\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 92, pp. 1--11.\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 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/92\hfil Existence results] {Existence results for nonlinear elliptic equations in bounded domains of $\mathbb{R}^n$} \author[M. Zribi\hfil EJDE-2006/92\hfilneg]{Malek Zribi} \address{Malek Zribi \\ 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 May 4, 2006. Published August 15, 2006.} \subjclass[2000]{34B27, 34J65} \keywords{Green function; elliptic equation; positive solutions} \begin{abstract} We establish existence results for the boundary-value problem $\Delta u+f(.,u)=0$ in a smooth bounded domain in $\mathbb{R}^n$ $(n\geq 2)$, where $f$ satisfies some appropriate conditions related to a Kato class. The proofs are based on various techniques related to potential theory. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \section{Introduction} Let $\Omega$ be a $C^{1,1}$ bounded domain in $\mathbb{R}^n$ $(n\geq 2)$. In this paper we study the existence and the asymptotic behaviour of bounded solutions to the nonlinear elliptic boundary-value problem $$\label{e1.1} \begin{gathered} \Delta u+f(.,u)=0\quad \text{in }\Omega \\ u>0,\quad \text{in }\Omega \\ u=g \quad\text{on } \partial \Omega, \end{gathered}$$ where $g$ is a nonnegative continuous function on $\partial \Omega$ and $f$ satisfies some convenient conditions. The question of existence of solutions of \eqref{e1.1} has been studied by several authors in both bounded and unbounded domains with various nonlinearities; see for example \cite% {b1,b2,d1,e1,k1,l1,l2,l3,m2,m3,m4,m5,s1,t1,y1,z1,z2} and references therein. Note that solutions of these problems are understood in distributional sense. Our tools are based essentially on some inequalities satisfied by the Green function $G(x,y)$ of $(-\Delta )$ in $\Omega$ which allow to some properties of functions belonging to the Kato class $K(\Omega )$ which contains properly the classical one; see \cite{a1,c1}. The class $K( \Omega )$ has been introduced in \cite{m5}, for $n\geq 3$ and \cite{m2,z1} for $n=2$ as follows. We denote by $\delta (x)$ the Euclidian distance between $x$ and $\partial \Omega$. \begin{definition} \label{def1} \rm A Borel measurable function $q$ in $\Omega$ belongs to the Kato class $K(\Omega )$ if $q$ satisfies $$\label{e1.2} \lim_{\alpha \to 0}\big(\sup_{x\in \Omega }\int_{\Omega \cap B(x,\alpha )}\frac{\delta (y)}{\delta (x)} G(x,y)| q(y)| dy\big)=0.$$ \end{definition} For the sake of simplicity we set $Hg$ the bounded continuous solution of the Dirichlet problem \begin{gather*} \Delta u=0\quad\text{in }\Omega \\ u=g\quad \text{on } \partial \Omega, \end{gather*} where $g$ is a nonnegative continuous function on $\partial \Omega$. We also refer to $Vf$ the potential of a measurable nonnegative function $f$, defined on $\Omega$ by \begin{equation*} Vf(x)=\int_{\Omega }G(x,y)f(y)dy. \end{equation*} Our plan in this paper is as follows. The section 2 is devoted to collect some preliminary results about the Green function $G(x,y)$ and the properties of the Kato class $K(\Omega )$. In section 3, we establish an existence result for \eqref{e1.1} where the combined effects of a singular and a sublinear term in the nonlinearity $f$ are considered. Our motivation in this section comes from paper \cite{s1}, where Shi and Yao investigated the existence of nonnegative solutions for the elliptic problem \begin{gather*} \Delta u+K(x)u^{-\gamma }+\lambda u^{\alpha }=0\quad \text{in }\Omega \\ u(x)>0\quad \text{in }\Omega \\ ,u=0\quad \text{on }\partial \Omega , \end{gather*}% where $\gamma$ and $\alpha$ in $(0,1)$ are two constants, $\lambda$ is a real parameter and $K$ is in $C^{0,\beta }(\overline{\Omega })$. Using this result. Sun and Li \cite{y1} gave a similar result in $\mathbb{R}^{n}$ ($% n\geq 2$). In fact they proved an existence result for the problem \begin{gather*} \Delta u+p(x)u^{-\gamma }+q(x)u^{\alpha }=0\quad \text{in }\mathbb{R}^{n} \\ u(x)>0,\quad x\in \mathbb{R}^{n} \\ u(x)\rightarrow 0,\quad \text{as }|x|\rightarrow \infty , \end{gather*}% where $\gamma$ and $\alpha$ in $(0,1)$ are two constants and $p,q$ are two nonnegative functions in $C_{\mathrm{loc}}^{\beta }(\mathbb{R}^{n})$ such that $p+q\neq 0$. The pure singular elliptic equation $$\label{e1.3} \Delta u+p(x)u^{-\gamma }=0,\quad \gamma >0,\; x\in D\subseteq \mathbb{R}^n$$ has been extensively studied for both bounded and unbounded domains $D$ in $% \mathbb{R}^n(n\geq 2)$. We refer to \cite{d1,e1,k1,l2,l3} and references therein) for various existence and uniqueness results related to solutions for equation \eqref{e1.3}. For more general situations M\^{a}agli and Zribi showed in \cite{m4} that the problem \begin{gather*} \Delta u+\varphi (.,u)=0,\quad x\in D \\ u=0 \quad\text{on } \partial D \\ \lim_{| x| \to \infty }u(x)=0,\quad \text{if $D$ is unbounded} \end{gather*} admits a unique positive solution if $\varphi$ is a nonnegative measurable function on $(0,\infty )$, which is nonincreasing and continuous with respect to the second variable and satisfies \begin{itemize} \item[(H0)] For all $c>0$, $\varphi (.,c)$ is in $K_{n}^{\infty }(D)$, where $K_{n}^{\infty }(D)$ is the classical Kato class; see \cite{z2}. \end{itemize} On the other hand, the problem \eqref{e1.1} with a sublinear term $f(.,u)$ have been studied in $\mathbb{R}^n$ by Brezis and Kamin in $[3]$. 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. Thus a natural question to ask is for more general singular and sublinear terms combined in the nonlinearity, whether or not \eqref{e1.1} has a solution which we aim to study in this section. In fact we are interested in solving the following problem (in the sense of distributions) $$\label{e1.4} \begin{gathered} \Delta u+\varphi (.,u)+\psi (.,u)=0,\quad \text{in }\Omega \\ u>0,\quad \text{in }\Omega \\ u=0 \quad \text{on } \partial \Omega\,. \end{gathered}$$ Here $\varphi$ and $\psi$ are required to satisfy the following hypotheses: \begin{itemize} \item[(H1)] $\varphi$ is a nonnegative Borel measurable function on $\Omega \times (0,\infty )$, continuous and nonincreasing with respect to the second variable. \item[(H2)] For all $c>0$, $x\to \varphi ( x,c\delta (x))$ is in $K(\Omega )$% . \item[(H3)] $\psi$ is a nonnegative Borel measurable function on $\Omega \times (0,\infty )$, continuous with respect to the second variable such that there exist a nontrivial nonnegative function $p$ and a nonnegative function $q\in K(\Omega )$ satisfying for $x\in \Omega$ and $t>0$, $$\label{e1.5} p(x)h(t)\leq \psi (x,t)\leq q(x)f(t),$$ where $h$ is a measurable nondecreasing function on $[0,\infty )$ satisfying $$\label{e1.6} \lim_{t\to 0^{+}}\frac{h(t)}{t}=+\infty$$ and $f$ is a nonnegative measurable function locally bounded on [ $0,\infty )$ satisfying $$\label{e1.7} \limsup_{t\to \infty }\frac{f(t) }{t}<\| Vq\| _{\infty }.$$ \end{itemize} Using a fixed point argument, we shall prove the following existence result. \begin{theorem} \label{thm1} Assume (H1)--(H3). Then the problem \eqref{e1.4} has a positive solution $u\in C_{b}(\Omega )$ such that for each $x\in \Omega$, \begin{equation*} a\delta (x)\leq u(x)\leq V(\varphi (.,a\delta ))(x)+bVq(x), \end{equation*} where $a,b$ are positive constants. \end{theorem} Typical examples of nonlinearities satisfying (H1)-(H3) are: \begin{gather*} \varphi (x,t)=p(x)(\delta (x))^{\gamma }t^{-\gamma }; \quad \gamma \geq 0, \\ \psi (x,t)=q(x)t^{\alpha }\log(1+t^{\beta }), \quad \alpha ,\beta \geq 0 \end{gather*} such that $\alpha +\beta <1$, where $p$ and $q$ are two nonnegative functions in $K(\Omega )$. In this section, using different techniques from those used by Shi and Yao \cite{s1}, we improve their results in the sense of distributional solutions. In section 4, we consider the nonlinearity $f(x,t)=-\varphi (x,t)$ and we suppose that $g$ is nontrivial, then using a potential theory approach we investigate an existence result and an uniqueness result for the problem $$\label{e1.8} \begin{gathered} \Delta u-\varphi (.,u)=0\quad \text{in }\Omega \\ u>0\quad \text{in }\Omega \\ u=g \quad\text{on } \partial \Omega, \end{gathered}$$ where $\varphi$ is required to satisfy the following three conditions: \begin{itemize} \item[(H4)] $\varphi$ is a nonnegative measurable function on $\Omega \times [ 0,\infty )$, continuous and nondecreasing with respect to the second variable. \item[(H5)] $\varphi (.,0)=0$. \item[(H6)] For all $c>0$, $\varphi (.,c)$ is in $K( \Omega )$. \end{itemize} Our main result is the following. \begin{theorem} \label{thm2} Assume (H4)-(H6). Then the problem \eqref{e1.8} has a unique positive solution $u$ such that $00$ such that \begin{equation*} f(x)\leq cg(x)\quad \text{for all }x\in S. \end{equation*} 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)\quad \text{for all }x\in S. \end{equation*} \section{Properties of the Green function and the Kato class} The existence results to prove, suggest collecting some estimates on the Green function $G$ and some properties of functions belonging to the Kato class $K(\Omega )$. The proofs of the following estimates and inequalities of $G$ can be found in \cite{m5} for $n\geq 3$ and \cite{z1} for $n=2$. \begin{proposition} \label{prop1} For each $x,y\in \Omega$, we have $$\label{e2.1} G(x,y)\sim \begin{cases} \frac{\delta (x)\delta (y)}{| x-y| ^{n-2}\big( | x-y| ^{2}+\delta (x)\delta (y)\big)} &\text{if }n\geq 3, \\ \log\big(1+\frac{\delta (x)\delta (y)}{| x-y| ^{2}}\big) &\text{if }n=2. \end{cases}$$ \end{proposition} \begin{corollary} \label{coro1} For $x,y\in \Omega$, $$\label{e2.2} \delta (x)\delta (y)\preceq G(x,y).$$ \end{corollary} \begin{theorem}[3G-Theorem] \label{thm3} There exists $C_{0}>0$ depending only on $\Omega$, such that for $x,y,z\in \Omega$, we have $$\label{e2.3} \frac{G(x,z)G(z,y)}{G(x,y)}\leq C_{0}\big[ \frac{\delta (z)}{\delta (x)} G(x,z)+\frac{\delta (z)}{\delta (y)}G(y,z)\big] .$$ \end{theorem} To recall some properties of the class $K(\Omega )$, we first give the following examples: \begin{enumerate} \item By \cite[Proposition 4]{m5}, the function $q(x)=1/ (\delta (x))^{\lambda }$ is in $K(\Omega )$ if and only if $\lambda <2$. \item By \cite[Proposition 3]{t1}, if $p>n/2$ and $\lambda <2-\frac{n}{p}$, then $L^{p}(\Omega )/(\delta (.))^{\lambda } \subset K(\Omega)$. \end{enumerate} The proof of the following Proposition can be found in \cite{m5,z1}. \begin{proposition} \label{prop2} Let $q$ be a nonnegative function in $K(\Omega )$. Then \begin{itemize} \item[(i)] $\| q\| <\infty$. \item[(ii)] The function $x\mapsto \delta (x) q(x)$ is in $L^{1}(\Omega )$. \item[(iii)] We have $$\label{e2.4} \delta (x)\preceq Vq(x).$$ \end{itemize} \end{proposition} For a fixed nonnegative function $q$ in $K(\Omega )$, we put \begin{equation*} \mathcal{M}_{q}:=\{ \varphi \in B(\Omega ),\text{ \ }| \varphi | \preceq q\} . \end{equation*} \begin{proposition} \label{prop3} Let $q$ be a nonnegative function in $K(\Omega )$, then the family of functions \begin{equation*} V(\mathcal{M}_{q})=\{ V\varphi :\varphi \in \mathcal{M}_{q}\} \end{equation*} is uniformly bounded and equicontinuous in $C_{0}(\Omega )$, and consequently it is relatively compact in $C_{0}(\Omega )$. \end{proposition} \begin{proof} The result holds by similar arguments as in \cite[proposition 3]{m5} and \cite[Proposition 8]{z1}. \end{proof} In the sequel, we use the notation \begin{equation*} \alpha _{q}:=\sup_{x,y\in \Omega }\int_{\Omega} \frac{ G(x,z)G(z,y)}{G(x,y)}% | q(z)| dz. \end{equation*} \begin{proposition}\label{prop4} %2.7 Let $q$ be a function in $K(\Omega )$ and $v$ be a nonnegative superharmonic function in $\Omega$. Then for each $x\in \Omega$, $$\label{e2.5} \int_{\Omega }G(x,y)v(y)| q(y)| dy\leq \alpha _{q}v(x)$$ and consequently, $\| q\| \leq \alpha _{q}\leq 2C_{0}\| q\|$, where $C_{0}$ is the constant given in \eqref{e2.3}. \end{proposition} For the proof of the above proposition, we refer the reader to \cite[Proposition 2]{t1}. \begin{corollary} \label{coro2} %\label{coro2.8} Let $q$ be a nonnegative function in $K(\Omega )$ and $v$ be a nonnegative superharmonic function in $\Omega$, then for each $x\in \Omega$ such that $v(x)<\infty$, we have \begin{equation*} \exp (-\alpha _{q})v(x)\leq (v-V_{q}(qv))(x)\leq v(x). \end{equation*} \end{corollary} \begin{proof} The upper inequality is trivial. For the lower one, we consider the function $\gamma (\lambda )=v(x)-\lambda V_{\lambda q}( qv)(x)$ for $\lambda \geq 0$. The function $\gamma$ is completely monotone on $[0,\infty )$ and so $\log\gamma$ is convex in $[0,\infty )$. This implies \begin{equation*} \gamma (0)\leq \gamma (1)\exp (-\frac{\gamma ^{\prime }(0)}{\gamma (0)}). \end{equation*} That is, \begin{equation*} v(x)\leq (v-V_{q}(qv))(x)\exp (\frac{V(qv)(x)}{v(x) }). \end{equation*} So, the result holds by \eqref{e2.5}. \end{proof} \section{First existence result} \begin{proof}[Proof of Theorem \protect\ref{thm1}] Assume (H1)-(H3). Using the Schauder fixed point theorem, we are going to construct a solution to problem \eqref{e1.4}. We note that by \eqref{e2.2} there exists a constant $\alpha _{1}>0$ such that for each $x,y\in \Omega$ $$\label{e3.1} \alpha _{1}\delta (x)\delta (y)\leq G(x,y).$$ Now, using (H3), there exists a compact $K$ of $\Omega$ such that \begin{equation*} 0<\alpha :=\int_{K}\delta (y)p(y)dy<\infty . \end{equation*} We put $\beta :=\min \{ \delta (x):x\in K\}$. Then from \eqref{e1.6}, we conclude that there exists $a>0$ such that $$\label{e3.2} \alpha _{1}\alpha h(a\beta )\geq a.$$ Furthermore, since $q\in K(\Omega )$, then by Proposition \ref{prop3} we have obviously that $\| Vq\| _{\infty }<\infty$. So taking $0<\eta <1/\|Vq\| _{\infty }$, we deduce by \eqref{e1.7} that there exists $\rho >0$ such that for $t\geq \rho$ we have $f(t)\leq \eta t$. Put $\gamma =\sup_{0\leq t\leq \rho }f(t)$. So we have that $$\label{e3.3} 0\leq f(t)\leq \eta t+\gamma ,\quad t\geq 0.$$ Next by (2.4), we note that there exists a constant $\alpha _{ \text{2}}>0$ such that $$\label{e3.4} \alpha _{2}\delta (x)\leq Vq(x),\quad \forall x\in \Omega .$$ >From (H2) and Proposition \ref{prop3}, we have that $\| V\varphi (.,a\delta )\| _{\infty }<\infty$. Hence, put \begin{equation*} b=\max \{ \frac{a}{\alpha_{2}}, \frac{\eta \| V\varphi (.,a\delta ) \| _{\infty }+\gamma }{1-\eta \| Vq\| _{\infty}}\} \end{equation*} and consider the closed convex set \begin{equation*} \Lambda =\{u\in C_{0}(\Omega ):a\delta (x)\leq u(x)\leq V\varphi (.,a\delta )(x)+bVq(x),\forall x\in \Omega \}. \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_{\Omega }G(x,y)[ \varphi (y,u(y))+\psi (y,u(y))] dy,\quad \forall x\in \Omega . \end{equation*} Let us prove that $T\Lambda \subset \Lambda$. Let $u\in \Lambda$ and $x\in \Omega$, then by (H1), (H3) and \eqref{e3.3} we have \begin{align*} Tu(x) & \leq V\varphi (.,a\delta )(x) +\int_{\Omega }G(x,y)q(y)f(u(y))dy \\ & \leq V\varphi (.,a\delta )(x)+\int_{\Omega }G(x,y)q(y)[ \eta u(y)+\gamma ] dy \\ &\leq V\varphi (.,a\delta )(x) +\int_{\Omega }G(x,y)q(y)[ \eta (\| V\varphi (.,a\delta )\| _{\infty }+b\| Vq\| _{\infty })+\gamma ] dy \\ & \leq V\varphi (.,a\delta )(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_{\Omega }G(x,y)\psi (y,u(y))dy \\ & \geq \alpha _{1}\delta (x)\int_{\Omega }\delta (y)p(y)h(a\delta (y))dy \\ & \geq \alpha _{1}\delta (x)h(a\beta )\int_{K}\delta (y)p(y)dy \\ & \geq \alpha _{1}\alpha h(a\beta )\delta (x) \\ & \geq a\delta (x). \end{align*} On the other hand, we have that for each $u\in \Lambda$, $$\label{e3.5} \varphi (.,u)\leq \varphi (.,a\delta )\text{ and } \psi (.,u)\leq [ \eta (\| V\varphi ( .,a\delta )\| +b\| Vq\| _{\infty })+\gamma ] q.$$ This implies by Proposition \ref{prop3} that $T\Lambda$ is relatively compact in $C_{0}(\Omega )$. 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 \Omega , \quad Tu_{k}(x)\to Tu(x)\quad \text{as } k\to \infty . \end{equation*} Now, since $T\Lambda$ is relatively compact in $C_{0}(\Omega )$, 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 $$\label{e3.6} u(x)=\int_{\Omega }G(x,y)[ \varphi (y,u(y))+\psi (y,u(y))] dy,\quad \forall x\in \Omega .$$ Finally, we need to prove that $u$ is solution of the problem \eqref{e1.4}. Since $q$ and $\varphi (.,a\delta )$ are in $K(\Omega )$, we deduce by \eqref{e3.5} and Proposition \ref{prop2}, that $y\mapsto \varphi (y,u(y))+\psi (y,u(y))\in L^{1}( \Omega )$. Moreover, since $u\in C_{0}(\Omega )$, we deduce from \eqref{e3.6}, that $V(\varphi (.,u)+\psi (.,u))\in L^{1}(\Omega )$. Hence $u$ satisfies in the sense of distributions the elliptic equation \begin{equation*} \Delta u+\varphi (.,u)+\psi (.,u)=0,\quad \text{in }\Omega . \end{equation*} This completes the proof. \end{proof} \begin{example} \label{exa3} \rm Let $\alpha ,\beta \geq 0$ such that $0\leq \alpha +\beta <1$, $\gamma >0$ and $p,q\in K^{+}(\Omega )$. Then the problem $$\label{e3.7} \begin{gathered} \Delta u+p(x)(u(x))^{-\gamma }(\delta (x))^{\gamma }+q(x)(u(x))^{\alpha }\log(1+(u(x)) ^{\beta })=0,\quad \text{in }\Omega \\ u>0,\quad \text{in }\Omega \end{gathered}$$ has a solution $u\in C_{0}(\Omega )$ satisfying $a\delta (x)\leq u(x)\leq Vp(x)+bVq(x)$, where $a,b>0$. \end{example} \begin{remark} \label{rmk1} \rm Taking in Example \ref{exa3} $\lambda <2$, $$p(x)=q(x)=\frac{1}{(\delta (x) ) ^{\lambda}},$$ we deduce from \cite{m5} that the solution of \eqref{e3.7} satisfies the following: \begin{itemize} \item[(i)] $u(x)\preceq (\delta (x))^{2-\lambda }$, if $1<\lambda <2$. \item[(ii)] $u(x)\preceq \delta (x)\log\frac{ (\sqrt{5}+1) d}{2\delta (x)}$, if $\lambda =1$, \item[(iii)] $u(x)\preceq \delta (x)$, if $\lambda<1$, where $d=\mathop{\rm diam}(\Omega )$. \end{itemize} Note that in Example \ref{exa3}, we have the result obtained by Shi and Yao \cite{s1}. \end{remark} \section{Second existence result} In this section, we shall prove Theorem \ref{thm2}. The proof is based on a comparison principle given by the following Lemma. For $u\in B(\Omega )$, put $u^{+}=\max (u,0)$. \begin{lemma} \label{lem1} Let $\varphi$ and $\psi$ satisfying (H4)-(H6). Assume that $\varphi \leq \psi$ on $\Omega \times \mathbb{R} _{+}$ and there exist continuous functions $u,v$ on $\Omega$ satisfying \begin{itemize} \item[(a)] $\Delta u-\varphi (.,u^{+})\leq \Delta v-\psi (.,v^{+})$ in $\Omega$ (in the distributional sense) \item[(b)] $u,v\in C_{b}(\Omega )$ \item[(c)] $u\geq v$ on $\partial \Omega$. \end{itemize} Then $u\geq v$ in $\Omega$. \end{lemma} \begin{proof} Suppose that the open set $D=\{ x\in \Omega :u(x)From the continuity of$\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 $satisfying \begin{equation*} u(x)=Hg(x)-V(\varphi (.,u^{+}))(x). \end{equation*} Finally, since$\varphi (.,u^{+})\in \mathcal{M}_{q}$, we conclude by Proposition \ref{prop2} that$u$satisfies in the sense of distributions the following \begin{gather*} \Delta u-\varphi (.,u^{+})=0 \\ \lim_{x\to \partial \Omega }u(x)=g. \end{gather*} Hence by (H5) and Lemma \ref{lem1}, we conclude that$u\geq 0$in$\Omega $and so it is a solution of \eqref{e1.8}. \end{proof} \begin{corollary} \label{coro3} Suppose that$\varphi $satisfies (H4)-(H6) and$g$is a nontrivial nonnegative continuous function in$\partial \Omega$. Suppose that there exists a function$q\in K(\Omega )$such that $$\label{e4.3} 0\leq \varphi (x,t)\leq q(x)t\quad \text{on }\Omega \times [0,\| g\| _{\infty }] .$$ Then the solution$u$of \eqref{e1.8} given by Theorem \ref{thm2} satisfies \begin{equation*} e^{-\alpha _{q}}Hg(x)\leq u(x)\leq Hg(x). \end{equation*} \end{corollary} \begin{proof} Since$usatisfies the integral equation \begin{equation*} u(x)=Hg(x)-V(\varphi (.,u))(x), \end{equation*} using \eqref{e4.1}, we obtain \begin{align*} u-V_{q}(qu)&=(Hg-V_{q}(qHg))-( V(\varphi (.,u))-V_{q}(qV(\varphi (.,u))) \\ &=(Hg-V_{q}(qHg))-V_{q}(\varphi (.,u)). \end{align*} That is, \begin{equation*} u=(Hg-V_{q}(qHg))+V_{q}(qu-\varphi (.,u)). \end{equation*} Now since00$and$q\in K^{+}(\Omega )$. 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