The existence of multi-peak positive solutions for nonlinear Kirchhoff equations

TL;DR

This paper proves the existence of multi-peak positive solutions for a nonlinear Kirchhoff equation with a small parameter, using Lyapunov-Schmidt reduction, extending previous results to this class of equations.

Contribution

It introduces new existence results for multi-peak solutions of Kirchhoff equations, extending prior work to a broader nonlinear setting.

Findings

Multi-peak solutions concentrate at critical points of Q(x).

Solutions exist for sufficiently small epsilon.

The method extends previous results to nonlinear Kirchhoff equations.

Abstract

In this work, we study the following Kirchhoff equation $$\begin{cases}-\left(\varepsilon^2 a+\varepsilon b\int_{\mathbb R^3}|\nabla u|^2\right)\Delta u +u =Q(x)u^{q-1},\quad u>0,\quad x\in {\mathbb{R}^{3}},\\u\to 0,\quad \text{as}\ |x|\to +\infty,\end{cases}$$ where $a,b>0$ are constants, $2<q<6$, and $\varepsilon>0$ is a parameter. Under some suitable assumptions on the function $Q(x)$, we obtain that the equation above has positive multi-peak solutions concentrating at a critical point of $Q(x)$ for $\varepsilon>0$ sufficiently small, by using the Lyapunov-Schmidt reduction method. We extend the result in (Discrete Contin. Dynam. Systems 6(2000), 39--50) to the nonlinear Kirchhoff equation.