Revista Matemática Iberoamericana


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Volume 27, Issue 1, 2011, pp. 253–271
DOI: 10.4171/RMI/635

Published online: 2011-04-30

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

David Ruiz[1] and Giusi Vaira[2]

(1) Universidad de Granada, Spain
(2) SISSA, Trieste, Italy

In this paper we consider the system in $\mathbb{R}^3$ \begin{equation} \left\{ \begin{array}{l} -\varepsilon^2 \Delta u + V(x)u + \phi(x)u = u^p, \\ -\Delta \phi = u^2, \end{array} \right. \end{equation} 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.

Keywords: Nonlinear analysis, Schrödinger-Poisson-Slater problem, variational methods, singular perturbation method, multi-bump solutions.

Ruiz David, Vaira Giusi: Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential. Rev. Mat. Iberoamericana 27 (2011), 253-271. doi: 10.4171/RMI/635