Cluster solutions for the Schrodinger-Poisson-Slater problem around a local minimum of the potential

TL;DR

This paper investigates the existence and properties of cluster solutions to the Schrödinger-Poisson-Slater problem in three-dimensional space, focusing on the behavior around local minima of the potential function.

Contribution

It introduces new methods to construct cluster solutions near local minima of the potential in the Schrödinger-Poisson-Slater system.

Findings

Existence of multi-peak solutions around potential minima

Asymptotic behavior of solutions as epsilon approaches zero

Characterization of solution concentration points

Abstract

In this paper we consider the system in $\R^3$ \label{problemadipartenza0} -\e^2\Delta u+V(x)u+\phi(x)u=u^{p},