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$$\{\begin{array}{l}-\end{array}{M}_1\left(ın{t}_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^{p}dx\right){\mathrm{\Delta }}_{p}u={\alpha }_{1}a(x)f_1(v)+{\beta }_{1}b(x)g_1(u),xın\mathrm{\Omega },-M_2\left(ın{t}_{\mathrm{\Omega }}|\mathrm{\nabla }v{|}^{q}dx\right){\mathrm{\Delta }}_{q}v={\alpha }_{2}c(x)f_2(u)+{\beta }_{2}d(x)g_2(v),xın\mathrm{\Omega },u=v=0,xın\partial \mathrm{\Omega },\text{endcases}$$where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$ with smooth boundary $\partial \mathrm{\Omega }$, $1<p,q<N$, $M_i:{\mathbb{R}}_0^{+}\to \mathbb{R}$, $i=1,2$,are continuous and nondecreasing functions, $a,b,c,dınC(\bar{\mathrm{\Omega }})$, and ${\alpha }_i$, ${\beta }_i$, $i=1,2$, are positive parameters.