About existence and regularity of positive solutions for a quasilinear Schrödinger equation with singular nonlinearity

Abstract

We establish the existence of positive solutions for the singular quasilinear Schrödinger equation \begin{equation*} \begin{cases} -\Delta u -\Delta (u^{2})u=h(x) u^{-\gamma} + f(x,u)& \mbox{in } \Omega,\\ u(x)=0&\mbox{on }\partial \Omega, \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^{N}~ (N\geq 3)$ is a bounded domain with smooth boundary $\partial \Omega$, $10$ almost everywhere in $\Omega$. The function $f$ may change sign on $\Omega$. By using the variational method and some analysis techniques, the necessary and sufficient condition for the existence of a solution is obtained.

Keywords

• strong singularity
• variational methods
• regularity
• fibering methods
• indefinite nonlinearity