Nontrivial weak solutions for nonlocal nonhomogeneous elliptic problems

Abstract

This article concerns the following class of nonlocal nonhomogeneous elliptic problems: $-{\left({\int }_{\mathrm{\Omega }}g(x,u)\mathrm{d}x\right)}^{\beta }\mathrm{div}\left(a\right(x\left)\mathrm{\nabla }u\right)=\left(g\right(x,u){)}^{\alpha }\mathrm{i}\mathrm{n}\mathrm{\Omega },u=0\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },$ where Ω is a bounded domain of ${\mathbb{R}}^n(n\ge 1)$, α, $\text{β}\in \mathbb{R}$, $a\left(x\right)$ is a matrix with variable coefficients and $g:\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$ is a positive function which are defined almost everywhere with respect to the variable x. By using the Schauder and Tychonoff fixed-point theorems, we establish two existence theorems of weak solutions for this problem according to two bundles of hypotheses on α, β and g.

COMMUNICATED BY:

• M. Mei

Keywords:

• Schauder and Tychonoff fixed-point theorems
• nonlocal nonhomogeneous elliptic problem
• homogeneous Dirichlet boundary condition
• weak solution
• existence

2010 Subject Classifications:

• 35A05
• 35J60
• 35J25

Disclosure statement

No potential conflict of interest was reported by the author(s).