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Existence and concentration of positive solutions for semilinear Schrödinger–Poisson systems in \({\mathbb{R}^{3}}\)

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An Erratum to this article was published on 06 November 2012

Abstract

In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger–Poisson system

$$\left\{\begin{array}{ll}-\varepsilon^{2}\Delta u + a(x)u + \lambda\phi(x)u = b(x)f(u), & x \in \mathbb{R}^{3},\\-\varepsilon^{2}\Delta\phi = u^{2}, \ u \in H^{1}(\mathbb{R}^{3}), &x \in \mathbb{R}^{3},\end{array}\right.$$

where ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that a(x) has at least one minimum and b(x) has at least one maximum. We first prove the existence of least energy solution (u ε , φ ε ) for λ ≠ 0 and ε > 0 sufficiently small. Then we show that u ε converges to the least energy solution of the associated limit problem and concentrates to some set. At the same time, some properties for the least energy solution are also considered. Finally, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.

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Authors and Affiliations

  1. Department of Mathematics, Jiangsu University, Zhenjiang, 212013, Jiangsu, People’s Republic of China

    Jun Wang & Lixin Tian

  2. Department of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Junxiang Xu & Fubao Zhang

Authors
  1. Jun Wang
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  2. Lixin Tian
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  3. Junxiang Xu
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  4. Fubao Zhang
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Correspondence to Jun Wang.

Additional information

Communicated by A. Malchiodi.

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Wang, J., Tian, L., Xu, J. et al. Existence and concentration of positive solutions for semilinear Schrödinger–Poisson systems in \({\mathbb{R}^{3}}\) . Calc. Var. 48, 243–273 (2013). https://doi.org/10.1007/s00526-012-0548-6

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  • DOI: https://doi.org/10.1007/s00526-012-0548-6

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