\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 192, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/??\hfil Positive solutions] {Positive solutions for classes of positone/semipositone systems with multiparameters} \author[R. S. Rodrigues \hfil EJDE-2013/192\hfilneg] {Rodrigo da Silva Rodrigues} % in alphabetical order \address{Rodrigo da Silva Rodrigues \newline Departamento de Matem\'atica, Universidade Federal de S\~ao Carlos, 13565-905, S\~ao Carlos, SP, Brasil} \email{rodrigosrodrigues@ig.com.br, rodrigo@dm.ufscar.br} \thanks{Submitted May 6, 2012. Published August 30, 2013.} \subjclass[2000]{34A38, 35B09} \keywords{Existence of positive solution; nonexistence of solution; \hfill\break\indent positone system; semipositone system} \begin{abstract} We study the existence and nonexistence of solution for a system involving p,q-Laplacian and nonlinearity with multiple parameteres. We use the method of lower and upper solutions for prove the existence of solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} We study the existence of solutions for the positone/semipositone system involving $p,q$-Laplacian $$\label{system} \begin{gathered} -\Delta_p u = \lambda f_1(x,u,v) +\mu g_1(x,u,v) \quad \text{in } \Omega, \\ -\Delta_q v = \lambda f_2(x,u,v) +\mu g_2(x,u,v) \quad \text{in } \Omega, \\ u=v=0 \quad \text{on } \partial \Omega, \end{gathered}$$ where $\Omega \subset \mathbb{R}^n$, $n\geq 1$, is a bounded domain with boundary $C^2$, and $f_i,g_i:\Omega\times (0,+\infty)\times (0,+\infty)\to \mathbb{R}$, $i=1,2$, are Carath\'eodory functions, $g_i$, $i=1,2$, are bounded on bounded sets. Moreover, we assume that there exists $h_i:\mathbb{R}\to\mathbb{R}$ continuous and nondecreasing such that $h_i(0)=0$, $0\leq h_i(s)\leq C(1+|s|^{r-1})$, for all $s\in \mathbb{R}$, $r=\min\{p,q\}$, $C>0$, $i=1,2$, and the maps $$\label{me-1} \begin{gathered} s\mapsto f_1(x,s,t)+h_1(s),\quad t\mapsto f_2(x,s,t)+h_1(t), \\ s\mapsto g_1(x,s,t)+h_2(s),\quad t\mapsto g_2(x,s,t)+h_2(t), \end{gathered}$$ are nondecreasing for almost everywhere $x\in \Omega$. Also, we will prove the nonexistence of nontrivial solution for system \eqref{system} in the positone case. In the scalar case, Castro, Hassanpour, and Shivaji in \cite{CHS}, using the lower and upper solutions method, focused their attention on a class of problems, so called semipositione problems, of the form \begin{gather*} -\Delta u = \lambda f(u) \quad \text{in } \Omega,\\ u=0 \quad\text{on } \partial \Omega, \end{gather*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $\lambda$ is a positive parameter, and $f:[0,\infty) \to \mathbb{R}$ is a monotone and continuous function satisfying the conditions $f(0)< 0$, $\lim_{s \to \infty} f(s)= +\infty$, and with the sublinear condition at infinity, $\lim_{s \to \infty} {f(s)}/{s}=0$. In 2008, Perera and Shivaji \cite{Perera-Shivaji} proved the existence of solutions for the problem \begin{gather*} -\Delta_p u = \lambda f(x,u) +\mu g(x,u) \quad \text{in } \Omega,\\ u = 0 \quad \text{on } \partial \Omega, \end{gather*} where $\Omega \subset \mathbb{R}^n$, $n\geq 1$, is a bounded domain with boundary $C^2$, and $f,g:\Omega\times (0,+\infty)\times (0,+\infty)\to \mathbb{R}$ are Carath\'eodory functions, $g$ is bounded on bounded sets and $|f(x,t)|\geq a_0$ for all $t\geq t_0$, where $a_0,t_0$ are positive constants. Moreover, the existence of solutions is assured for $\lambda\geq \lambda_0$ and small $0<|\mu|\leq\mu_0$, for some $\lambda_0>0$ and $\mu_0=\mu(\lambda_0)>0$. Many authors have studied the existence of positive solutions for elliptic systems, due to the great number of applications in reaction-diffusion problems, in fluid mechanics, in newtonian fluids, glaciology, population dynamics, etc; see \cite{Drabek, Fleckinger} and references therein. Hai and Shivaji \cite{HS} applied the lower and upper solutions method for obtaining the existence of solution for the semipositone system $$\label{PQ} \begin{gathered} -\Delta_p u =\lambda f_1(v) \quad \text{in } \Omega, \\ -\Delta_p v =\lambda f_2(u) \quad \text{in } \Omega, \\ u = v= 0 \quad \text{on } \partial\Omega, \end{gathered}$$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ with smooth boundary, $\lambda$ is a positive parameter, and $f_1,f_2:[0,\infty) \to \mathbb{R}$ are monotone and continuous functions satisfying conditions $f_i(0)< 0$, ${\lim_{s \to +\infty}} f_i(s)= +\infty$, $i=1,2$, and $$\label{cod} \lim_{s \to + \infty}\frac{f_1(M(f_2(s))^{1/(p-1)})}{s^{p-1}}=0 \quad \text{for all } M > 0.$$ While, Chhetri, Hai, and Shivaji \cite{HCS} proved an existence result for system \eqref{PQ} with the condition $$\lim_{s \to +\infty} \frac{ \max{\{f_1(s), f_2(s)\}} }{s^{p-1}}=0,$$ instead of \eqref{cod}. In 2007, Ali and Shivaji \cite{JShi} obtained a positive solution for the system $$\label{JS} \begin{gathered} -\Delta_p u = \lambda_1 f_1(v)+ \mu_1 g_1(u) \quad \text{in } \Omega, \\ -\Delta_q v = \lambda_2 f_2(u)+\mu_2 g_2(v) \quad \text{in }\Omega, \\ u = v = 0 \quad \text{on } \partial\Omega, \end{gathered}$$ when $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $\lambda_i,\mu_i$, $i=1,2$, are nonnegative parameters with $\lambda_1+\mu_1$ and $\lambda_2+\mu_2$ large and $$\lim_{x\to +\infty} \frac{f_1(M[f_2(x)]^{1/q-1})}{x^{p-1}}=0,$$ for all $M>0$, $\lim_{x\to +\infty}\frac{g_1(x)}{x^{p-1}}=0$, and $\lim_{x\to +\infty}\frac{g_2(x)}{x^{q-1}}=0$. Our first result deal with the existence of solution for \eqref{system} which has $p,q$-Laplacian operators and nonautonomous nonlinearity with multiple parameters. Note that, we make no suppositions about the signs of $g_1(x,0,0)$ and $g_2(x,0,0)$, and hence can occur the positone case: $\lambda f_i(x,0,0)+\mu g_i(x,0,0)\geq 0$, $i=1,2$; the semipositone case: $\lambda f_i(x,0,0)+\mu g_i(x,0,0)< 0$, $i=1,2$; the case $\lambda f_1(x,0,0)+\mu g_1(x,0,0)\geq 0$ and $\lambda f_2(x,0,0)+\mu g_2(x,0,0) < 0$; or the case $\lambda f_1(x,0,0)+\mu g_1(x,0,0) < 0$ and $\lambda f_2(x,0,0)+\mu g_2(x,0,0) \geq 0$; for almost everywhere $x\in \Omega$. \begin{theorem}\label{theorem1} Consider the system \eqref{system} assuming \eqref{me-1}, and that there exist $a_0, \gamma, \delta>0$ and $\alpha, \beta\geq 0$ such that $0\leq \alpha0$, and $$\label{hip-existence} | f_1(x,s,t) |\leq a_0 |s|^\alpha|t|^\gamma,\quad | f_2(x,s,t) |\leq a_0 |s|^\delta|t|^\beta,$$ for all $s,t\in (0,+\infty)$ and $x\in \Omega$. In addition, suppose there exist $a_1>0$, $a_2>0$, and $R>0$ such that $$\label{hip-limite} f_i(x,s,t) \geq a_1,\quad\text{for i=1,2, and all s>R, t>R},$$ and $$\label{hip-limite1} f_i(x,s,t) \geq-a_2,\quad\text{for i=1,2, and all s,t\in(0,+\infty)},$$ uniformly in $x\in \Omega$. Then, there exists $\lambda_0>0$ such that for each $\lambda > \lambda_0$, there exists $\mu_0=\mu_0(\lambda)>0$ for which system \eqref{system} has a solution $(u,v)\in C^{1,\rho_1}(\Omega)\times C^{1,\rho_2}(\Omega)$ for some $\rho_1,\rho_2>0$, where each component is positive, whenever $|\mu|\leq \mu_0$. \end{theorem} Let $\lambda_p>0$ and $\lambda_q>0$ be the first eigenvalue of $p$-Laplacian and $q$-Laplacian, respectively, where $\phi_p\in C^{1,\alpha_p}(\Omega)$ and $\phi_q\in C^{1,\alpha_q}(\Omega)$ are the respective positive eigenfunctions (see \cite{Mabel}). Chen \cite{chen} proved the nonexistence of nontrivial solution for the system \begin{gather*} -\Delta_p u = \lambda u^\alpha v^\gamma, \quad \text{in } \Omega,\\ -\Delta_q v = \lambda u^\delta v^\beta, \quad \text{in } \Omega,\\ u = v = 0 \quad \text{on } \Omega, \end{gather*} when $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $p\gamma =q(p-1-\alpha)$, $(p-1-\alpha)(q-1-\beta)-\gamma\delta=0$, and $0<\lambda<\lambda_0$ where $\lambda_0=\min\{\lambda_p, \lambda_q\}$ (see also \cite{Rodrigo}). We note that due to Young's inequality we have $u^{\alpha+1}v^\gamma \leq \frac{1+\alpha}{p}u^p + \frac{p-1-\alpha}{p}v^q,\quad u^{\delta}v^{\beta+1} \leq \frac{q-1-\beta}{q}u^p + \frac{\beta+1}{q}v^q.$ Now, we will enunciated the nonexistence theorem for the system \eqref{system}, improving the result proved by Chen in \cite{chen}. \begin{theorem}\label{nonexistence} Suppose that there exist $k_i>0$, $i=1,\dots, 8$, such that $$\label{hip-nonexistence} \begin{gathered} | f_1(x,s,t)s |\leq \left(k_1|s|^p+k_2|t|^q \right),\quad | f_2(x,s,t)t |\leq \left(k_3|s|^p+k_4|t|^q\right), \\ | g_1(x,s,t)s |\leq \left(k_5|s|^p+k_6|t|^q \right), \quad | g_2(x,s,t)t |\leq \left(k_7|s|^p+k_8|t|^q\right), \end{gathered}$$ for all $x\in \Omega$ and $s,t\in (0,+\infty)$. Then \eqref{system} does not possess nontrivial solutions, for all $\lambda, \mu$ satisfying $$\label{cond. teo nonexist} |\lambda|(k_1+k_3) +|\mu|(k_5+k_7) < \lambda_{p},\quad |\lambda|(k_2+k_4) +|\mu|(k_6+k_8) < \lambda_{q}.$$ \end{theorem} \begin{remark} \label{rmk1.1} \rm The typical functions considered in Theorem \ref{theorem1} are as follows: $f_1(x,s,t)=A(x) s^{\alpha}t^{\gamma},\quad f_2(x,s,t)=B(x)s^{\delta}t^{\beta},$ where $A(x),B(x)$ are continuous functions on $\Omega$ satisfying $\inf_{x\in\Omega}A(x)>0$ and $\sup_{x\in\Omega}A(x)<+\infty$, $\inf_{x\in\Omega}B(x)>0$, and $\sup_{x\in\Omega}B(x)<+\infty$ for all $x\in\Omega$, $0\leq \alpha0$, and $g_1(x,s,t)$ and $g_2(x,s,t)$ are any continuous functions on $\overline{\Omega}\times [0,+\infty) \times [0,+\infty)$ with $g_1(x,s,t)$ nondecreasing in variable $s$ and $g_2(x,s,t)$ nondecreasing in variable $t$. \end{remark} \begin{remark} \label{rmk1.2} \rm Theorem \ref{nonexistence} can be applied for functions of the form \begin{gather*} f_1(x,s,t)={ \sum_{i=1}^m} a_is^{\alpha_{1,i}}t^{\gamma_{1,i}},\quad f_2(x,s,t)={ \sum_{i=1}^m} b_is^{\delta_{1,i}}t^{\beta_{1,i}} \\ g_1(x,s,t)={ \sum_{i=1}^m} c_is^{\alpha_{2,i}}t^{\gamma_{2,i}},\quad g_2(x,s,t)={ \sum_{i=1}^m} d_is^{\delta_{2,i}}t^{\beta_{2,i}}, \end{gather*} with $a_i,b_i,c_i,d_i \geq 0$, $p\gamma_{j,i}=q(p-1-\alpha_{j,i})$, and $(p-1-\alpha_{j,i})(q-1-\beta_{j,i})=\gamma_{j,i}\delta_{j,i}$, for $j=1,2$ and $i=1,\cdots,m$. \end{remark} Theorems \ref{theorem1} and Theorem \ref{nonexistence} will be proved in the next sections. \section{Proof of Theorem \ref{theorem1}}\label{existence} We prove Theorem \ref{theorem1} by using a general method of lower and upper-solutions. This method, in the scalar situation, has been used by many authors, for instance \cite{Leonelo} and \cite{Drabek}. The proof for the system case can be found in \cite{Rodrigo}. \subsection{Upper-solution}\label{upper} First of all, we will prove that \eqref{system} possesses a upper-solution. Consider $e_i\in C^{1,\alpha_i}(\overline{\Omega})$, with $\alpha_i>0$, $i=1,2$, where $(e_1,e_2)$ is a solution of \eqref{system} with $f_1(x,u,v)=\frac{1}{\lambda}$, $f_2(x,u,v)=\frac{1}{\lambda}$, and $g_1(x,u,v)=g_2(x,u,v)=0$, and each component is positive. \noindent{\bf Claim.} Since $\delta>0$, $\gamma>0$, $0\leq \alpha 0$, there exist $s_1$ and $s_2$ such that $$\label{super-0} s_1>\frac{1}{p-1},\quad s_2>\frac{1}{q-1},\quad \frac{\delta}{q-1-\beta}<\frac{s_2}{s_1}<\frac{p-1-\alpha}{\gamma}\,.$$ In fact, since $0<\frac{\delta}{q-1-\beta}<\frac{p-1-\alpha}{\gamma},$ there exist $k>0$ such that $\frac{\delta}{q-1-\beta}0 satisfying \vartheta(\epsilon)>\frac{1}{q-1} for all \epsilon>\epsilon_0. Fixed \epsilon>\epsilon_0, we define s_1=\frac{1}{p-1}+\epsilon and s_2=\vartheta(\epsilon)=ks_1. Then, s_1 >\frac{1}{p-1}, s_2>\frac{1}{q-1}, and \frac{s_1}{s_2} =k, which proves the claim. Then, by using \eqref{super-0}, we obtain \lambda_0>0 such that $$\label{super-1} a_{\lambda} := \max\{a_0 \lambda^{s_1(\alpha-p+1)+s_2\gamma},\; a_0 \lambda^{s_1\delta+s_2(\beta-q+1)}\}<1,$$ for all \lambda > \lambda_0. Moreover, there exist A and B positive constants satisfying $$\label{super-2} A^{p-1} = \lambda A^{\alpha} l^{\alpha}B^{\gamma}L^{\gamma} \text{ and } B^{q-1} = \lambda A^{\delta}l^{\delta}B^{\beta}L^{\beta},$$ where l=\|e_1\|_{\infty} and L=\|e_2\|_{\infty}. For a fixed \lambda>\lambda_0, we define \[ (\bar{u}(x),\bar{v}(x)):=\left(\lambda^{s_1} A e_1(x), \lambda^{s_2} B e_2(x)\right).$ Note that $\bar{u}\in C^{1,\alpha_1}(\overline{\Omega})$ and $\bar{v}\in C^{1,\alpha_2}(\overline{\Omega})$. Let $w \in W_0^{1,p}(\Omega)$ with $w(x)\geq 0$ for a.e. (almost everywhere) $x\in \Omega$. Then $$\label{e1} {\int_{\Omega}{ |\nabla \bar{u}|^{p-2}\nabla \bar{u} \nabla w}} \,dx = \lambda^{s_1 (p-1)} A^{p-1}{\int_{\Omega}{ w}} \,dx$$ and, for $z \in W_0^{1,q}(\Omega)$ with $z(x)\geq 0$ for a.e. $x\in \Omega$, $$\label{e2} {\int_{\Omega}{ |\nabla \bar{v}|^{q-2}\nabla \bar{v} \nabla z}} \,dx =\lambda^{s_2 (q-1)} B^{q-1}{\int_{\Omega}{ z}} \,dx.$$ On the other hand, by using \eqref{hip-existence}, \eqref{super-1}, and \eqref{super-2}, we have \label{super-3} \begin{aligned} \lambda f_1(x,\bar{u}(x),\bar{v}(x)) & \leq \lambda a_0 \lambda^{s_1\alpha} A^{\alpha} l^{\alpha} \lambda^{s_2\gamma} B^{\gamma}L^{\gamma}\\ & = \lambda a_0 \lambda^{s_1(\alpha-p+1)+s_2\gamma} \lambda^{s_1(p-1)}A^{\alpha} l^{\alpha} B^{\gamma}L^{\gamma} \\ & \leq a_{\lambda} \lambda^{s_1(p-1)} A^{p-1} \end{aligned} and $$\label{super-33} \lambda f_2(x,\bar{u}(x),\bar{v}(x)) \leq a_{\lambda} \lambda^{s_2(q-1)} B^{q-1}.$$ But, as $a_\lambda<1$ for $\lambda > \lambda_0$, there exists $c_\lambda>0$ such that $$\label{super-4} a_\lambda \lambda^{s_1(p-1)}A^{p-1} +c_\lambda \leq \lambda^{s_1(p-1)} A^{p-1},\quad a_\lambda \lambda^{s_2(q-1)}B^{q-1} +c_\lambda \leq \lambda^{s_2(q-1)} B^{q-1}.$$ Also, since that $g_i$, $i=1,2$, are bounded on bounded sets, there exists $\mu_0=\mu_0(\lambda)>0$ such that $$\label{super-5} |\mu| |g_1(x,\bar{u}(x),\bar{v}(x))|\leq c_\lambda,\quad |\mu| |g_2(x,\bar{u}(x),\bar{v}(x))|\leq c_\lambda$$ for all $|\mu| <\mu_0$. Then, by \eqref{super-3}, \eqref{super-4}, and \eqref{super-5} we obtain \label{super-6} \begin{aligned} &\lambda f_1(x,\bar{u}(x),\bar{v}(x)) + \mu g_1(x,\bar{u}(x),\bar{v}(x))\\ & \leq a_{\lambda} \lambda^{s_1(p-1)} A^{p-1} + |\mu g_1(x,\bar{u}(x),\bar{v}(x))| \\ & \leq a_{\lambda} \lambda^{s_1(p-1)} A^{p-1}+c_\lambda \\ & \leq \lambda^{s_1(p-1)} A^{p-1}\,. \end{aligned} From \eqref{super-33}, \eqref{super-4}, and \eqref{super-5}, we obtain $$\label{super-66} \lambda f_2(x,\bar{u}(x),\bar{v}(x)) + \mu g_2(x,\bar{u}(x),\bar{v}(x)) \leq \lambda^{s_2(q-1)} B^{q-1},$$ for all $|\mu| <\mu_0$. Hence, by \eqref{e1} and \eqref{super-6}, we conclude that $$\label{super-7} {\int_{\Omega}{|\nabla \bar{u}|^{p-2}\nabla \bar{u} \nabla w}}\,dx \geq \lambda {\int_{\Omega}} f_1(x,\bar{u}(x),\bar{v}(x)) w \,dx + \mu {\int_{\Omega}} g_1(x,\bar{u}(x),\bar{v}(x)) w \,dx.$$ Analogously, from \eqref{e2} and \eqref{super-66}, we obtain $$\label{super-8} {\int_{\Omega}{ |\nabla \bar{v}|^{q-2}\nabla \bar{v} \nabla z}}\,dx \geq \lambda {\int_{\Omega}} f_2(x,\bar{u}(x),\bar{v}(x)) z \,dx + \mu {\int_{\Omega}} g_2(x,\bar{u}(x),\bar{v}(x)) z \,dx.$$ Thus, from \eqref{super-7} and \eqref{super-8}, we see that $(\bar{u},\bar{v})$ is a upper-solution of \eqref{system} with $\bar{u}\in C^{1,\alpha_1}(\overline{\Omega})$ and $\bar{v}\in C^{1,\alpha_2}(\overline{\Omega})$. \subsection{Lower-solution}\label{lower} In this subsetion, we prove that \eqref{system} possesses a lower-solution. Let us fix $\xi$ and $\eta$ such that $$\label{sub-0} 1<\xi<\frac{p}{p-1},\quad 1<\eta<\frac{q}{q-1}\cdot$$ From \eqref{hip-limite} and \eqref{hip-limite1} we have $a_1>0$, $a_2>0$, and $R>0$ such that \begin{gather}\label{hip-cons-limite} f_i(x,s,t)\geq a_1,\quad\text{for $i=1,2$ an all $s>R$ $t>R$},\\ \label{sub-1} f_i(x,s,t) \geq -a_2,\quad\text{for $i=1,2$ and all $s,t\in (0,+\infty)$}, \end{gather} uniformly in $x\in\Omega$. Consider $\lambda_p$ the eigenvalue associated to positive eigenfunction $\varphi_p$ of the problem of eigenvalue of $p$-Laplacian operator, and $\lambda_q$ the eigenvalue associated with positive eigenfunction $\varphi_q$ of the problem of eigenvalue of $q$-Laplacian operator. We take $a_3$ and $a_4$ positive constants satisfying $$\label{sub-2} a_3 >2\frac{\lambda_p (a_2+1) \xi^{p-1}}{a_1},\quad a_4 >2\frac{\lambda_q (a_2+1) \eta^{q-1}}{a_1},$$ and define $(\underline{u}(x), \underline{v}(x)): = (c_\lambda \varphi^{\xi}_p(x), d_\lambda \varphi^{\eta}_q(x)),$ where $$\label{sub-4} c_\lambda = \left(\frac{\lambda a_2 +1}{a_3}\right)^{\frac{1}{p-1}},\quad d_\lambda = \left(\frac{\lambda a_2 +1}{a_4}\right)^{\frac{1}{q-1}}.$$ Thus, for $w \in W_0^{1,p}(\Omega)$ and $z \in W_0^{1,q}(\Omega)$ with $w(x)\geq 0$ and $z(x)\geq 0$ for a.e. $x\in \Omega$, we obtain \label{sub-5} \begin{aligned} &{\int_\Omega} |\nabla \underline{u}|^{p-2} \nabla\underline{u} \nabla w \, dx \\ &= c_\lambda^{p-1} \xi^{p-1} {\int_\Omega} \left[\lambda_p \varphi_p^{\xi(p-1)}-(\xi-1)(p-1)\varphi_p^{(\xi-1)(p-1)-1} |\nabla \varphi_p|^p \right] w \, dx \end{aligned} and \label{sub-6} \begin{aligned} &{\int_\Omega} |\nabla \underline{v}|^{q-2} \nabla \underline{v} \nabla z \,dx \\ &= d_\lambda^{q-1} \eta^{q-1} {\int_\Omega} \left[\lambda_q \varphi_q^{\eta(q-1)}-(\eta-1)(q-1)\varphi_q^{(\eta-1)(q-1)-1} |\nabla \varphi_q|^q \right] z dx. \end{aligned} We know that $\varphi_p, \varphi_q>0$ in $\Omega$ and $|\nabla \varphi_p|, |\nabla \varphi_q|\geq\sigma$ on $\partial \Omega$ for some $\sigma>0$. Also, we can suppose that $\|\varphi_p\|_{\infty}=\|\varphi_q\|_{\infty}=1$. Furthermore, by using \eqref{sub-0}, it is easy to prove that there exists $\zeta>0$ such that \begin{gather}\label{sub-7} \lambda_p \varphi_{p}^{\xi(p-1)} -(\xi-1)(p-1) \varphi_{p}^{(\xi-1)(p-1)-1} | \nabla \varphi_p|^{p} \leq -a_3, \\ \label{sub-711} \lambda_q \varphi_{q}^{\eta(q-1)} -(\eta-1)(q-1) \varphi_{q}^{(\eta-1)(q-1)-1} | \nabla \varphi_q|^{q} \leq -a_4, \end{gather} in $\Omega_\zeta := \{x\in \Omega: \operatorname{dist}(x,\partial \Omega)\leq \zeta \}$. But, we have by \eqref{sub-0}, \eqref{sub-1}, and \eqref{sub-4} that $$\label{sub-8} -c_\lambda^{p-1} \xi^{p-1} a_3 = -(\lambda a_2+1)\xi^{p-1} \leq -(\lambda a_2+1) \leq \lambda f_1(x,\underline{u},\underline{v}) -1$$ and $$\label{sub-9} -d_\lambda^{q-1} \eta^{q-1} a_4 \leq \lambda f_2(x,\underline{u},\underline{v}) -1,$$ for all $x\in \Omega$. Therefore, from \eqref{sub-7}, \eqref{sub-711}, \eqref{sub-8}, and \eqref{sub-9}, we obtain $$\label{sub-10} c_\lambda^{p-1} \xi^{p-1}\left[\lambda_p \varphi_{p}^{\xi(p-1)} -(\xi-1)(p-1) \varphi_{p}^{(\xi-1)(p-1)-1} | \nabla \varphi_p|^{p}\right] \leq \lambda f_1(x,\underline{u},\underline{v}) -1$$ and $$\label{sub-101} d_\lambda^{q-1} \eta^{q-1}\left[\lambda_q \varphi_{q}^{\eta(q-1)} -(\eta-1)(q-1) \varphi_{q}^{(\eta-1)(q-1)-1} | \nabla \varphi_q|^{q}\right] \leq \lambda f_2(x,\underline{u},\underline{v}) -1,$$ in $\Omega_\zeta := \{x\in \Omega: \operatorname{dist}(x,\partial \Omega) \leq \zeta \}$. On the other hand, there exists $a_5>0$ such that $\varphi_p(x),\varphi_q(x)\geq a_5$ for all $x\in\Omega \setminus\Omega_\zeta$. Then, if $\lambda_0>0$ is provided of proof of existence of upper-solution, and by taking $\lambda_0 > 0$ greater than one, if necessary, we can suppose $\lambda_0 \geq \max\{1, \frac{2}{a_1}, \frac{R^{p-1}a_5^{-\xi(p-1)}a_3^{-1}}{a_2}, \frac{R^{q-1}a_5^{-\eta(q-1)}a_4^{-1}}{a_2}\}>0.$ Thus $\underline{u}(x)=c_\lambda \varphi^\xi_p(x)\geq c_\lambda a_5^\xi>R, \quad \underline{v}(x)=d_\lambda \varphi^\xi_p(x)\geq d_\lambda a_5^\eta>R,$ for all $x\in \Omega\setminus\Omega_\zeta$ and $\lambda>\lambda_0$. Therefore, by \eqref{hip-cons-limite}, we have $$\label{sub-341} \lambda f_1(x,\underline{u}(x),\underline{v}(x)) -1\geq \lambda a_1-1,\quad \lambda f_2(x,\underline{u}(x),\underline{v}(x)) -1\geq \lambda a_1-1$$ for all $x\in \Omega\setminus\Omega_\zeta$ and $\lambda>\lambda_0$. \noindent{\bf Claim.} By \eqref{sub-2} and $\lambda>\lambda_0\geq \max\{1, \frac{2}{a_1}, \frac{R^{p-1}a_5^{-\xi(p-1)}a_3^{-1}}{a_2}, \frac{R^{q-1}a_5^{-\eta(q-1)}a_4^{-1}}{a_2}\}$, we have $$\label{sub-34} a_3 > \frac{\lambda_p \xi^{p-1} (\lambda a_2 +1)}{\lambda a_1 -1} \text{\, and\, } a_4 > \frac{\lambda_q \eta^{q-1} (\lambda a_2 +1)}{\lambda a_1 -1}\cdot$$ In fact, since that $\lambda>\frac{2}{a_1}$, we obtain $a_1-\frac{1}{\lambda}>a_1-\frac{a_1}{2}=\frac{a_1}{2},$ so, as $\lambda >1$ and by \eqref{sub-2}, \begin{align*} \frac{\lambda_p \xi^{p-1} (\lambda a_2 +1)}{\lambda a_1 -1} & = \frac{\lambda_p \xi^{p-1} (a_2 +\frac{1}{\lambda})}{ a_1 -\frac{1}{\lambda}} \\ & < \frac{\lambda_p \xi^{p-1} (a_2 +1)}{ a_1 -\frac{1}{\lambda}} \\ & < \frac{\lambda_p \xi^{p-1} (a_2 +1)}{ \frac{a_1}{2}} \\ & = \frac{2\lambda_p (a_2+1) \xi^{p-1}}{a_1} \frac{\lambda_q \eta^{q-1} (\lambda a_2 +1)}{\lambda a_1 -1}, \] which prove the claim. Then, from \eqref{sub-5}, \eqref{sub-341}, and \eqref{sub-34}, we achieve \label{sub-20} \begin{aligned} &c_\lambda^{p-1} \xi^{p-1} \big[ \lambda_p \varphi_p^{\xi(p-1)} - (\xi-1)(p-1)\varphi_p^{(\xi-1)(p-1)-1}|\nabla\varphi_p|^{p}\big](x) \\ &\leq c_\lambda^{p-1} \xi^{p-1} \lambda_p \varphi_p^{\xi(p-1)}(x) \\ &\leq \lambda_p c_\lambda^{p-1} \xi^{p-1} \\ &\leq \lambda_p \frac{\lambda a_2 +1}{a_3} \xi^{p-1} \\ & \leq \lambda a_1-1 \\ & \leq \lambda f_1(x,\underline{u}(x),\underline{v}(x)) -1 \end{aligned} and, by \eqref{sub-6}, \eqref{sub-341}, and \eqref{sub-34}, \label{sub-21} \begin{aligned} &d_\lambda^{q-1} \eta^{q-1} \big[ \lambda_q \varphi_q^{\eta(q-1)} - (\eta-1)(q-1)\varphi_q^{(\eta-1)(q-1)-1}|\nabla\varphi_q|^{q}\big](x) \\ & \leq \lambda_q \frac{\lambda a_2 +1}{a_4} \eta^{q-1} \\ & \leq \lambda f_2(x,\underline{u}(x),\underline{v}(x)) -1, \end{aligned} for all $x\in \Omega\setminus\Omega_\zeta$. Thus, by combining \eqref{sub-10}, \eqref{sub-101}, \eqref{sub-20}, and \eqref{sub-21}, we obtain \label{sub-200} \begin{aligned} &c_\lambda^{p-1} \xi^{p-1} \left[ \lambda_p \varphi_p^{\xi(p-1)} - (\xi-1)(p-1)\varphi_p^{(\xi-1)(p-1)-1}|\nabla\varphi_p|^{p}\right](x) \\ & \leq \lambda f_1(x,\underline{u}(x),\underline{v}(x)) -1 \end{aligned} and \label{sub-210} \begin{aligned} &d_\lambda^{q-1} \eta^{q-1} \left[ \lambda_q \varphi_q^{\eta(q-1)} - (\eta-1)(q-1)\varphi_q^{(\eta-1)(q-1)-1}|\nabla\varphi_q|^{q}\right](x) \\ & \leq \lambda f_2(x,\underline{u}(x),\underline{v}(x)) -1, \end{aligned} for all $\lambda > \lambda_0$ and $x\in\Omega$. Moreover, if $\mu_0=\mu_0(\lambda)>0$ is provided of proof of existence of upper-solution; for each $\lambda >\lambda_0$, since that $g_i$, $i=1,2$, are bounded on bounded sets, replacing $\mu_0>0$ by another smaller, if necessary, we have $$\label{sub-22} |\mu| |g_1(x,\underline{u}(x),\underline{v}(x))|\leq 1,\quad |\mu| |g_2(x,\underline{u}(x),\underline{v}(x))|\leq 1$$ for all $|\mu| <\mu_0$. Therefore, by \eqref{sub-22} it follows that \begin{gather}\label{sub-23} \lambda f_1(x,\underline{u}(x),\underline{v}(x)) - 1 \leq \lambda f_1(x,\underline{u}(x),\underline{v}(x)) + \mu g_1(x,\underline{u}(x),\underline{v}(x)),\\ \label{sub-24} \lambda f_2(x,\underline{u}(x),\underline{v}(x)) - 1 \leq \lambda f_2(x,\underline{u}(x),\underline{v}(x)) + \mu g_2(x,\underline{u}(x), \underline{v}(x)), \end{gather} for all $|\mu|<\mu_0$ and $x\in \Omega$. Hence, substituting \eqref{sub-23} and \eqref{sub-24} in \eqref{sub-200} and \eqref{sub-210}, respectively, and by using \eqref{sub-5} and \eqref{sub-6} , we achieve \begin{aligned} {\int_{\Omega}{ |\nabla \underline{u}|^{p-2}\nabla \underline{u} \nabla w}}dx & \leq \lambda {\int_{\Omega}} f_1(x,\underline{u}(x),\underline{v}(x)) w dx \\ & \quad + \mu {\int_{\Omega}} g_1(x,\underline{u}(x),\underline{v}(x)) w dx \end{aligned} and \begin{aligned} {\int_{\Omega}{ |\nabla \underline{v}|^{q-2}\nabla \underline{v} \nabla z}}dx & \leq \lambda {\int_{\Omega}} f_2(x,\underline{u}(x),\underline{v}(x)) z dx \\ &\quad + \mu {\int_{\Omega}} g_2(x,\underline{u}(x),\underline{v}(x)) z dx, \end{aligned} so, we conclude that $(\underline{u},\underline{v})$ is a lower-solution of \eqref{system} with $\underline{u}, \underline{v}\in C^{1}(\Omega)$. \subsection{Proof of Theorem \ref{theorem1}} In subsections \ref{upper} and \ref{lower} we proved that there exists $\lambda_0>0$ such that for each $\lambda>\lambda_0$ there exist $\mu_0=\mu_0(\lambda)>0$ and $(\bar{u},\bar{v})$, $(\underline{u},\underline{v})$ that are upper-solution and lower-solution, respectively, of system \eqref{system}, with $\bar{u}\in C^{1,\alpha_1}(\overline{\Omega})$, $\overline{v}\in C^{1,\alpha_2}(\overline{\Omega})$, and $\underline{u}, \underline{v}\in C^{1}(\Omega)$, whenever $|\mu|<\mu_0$. Let $w\in W_0^{1,p}(\Omega)$ and $z\in W_0^{1,q}(\Omega)$ satisfy $w,z\geq 0$ for a.e. in $\Omega$. Then, from \eqref{sub-2}, \eqref{sub-10}, and \eqref{sub-20}, we have \label{f1} \begin{aligned} {\int_{\Omega}{ |\nabla \underline{u}|^{p-2}\nabla \underline{u} \nabla w}}dx & \leq \lambda_p \frac{(\lambda a_2+1)}{a_3}\xi^{p-1} {\int_{\Omega}} w dx \\ & \leq \lambda \frac{a_2+\frac{1}{\lambda}}{a_2+1} \frac{a_1}{2} {\int_{\Omega}} w dx \\ & \leq \lambda \frac{a_1}{2} {\int_{\Omega}} w dx\,. \end{aligned} By \eqref{sub-2}, \eqref{sub-101}, and \eqref{sub-21}, we have $$\label{f2} {\int_{\Omega}{ |\nabla \underline{v}|^{q-2}\nabla \underline{v} \nabla z}}dx \leq \lambda \frac{a_1}{2} {\int_{\Omega}} z dx.$$ However, since that $s_1(p-1)>1$ and $s_2(q-1)>1$, changing $\lambda_0>0$ by another greater than 1, if necessary, we can suppose that $$\label{f3} \lambda \frac{a_1}{2} \leq \min\{\lambda^{s_1(p-1)}A^{p-1},\; \lambda^{s_2(q-1)}B^{q-1}\}$$ for all $\lambda\geq \lambda_0$. Hence, from \eqref{e1}, \eqref{f1}, and \eqref{f3}, we conclude that $${\int_{\Omega}} |\nabla \underline{u}|^{p-2}\nabla \underline{u} \nabla w\,dx \leq {\int_{\Omega}} |\nabla \bar{u}|^{p-2}\nabla \bar{u} \nabla w\,dx$$ and by \eqref{e2}, \eqref{f2}, and \eqref{f3}, $${\int_{\Omega}} |\nabla \underline{v}|^{q-2}\nabla \underline{v} \nabla z dx \leq {\int_{\Omega}} |\nabla \bar{v}|^{q-2}\nabla \bar{v} \nabla z dx,$$ so, by the weak comparison principle (see \cite[Lemma 2.2]{Drabek}), we obtain $\underline{u}\leq \bar{u}$ and $\underline{v}\leq \bar{v}$ for all $x\in \Omega$. Thus, by using \eqref{me-1}, we obtain by the standard theorem of lower and upper solution (see \cite[Theorem 2.4]{Rodrigo}) a solution $(u,v)\in W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega)$ of system \eqref{system} with $\underline{u}\leq u \leq \bar{u}$ and $\underline{v}\leq v \leq \bar{v}$ for almost everywhere in $\Omega$. In particular, we see that $u,v \in L^{\infty}(\Omega)$ and $u(x)>0$, $v(x)>0$ for a.e. $x\in \Omega$. Then, by \cite[Theorem 1]{Tolksdorf}, we obtain $u\in C^{1,\rho_1}(\Omega)$ and $v\in C^{1,\rho_2}(\Omega)$ for some $\rho_1,\rho_2>0$, so $u(x)>0$, $v(x)>0$ for all $x\in \Omega$. \section{Proof of Theorem \ref{nonexistence}}\label{inexistence} Supposing by contradiction that there exists a nontrivial solution $(u,v)$ of \eqref{system}, for some $\lambda, \mu$ satisfying \eqref{cond. teo nonexist}, then by variational characterization of $\lambda_p$ and $\lambda_q$, we achieve \label{Eq nonexistence 1} \begin{aligned} \lambda_p {{\int_{\Omega}|u|^p}}dx &\leq {{\int_{\Omega}|\nabla u|^p}}dx \\ &\leq {{\int_{\Omega}} \left[(|\lambda|k_1+|\mu|k_5)|u|^{p}+ (|\lambda|k_2+|\mu|k_6) |v|^{q}\right]}dx \end{aligned} and similarly $$\label{Eq nonexistence 2} \lambda_q {\int_{\Omega}} |v|^qdx \leq {\int_{\Omega}} \left[(|\lambda|k_3+|\mu|k_7)|u|^{p}+ (|\lambda|k_4+|\mu|k_8) |v|^{q}\right]dx.$$ From \eqref{Eq nonexistence 1} and \eqref{Eq nonexistence 2}, we have \begin{align*} 0&<\left\{\lambda_p- [|\lambda|(k_1+k_3)+|\mu|(k_5+k_7)]\right\} {{\int_{\Omega}|u|^p}}dx\\ &\quad +\left\{\lambda_q- [|\lambda|(k_2+k_4)+|\mu|(k_6+k_8)]\right\} {{\int_{\Omega}|v|^q}}dx \leq 0, \end{align*} which is a contradiction. \begin{thebibliography}{00} \bibitem{JShi} J. 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