A perturbed Kirchhoff problem with critical exponent

ABSTRACT

In this paper, we consider a class of Kirchhoff problems as follows (1) $-\left(a+b{\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }u{|}^2\right)\mathrm{\Delta }u=(1+\epsilon K(x\left)\right)u^{2^{\ast }-1},u>0\mathrm{i}\mathrm{n}{\mathbb{R}}^N,$(1) where a, b>0 are given constants, ϵ is a small parameter and $2^{\ast }=2N/\left(N-2\right),N\ge 3$. We first prove the nondegeneracy of positive solutions to Equation (1) when $\epsilon =0$. Then, as an application, using a perturbed argument and finite-dimensional reduction method, we prove the existence of positive solutions of Equation (1) for ϵ small.

COMMUNICATED BY:

• S. Leonardi

Keywords:

• Kirchhoff problem
• perturbed argument
• finite-dimensional reduction

2010 AMS Subject Classifications:

• 35A01
• 35J60

Acknowledgments

The authors would like to thank Professor Shuangjie Peng very much for stimulating discussions and helpful suggestions on the present paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of Q. Li was supported by the excellent doctorial dissertation cultivation [grant number 2018YBZZ067] from Central China Normal University.