About existence and regularity of positive solutions for a Quasilinear Schr\"odinger equation with singular nonlinearity

TL;DR

This paper investigates the existence and regularity of positive solutions for a singular quasilinear Schrödinger equation with nonlinearities that may be sublinear or critical, establishing optimal conditions on data for solutions to exist.

Contribution

It introduces an optimal compatibility condition on the data and parameters ensuring the existence of solutions to a complex singular quasilinear Schrödinger problem.

Findings

Established an optimal condition on (h(x), γ) for solution existence.

Proved existence of H_0^1 solutions under the derived condition.

Extended understanding of singular nonlinear Schrödinger equations with variable nonlinearities.

Abstract

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.