Nondegeneracy of positive solutions to a Kirchhoff problem with critical Sobolev growth

TL;DR

This paper proves the uniqueness and nondegeneracy of positive solutions to a Kirchhoff equation with critical Sobolev growth, which is important for understanding the solution structure and stability in related nonlinear PDEs.

Contribution

It establishes the first rigorous proof of nondegeneracy for positive solutions to a Kirchhoff problem with critical Sobolev growth, advancing the theoretical understanding of such equations.

Findings

Proved uniqueness of positive solutions.

Established nondegeneracy of solutions.

Potential applications in singular perturbation problems.

Abstract

In this paper, we prove uniqueness and nondegeneracy of positive solutions to the following Kirchhoff equations with critical growth \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u=u^{5}, & u>0 & \text{in }\mathbb{R}^{3},\end{eqnarray*} where $a,b>0$ are positive constants. This result has potential applications in singular perturbation problems concerning Kirchhoff equaitons.