Revista Matemática Iberoamericana

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

David Ruiz and Giusi Vaira

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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.

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Rev. Mat. Iberoamericana, Volume 27, Number 1 (2011), 253-271.

First available in Project Euclid: 4 February 2011

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Primary: 35B40: Asymptotic behavior of solutions 35J20: Variational methods for second-order elliptic equations 35J55

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), no. 1, 253--271.

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