This paper deals with the existence of positive solution for the singular quasilinear Schr\"odinger equation $-\Delta u -\Delta (u^{ 2})u=h(x) u^{ -\gamma} + f(x,u)~\mbox{ in} ~ \Omega,$ where $\gamma > 1$, $\Omega \subset \mathbb{ R}^{ N}, (N\geq 3)$ 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.
Categories: math.AP