Multiplicity results for nonhomogeneous elliptic equations with singular nonlinearities

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

$ \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. $

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 : = {\rm{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 [1], we prove existence of three positive solutions in the positive cone of $ C_\delta(\overline{\Omega}) $ and in a certain range of $ \lambda $.