Non-degeneracy of single-peak solutions to a Kirchhoff equation

ABSTRACT

In this paper, we are concerned with the following singular perturbation Kirchhoff equation $-({\epsilon }^{2}a+\epsilon b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^2\mathrm{d}y)\mathrm{\Delta }u+V(y)u=|u{|}^{p-1}u,u\in H^1({\mathbb{R}}^3),$where constants $a,b,\epsilon >0$ and 1<p<5. By using local Pohozaev identity and blow-up analysis, we derive the non-degeneracy of the single-peak solutions to the above problem under some suitable condition on $V(y)$ for small ε.

KEYWORDS:

• Kirchhoff equation
• single-peak solution
• non-degeneracy
• local Pohozaev identity

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35A01
• 35A02
• 35B25
• 35J20
• 35J60

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partially supported by NSFC [grant number 11601194].