Uniqueness and nondegeneracy of positive solutions to a class of Kirchhoff equations in $\mathbb{R}^3$

TL;DR

This paper proves the uniqueness and nondegeneracy of positive solutions to a class of nonlocal Kirchhoff equations in three-dimensional space, filling a gap in the mathematical understanding of such equations.

Contribution

It establishes the first known results on the uniqueness and nondegeneracy of positive solutions for Kirchhoff equations, using new insights and methods adapted from classical Schrödinger equations.

Findings

Proves uniqueness of positive solutions.

Shows nondegeneracy of these solutions.

Introduces new techniques to handle nonlocality.

Abstract

In this paper, we establish a type of uniqueness and nondegeneracy results for positive solutions to the following nonlocal Kirchhoff equations \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\text{d} x\right)\Delta u+u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b$ are positive constants and $1<p<5$. Before this paper, it seems that there have no this type of results even on positive ground states solutions to Kirchhoff type equations, much less on general positive solutions. To overcome the difficulty brought by the nonlocality, some new observation on Kirchhoff equations is found, and some related theories on classical Schr\"odinger equations are applied.