Existence of positive solutions for a class of nonlocal elliptic systems with multiple parameters

Abstract

In this paper, we study the existence of positive solutions to the following nonlocal elliptic systems$$\cases- M_1\left(ınt_\Omega |\nabla u|^p\,dx\right)\Delta_p u = \alpha_1 a(x)f_1(v) + \beta_1b(x)g_1(u), \quad x ın \Omega, - M_2\left(ınt_\Omega |\nabla v|^q\,dx\right)\Delta_q v = \alpha_2 c(x)f_2(u) + \beta_2d(x)g_2(v), \quad x ın \Omega, u = v = 0, \quad x ın \partial\Omega,\endcases$$where $\Omega$ is a bounded domain in $\Bbb{R}^N$ with smooth boundary $\partial\Omega$, $1<p,q<N$, $M_i : \Bbb{R}^+_0 \to \Bbb{R}$, $i=1,2$,are continuous and nondecreasing functions, $a,b,c,d ın C(\overline\Omega)$, and $\alpha_i$, $\beta_i$, $i=1,2$, are positive parameters.