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 & {}= λ\frac{f(u)}{u^γ}, \ u>0 && \quad\mbox{ in } \, Ω, u & {}= 0 && \quad\mbox{ on } \partialΩ, \end{alignedat} \right. \end{equation*} where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ with $C^2$ boundary, $N\ge 1$, $\lambda >0$ 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 $\lambda$.

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