Multiple positive solutions for a Schr\"odinger-Poisson-Slater system

TL;DR

This paper proves the existence of multiple positive solutions for a Schrödinger-Poisson-Slater system in bounded domains, especially when the nonlinearity exponent is close to the critical Sobolev exponent, linking solution count to topological complexity.

Contribution

It establishes the existence of multiple positive solutions for the system near the critical Sobolev exponent, relating the number of solutions to the domain's topological category.

Findings

Number of solutions exceeds the domain's Ljusternik-Schnirelmann category.

Solutions exist when the nonlinearity exponent is near the critical value.

The results connect topological properties of the domain with solution multiplicity.

Abstract

In this paper we investigate the existence of positive solutions to the following Schr\"odinger-Poisson-Slater system [c]{ll} - \Delta u+ u + \lambda\phi u=|u|^{p-2}u & \text{in} \Omega -\Delta\phi= u^{2} & \text{in} \Omega u=\phi=0 & \text{on} \partial\Omega. where $\Omega$ is a bounded domain in $\mathbf{R}^{3},\lambda$ is a fixed positive parameter and $p<2^{*}=\frac{2N}{N-2}$. We prove that if $p$ is "near" the critical Sobolev exponent $2^*$, then the number of positive solutions is greater then the Ljusternik-Schnirelmann category of $\Omega$.