Electron. J. Diff. Equ., Vol. 2015 (2015), No. 03, pp. 1-9.

Existence and concentration of solutions for sublinear fourth-order elliptic equations

Wen Zhang, Xianhua Tang, Jian Zhang

Abstract:
This article concerns the fourth-order elliptic equations
$$\displaylines{ \Delta^{2}u-\Delta u+\lambda V(x)u=f(x, u), \quad x\in \mathbb{R}^N,\cr u\in H^{2}(\mathbb{R}^N), }$$
where $\lambda >0$ is a parameter, $V\in C(\mathbb{R}^N)$ and $V^{-1}(0)$ has nonempty interior. Under some mild assumptions, we establish the existence of nontrivial solutions. Moreover, the concentration of solutions is also explored on the set $V^{-1}(0)$ as $\lambda\to\infty$.

Submitted November 23, 2014. Published January 5, 2015.
Math Subject Classifications: 35J35, 35J60.
Key Words: Fourth-order elliptic equations; variational method; concentration.

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Wen Zhang
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: zwmath2011@163.com
Xianhua Tang
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: tangxh@mail.csu.edu.cn
Jian Zhang
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: zhangjian433130@163.com

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