Multi-peak positive solutions to a class of Kirchhoff equations

TL;DR

This paper establishes the existence of multi-peak positive solutions for a class of nonlocal Kirchhoff equations in three dimensions, using Lyapunov-Schmidt reduction, addressing a gap in the literature on multi-peak solutions.

Contribution

It introduces the first results on multi-peak solutions for Kirchhoff problems, overcoming challenges posed by the nonlocal term and the system nature of the unperturbed problem.

Findings

Existence of multi-peak solutions for small epsilon.

Application of Lyapunov-Schmidt reduction method.

Addresses nonlocal term difficulties in Kirchhoff equations.

Abstract

In the present paper, we consider the nonlocal Kirchhoff problem \begin{eqnarray*} -\left(\epsilon^2a+\epsilon b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u+V(x)u=u^{p},\,\,\,u>0 & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b>0$, $1<p<5$ are constants, $\epsilon>0$ is a parameter. Under some mild assumptions on the function $V$, we obtain multi-peak solutions for $\epsilon$ sufficiently small by Lyapunov-Schmidt reduction method. Even though many results on single peak solutions to singularly perturbed Kirchhoff problems have been derived in the literature by various methods, there exist no results on multi-peak solutions before this paper, due to some difficulties caused by the nonlocal term $\left(\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u$. A remarkable new feature of this problem is that the corresponding unperturbed problem turns out to be a system of partial differential equations, but not a single Kirchhoff equation, which is quite different from most of elliptic singular perturbation problems.