_{q}( qu) ( x) \leq h( x) . \] Moreover, by \eqref{e4.2} and Corollary \ref{coro2.9}, we obtain \[ Tu( x) \geq h( x) -V_{q}( qh) (x) \geq \exp (-\alpha _{q})h( x) . \] Next, we prove that the operator $T$ is nondecreasing on $\Lambda $. Let $u,v\in \Lambda $ such that $u\leq v$, then from \eqref{e4.1} we have \begin{equation*} Tv-Tu=V_{q}[ ( q-\varphi ( .,v) ) v-( q-\varphi ( .,u) ) u] \geq 0. \end{equation*} Now, we consider the sequence $( u_{j}) $ defined by $u_{0}=h-V_{q}( qh) $ and $u_{j+1}=Tu_{j}$ for $j\in \mathbb{N}$. Then since $\Lambda $ is invariant under $T$, we obtain obviously that $% u_{1}=Tu_{0}\geq $ $u_{0}$ and so from the monotonicity of $T$, we deduce that \begin{equation*} u_{0}\leq u_{1}\leq \dots \leq u_{j}\leq h. \end{equation*} Hence by \eqref{e4.1} and the dominated convergence theorem, we deduce that the sequence $( u_{j}) $ converges to a function $u\in \Lambda $, which satisfies \begin{equation*} u( x) =h( x) -V_{q}( qh) ( x) +V_{q}[ ( q-\varphi ( .,u) ) u] ( x). \end{equation*} Or, equivalently \begin{equation*} u-V_{q}( qu) =(h-V_{q}(qh))-V_{q}(u\varphi ( .,u) ). \end{equation*} Applying the operator $( I+V( q.) ) $ on both sides of the above equality and using \eqref{e1.15}, we deduce that $u$ satisfies \begin{equation} \label{e4.3} u=h-V( u\varphi ( .,u) ) . \end{equation} Finally, we need to verify that $u$ is a positive continuous solution for the problem \eqref{e1.12}. Indeed, from \eqref{e4.2}, we have \begin{equation} \label{e4.4} u\varphi ( .,u) \leq qh\leq cq\omega . \end{equation} This implies by Proposition \ref{prop2.4} that either $u$ and $u\varphi (.,u) $ are in $L_{\rm loc}^{1}( \mathbb{R}_{+}^{n}) $. Furthermore, from \eqref{e4.4}, we have that $\frac{1}{\omega }u\varphi (.,u) \in \mathcal{M}_{q}$. Which implies by Lemma \ref{lem4.1} that $\frac{1}{\omega }V(u\varphi ( .,u) )\in C_{0}(\overline{\mathbb{R}_{+}^{n}})$. In particular, we have $V(u\varphi ( .,u) )\in L_{\rm loc}^{1}(\mathbb{R}_{+}^{n}) $. Hence, by \eqref{e4.3}, we obtain that $u$ is continuous on $\mathbb{R}_{+}^{n}$ and satisfies (in the sense of distributions) the elliptic differential equation \begin{equation*} \Delta u-u\varphi ( .,u) =0\text{ \ in \ }\mathbb{R}_{+}^{n}. \end{equation*} On the other hand, since $\frac{1}{\omega }V(u\varphi ( .,u) )\in C_{0}(\overline{\mathbb{R}_{+}^{n}})$ and $Hg( x) $ is bounded on $\mathbb{R}_{+}^{n}$ and satisfies $\lim_{x_{n}\to 0}Hg( x) =g(x')$, we deduce easily that $% \lim_{x_{n}\to 0}u( x) =( 1-\alpha ) g(x')$ and $\lim_{x_{n}\to +\infty }\frac{u(x)}{x_{n}}=\alpha \lambda $. This completes the proof. \end{proof} \begin{example}\label{ex4.3} \rm Let $\gamma >1$, $\alpha \in [ 0,1] $, $\beta>0$ and $\lambda <2<\mu $. Let $g$ be a nontrivial nonnegative bounded continuous function in $\mathbb{R}^{n-1}$ and $p\in B^{+}( \mathbb{R}_{+}^{n}) $ satisfying \[ p(x)\preceq \frac{1}{(| x| +1) ^{\mu -\lambda }x_{n}^{\lambda }(x_{n}+1)^{\gamma -1}}. \] Then the problem \begin{gather*} \Delta u-p(x)u^{\gamma }(x)=0,\quad\text{in }\mathbb{R}_{+}^{n}\\ \lim_{x_{n}\to 0}u(x)=( 1-\alpha ) g(x'), \\ \lim_{x_{n}\to +\infty }\frac{u(x)}{x_{n}}=\alpha \beta , \end{gather*} (in the sense of distributions) has a continuous positive solution $u$ satisfying \begin{equation*} u(x)\sim \alpha \beta x_{n}+( 1-\alpha ) Hg( x). \end{equation*} \end{example} \section{Proof of Theorem \ref{thm1.5}} In this section, we need the following standard Lemma. For $u\in B(\mathbb{R}_{+}^{n}) $, put $u^{+}=\max (u,0)$. \begin{lemma} \label{lem5.1} Let $\varphi $ and $\psi $ satisfy (H6)--(H8). Assume that $\varphi \leq \psi $ on $\mathbb{R}_{+}^{n}\times \mathbb{R}_{+}$ and there exist continuous functions $u,v$ on $\mathbb{R}_{+}^{n}$ satisfying \begin{itemize} \item[(a)] $\Delta u-\varphi ( .,u^{+}) =0=\Delta v-\psi ( .,v^{+}) $ in $\mathbb{R}_{+}^{n}$ \item[(b)] $u,v\in C_{b}( \mathbb{R}_{+}^{n}) $ \item[(c)] $u\geq v$ on $\partial \mathbb{R}_{+}^{n}$. \end{itemize} Then $u\geq v$ in $\mathbb{R}_{+}^{n}$. \end{lemma} \begin{proof}[Proof of Theorem \ref{thm1.5}] An immediate consequence of the comparison principle in Lemma \ref{lem5.1} is that problem \eqref{e1.13} has at most one solution in $\mathbb{R}_{+}^{n}$. The existence of a such solution is assured by the Schauder fixed point Theorem. Indeed, to construct the solution, we consider the convex set \begin{equation*} \Lambda =\{ u\in C_{b}( \mathbb{R}_{+}^{n}) :u\leq \| g\|_{\infty }\} . \end{equation*} We define the integral operator $T$ on $\Lambda $ by \begin{equation*} Tu(x)=Hg(x)-V(\varphi (.,u^{+}))(x). \end{equation*} Since $Hg(x)\leq \| g\| _{\infty }$, for $x\in \mathbb{R}_{+}^{n}, $ we deduce that for each $u\in \Lambda $, \begin{equation*} Tu\leq \| g\| _{\infty }\text{, in }\mathbb{R}_{+}^{n}. \end{equation*} Furthermore, putting $q=\varphi (.,\| g\| _{\infty })$, we have by (H8) that $q\in K^{\infty }(\mathbb{R}_{+}^{n})$. So by (H6), we deduce that $V(\varphi (.,u^{+}))\in V(\mathcal{M}_{q}) $. This together with the fact that $Hg\in C_{b}( \mathbb{R}_{+}^{n}) $ imply by Proposition \ref{prop2.5} that $T\Lambda $ is relatively compact in $C_{b}( \mathbb{R}_{+}^{n}) $ and in particular $T\Lambda \subset \Lambda $. From the continuity of $\varphi $ with respect to the second variable, we deduce that $T$ is continuous in $\Lambda $ and so it is a compact operator from $\Lambda $ to itself. Then by the Schauder fixed point Theorem, we deduce that there exists a function $u\in \Lambda $ satisfying \begin{equation*} u(x)=Hg(x)-V(\varphi (.,u^{+}))(x). \end{equation*} This implies, using Proposition \ref{prop2.4} and the fact that $V(\varphi (.,u^{+}))\in C_{0}( \mathbb{R}_{+}^{n}) $, that $u$ satisfies in the sense of distributions \begin{gather*} \Delta u-\varphi ( .,u^{+}) =0 \\ \lim_{x_{n}\to 0}u(x)=g(x'). \end{gather*} Hence by (H7) and Lemma \ref{lem5.1}, we conclude that $u\geq 0$ in $\mathbb{R}_{+}^{n}$. This completes the proof. \end{proof} \begin{corollary} \label{coro5.2} Let $\varphi $ satisfying (H6)--(H8) and $g$ be a nontrivial nonnegative bounded continuous function in $\mathbb{R}^{n-1}$. Suppose that there exists a function $q\in K^{\infty }(\mathbb{R}_{+}^{n})$ such that \begin{equation} 0\leq \varphi ( x,t) \leq q( x) t\quad \text{on } \mathbb{R}_{+}^{n}\times [ 0,\| g\| _{\infty }] . \end{equation} Then the solution $u$ of \eqref{e1.13} given by Theorem \ref{thm1.5} satisfies \begin{equation*} e^{-\alpha _{q}}Hg(x)\leq u(x)\leq Hg(x). \end{equation*} \end{corollary} \begin{proof} Since $u$ satisfies the integral equation \begin{equation*} u(x)=Hg(x)-V(\varphi (.,u))(x), \end{equation*} using (1.15), we obtain \begin{align*} u-V_{q}( qu) =( Hg-V_{q}( qHg) ) -( V(\varphi (.,u))-V_{q}( qV(\varphi (.,u)) ) \\ =( Hg-V_{q}( qHg) ) -V_{q}( \varphi (.,u)) . \end{align*} That is, \begin{equation*} u=( Hg-V_{q}( qHg) ) +V_{q}( qu-\varphi (.,u)) . \end{equation*} Now since $00$ and $q\in K^{\infty }(\mathbb{R}_{+}^{n})$. Put $\varphi ( x,t) =q( x) t^{\sigma }$. Then the problem \begin{gather*} \Delta u-q( x) u^{\sigma }=0,\quad \text{in }\mathbb{R}_{+}^{n} \\ \lim_{x_{n}\to 0}u(x)=g(x') \end{gather*} (in the sense of distributions) has a positive bounded continuous solution $u$ in $\mathbb{R}_{+}^{n}$ satisfying \begin{equation*} 0\leq Hg(x)-u(x)\leq \| g\| _{\infty }^{\sigma }Vq(x). \end{equation*} Furthermore, if $\sigma \geq 1$, we have by Corollary \ref{coro5.2} that for each $x\in \mathbb{R}_{+}^{n}$ \begin{equation*} e^{-\alpha _{q}}Hg(x)\leq u(x)\leq Hg(x). \end{equation*} \end{example} \subsection*{Acknowledgement} The authors want to thank the referee for his/her useful suggestions. \begin{thebibliography}{99} \bibitem{a1} Armitage, D. H., Gardiner, S. 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