Electron. J. Diff. Equ., Vol. 2015 (2015), No. 118, pp. 1-11.

Positive ground state solutions to Schrodinger-Poisson systems with a negative non-local term

Yan-Ping Gao, Sheng-Long Yu, Chun-Lei Tang

Abstract:
In this article, we study the Schrodinger-Poisson system
$$\displaylines{ -\Delta u+u-\lambda K(x)\phi(x)u=a(x)|u|^{p-1}u, \quad x\in\mathbb{R}^3, \cr -\Delta\phi=K(x)u^{2},\quad x\in\mathbb{R}^3, }$$
with $p\in(1,5)$. Assume that $a:\mathbb{R}^3\to \mathbb{R^{+}}$ and $K:\mathbb{R}^3\to \mathbb{R^{+}}$ are nonnegative functions and satisfy suitable assumptions, but not requiring any symmetry property on them, we prove the existence of a positive ground state solution resolved by the variational methods.

Submitted January 20, 2015. Published April 30, 2015.
Math Subject Classifications: 35J47, 35J50, 35J99.
Key Words: Schrodinger-Poisson system; ground state solution; variational methods.

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Yan-Ping Gao
School of Mathematics and Statistics
Southwest University
Chongqing 400715, China
email: gao0807@swu.edu.cn
Sheng-Long Yu
School of Mathematics and Statistics
Southwest University
Chongqing 400715, China
email: ysl345827434@163.com
Chun-Lei Tang
School of Mathematics and Statistics
Southwest University
Chongqing 400715, China
Phone +86 23 68253135, Fax +86 23 68253135
email: tangcl@swu.edu.cn

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