Electron. J. Differential Equations, Vol. 2017 (2017), No. 164, pp. 1-13.

Ground state solutions for Hamiltonian elliptic system with sign-changing potential

Wen Zhang, Xiaoliang Xie, Heilong Mi

Abstract:
This article concerns the Hamiltonian elliptic system
$$\displaylines{ -\Delta u +V(x)u=H_{v}(x, u, v),\quad x\in \mathbb{R}^N, \cr -\Delta v +V(x)v=H_{u}(x, u, v),\quad x\in \mathbb{R}^N, \cr u(x)\to 0,\quad v(x)\to 0, \quad \text{as } |x|\to \infty, }$$
where $z=(u,v): \mathbb{R}^{N}\to\mathbb{R}\times\mathbb{R}$, $N\geq 3$ and the potential V(x) is allowed to be sign-changing. Under weak superquadratic assumptions for the nonlinearities, by applying the variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou, we obtain the existence of nontrivial and ground state solutions.

Submitted March 21, 2017. Published July 4, 2017.
Math Subject Classifications: 35J50, 35J55.
Key Words: Hamiltonian elliptic system; superquadratic; sign-changing potential; generalized weak linking theorem.

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Wen Zhang
School of Mathematics and Statistics
Hunan University of Commerce
Changsha, 410205 Hunan, China
email: zwmath2011@163.com
Xiaoliang Xie
School of Mathematics and Statistics and
Key Laboratory of Hunan Province for Mobile Business Intelligence
Hunan University of Commerce
Changsha, 410205 Hunan, China
email: xiexiaoliangmath@163.com
Heilong Mi
School of Mathematics and Statistics
Hunan University of Commerce
Changsha, 410205 Hunan, China
email: 691473547@qq.com

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