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On the Schrödinger–Poisson–Slater System: Behavior of Minimizers, Radial and Nonradial Cases

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Abstract

This paper is motivated by the study of a version of the so-called Schrödinger–Poisson–Slater problem:

$$- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,$$

where \({u \in H^{1}(\mathbb {R}^3)}\). We are concerned mostly with \({p \in (2,3)}\). The behavior of radial minimizers motivates the study of the static case ω = 0. Among other things, we obtain a general lower bound for the Coulomb energy, which could be useful in other frameworks. The radial and nonradial cases turn out to yield essentially different situations.

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Authors and Affiliations

  1. Dpto. Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Avda. Fuentenueva s/n., 18071, Granada, Spain

    David Ruiz

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  1. David Ruiz
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Correspondence to David Ruiz.

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Communicated by P.-L. Lions

The author has been supported by the Spanish Ministry of Science and Technology under Grant MTM2005-01331 and by J. Andalucía (FQM 116).

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Ruiz, D. On the Schrödinger–Poisson–Slater System: Behavior of Minimizers, Radial and Nonradial Cases. Arch Rational Mech Anal 198, 349–368 (2010). https://doi.org/10.1007/s00205-010-0299-5

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