Existence of multiple solutions to an elliptic problem with measure data

TL;DR

This paper establishes the existence of multiple solutions for a nonlinear elliptic PDE involving measure data, expanding understanding of solution multiplicity under complex conditions.

Contribution

It proves the existence of multiple solutions for a class of elliptic equations with measure data, a novel result in the context of nonlinear PDEs with singular measures.

Findings

Multiple nontrivial solutions are proven to exist.

The results apply to equations with measure data and nonlinear terms.

The work extends previous results to more general measure and nonlinear conditions.

Abstract

In this paper we prove the existence of multiple nontrivial solutions of the following equation. \begin{align*} \begin{split} -\Delta_{p}u & = \lambda |u|^{q-2}u+f(x,u)+\mu\,\,\mbox{in}\,\,\Omega, u & = 0\,\, \mbox{on}\,\, \partial\Omega; \end{split} \end{align*} where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain with $N \geq 3$, $1 < q^{\prime} < q < p-1; \; \lambda,\;$ and $f$ satisfies certain conditions, $\mu>0$ is a Radon measure, $q^{\prime}=\frac{q}{q-1}$ is the conjugate of $q$.