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

### David Ruiz

Universidad de Granada, Spain### Giusi Vaira

SISSA, Trieste, Italy

## Abstract

In this paper we consider the system in $\mathbb{R}^3$

$\left\{ \begin{array}{l} -\varepsilon^2 \Delta u + V(x)u + \phi(x)u = u^p, \\ -\Delta \phi = u^2, \end{array} \right.$

for $p\in (1,5)$. We prove the existence of multi-bump solutions whose bumps concentrate around a local minimum of the potential $V(x)$. We point out that such solutions do not exist in the framework of the usual Nonlinear Schrödinger Equation.

## Cite this article

David Ruiz, Giusi Vaira, Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential. Rev. Mat. Iberoam. 27 (2011), no. 1, pp. 253–271

DOI 10.4171/RMI/635