About existence and regularity of positive solutions for a Quasilinear Schrödinger equation with singular nonlinearity
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 $γ> 1$, $Ω\subset \mathbb{R}^{N}, (N\geq 3)$ is a bounded smooth domain, $0<h\in L^{1}(Ω)$, $f$ 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 $(h(x),γ)$, which is optimal for the existence of $H_{0}^{1}$-solution for this problem.
math.AP↗