Electron. J. Differential Equations, Vol. 2016 (2016), No. 276, pp. 1-8.

Multiple solutions for Schr\"odinger-Poisson systems with sign-changing potential and critical nonlinearity

Liuyang Shao, Haibo Chen

Abstract:
In this article, we study the Schr\"odinger-Poisson system
$$\displaylines{ -\Delta u+V(x)u+k(x)\phi(x)u=h_1(x)|u|^{4}u+\mu h_{2}(x)u+h_3(x), \quad\text{in } \mathbb{R}^3, \cr -\Delta \phi(x)=k(x)u^2 , \quad\text{in } \mathbb{R}^3, }$$
where $h_1(x),h_{2}(x),h_3(x), V(x)$ are allowed to be sign-changing and $\mu>0$ is a parameter. Under some appropriate assumptions on V(x), we obtain the existence of two different solutions for the above system via variational methods.

Submitted March 12, 2016. Published October 17, 2016.
Math Subject Classifications: 35B38, 35G99.
Key Words: Schrodinger-Poisson system; variational methods; mountain pass theorem; Ekeland's variational principle.

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Liuyang Shao
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: sliuyang316@163.com
Haibo Chen
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: math_chb@163.com

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