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

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, 20$ 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.

Keywords

• Nehari-Pohozaev manifold
• nonlocal problem
• positive solution
• concentration property