Existence results for a class of Kirchhoff-type systems with combined nonlinear effects

Abstract

UDC 517.9
We study the existence of positive solutions for a nonlinear system $$M_1 \bigl(\int_{\Omega} |\nabla u|^p dx\bigr)\mathrm{d}\mathrm{i}\mathrm{v} (| x|^{ap}| \nabla u|^{p-2}\nabla u) = \lambda | x| (a+1)p+c_1f(u, \upsilon ),\; x \in \Omega ,$$ $$M2 \bigl( \int_{ \Omega }| \nabla v| q dx \bigr)\mathrm{d}\mathrm{i}\mathrm{v} (| x| bq|\nabla \upsilon | q 2\nabla \upsilon ) = \lambda | x| (b+1)q+c_2g(u, \upsilon ),\; x \in \Omega ,$$ $$u = \upsilon = 0, x \in \partial \Omega ,$$ where $\Omega$ is a bounded smooth domain in $R^N$ with $0 \in \Omega,\; 1 < p, q < N, 0 \leq a < \cfrac{N-p}{p}, 0 \leq b < \cfrac{N-q}{p},$ а $c_1, c_2, \lambda$ are positive parameters. Here, $M_1,M_2, f$, and g satisfy certain conditions. We use the method of sub- and supersolutions to establish our results.