Existence and Concentration Results for the General Kirchhoff Type Equations

Abstract

We consider the following singularly perturbed Kirchhoff type equations $$-{\epsilon }^{2}M\left({\epsilon }^{2-N}{\int }_{{\mathbb{R}}^N}|\nabla u{|}^{2}dx\right)\Delta u+V(x)u=|u{|}^{p-2}uin{\mathbb{R}}^N,u\in H^1({\mathbb{R}}^N),N\ge 1,$$ where $M\in C([0,\infty ))$ and $V\in C({\mathbb{R}}^N)$ are given functions. Under very mild assumptions on $M$, we prove the existence of single-peak or multi-peak solution $u_{\epsilon }$ for above problem, concentrating around topologically stable critical points of $V$, by a direct corresponding argument. This gives an affirmative answer to an open problem raised by Figueiredo et al. in 2014 [ARMA,213].

18 pages