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, 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.
1
Central China Normal University, School of Mathematics and Statistics
Hong Chen; Qiaoqiao Hua. Solutions with multiple peaks for nonlinear Kirchhoff equations on R^3. Annales Fennici Mathematici, Tome 48 (2023) no. 2, pp. 537-566. doi: 10.54330/afm.131900
@article{AFM_2023_48_2_a5,
author = {Hong Chen and Qiaoqiao Hua},
title = {Solutions with multiple peaks for nonlinear {Kirchhoff} equations on {R^3}},
journal = {Annales Fennici Mathematici},
pages = {537--566},
year = {2023},
volume = {48},
number = {2},
doi = {10.54330/afm.131900},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.54330/afm.131900/}
}
TY - JOUR
AU - Hong Chen
AU - Qiaoqiao Hua
TI - Solutions with multiple peaks for nonlinear Kirchhoff equations on R^3
JO - Annales Fennici Mathematici
PY - 2023
SP - 537
EP - 566
VL - 48
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.54330/afm.131900/
DO - 10.54330/afm.131900
LA - en
ID - AFM_2023_48_2_a5
ER -
%0 Journal Article
%A Hong Chen
%A Qiaoqiao Hua
%T Solutions with multiple peaks for nonlinear Kirchhoff equations on R^3
%J Annales Fennici Mathematici
%D 2023
%P 537-566
%V 48
%N 2
%U http://geodesic.mathdoc.fr/articles/10.54330/afm.131900/
%R 10.54330/afm.131900
%G en
%F AFM_2023_48_2_a5
