Existence and Concentration Results for the General Kirchhoff Type Equations

TL;DR

This paper proves the existence and concentration of solutions for a class of singularly perturbed Kirchhoff equations, addressing an open problem and demonstrating solutions concentrate around stable critical points of the potential.

Contribution

It establishes existence and concentration results for Kirchhoff equations with mild assumptions, solving an open problem from 2014.

Findings

Existence of single-peak and multi-peak solutions.

Solutions concentrate around stable critical points of V.

Addresses an open problem in the field.

Abstract

We consider the following singularly perturbed Kirchhoff type equations $$-\varepsilon^2 M\left(\varepsilon^{2-N}\int_{\R^N}|\nabla u|^2 dx\right)\Delta u +V(x)u=|u|^{p-2}u~\hbox{in}~\R^N, u\in H^1(\R^N),N\geq 1,$$ where $M\in C([0,\infty))$ and $V\in C(\R^N)$ 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. in 2014 [ARMA,213].