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

Fan, Haining

Electronic journal of differential equations, Tome 2016 (2016)

Résumé

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.

Détail

Classification : 35A15, 35B33, 35J62
Keywords: Nehari-pohozaev manifold, nonlocal problem, positive solution, concentration property