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Existence criteria of ground state solutions for Schrödinger-Poisson systems with a vanishing potential

  • * Corresponding author: Xianhua Tang

    * Corresponding author: Xianhua Tang
This work is partially supported by the National Natural Science Foundation of China (No: 11571370)
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  • In this paper, we consider the following Schrödinger-Poisson system

    $ \begin{equation*} \left\{ \begin{array}{ll} -\triangle u+u+K(x)\phi(x)u = a(x)|u|^{p-2}u, \ \ \ \ x\in { \mathbb{R}}^{3},\\ -\triangle \phi = K(x)u^2, \ \ \ \ x\in { \mathbb{R}}^{3}, \end{array} \right. \end{equation*} $

    where $ p\in [4,6) $, $ a(x)\ge \lim_{|x|\to\infty}a(x) = a_{\infty}>0 $ and $ \lim_{|x|\to\infty}K(x) = 0 $. Lack of any symmetry property of $ a $ and $ K $, we present some new sufficient conditions to guarantee the existence of a positive ground state solution of above system. Our results extend and complement the ones of [G. Cerami, G. Vaira, J. Differential Equations 248 (2010)] in which $ p\in (4,6) $, $ a $ and $ K $ need to satisfy additional integrability conditions.

    Mathematics Subject Classification: Primary: 35J20; Secondary: 35Q55.


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