Electron. J. Differential Equations, Vol. 2017 (2017), No. 250, pp. 1-15.

Multiplicity and concentration of solutions for fourth-order elliptic equations with mixed nonlinearity

Wen Zhang, Xianhua Tang, Jian Zhang, Zhiming Luo

Abstract:
This article concerns the fourth-order elliptic equation
$$\displaylines{ \Delta^2u-\Delta u+\lambda V(x)u=f(x, u)+\mu \xi(x)|u|^{p-2}u, \quad x\in \mathbb{R}^{N},\cr u\in H^2(\mathbb{R}^{N}), }$$
where $\lambda >0$ is a parameter, $V\in C(\mathbb{R}^{N},\mathbb{R})$ and $V^{-1}(0)$ has nonempty interior. Under some mild assumptions, we establish the existence of two nontrivial solutions. Moreover, the concentration of these solutions is explored on the set $V^{-1}(0)$ as $\lambda\to\infty$. As an application, we give the similar results and concentration phenomenona for the above problem with concave and convex nonlinearities.

Submitted June 16, 2017. Published October 10, 2017.
Math Subject Classifications: 35J35, 35J60.
Key Words: Fourth-order elliptic equations; concentration; mixed nonlinearity; concave-convex nonlinearity; variational methods.

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Wen Zhang
School of Mathematics and Statistics
Hunan University of Commerce
Changsha, 410205 Hunan, China
email: zwmathcsu@163.com
Xianhua Tang
School of Mathematics and Statistics
Central South University
Changsha, Hunan 410083, China
email: tangxh@mail.csu.edu.cn
Jian Zhang
School of Mathematics and Statistics
Hunan University of Commerce
Changsha, 410205 Hunan, China
email: zhangjian433130@163.com
Zhiming Luo
School of Mathematics and Statistics
Hunan University of Commerce
Changsha, 410205 Hunan, China
email: zhmluo2007@163.com

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