Multiplicity results for nonhomogeneous elliptic equations with singular nonlinearities

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 $\lambda$.