Electron. J. Diff. Equ., Vol. 2016 (2016), No. 178, pp. 1-13.

Kirchhoff type problems with potential well and indefinite potential

Yuanze Wu, Yisheng Huang, Zeng Liu

Abstract:
In this article, we study the Kirchhoff type problem
$$\displaylines{ -\Big(\alpha\int_{\mathbb{R}^3}|\nabla u|^2dx+1\Big)\Delta u +(\lambda a(x)+a_0)u=|u|^{p-2}u \quad\text{in }\mathbb{R}^3,\cr u\in H^1(\mathbb{R}^3), }$$
where $4<p<6$, $\alpha$ and $\lambda$ are two positive parameters, $a_0\in\mathbb{R}$ is a (possibly negative) constant and $a(x)\geq0$ is the potential well. Using the variational method, we show the existence of nontrivial solutions. We also obtain the concentration behavior of the solutions as $\lambda\to+\infty$.

Submitted February 19, 2016. Published July 6, 2016.
Math Subject Classifications: 35B38, 35B40, 35J10, 35J20.
Key Words: Kirchhoff type problem; indefinite potential; potential well; variational method.

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Yuanze Wu
College of Sciences
China University of Mining and Technology
Xuzhou 221116, China
email: wuyz850306@cumt.edu.cn
Yisheng Huang
Department of Mathematics
Soochow University, Suzhou 215006, China
email: yishengh@suda.edu.cn
Zeng Liu
Department of Mathematics
Suzhou University of Science and Technology
Suzhou 215009, China
email: luckliuz@163.com

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