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

Abstract

This paper deals with the existence of positive solution for the singular quasilinear Schrödinger equation $-Δu -Δ(u^{2})u=h(x) u^{-γ} + f(x,u)~\mbox{in} ~ Ω,$ where $\gamma >1$, $\Omega \subset {\mathbb{R}}^N,(N\ge 3)$ is a bounded smooth domain, $0<h\in L^1(\Omega )$, $f$ is a measurable function that can change signal and can be sublinear or has critical growth. Inspired by Sun \cite{Y} we derive a compatible condition on the couple $(h(x),\gamma )$, which is optimal for the existence of $H_0^1$-solution for this problem.