Electron. J. Differential Equations, Vol. 2016 (2016), No. 317, pp. 1-9.

Sign-changing solutions for asymptotically linear Schrodinger equation in bounded domains

Sitong Chen, Yinbin Li, Xianhua Tang

Abstract:
In this article we study the Schrodinger equation
$$ -\Delta u=f(x,u),\quad x\in\Omega, \quad u\in H_0^1(\Omega), $$
where $\Omega$ is a bounded domain in $\mathbb{R}^N$ and f(x,u) is asymptotically linear at infinity with respect to u. Inspired by the works of Salvatore [14] on sign-changing solutions, in which $f(x,u)$ is asymptotically linear at zero with respect to u, we prove, via the constraint variational method and the quantitative deformation lemma, that the equation possesses one sign-changing solution with exactly two nodal domains.

Submitted June 22, 2016. Published December 14, 2016.
Math Subject Classifications: 35J10, 35J20.
Key Words: Schrodinger equation; sign-changing solutions; asymptotically linear.

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Sitong Chen
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: mathsitongchen@163.com
Yinbin Li
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: liyinbin1991@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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