\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 69, 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 2016 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2016/69\hfil $(p,q)$-Kirchhoff type systems] {Existence of solutions for singular $(p,q)$-Kirchhoff type systems with multiple parameters} \author[S. H. Rasouli \hfil EJDE-2016/69\hfilneg] {Sayyed Hashem Rasouli} \address{Sayyed Hashem Rasouli \newline Department of Mathematics, Faculty of Basic Sciences, Babol University of Technology, Babol, Iran} \email{s.h.rasouli@nit.ac.ir} \thanks{Submitted Mar 18, 2015. Published March 14, 2016.} \subjclass[2010]{35J55, 35J65} \keywords{$(p,q)$-Kirchhoff type system; subsolution; supersolution} \begin{abstract} This article concerns the existence of positive solutions for singular $(p,q)$-Kirchhoff type systems with multiple parameters. Our approach is based on the method of sub- and super-solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} In this article, we are interested in the existence of positive solutions for the singular $(p,q)$-Kirchhoff type system \begin{gather} -M_1\Big(\int_{\Omega}|\nabla u|^{p}dx\Big)\Delta_{p} u = a(x)\Big[\alpha_1 \Big(f(v)-\frac{1}{u^{\eta}}\Big)+\beta_1\Big(h(u) -\frac{1}{u^{\eta}}\Big)\Big], \quad x\in \Omega, \nonumber\\ -M_2\Big(\int_{\Omega}|\nabla v|^{q}dx\Big)\Delta_q v =b(x)\Big[\alpha_2 \Big(g(u)-\frac{1}{v^{\theta}}\Big)+\beta_2\Big(k(v) -\frac{1}{v^{\theta}}\Big)\Big], \quad x\in \Omega, \nonumber \\ u = v = 0 , \quad x\in\partial \Omega, \label{e1} \end{gather} where $M_{i}:\mathbb{R}^{+}\to \mathbb{R}^{+}$, $i=1,2$ are two continuous and increasing functions such that $M_{i}(t)\geq m_{i}>0$ for all $t\in \mathbb{R}^{+}$, $\Delta_{r}z=div(|\nabla z|^{r-2}\nabla z)$, for $r>1$ denotes the r-Laplacian operator, $\alpha_1,\alpha_2,\beta_1,\beta_2$ are positive parameters, $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $n\geq1$ with sufficiently smooth boundary and $\eta ,\theta\in (0,1)$. Here $a(x), b(x)\in C(\overline{\Omega})$ are weight functions such that $a(x)\geq a_0>0$, $b(x)\geq b_0>0$ for all $x\in \overline{\Omega}$, $f,g,h,k\in C([0,\infty)$ are nondecreasing functions and $f(0),g(0),h(0),k(0)>0$. Problem \eqref{e1} is called nonlocal because of the term $-M(\int_{\Omega}|\nabla u|^{r}dx)$ which implies that the first two equations in \eqref{e1} are no longer pointwise equalities. This phenomenon causes some mathematical difficulties which makes the study of such a class of problem particularly interesting. Also, such a problem has physical motivation. Moreover, system \eqref{e1} is related to the stationary version of the Kirchhoff equation \begin{equation} \label{e2} \rho \frac {\partial^2u}{\partial t^2}-\Big(\frac {P_0}{h} + \frac {E}{2L}\int_0^{L}|\frac {\partial u}{\partial x}|^2dx\Big)\frac {\partial^2u}{\partial x^2}=0 \end{equation} presented by Kirchhoff \cite{Gk}. This equation extends the classical d'Alembert's wave equation by considering the effects of the changes in the length of the strings during the vibrations. The parameters in \eqref{e2} have the following meanings: $L$ is the length of the string, $h$ is the area of cross section, $E$ is the Young’s modulus of the material, $\rho$ is the mass density, and $P_0$ is the initial tension. When an elastic string with fixed ends is subjected to transverse vibrations, its length varies with the time: this introduces changes of the tension in the string. This induced Kirchhoff to propose a nonlinear correction of the classical D'Alembert's equation. Later on, Woinowsky-Krieger (Nash-Modeer) incorporated this correction in the classical Euler-Bernoulli equation for the beam (plate) with hinged ends. See, for example, \cite{aa1,aa2} and the references therein. Nonlocal problems also appear in other fields: for example, biological systems where $u$ and $v$ describe a process which depends on the average of itself (for instance, population density). See \cite{coa-fjsac,coa-fjsac-tfm,fjsac-sdbm,tfm,kp-zz} and the references therein. In recent years, problems involving Kirchhoff type operators have been studied in many papers, we refer to \cite{gaa-ntc-ss, bc, yl-fl-js, gmf, lw, js-sl, ntc}, in which the authors have used different methods to prove the existence of solutions. Let $F(s,t)= (f(t)-\frac{1}{s^{\eta}})+(h(s)-\frac{1}{s^{\eta}})$, and $G(s,t)= (g(s)-\frac{1}{t^{\eta}})+(k(t)-\frac{1}{t^{\eta}})$. Then $\lim_{(s,t)\to (0,0)} F(s,t) = -\infty=\lim_{(s,t)\to (0,0)} G(s,t)$, and hence we refer to \eqref{e1} as an infinite semipositone problem. See \cite{ali-shi1}, where the authors studied the corresponding non-singular finite system when $M_1(t)=M_2(t)\equiv 1$, and $a(x)=b(x)\equiv 1$. It is well documented that the study of positive solutions to such semipositone problems is mathematically very challenging \cite{hb-lac-ln}, \cite{pll}. In this paper, we study the even more challenging semipositone system with $\lim_{(s,t)\to (0,0)} F(s,t) = -\infty=\lim_{(s,t)\to (0,0)} G(s,t)$. We do not need the boundedness of the Kirchhoff functions $M_1,M_2$, as in \cite{bc-xw-jl}. Using the sub and supersolutions techniques, we prove the existence of positive solutions to the system \eqref{e1}. To our best knowledge, this is an interesting and new research topic for singular $(p,q)$-Kirchhoff type systems. One can refer to \cite{ekl-rs, ekl-rs-jy, ekl-rs-jy1} for some recent existence results of infinite semipositone systems. To precisely state our existence result we consider the eigenvalue problem \begin{equation} \label{e3} \begin{gathered} -\Delta_{r} \phi = \lambda\,|\phi|^{r-2}\phi, \quad x\in \Omega,\\ \phi = 0 , \quad x\in\partial \Omega. \end{gathered} \end{equation} Let $\phi_{1,r}$ be the eigenfunction corresponding to the first eigenvalue $\lambda_{1,r}$ of \eqref{e3} such that $\phi_{1,r}(x)>0$ in $\Omega$, and $\|\phi_{1,r}\|_{\infty}=1$ for $r=p, q$. Let $m,\sigma,\delta>0$ be such that \begin{gather} \label{e4} \sigma\leq\phi_{1,r}^{\frac{r}{r-1+s}}\leq 1, \quad x\in\Omega-\overline{\Omega_\delta}, \\ \label{e5} |\nabla \phi_{1,r}|^{r} \geq m , \quad x\in \overline{\Omega_\delta}, \end{gather} for $r=p, q$, and $s=\eta, \theta$, where $\overline{\Omega_\delta}:=\{x\in\Omega|d(x,\partial\Omega)\leq \delta\} $. (This is possible since $|\nabla \phi_{1,r}|^{r}\neq 0$ on $\partial \Omega$ while $\phi_{1,r}=0$ on $\partial \Omega$ for $r=p, q)$. We will also consider the unique solution $\zeta_{r}\in W_0^{1,r}(\Omega)$ of the boundary-value problem \begin{gather*} -\Delta_{r}\zeta_{r} = 1, \quad x\in \Omega,\\ \zeta_{r}=0 , \quad x\in\partial \Omega. \end{gather*} It is known that $\zeta_{r}>0 $ in $\Omega$ and $ \frac{\partial \zeta_{r}}{\partial n}<0 $ on $\partial\Omega$. \section{Existence of solutions} In this section, we shall establish our existence result via the method of sub- super-solution \cite{ali-shi1}. For the system \begin{gather*} -M_1\Big(\int_{\Omega}|\nabla u|^{p}dx\Big)\Delta_{p} u = h_1(x,u,v), \quad x\in \Omega,\\ -M_2\Big(\int_{\Omega}|\nabla v|^{q}dx\Big)\Delta_q v = h_2(x,u,v), \quad x\in \Omega,\\ u = v = 0 , \quad x\in\partial \Omega, \end{gather*} a pair of functions $(\psi_1,\psi_2)\in W^{1,p}\cap C(\overline{\Omega})\times W^{1,q}\cap C(\overline{\Omega})$ and $(z_1,z_2)\in W^{1,p}\cap C(\overline{\Omega})\times W^{1,q}\cap C(\overline{\Omega})$ are called a subsolution and supersolution if they satisfy $(\psi_1,\psi_2)=(0,0)=(z_1,z_2)$ on $\partial\Omega$, \begin{gather*} M_1\Big(\int_{\Omega}|\nabla \psi_1|^{p}dx\Big) \int_{\Omega}\,|\nabla\psi_1|^{p-2}\nabla \psi_1\cdot\nabla w\,dx\leq \int_{\Omega}\,h_1(x,\psi_1,\psi_2)w\,dx, \\ M_2\Big(\int_{\Omega}|\nabla \psi_2|^{q}dx\Big) \int_{\Omega}\,|\nabla\psi_2|^{q-2}\nabla \psi_2\cdot\nabla w\,dx\leq \int_{\Omega}\,h_2(x,\psi_1,\psi_2)w\,dx \end{gather*} and \begin{gather*} M_1\Big(\int_{\Omega}|\nabla z_1|^{p}dx\Big) \int_{\Omega}\,|\nabla z_1|^{p-2}\nabla z_1\cdot\nabla w\,dx\geq \int_{\Omega}\,h_1(x,z_1,z_2)w\,dx, \\ M_2\Big(\int_{\Omega}|\nabla z_2|^{q}dx\Big) \int_{\Omega}\,|\nabla z_2|^{q-2}\nabla z_2\cdot\nabla w\,dx\geq \int_{\Omega}\,h_2(x,z_1,z_2)w\,dx, \end{gather*} for all $ w\in W=\{w\in C_0^{\infty}(\Omega)| w\geq 0, x\in \Omega\}$. Then the following result holds. \begin{lemma}[\cite{ ntc}] \label{lem2.1} Suppose there exist sub- and super-solutions $(\psi_1,\psi_2)$ and $(z_1,z_2)$ respectively of \eqref{e1} such that $(\psi_1,\psi_2)\leq (z_1,z_2)$. Then \eqref{e1} has a solution $ (u,v)$ such that $(u,v)$ $\in[(\psi_1,\psi_2),(z_1,z_2)]$. \end{lemma} We use the following hypotheses: \begin{itemize} \item[(H1)] $f,g,h,k\in C([0,\infty)$ are nondecreasing functions such that $f(0)>0$, $g(0)>0$, $h(0)>0$, $k(0)>0$, \begin{gather*} \lim_{s\to+\infty}f(s)=\lim_{s\to+\infty}h(s) =\lim_{s\to+\infty}g(s)=\lim_{s\to+\infty}k(s)=+\infty, \\ \lim_{s\to+\infty}\frac{h(s)}{s^{p-1}}=\lim_{s\to+\infty}\frac{k(s)}{s^{q-1}}=0. \end{gather*} \item[(H2)] for all $ A>0$, \[ \lim_{s\to\infty}\frac{f(Ag(s)^{\frac{1}{q-1}})}{s^{p-1+\eta}}=0. \] \end{itemize} Our main result read as follows. \begin{theorem} \label{thm2.2} Let {\rm (H1)--(H2)} hold. Then \eqref{e1} has a large positive solution $(u,v)$ provided $\alpha_1+\beta_1$ and $\alpha_2+\beta_2$ are large. \end{theorem} \begin{proof} Let $\gamma_0=\min\{f(0),h(0)\}>0$ and \begin{align*} \gamma_1=\min \Big\{& f\Big((\alpha_2 +\beta_2)^{r_2}(\frac{b_0}{m_2})^{\frac{1}{q-1}}(\frac{q-1+\theta}{q})\sigma \Big),\\ &h\Big((\alpha_1 +\beta_1)^{r_1}(\frac {a_0}{m_1})^{\frac{1}{p-1}}(\frac{p-1+\eta}{p})\sigma \Big) \Big\}. \end{align*} For fixed $r_1\in (\frac{1}{p-1+\eta},\frac{1}{p-1})$ and $r_2\in (\frac{1}{q-1+\theta},\frac{1}{q-1})$, we shall verify that with \begin{gather*} \psi_1=\frac{(\alpha_1+\beta_1)^{r_1}(p-1+\eta)}{p} (\frac{a_0}{m_1})^{\frac{1}{p-1}} \phi_{1,p}^{\frac{p}{p-1+\eta}}, \\ \psi_2= \frac{(\alpha_2+\beta_2)^{r_2} (q-1+\theta)}{q}(\frac{b_0}{m_2})^{\frac{1}{q-1}} \phi_{1,q}^{\frac{q}{q-1+\theta}}, \end{gather*} and $(\psi_1,\psi_2)$is a sub-solution of \eqref{e1}. Let $w\in W$. Then a calculation shows that $$ \nabla\psi_1=(\alpha_1+\beta_1)^{r_1}(\frac{a_0}{m_1}) ^{\frac{1}{p-1}}\phi_{1,p}^{\frac{1-\eta}{p-1+\eta}}\nabla\phi_{1,p}, $$ and we have \begin{align*} &M_1\Big(\int_{\Omega}|\nabla \psi_1|^{p}dx\Big)\int_{\Omega}\,|\nabla\psi_1|^{p-2}\nabla \psi_1\cdot\nabla w\,dx\\ &=\frac{a_0(\alpha_1+\beta_1)^{(p-1)r_1}}{m_1}M_1\Big(\int_{\Omega}|\nabla \psi_1|^{p}dx\Big)\\ &\quad\times \int_{\Omega}\phi_{1,p}^{1-\frac{\eta p}{p-1+\eta}}|\nabla\phi_{1,p}|^{p-2}\nabla \phi_{1,p}\nabla w dx \\ &=\frac{a_0(\alpha_1+\beta_1)^{(p-1)r_1}}{m_1}M_1\Big(\int_{\Omega}|\nabla \psi_1|^{p}dx\Big) \\ &\quad \int_{\Omega}\,|\nabla\phi_{1,p}|^{p-2}\nabla \phi_{1,p}\big\{\nabla(\phi_{1,p}^{1-\frac{\eta p}{p-1+\eta}}w)-w\nabla(\phi_{1,p}^{1-\frac{\eta p}{p-1+\eta}})\big\}dx \\ &=\frac{a_0(\alpha_1+\beta_1)^{(p-1)r_1}}{m_1}M_1\Big(\int_{\Omega}|\nabla \psi_1|^{p}dx\Big)\\ &\quad\times \Big\{\int_{\Omega}\,\big[\lambda_{1,p}\phi_{1,p}^{p-\frac{\eta p}{p-1+\eta}}-|\nabla \phi_{1,p}|^{p-2}\nabla\phi_{1,p}\nabla(\phi_{1,p}^{1-\frac{\eta p }{p-1+\eta}})\big]w dx\Big\} \\ &=\frac{a_0(\alpha_1+\beta_1)^{(p-1)r_1}}{m_1}M_1\Big(\int_{\Omega}|\nabla \psi_1|^{p}dx\Big)\\ &\quad \times \Big\{\int_{\Omega}\,\big[\lambda_{1,p}\phi_{1,p}^{p-\frac{\eta p}{p-1+\eta}}-|\nabla \phi_{1,p}|^{p}({1-\frac{\eta p}{p-1+\eta}})\phi_{1,p}^{-\frac{\eta p}{p-1+\eta}}\big]w dx\Big\} \\ &\leq a_0(\alpha_1+\beta_1)^{(p-1)r_1}\Big\{\int_{\Omega} \big[\lambda_{1,p}\phi_{1,p}^{\frac{p (p-1)}{p-1+\eta}}-\frac{(1-\eta)(p-1)}{p-1+\eta}\frac{|\nabla \phi_{1,p}|^{p}}{\phi_{1,p}^{\frac{\eta p}{p-1+\eta}}}\big]w dx\Big\}. \end{align*} Similarly \begin{align*} &M_2\Big(\int_{\Omega}|\nabla \psi_2|^{q}dx\Big)\int_{\Omega}\,|\nabla\psi_2|^{q-2}\nabla \psi_2\cdot\nabla w\,dx\\ &\leq b_0(\alpha_2+\beta_2)^{(q-1)r_2} \Big\{\int_{\Omega}\,\big[\lambda_{1,q}\phi_{1,q}^{\frac{q (q-1)}{q-1+\theta}}-\frac{(1-\theta)(q-1)}{q-1+\theta}\frac{|\nabla \phi_{1,q}|^{q}}{\phi_{1,q}^{\frac{\theta q}{q-1+\theta}}}\big]w dx\Big\}. \end{align*} Thus $(\psi_1,\psi_2)$ is a sub-solution if \begin{align*} &a_0(\alpha_1 +\beta_1)^{r_1}\big\{\lambda_{1,p}\phi_{1,p}^{\frac{p(p-1)}{p-1+\eta}} -\frac{(1-\eta)(p-1)}{p-1+\eta}\frac{|\nabla \phi_{1,p}|^{p}}{\phi_{1,p}^{\frac{\eta p}{p-1+\theta}}}\big\}\\ & \leq a(x)\Big[\alpha_1 \Big( f(\psi_2)-\frac{1}{\psi_1^{\eta}}\Big)+\beta_1 \Big( h(\psi_1)-\frac{1}{\psi_1^{\eta}}\Big)\Big]. \end{align*} and \begin{align*} &b_0(\alpha_2 +\beta_2)^{r_2}\big\{\lambda_{1,q}\phi_{1,q}^{\frac{q(q-1)}{q-1+\theta}} -\frac{(1-\theta)(q-1)}{q-1+\theta}\frac{|\nabla \phi_{1,q}|^{q}}{\phi_{1,q}^{\frac{\theta q}{q-1+\theta}}}\big\}\\ &\leq b(x)\Big[\alpha_2 \Big( g(\psi_1)-\frac{1}{\psi_2^{\theta}}\Big)+\beta_2 \Big( k(\psi_2)-\frac{1}{\psi_2^{\theta}}\Big)\Big]. \end{align*} First we consider the case when $ x\in\overline{\Omega_{\delta}}$. Since $1-(p-1)r_1-r_1\eta<0$, for $\alpha_1+\beta_1\gg 1$, we have \begin{align*} &-(\alpha_1 +\beta_1)^{(p-1)r_1}\frac{(1-\eta)(p-1)}{p-1+\eta} \frac{|\nabla\phi_{1,p}|^{p}}{\phi_{1,p}^{\frac{p\eta}{p-1+\eta}}}\\ &\leq (\alpha_1 +\beta_1)\Big[-\frac{1}{\Big((\alpha_1 +\beta_1)^{r_1}(\frac{a_0}{m_1})^{\frac{1}{p-1}} (\frac{p-1+\eta}{p})\phi_{1,p}^{\frac{p}{p-1+\eta}}\Big)^{\eta}}\Big]. \end{align*} Also in $\bar{\Omega}_{\delta}$ (in fact in $\Omega$), since $(p-1)r_1<1$, if $\alpha_1+\beta_1\gg 1$ and $\alpha_2+\beta_2\gg 1$, \begin{align*} (\alpha_1 +\beta_1)^{(p-1)r_1}\,\lambda_{1,p}\phi_{1,p}^{\frac{p(p-1)}{p-1+\eta}} &\leq (\alpha_1 +\beta_1)\gamma_0\\ &=\alpha_1 \gamma_0+\beta_1)\gamma_0\\ &\leq \alpha_1 f\Big((\alpha_2 +\beta_2)^{r_2}(\frac{b_0}{m_2})^{\frac{1}{q-1}}(\frac{q-1+\theta}{q}) \phi_{1,q}^{\frac{q}{q-1+\theta}}\Big)\\ &\quad +\beta_1h\Big((\alpha_1 +\beta_1)^{r_1}(\frac{a_0}{m_1})^{\frac{1}{p-1}} (\frac{p-1+\eta}{p})\phi_{1,p}^{\frac{p}{p-1+\eta}}\Big). \end{align*} It follows that in $\overline{\Omega_{\delta}}$ for $\alpha_1+\beta_1\gg 1$ and $\alpha_2+\beta_2\gg 1$, we have \begin{align*} &a_0(\alpha_1 +\beta_1)^{r_1}\Big[\lambda_{1,p}\phi_{1,p} ^{\frac{p(p-1)}{p-1+\eta}}-\frac{(1-\eta)(p-1)}{p-1+\eta}\frac{|\nabla \phi_{1,p}|^{p}}{\phi_{1,p}^{\frac{\eta p}{p-1+\eta}}}\Big]\\ &= a_0\Big[(\alpha_1 +\beta_1)^{r_1}\lambda_{1,p}\phi_{1,p}^{\frac{p(p-1)}{p-1+\eta}}-(\alpha_1 +\beta_1)^{r_1}\frac{(1-\eta)(p-1)}{p-1+\eta}\frac{|\nabla \phi_{1,p}|^{p}}{\phi_{1,p}^{\frac{\eta p}{p-1+\eta}}}\Big]\\ &\leq a(x)\Big[\alpha_1 f\Big((\alpha_2 +\beta_2)^{r_2}(\frac{b_0}{m_2})^{\frac{1}{q-1}}(\frac{q-1+\theta}{q})\phi_{1,q}^{\frac{q}{q-1+\theta}}\Big) +\beta_1h\Big((\alpha_1 \\ &\quad +\beta_1)^{r_1}(\frac{a_0}{m_1})^{\frac{1}{p-1}} (\frac{p-1+\eta}{p})\phi_{1,p}^{\frac{p}{p-1+\eta}}\Big)\\ &\quad -\frac{(\alpha_1 +\beta_1)} {\big((\alpha_1 +\beta_1)^{r_1}(\frac{a_0}{m_1})^{\frac{1}{p-1}}(\frac{p-1+\eta}{p}) \phi_{1,p}^{\frac{p}{p-1+\eta}}\big)^{\eta}}\Big]\\ &= a(x)\Big[\alpha_1\Big( f(\psi_2)-\frac{1}{\psi_1^{\eta}}\Big)+\beta_1 \Big( h(\psi_1)-\frac{1}{\psi_1^{\eta}}\Big)\Big]. \end{align*} On the other hand, on $\Omega-\overline{\Omega_{\delta}}$, we have $\sigma\leq\phi_{1,r}^{\frac{r}{r-1+s}}\leq 1$, for $r= p,q$ and $s=\eta,\theta$. Also, since $(p-1)r_1<1$, for $\alpha_1+\beta_1\gg 1$ and $\alpha_2+\beta_2\gg 1$, \begin{align*} &a_0(\alpha_1 +\beta_1)^{r_1}\Big[\lambda_{1,p}\phi_{1,p}^{\frac{p(p-1)}{p-1+\eta}} -\frac{(1-\eta)(p-1)}{p-1+\eta}\frac{|\nabla \phi_{1,p}|^{p}}{\phi_{1,p}^{\frac{\eta p}{p-1+\eta}}}\Big]\\ &\leq a(x)(\alpha_1 +\beta_1)^{r_1}\lambda_{1,p}\phi_{1,p}^{\frac{p(p-1)}{p-1+\eta}}\\ &\leq a(x)(\alpha_1 +\beta_1)\Big(\gamma_1- \frac{1}{\Big((\alpha_1 +\beta_1)^{r_1}(\frac{a_0}{m_1})^{\frac{1}{p-1}}(\frac{p-1+\eta}{p}) \sigma \Big)^{\eta}} \Big)\\ &= a(x)\Big[\alpha_1\Big(\gamma_1- \frac{1}{\Big((\alpha_1 +\beta_1)^{r_1}(\frac{a_0}{m_1})^{\frac{1}{p-1}}(\frac{p-1+\eta}{p})\sigma \Big)^{\eta}} \Big)\\ &\quad +\beta_1\Big(\gamma_1-\frac{1}{\Big((\alpha_1 +\beta_1)^{r_1}(\frac{a_0}{m_1})^{\frac{1}{p-1}}(\frac{p-1+\eta}{p})\sigma \Big)^{\eta}} \Big)\Big]\\ &\leq a(x)\Big\{\alpha_1\Big[f\Big((\alpha_2 +\beta_2)^{r_2}(\frac{b_0}{m_2})^{\frac{1}{q-1}}(\frac{q-1+\theta}{q})\sigma \Big)\\ &\quad - \frac{1}{\Big((\alpha_1 +\beta_1)^{r_1}(\frac{a_0}{m_1})^{\frac{1}{p-1}} (\frac{p-1+\eta}{p})\sigma \Big)^{\eta}} \Big]\\ &\quad +\beta_1\Big[h\Big((\alpha_1 +\beta_1)^{r_1}(\frac{a_0}{m_1})^{\frac{1}{p-1}}(\frac{p-1+\eta}{p}) \sigma\Big)\\ &\quad - \frac{1}{\Big((\alpha_1 +\beta_1)^{r_1}(\frac{a_0}{m_1})^{\frac{1}{p-1}} (\frac{p-1+\eta}{p})\sigma\Big)^{\eta}} \Big]\Big\}\\ &\leq a(x)\Big\{\alpha_1\Big[f\Big((\alpha_2 +\beta_2)^{r_2}(\frac{b_0}{m_2})^{\frac{1}{q-1}}(\frac{q-1+\theta}{q}) \phi_{1,q}^{\frac{q}{q-1+\theta}}\Big) \\ &\quad - \frac{1}{\big((\alpha_1 +\beta_1)^{r_1}(\frac{a_0}{m_1})^{\frac{1}{p-1}}(\frac{p-1+\eta}{p}) \phi_{1,p}^{\frac{p}{p-1+\eta}} \big)^{\eta}}\Big]\\ &\quad +\beta_1\Big[h\Big((\alpha_1 +\beta_1)^{r_1}(\frac{a_0}{m_1})^{\frac{1}{p-1}} (\frac{p-1+\eta}{p})\phi_{1,p}^{\frac{p}{p-1+\eta}}\Big) \\ &\quad - \frac{1}{\Big((\alpha_1 +\beta_1)^{r_1}(\frac{a_0}{m_1})^{\frac{1}{p-1}} (\frac{p-1+\eta}{p})\phi_{1,p}^{\frac{p}{p-1+\eta}}\Big)^{\eta}} \Big]\Big\}\\ &= a(x)\Big[\alpha_1\Big( f(\psi_2)-\frac{1}{\psi_1^{\eta}}\Big)+\beta_1 \Big( h(\psi_1)-\frac{1}{\psi_1^{\eta}}\Big)\Big]. \end{align*} Hence, if $\alpha_1+\beta_1\gg 1$ and $\alpha_2+\beta_2\gg 1$, we see that \begin{align*} & M_1\Big(\int_{\Omega}|\nabla \psi_1|^{p}dx\Big) \int_{\Omega}\,|\nabla\psi_1|^{p-2}\nabla \psi_1\cdot\nabla w\,dx\\ &\leq \int_{\Omega}\,a(x)\Big[\alpha_1\Big( f(\psi_2)-\frac{1}{\psi_1^{\eta}}\Big)+\beta_1 \Big( h(\psi_1)-\frac{1}{\psi_1^{\eta}}\Big)\Big]w\,dx. \end{align*} Similarly, for $\alpha_1+\beta_1\gg 1$ and $\alpha_2+\beta_2\gg 1$, we get \begin{align*} & M_2\Big(\int_{\Omega}|\nabla \psi_2|^{q}dx\Big) \int_{\Omega}\,|\nabla\psi_2|^{q-2}\nabla \psi_2\cdot\nabla w\,dx\\ &\leq \int_{\Omega}\,b(x)\Big[\alpha_2 \Big( g(\psi_1)-\frac{1}{\psi_2^{\theta}}\Big)+\beta_2 \Big( k(\psi_2)-\frac{1}{\psi_2^{\theta}}\Big)\Big]w\,dx. \end{align*} This means that, $(\psi_1,\psi_2)$ is a positive subsolution of \eqref{e1}. Now, we construct a supersolution $(z_1,z_2)\geq(\psi_1,\psi_2)$. By (H1) and (H2) we can choose $C\gg 1$ so that \begin{align*} \frac{m_1}{\|a\|_{\infty}} \geq \frac{\alpha_1f\Big([\frac{\|b\|_{\infty}(\alpha_2+\beta_2)}{m_2}]^{\frac{1}{q-1}} [g(C \|\zeta_{p}\|_{\infty})]^{\frac{1}{q-1}}\|\zeta_q\|_{\infty}\Big) +\beta_1h\big(C\|\zeta_{p}\|_{\infty} \big)}{C^{p-1}}. \end{align*} Let $$ (z_1,z_2)=\Big(C \zeta_{p}\,,\,\Big[\frac{\|b\|_{\infty}(\alpha_2+\beta_2)} {m_2}\Big]^{\frac{1}{q-1}}[g(C\|\zeta_{p}\|_{\infty})]^{\frac{1}{q-1}} \zeta_q\Big). $$ We shall show that $(z_1,z_2)$ is a supersolution of \eqref{e1}. Then \begin{align*} &M_1\Big(\int_{\Omega}|\nabla z_1|^{p}dx\Big)\int_{\Omega}\,|\nabla z_1|^{p-2}\nabla z_1\cdot\nabla w\,dx\\ &= C^{p-1}M_1\Big(\int_{\Omega}|\nabla z_1|^{p}dx\Big)\int_{\Omega}\,|\nabla \zeta_{p}|^{p-2}\nabla \zeta_{p}\cdot\nabla w\,dx\\ &= C^{p-1}M_1\Big(\int_{\Omega}|\nabla z_1|^{p}dx\Big)\int_{\Omega}w\,dx\\ &\geq m_1C^{p-1}\int_{\Omega}w\,dx\\ &\geq \|a\|_{\infty}\int_{\Omega}\,\Big[\alpha_1f \Big(\Big[\frac{\|b\|_{\infty}(\alpha_2+\beta_2)}{m_2}\Big]^{\frac{1}{q-1}}[g(C \|\zeta_{p}\|_{\infty})]^{\frac{1}{q-1}}\|\zeta_q\|_{\infty}\Big)\\ &\quad +\beta_1h\big(C\|\zeta_{p}\|_{\infty} \big)\Big]w dx\\ &\geq \int_{\Omega}\,a(x)\Big[\alpha_1\Big(f(z_2)-\frac{1}{z_1^{\eta}}\Big) +\beta_1 \Big( h(z_1)-\frac{1}{z_1^{\eta}}\Big)\Big]w\,dx. \end{align*} Again by (H2) for $C$ large enough we have $$ g(C\|\zeta(x)\|_{\infty})\geq k\Big(\Big[\frac{\|b\|_{\infty}(\alpha_2+\beta_2)}{m_2}\Big]^{\frac{1}{q-1}}[g(C \|\zeta_{p}\|_{\infty})]^{\frac{1}{q-1}}\|\zeta_q\|_{\infty}\Big). $$ Hence \begin{align*} &M_2\Big(\int_{\Omega}|\nabla z_2|^{q}dx\Big)\int_{\Omega}\,|\nabla z_2|^{q-2}\nabla z_2\cdot\nabla w\,dx\\ &= \frac{\|b\|_{\infty}(\alpha_2+\beta_2)}{m_2}M_2\Big(\int_{\Omega}|\nabla z_2|^{q}dx\Big)\int_{\Omega}\,g( C\|\zeta_{p}\|_{\infty})w\, dx \\ &\geq \int_{\Omega}\,b(x)\Big\{\alpha_2g( C\|\zeta_{p}\|_{\infty}) +\beta_2g( C\|\zeta_{p}\|_{\infty})\Big\}w\, dx\\ &\geq \int_{\Omega}\,b(x)\Big\{\alpha_2g( C\|\zeta_{p}\|_{\infty}) \\ &\quad +\beta_2k \Big(\Big[\frac{\|b\|_{\infty}(\alpha_2+\beta_2)}{m_2} \Big]^{\frac{1}{q-1}}[g(C \|\zeta_{p}\|_{\infty})]^{\frac{1}{q-1}} \|\zeta_q\|_{\infty}\Big\}w\, dx\\ &\geq \int_{\Omega}\,b(x)\Big(\alpha_2 g(z_1)+\beta_2 k(z_2)\Big)w\,dx\\ &\geq \int_{\Omega}\,b(x)\Big[\alpha_2\Big( g(z_1)-\frac{1}{z_2^{\theta}}\Big)+\beta_2 \Big( k(z_2)-\frac{1}{z_2^{\theta}}\Big)\Big]w\,dx. \end{align*} i.e., $(z_1,z_2)$ is a supersolution of \eqref{e1}. 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