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Semi-classical solutions for Kirchhoff type problem with a critical frequency

Published online by Cambridge University Press:  27 May 2020

Qilin Xie
Affiliation:
School of Applied Mathematics, Guangdong University of Technology, Guangzhou, Guangdong510006, China (xieqilinsxdt@163.com, xieql@gdut.edu.cn)
Xu Zhang
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha, Hunan410083, China (darkblue1121@163.com)

Abstract

In the present paper, we consider the following Kirchhoff type problem

$$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$
where b > 0, p ∈ (4, 6), the potential $V\in C(\mathbb R^3,\mathbb R)$ and ɛ is a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e., $\inf _{\mathbb R^N} V=0$. As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.

Type
Research Article
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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