Multiplicity results for nonhomogeneous elliptic equations with singular nonlinearities

TL;DR

This paper proves the existence of multiple positive solutions for a class of nonhomogeneous elliptic equations with singular nonlinearities and nonstandard growth, using fixed point theorems and sub-super solution methods.

Contribution

It introduces a novel approach to establish multiple solutions for elliptic problems with singular nonlinearities and nonhomogeneous operators of p-q type.

Findings

Established existence of three positive solutions within a certain parameter range.

Constructed two pairs of strict sub and super solutions for the problem.

Applied fixed point theorems to prove solution multiplicity.

Abstract

This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of $p$-$q$ type and singular nonlinearities \begin{equation*} \left\{ \begin{alignedat}{2} {} - \mathcal{L}_{p,q} u & {}= \lambda \frac{f(u)}{u^\gamma}, \ u>0 && \quad\mbox{ in } \, \Omega, u & {}= 0 && \quad\mbox{ on } \partial\Omega, \end{alignedat} \right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary, $N \geq 1$, $\lambda >0$ is a real parameter, $$\mathcal{L}_{p,q} u := div(|\nabla u|^{p-2} \nabla u + |\nabla u|^{q-2} \nabla u),$$ $1<p<q< \infty$, $\gamma \in (0,1)$, and $f$ is a continuous nondecreasing map satisfying suitable conditions. By constructing two distinctive pairs of strict sub and super solution, and using fixed point theorems by Amann , we prove existence of three positive solutions in the positive cone of $C_\delta(\overline{\Omega})$ and in a certain range of $\lambda$.