Electron. J. Diff. Equ., Vol. 2015 (2015), No. 213, pp. 1-15.

Asymptotically linear Schrodinger equation with zero on the boundary of the spectrum

Dongdong Qin, Xianhua Tang

Abstract:
This article concerns the Schrodinger equation
$$\displaylines{ -\Delta u+V(x)u=f(x, u), \quad \text{for } x\in\mathbb{R}^N,\cr u(x)\to 0, \quad \text{as } |x| \to \infty, }$$
where V and f are periodic in x, and 0 is a boundary point of the spectrum $\sigma(-\Delta+V)$. Assuming that f(x,u) is asymptotically linear as $|u|\to\infty$, existence of a ground state solution is established using some new techniques.

Submitted January 18, 2015. Published August 17, 2015.
Math Subject Classifications: 35J20, 35J60, 35Q55.
Key Words: Schrodinger equation; strongly indefinite functional; spectrum point zero; asymptotically linear; ground states solution.

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Dongdong Qin
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: qindd132@163.com
Xianhua Tang
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: tangxh@mail.csu.edu.cn

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