Multiplicity of solutions to an elliptic problem with singularity and measure data

TL;DR

This paper establishes the existence of multiple positive solutions for a nonlinear elliptic PDE involving singularities and measure data, extending understanding of solution multiplicity in complex boundary value problems.

Contribution

It proves the existence of multiple solutions to a singular elliptic problem with measure data, a novel extension in the study of nonlinear PDEs with singularities.

Findings

Multiple nontrivial solutions exist for the problem.

Solutions are positive and satisfy boundary conditions.

The results apply to measure data and singular nonlinearities.

Abstract

In this paper, we prove the existence of multiple nontrivial solutions of the following equation. \begin{align*} \begin{split} -\Delta_{p}u & = \frac{\lambda}{u^{\gamma}}+g(u)+\mu~\mbox{in}\,\,\Omega, u & = 0\,\, \mbox{on}\,\, \partial\Omega, u&>0 \,\,\mbox{in}\,\,\Omega, \end{split} \end{align*} where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain with $N \geq 3$, $1 < p-1 < q$ , $ \lambda>0$, $\gamma>0$, $g$ satisfies certain conditions, $\mu\geq 0$ is a bounded Radon measure.