Multiplicity of solutions for a Dirichlet problem with a singular and a supercritical nonlinearities

Abstract

For a bounded, open set $\Omega\subset\mathbb{R}^N$ and depending on $\lambda>0$, we study the multiplicity of solutions of \begin{equation*} \begin{cases} u>0 \text{ in }\;\Omega\;, \\ -\div (M(x)\nabla u)=\frac{\lambda}{\;u^\gamma\;}+ u^{p} \text{ in }\;\Omega, \\ u=0 \text{ on }\;\partial\Omega, \end{cases} \end{equation*} where $M(x)$ is a symmetric, bounded, and elliptic matrix and $0 <\gamma <1 <p <\frac{N+2}{N-2}$.