\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 177, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/177\hfil Semilinear elliptic systems in the half-space] {Positive bounded solutions for semilinear elliptic systems with indefinite weights in the half-space} \author[R. Alsaedi, H. M\^{a}agli, V. R\u{a}dulescu, N. Zeddini \hfil EJDE-2015/177\hfilneg] {Ramzi Alsaedi, Habib M\^{a}agli, Vicen\c tiu R\u{a}dulescu, Noureddine Zeddini} \address{ Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia} \email[R. Alsaedi]{ramzialsaedi@yahoo.co.uk} \email[H. M\^{a}agli]{abobaker@kau.edu.sa} \email[V. R\u{a}dulescu]{vicentiu.radulescu@imar.ro} \email[N. Zeddini]{nalzeddini@kau.edu.sa} \thanks{Submitted April 13, 2015. Published June 26, 2015.} \subjclass[2010]{35B09, 35J47, 35J57, 58C30} \keywords{Semilinear elliptic system; Leray-Schauder fixed point theorem; \hfill\break\indent positive bounded solution; indefinite potential} \begin{abstract} In this article, we study the existence and nonexistence of positive bounded solutions of the Dirichlet problem \begin{gather*} -\Delta u=\lambda p(x)f(u,v),\quad \text{in } {\mathbb{R}}_+^n,\\ -\Delta v=\lambda q(x)g(u,v), \quad \text{in } {\mathbb{R}}_+^n,\\ u=v=0\quad \text{on }\partial {\mathbb{R}}_+^n,\\ \lim_{|x|\to \infty}u(x)=\lim_{|x|\to \infty}v(x)=0, \end{gather*} where ${\mathbb{R}}_+^n=\{x=(x_1,x_2,\dots, x_n)\in {\mathbb{R}}^n: x_n>0\}$ ($n\geq3$) is the upper half-space and $\lambda$ is a positive parameter. The potential functions $p,q$ are not necessarily bounded, they may change sign and the functions $f,g:\mathbb{R}^2 \to \mathbb{R}$ are continuous. By applying the Leray-Schauder fixed point theorem, we establish the existence of positive solutions for $\lambda$ sufficiently small when $f(0,0)>0$ and $g(0,0)>0$. Some nonexistence results of positive bounded solutions are also given either if $\lambda$ is sufficiently small or if $\lambda$ is large enough. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} This paper deals with the existence of positive continuous solutions (in the sense of distributions) for the semilinear elliptic system \begin{equation} \label{S} \begin{gathered} -\Delta u=\lambda p(x)f(u,v), \quad \text{in } {\mathbb{R}}_+^n, \\ -\Delta v=\lambda q(x)g(u,v),\quad \text{in } {\mathbb{R}}_+^n,\\ u=v=0\quad \text{on }\partial {\mathbb{R}}_+^n,\\ \lim_{|x|\to \infty}u(x)=\lim_{|x|\to \infty}v(x)=0, \end{gathered} \end{equation} where ${\mathbb{R}}_+^n=\{x=(x_1,x_2,\dots, x_n)\in {\mathbb{R}}^n: \; x_n>0\}$ ($n\geq3$) is the upper half-space. We assume that the potentials $p,q$ are sign-changing functions belonging to the Kato class $K^{\infty}({\mathbb{R}}_+^n)$ introduced and studied in \cite{BM}, and the functions $f,g$ satisfy the following hypothesis: \begin{itemize} \item[(H1)] $f, g:\mathbb{R}^2 \to \mathbb{R}$ are continuous with $f(0,0)>0$ and $g(0,0)>0$. \end{itemize} In recent years, a good amount of research is established for reaction-diffusion systems. Reaction-diffusions systems model many phenomena in biology, ecology, combustion theory, chemical reactors, population dynamics etc. The case $p(x)=q(x)=1$ has been considered as a typical example in bounded regular domains in ${\mathbb{R}}^n$ and many existence results where established by variational methods, topological methods and the method of sub- and super-solutions (see \cite{Dal,HulV,FigMR,gmrz,radbook}). Recently, Chen \cite{RC} studied the existence of positive solutions for the system \begin{equation} \label{S2} \begin{gathered} -\Delta u=\lambda p(x)f_1(v), \quad \text{in } D, \\ -\Delta v=\lambda q(x)g_1(v),\quad \text{in } D,\\ u=v=0\quad \text{on }\partial D, \end{gathered} \end{equation} where $D$ is a bounded domain. He assumed that $p,q $ are continuous in $\overline{D}$ and there exist positive constants $\mu_1$, $\mu_2$ such that \begin{gather*} \int_D G_D(x,y)p_+(y)\,dy > (1+\mu_1) \int_D G_D(x,y)p_-(y)\,dy \quad \forall x \in D,\\ \int_D G_D(x,y)q_+(y)\,dy > (1+\mu_2) \int_D G_D(x,y)q_-(y)\,dy \quad \forall x \in D , \end{gather*} where $G_D(x,y)$ is the Green's function of the Dirichlet Laplacian in $D$. Here $p^+,\,q^+$ are the positive parts of $p$ and $q$, while $p_-,\, q_-$ are the negative ones. Chen \cite{RC} showed that if $f_1,g_1:[0,\infty) \to {\mathbb{R}}$ are continuous with $f_1(0)>0$, $g_1(0)>0$ and $p,q$ are nonzero continuous functions on ${\overline{D}}$ satisfying the above integral conditions, then there exists a positive number $\lambda^{\star}$ such that problem \eqref{S2} has a positive solution for small positive values of the parameter, namely if $0<\lambda<\lambda^{\star}$. We note that when $f_1,g_1$ are nonnegative nondecreasing continuous functions, $p(x)\leq 0$ in ${\mathbb{R}}_+^n$ and $q(x)\leq 0$ in ${\mathbb{R}}_+^n$, system \eqref{S2} was studied in \cite{Zed} in the half-space ${\mathbb{R}}_+^n$ with nontrivial nonnegative boundary and infinity data. In this framework, the existence of positive solutions for \eqref{S2} is established for small perturbations, that is, whenever $\lambda$ is a small positive real number. Our aim in this article is to study these systems in the case where the domain is the half-space ${\mathbb{R}}_+^n$ and the functions $p,q$ are not necessarily continuous in $\overline{{\mathbb{R}}_+^n}$. Indeed $p,q$ may be singular on the boundary of ${\mathbb{R}}_+^n$. More precisely, we establish the existence of a positive bounded solution for \eqref{S} in the case where $f(0,0)>0$, $g(0,0)>0$ and the functions $p,q$ belong to the Kato class introduced and studied in \cite{BM} and satisfy the following hypothesis: \begin{itemize} \item[(H2)] there exist positive numbers $\mu_1,\mu_2$ such that \begin{gather*} \int_{{\mathbb{R}}_+^n} G(x,y)p_+(y)\,dy > (1+\mu_1) \int_{{\mathbb{R}}_+^n} G(x,y)p_-(y)\,dy \quad \forall x \in {\mathbb{R}}_+^n,\\ \int_{{\mathbb{R}}_+^n} G(x,y)q_+(y)\,dy > (1+\mu_2) \int_{{\mathbb{R}}_+^n} G(x,y)q_-(y)\,dy \quad \forall x \in {\mathbb{R}}_+^n , \end{gather*} \end{itemize} where $G(x,y)$ is the Green function of the Dirichlet Laplacian in the half space ${\mathbb{R}}_+^n$. Two nonexistence results of positive bounded solutions will be established in this paper. To this aim, we recall in the sequel some notations and properties of the Kato class, cf. \cite{BM}. \begin{definition} \label{def1} \rm A Borel measurable function $k$ in ${\mathbb{R}}_+^n$ belongs to the Kato class $K^{\infty}({\mathbb{R}}_+^n)$ if $$ \lim_{r\to0}\sup_{x\in {\mathbb{R}}_+^n } \int_{{\mathbb{R}}_+^n\cap B(x,r)}\frac{y_n}{x_n}G(x,y)|k(y)|dy=0 $$ and $$ \lim_{M\to\infty}\sup_{x\in {\mathbb{R}}_+^n}\int_{{\mathbb{R}}_+^n \cap \{|y|\geq M\}}\frac{y_n}{x_n}G(x,y)|k(y)|dy=0\,, $$ where $$ G(x,y)=\frac{\Gamma(\frac{n}{2}-1)}{4 {\pi}^{n/2}} \Big[\frac{1}{|x-y|^{n-2}}-\frac{1}{\left(|x-y|^2+4x_ny_n\right) ^{\frac{n-2}{2}}}\Big] $$ is the Green function of the Dirichlet Laplacian in ${\mathbb{R}}_+^n$. \end{definition} Next, we give some examples of functions belonging to $K^{\infty}({\mathbb{R}}_+^n)$. \begin{example} \label{examp1.1} \rm Let $\lambda,\mu\in {\mathbb{R}}$ and put $q(y)=\frac{1}{(|y|+1)^{\mu -\lambda}y_n^{\lambda}}$ for $y\in {\mathbb{R}}^n_+$. Then $$ q\in K^{\infty}({\mathbb{R}}^n_+) \text{ if and only if } \lambda<2<\mu. $$ \end{example} For any nonnegative Borel measurable function $\varphi$ in ${\mathbb{R}}_+^n$, we denote by $V \varphi$ the Green potential of $\varphi$: $$ V \varphi(x)=\int_{{\mathbb{R}}_+^n} G(x,y)\varphi(y)dy,\quad \forall x\in {\mathbb{R}}_+^n. $$ Recall that if $\varphi \in L^1_{\rm loc}({\mathbb{R}}_+^n)$ and $V \varphi \in L^1_{\rm loc}({\mathbb{R}}_+^n)$, then we have in the distributional sense (see \cite[p. 52]{CZ}) \begin{equation} \label{deltaVf} \Delta (V \varphi)=-\varphi \; \text{ in } {\mathbb{R}}_+^n. \end{equation} The first result establishes the existence of bounded positive solutions in case of small perturbations, that is, if $\lambda$ is a small positive parameter. \begin{theorem} \label{mainresult1} Let $p$, $q$ be in the Kato class $K^{\infty}({\mathbb{R}}_+^n)$ and assume that {\rm (H1)--(H2)} are satisfied. Then there exists $\lambda_0>0$ such that for each $\lambda\in(0,\lambda_0)$, problem \eqref{S} has a positive continuous solution in ${\mathbb{R}}_+^n$. \end{theorem} The first nonexistence result of positive bounded solutions is in relationship with the previous theorem and concerns a particular class of functions $f$ and $g$ with linear growth and vanishing at the origin. \begin{theorem} \label{mainresult2} Let $p,q$ be nontrivial functions in the Kato class $K^{\infty}({\mathbb{R}}_+^n)$. Assume that the functions $f,\, g\,:{\mathbb{R}}^2 \to {\mathbb{R}}$ are measurable and there exists a positive constant $M$ such that for all $u\,,\,v$ \begin{eqnarray*} |f(u,v)|\leq M(|u|+|v|)\,\\ |g(u,v)|\leq M(|u|+|v|). \end{eqnarray*} Then there exists $\lambda_0>0$ such that problem \eqref{S} has no positive bounded continuous solution in ${\mathbb{R}}_+^n$ for each $\lambda\in(0,\lambda_0)$. \end{theorem} The second nonexistence result is established for $\lambda$ sufficiently large. \begin{theorem} \label{mainresult3} Let $p,q \in K^{\infty}({\mathbb{R}}_+^n)$ and let $f(u,v)=f(v)$, $g(u,v)=g(u)$. Assume that the following hypotheses are fulfilled: \begin{itemize} \item[(H3)] there exist an open ball $B\subset {\mathbb{R}}_+^n$ and a positive number $\varepsilon$ such that $$ p(x),q(x)\geq \varepsilon \quad\text{ a.e. } x \in B. $$ \item[(H4)] $f,g : [0,\infty)\to [0, \infty)$ are continuous and there exists a positive number $m$ such that $f(v)+g(u)\geq m(u+v)$ for all $u,v>0$. \end{itemize} Then there exists a positive number $\lambda_0$ such that problem \eqref{S} has no positive bounded continuous solution in ${\mathbb{R}}_+^n$ for each $\lambda>\lambda_0$. \end{theorem} Throughout this article, we denote by $B({\mathbb{R}}_+^n)$ the set of Borel measurable functions in ${\mathbb{R}}_+^n$ and by $C_0({\mathbb{R}}_+^n)$ the set of continuous functions satisfying $$ \lim_{x\to \partial {\mathbb{R}}_+^n } u(x)=\lim_{|x|\to \infty}u(x)=0. $$ Finally, for a bounded real function $\omega$ defined on a set $S$ we denote ${\|\omega\|}_{\infty}=\sup_{x\in S}|\omega(x)|$. \section{Proof of main results} We start this section with the following continuity property. We refer to \cite{BM} for more details. \begin{proposition} \label{prop1} Let $\varphi$ be a nonnegative function in $K^{\infty}({\mathbb{R}}_+^n)$. Then the following properties hold. \begin{itemize} \item[(i)] The function $y \to \frac{y_n}{(1+|y|)^{n}} \varphi(y)$ is in $L^1({\mathbb{R}}_+^n)$, hence $\varphi \in L^1_{\rm loc}({\mathbb{R}}_+^n)$. \item[(ii)] $V \varphi \in C_0({\mathbb{R}}_+^n)$. \item[(iii)] Let $h_0$ be a positive harmonic function in ${\mathbb{R}}_+^n$ which is continuous and bounded in $\overline{{\mathbb{R}}_+^n}$. Then the family of functions $$ \Big\{\int_{{\mathbb{R}}_+^n} G(.,y)h_0(y)p(y)dy: |p|\leq \varphi\Big\} $$ is relatively compact in $C_0({\mathbb{R}}_+^n)$. \end{itemize} \end{proposition} Next, we recall the Leray-Schauder fixed point theorem. \begin{lemma} \label{lem1} Let $X$ be a Banach space with norm $\|\cdot\|$ and $x_0$ be a point of $X$. Suppose that $T:X \times[0,1]\to {X}$ is continuous and compact with $T(x,0)= x_0$ for each $x\in X$, and that there exists a fixed constant $M>0$ such that each solution $(x, \sigma) \in X\times [0,1]$ of the $T(x,\sigma)=x$ satisfies $\|x\|\leq M$. Then $T(.,1)$ has a fixed point. \end{lemma} Using this fixed point property, we obtain the following general existence result. \begin{lemma} \label{lem2} Suppose that $p$ and $q$ are in the Kato class $K({\mathbb{R}}_+^n)$ and $f,g$ are continuous and bounded from ${\mathbb{R}}^2$ to ${\mathbb{R}}$. Then for every $\lambda\in (0,\infty)$, problem \eqref{S} has a solution $(u_\lambda,v_\lambda)\in C_0({\mathbb{R}}_+^n)\times C_0({\mathbb{R}}_+^n)$. \end{lemma} \begin{proof} For $\lambda \in {\mathbb{R}}$, we consider the operator $$ T_{\lambda}:C_0({\mathbb{R}}_+^n)\times C_0({\mathbb{R}}_+^n) \times[0,1]\to C_0({\mathbb{R}}_+^n)\times C_0({\mathbb{R}}_+^n) $$ defined by $$ T_{\lambda}((u,v),\sigma)=(\sigma \lambda V( pf(u,v)),\sigma \lambda V( qg(u,v))). $$ By Proposition \ref{prop1}, the operator $T_{\lambda}$ is well defined, continuous, compact and $$ T_{\lambda}((u,v),0)=(0,0):=x_0\in C_0({\mathbb{R}}_+^n) \times C_0({\mathbb{R}}_+^n). $$ Let $(u,v)\in C_0({\mathbb{R}}_+^n)\times C_0({\mathbb{R}}_+^n)$ and $ \sigma \in [0,1]$ such that $T_{\lambda}((u,v),\sigma)=(u,v)$. Then, since $f,g$ are bounded and $p$, $q$ are in $K^{\infty}({\mathbb{R}}_+^n)$ we deduce by using Proposition \ref{prop1} that \begin{align*} \max(\|u\|_{\infty},\|v\|_{\infty}) &= \sigma \, \lambda \max(\|V(p f(u,v))\|_\infty,\|V(q g(u,v))\|_\infty) \\ &\leq \lambda \max(\|Vp\|_{\infty}\|f\|_{\infty},\|Vq\|_{\infty}\|g\|_{\infty}) =M. \end{align*} Applying the Leray-Schauder fixed point theorem, the operator $T_\lambda(.,1)$ has a fixed point, hence there exists $(u,v)\in C_0({\mathbb{R}}_+^n)\times C_0({\mathbb{R}}_+^n)$ such that $$ (u,v)=(\lambda V(p\,f(u,v)),\lambda V( q\,g(u,v))). $$ So, using \eqref{deltaVf} and Proposition \ref{prop1}, we deduce that $(u,v)$ is a solution of system \eqref{S}. \end{proof} \begin{proof}[Proof of Theorem \ref{mainresult1}] Fix a large number $M>0$ and an infinitely continuously differentiable function $\psi$ with compact support on ${\mathbb{R}}^2$ such that $\psi=1$ in the open ball with center $0$ and radius $M$ and $\psi=0$ on the exterior of the ball with center $0$ and radius $2M$. Define the bounded functions $\widetilde{f}$, $\widetilde{g}$ on ${\mathbb{R}}^2$ by $$ \widetilde{f}(u,v)=\psi (u,v) f(u,v)\quad \text{and}\quad \widetilde{g}(u,v)= \psi (u,v) g(u,v). $$ By Lemma \ref{lem2}, the Dirichlet problem \begin{equation} \label{S3} \begin{gathered} -\Delta u=\lambda p(x)\widetilde{f}(u,v), \quad \text{in } {\mathbb{R}}_+^n, \\ -\Delta v=\lambda q(x)\widetilde{g}(u,v),\quad \text{in } {\mathbb{R}}_+^n,\\ u=v=0\quad \text{on }\partial {\mathbb{R}}_+^n,\\ \lim_{|x|\to \infty}u(x)=\lim_{|x|\to \infty}v(x)=0, \end{gathered} \end{equation} has a solution $(u_\lambda,v_\lambda)\in C_0({\mathbb{R}}_+^n) \times C_0({\mathbb{R}}_+^n)$ satisfying $$ (u_\lambda,v_\lambda)=(\lambda V(p \widetilde{f}(u_\lambda,v_\lambda)) \lambda V(q\widetilde{g}(u_\lambda,v_\lambda))). $$ Moreover, we have \begin{equation} \label{eqstar} \max(\|u_\lambda\|_{\infty},\|v_\lambda\|_{\infty}) \leq\lambda\max(\|Vp\|_\infty\|\widetilde{f}\|_{\infty}, \|Vq\|_{\infty}\|\widetilde{g}\|_{\infty}), \end{equation} Put $\mu=\min(\mu_1,\mu_2)$ and consider $\gamma\in(0,\frac{\mu}{2+\mu})$. Since $\widetilde{f}$ and $\widetilde{g}$ are continuous, then there exists $\delta\in(0,M)$ such that if $\max(|\zeta|,|\xi|)<\delta$, we have \begin{gather*} \widetilde{f}(0,0)(1-\gamma)<\widetilde{f}(\zeta,\xi)<\widetilde{f}(0,0)(1+\gamma),\\ \widetilde{g}(0,0)(1-\gamma)<\widetilde{g}(\zeta,\xi)<\widetilde{g}(0,0)(1+\gamma). \end{gather*} Using relation \eqref{eqstar}, we deduce that there exists $\lambda_0>0$ such that $\|u_\lambda\|_{\infty}<\delta$ and $\|v_\lambda\|_{\infty}<\delta$ for any $\lambda \in(0,\lambda_0)$. This together with the fact that $0<\delta \lambda f(0,0)(1-\gamma) V(p_+)-\lambda f(0,0)(1+\gamma)V(p_-) \\ &> \lambda f(0,0)[(1-\gamma)(1+\mu_1)-(1+\gamma)] V(p_-)\\ &> \lambda f(0,0)(1-\gamma)\big[1+\mu_1-\frac{1+\gamma}{1-\gamma}\big]V(p_-)\\ &> \lambda f(0,0)(1-\gamma)\big[1+\mu-\frac{1+\gamma}{1-\gamma}\big]V(p_-). \end{align*} Now, since $\gamma\in(0,\frac{\mu}{2+\mu})$, then $1+\mu-\frac{1+\gamma}{1-\gamma}>0$ and it follows that $$ \lambda f(0,0)(1-\gamma)\big[1+\mu-\frac{1+\gamma}{1-\gamma}\big]V(p_-)\geq 0. $$ Consequently, for each $\lambda \in (0,\lambda_0)$ and for each $x\in {\mathbb{R}}_+^n$ we have $u_\lambda(x)>0$. Similarly, we obtain $v_\lambda(x)>0$ for each $x\in {\mathbb{R}}_+^n$. \end{proof} \begin{proof}[Proof of Theorem \ref{mainresult2}] Suppose that problem \eqref{S} has a bounded positive solution $(u,v)$ for all $\lambda>0$. Then $f(u,v)$ and $g(u,v)$ are bounded. Put $\widetilde{u}=\lambda V(p\,f(u,v))$ and $\widetilde{v}=\lambda V(q\,g(u,v))$. Since $f(u,v)$ and $g(u,v)$ are bounded, it follows that $\widetilde{u}\,,\,\widetilde{v} \in C_0({\mathbb{R}}_+^n)$. The the functions $z=u-\widetilde{u}$ and $\omega=v-\widetilde{v}$ are harmonic in the distributional sense and continuous in ${\mathbb{R}}_+^n$, so they are harmonic in the classical sense. Moreover, since $u=\widetilde{u}=v=\widetilde{v}=0$ on $\partial {\mathbb{R}}_+^n$ and $\lim_{|x|\to \infty}u(x)=\lim_{|x|\to \infty}v(x)=0$, then $u=\widetilde{u}$ and $v=\widetilde{v}$ in ${\mathbb{R}}_+^n$. It follows that \begin{gather*} \|u\|_{\infty}\leq \lambda\,V(|p|f(u,v)) \leq \lambda M \|V(|p|)\|_{\infty}\,(\|u\|_{\infty}+\|v\|_{\infty})\,,\\ \|v\|_{\infty}\leq \lambda\,V(|q|g(u,v)) \leq \lambda M \|V(|q|)\|_{\infty}\,(\|u\|_{\infty}+\|v\|_{\infty}). \end{gather*} By adding these inequalities, we obtain \[ (\|u\|_{\infty}+\|v\|_{\infty})\leq \lambda M\, [\|V(|p|)\|_{\infty} +\|V(|q|)\|_{\infty}]\,\,(\|u\|_{\infty}+\|v\|_{\infty}). \] This gives a contradiction if $\lambda M [\|V(|p|)\|_{\infty}+\|V(|q|)\|_{\infty}]<1$. \end{proof} \begin{proof}[Proof of Theorem \ref{mainresult3}] Without loss of generality, we assume that $\overline{B}\subset \Omega$. We first note that the assumption (H4) implies that $$ f(v)\geq m v\quad\text{for all }v>0 $$ or $$ g(u)\geq mu\quad \text{for all }u>0. $$ Suppose that $f(v)\geq mv$ for all $v>0$. We distinguish the following situations. \smallskip \noindent\textbf{Case 1.} $f(0)=0$. Then it follows from (H4) that $$ g(u)\geq mu\quad \text{for }u>0. $$ Suppose that $(u,v)$ is a positive solution of \eqref{S}. It follows that \begin{equation} \label{eq3.1} -\Delta u=\lambda a(x)f(v)\geq \lambda \varepsilon mv\quad \text{in } B. \end{equation} Let $\widetilde{\lambda}_1$ be the first eigenvalue of $-\Delta$ in $B$ with Dirichlet boundary conditions, and $\phi_1$ be the corresponding normalized positive eigenfunction. Let $\delta>0$ be the largest number so that \begin{equation}\label{eq3.2} v\geq \delta\phi_1\quad \text{in } B. \end{equation} Then we have from \eqref{eq3.1} and \eqref{eq3.2} that \[ -\Delta v \geq \lambda \varepsilon m \delta \phi_1\quad \text{in } B, \] and therefore by the weak comparison principle \begin{equation}\label{eq3.3} u\geq \frac{\lambda \varepsilon m}{\widetilde{\lambda}_1}\delta\phi_1\quad \text{in } B. \end{equation} Therefore, \[ -\Delta v \geq \lambda \varepsilon m u \geq \frac{(\lambda \varepsilon m)^2}{\widetilde{\lambda}_1}\delta \phi_1\, \quad \text{in } B. \] Using by the weak comparison principle we obtain \[ v\geq \Big(\frac{\lambda \varepsilon m}{\widetilde{\lambda}_1}\Big)^2 \delta \phi_1 \quad \text{ in } B. \] This contradicts the maximality of $\delta$ for $\lambda$ large enough. \smallskip \noindent\textbf{Case 2.} $f(0)>0$. Then there exists $\delta_0>0$ such that \[ f(t)\geq \delta_0 \quad \text{for all } t\geq 0. \] Hence $ -\Delta u\geq \lambda \varepsilon \delta_0 $ in $B$, from which it follows that \begin{equation}\label{eq3.4} u\geq (\lambda \varepsilon \delta_0) \widetilde{\Phi}\quad \text{in } B, \end{equation} where $\widetilde{\Phi}$ satisfies \[ -\Delta \widetilde{\Phi}=1 \quad\text{in } B,\quad \widetilde{\Phi}=0\quad \text{on } \partial B. \] Let $D$ be an open set such that $\overline{D}\subset B$ and let $c>0$ such that \begin{equation}\label{eq3.5} \widetilde{\Phi}\geq c \quad \text{in } \overline{D}. \end{equation} Suppose $m \lambda \varepsilon \delta_0 c>2f(0)$. Relations \eqref{eq3.4} and \eqref{eq3.5} yield \[ m u\geq m \lambda \varepsilon \delta_0 c>2 f(0), \] which implies \[ g(u)\geq m u-f(0)\geq \frac{m}{2} u\quad \text{in } D. \] Using the same arguments as in Case 1 in $D$, we obtain a contradiction if $\lambda$ is large enough. 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