Schr\"odinger-Poisson equations with singular potentials in $R^3$

TL;DR

This paper investigates the existence and boundedness of positive solutions to a Schrödinger-Poisson system with singular potentials in three-dimensional space, extending understanding of such systems with specific parameter ranges.

Contribution

It establishes existence and $L^{ abla}$ estimates for positive solutions of Schrödinger-Poisson equations with singular potentials in $\,R^3$, considering new parameter conditions.

Findings

Existence of positive solutions under specified conditions.

$L^{ abla}$ estimates for solutions are derived.

Results extend previous work to singular potentials with $\,rac{1}{|y|^ ext{ extalpha}}$ term.

Abstract

The existence and $L^{\infty}$ estimate of positive solutions are discussed for the following Schr\"{o}dinger-Poisson system {ll} -\Delta u +(\lambda+\frac{1}{|y|^\alpha})u+\phi (x) u =|u|^{p-1}u, x=(y,z)\in \mathbb{R}^2\times\mathbb{R}, -\Delta\phi = u^2,\ \lim\limits_{|x|\rightarrow +\infty}\phi(x)=0, \hfill y=(x_1,x_2) \in \mathbb{R}^2 with |y|=\sqrt{x_1^2+x_2^2}, where $\lambda\geqslant0$, $\alpha\in[0,8)$ and $\max\{2,\frac{2+\alpha}{2}\}<p<5$.