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Multi-bump solutions of Schrödinger–Poisson equations with steep potential well

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Abstract

In this paper, we study a system of Schrödinger–Poisson equation

$$\left\{\begin{array}{ll}-\Delta u+(\lambda a(x)+a_0(x))u+K(x)\phi u=|u|^{p-2}u, & \quad x \in\mathbb{R}^3, \\- \Delta \phi=K(x)u^2,& \quad x \in \mathbb{R}^3 \end{array} \right.$$

where \({p \in (4,6)}\) and \({\lambda}\) is a parameter. We require that \({a(x) \geq 0}\) and has a bounded potential well \({\Omega = a^{-1}(0)}\). Combining this with other suitable assumptions on Ω, a 0 and K, we obtain the existence of multi-bump-type solution \({u_\lambda}\) when \({\lambda}\) is large via variational methods.

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Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China

    Xu Zhang & Shiwang Ma

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  1. Xu Zhang
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  2. Shiwang Ma
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Correspondence to Shiwang Ma.

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Research supported by the Specialized Fund for the Doctoral Program of Higher Education and the National Natural Science Foundation of China.

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Zhang, X., Ma, S. Multi-bump solutions of Schrödinger–Poisson equations with steep potential well. Z. Angew. Math. Phys. 66, 1615–1631 (2015). https://doi.org/10.1007/s00033-014-0490-x

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  • DOI: https://doi.org/10.1007/s00033-014-0490-x

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