JournalsrmiVol. 27, No. 1pp. 253–271

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

In this paper we consider the system in R3\mathbb{R}^3

{ε2Δu+V(x)u+ϕ(x)u=up,Δϕ=u2,\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(1,5)p\in (1,5). We prove the existence of multi-bump solutions whose bumps concentrate around a local minimum of the potential V(x)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