Positive solutions for a class of nonlocal problems with possibly singular nonlinearity

TL;DR

This paper investigates elliptic boundary value problems with nonlocal nonlinearities that may be singular, establishing conditions for the existence of multiple positive solutions using fixed point methods.

Contribution

It introduces a novel approach to handle nonlocal problems with possibly singular nonlinearities, demonstrating the existence of multiple positive solutions.

Findings

Existence of multiple positive solutions under certain conditions

Handling of singularities in the nonlocal nonlinear term

Application of fixed point methods to nonlocal elliptic problems

Abstract

We study a class of elliptic problems with homogeneous Dirichlet boundary condition and a nonlinear reaction term $f$ which is nonlocal depending on the $L^{p}$-norm of the unknown function. The nonlinearity $f$ can make the problem degenerate since it may even have multiple singularities in the nonlocal variable. We use fixed point arguments for an appropriately defined solution map to produce multiplicity of classical positive solutions with ordered norms.