Electron. J. Diff. Equ., Vol. 2016 (2016), No. 05, pp. 1-18.

Existence and concentration of ground state solutions for a Kirchhoff type problem

Haining Fan

Abstract:
This article concerns the Kirchhoff type problem
$$\displaylines{ -\Big(\varepsilon^2a+\varepsilon b\int_{\mathbb{R}^3} |\nabla u|^2dx\Big)\Delta u +V(x)u= K(x)|u|^{p-1}u,\quad x\in \mathbb{R}^3,\cr u\in H^1(\mathbb{R}^3), }$$
where a,b are positive constants, 2< p < 5, $\varepsilon>0$ is a small parameter, and $V(x),K(x)\in C^1(\mathbb{R}^3)$. Under certain assumptions on the non-constant potentials V(x) and K(x), we prove the existence and concentration properties of a positive ground state solution as $\varepsilon\to 0$. Our main tool is a Nehari-Pohozaev manifold.

Submitted July 6, 2015. Published January 4, 2016.
Math Subject Classifications: 35A15, 35B33, 35J62.
Key Words: Nehari-Pohozaev manifold; nonlocal problem; positive solution; concentration property.

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Haining Fan
School of Sciences
China University of Mining and Technology
Xuzhou 221116, China
email: fanhaining888@163.com

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