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

2022/06/28 by Chen, Hong, Hua, Qiaoqiao
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2206.13777

Abstract

In this work, we study the following Kirchhoff equation \begincases-(ε² a+ε b∫_{\mathbb R³}|∇ u|²)Δu +u =Q(x)u^q-1, ugt;0, x∈ ℝ³,
u→ 0, as |x|→ +∞,\endcases where a,b>0 are constants, 20 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 ε>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.

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