Solutions with multiple peaks for nonlinear Kirchhoff equations on R^3

Hong Chen¹ ; Qiaoqiao Hua¹

¹ Central China Normal University, School of Mathematics and Statistics

Annales Fennici Mathematici, Tome 48 (2023) no. 2, pp. 537-566.

Résumé

In this paper, we mainly investigate the following nonlinear Kirchhoff equation $-\left(\epsilon^2 a+\epsilon b\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u +u =Q(x)u^{q-1}$, $u>0$, $x\in\mathbb{R}^{3}$, $u\to 0$, as $|x|\to +\infty$,   where $a,b>0$ are constants, $2<q<6$, and $\epsilon>0$ 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 $\epsilon>0$ sufficiently small, by using the finite dimensional reduction method. Different from the local Schrödinger problem, here the corresponding limit problem is a system. Moreover, the nonlocal term brings some new difficulties which involve some technical and complicated estimates.

Détail

DOI : 10.54330/afm.131900

Keywords: Kirchhoff equations, multi-peak positive solutions, the finite dimensional reduction method