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

Abstract

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.