Existence and Concentration Results for the General Kirchhoff-Type Equations

Abstract

We consider the following singularly perturbed Kirchhoff-type equations $$\begin{aligned} -\varepsilon ^2 M\left( \varepsilon ^{2-N}\int _{{\mathbb {R}}^N}|\nabla u|^2 \textrm{d}x\right) \Delta u +V(x)u=|u|^{p-2}u~\hbox {in}~{\mathbb {R}}^N, u\in H^1({\mathbb {R}}^N),N\ge 1, \end{aligned}$$ 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 $$u_\varepsilon $$ 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. (Arch Ration Mech Anal 213(3):931–979, 2014)