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 \begin{document}$ p $\end{document}-\begin{document}$ q $\end{document} type and singular nonlinearities

\begin{document}$ \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{document}

where \begin{document}$ \Omega $\end{document} is a bounded domain in \begin{document}$ \mathbb{R}^N $\end{document} with \begin{document}$ C^2 $\end{document} boundary, \begin{document}$ N \geq 1 $\end{document}, \begin{document}$ \lambda >0 $\end{document} is a real parameter,

\begin{document}$ \mathcal{L}_{p,q} u : = {\rm{div}}(|\nabla u|^{p-2} \nabla u + |\nabla u|^{q-2} \nabla u), $\end{document}

\begin{document}$ 1, \begin{document}$ \gamma \in (0,1) $\end{document}, and \begin{document}$ f $\end{document} is a continuous nondecreasing map satisfying suitable conditions. By constructing two distinctive pairs of strict sub and super solution, and using fixed point theorems by Amann [1], we prove existence of three positive solutions in the positive cone of \begin{document}$ C_\delta(\overline{\Omega}) $\end{document} and in a certain range of \begin{document}$ \lambda $\end{document}.