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Critical fractional Kirchhoff problems: Uniqueness and Nondegeneracy

Zhipeng Yang, Yuanyang Yu

TL;DR

This work studies a critical fractional Kirchhoff equation on $\mathbb{R}^N$ with a nonlocal coefficient depending on the gradient energy. It proves the uniqueness of the positive ground state up to translation and a full nondegeneracy property of the linearized operator, by relating the Kirchhoff problem to the standard fractional critical problem through a scaling parameter and exploiting the bubble profile for $(-\Delta)^s u = u^{2^*_s-1}$. The key contributions are a scaling-based uniqueness reduction and a complete angular-momentum–decomposed nondegeneracy analysis, which underpin singular perturbation methods for fractional Kirchhoff models and extend known results to the critical, high-dimensional fractional setting. These results enhance understanding of the qualitative behavior and stability of ground states in nonlocal nonlinear PDEs and provide a framework for further perturbation studies.

Abstract

In this paper, we consider the following critical fractional Kirchhoff equation \begin{equation*} \Big(a+b{\int_{\mathbb{R}^{N}}}|(-Δ)^{\frac{s}{2}}u|^2dx\Big)(-Δ)^su=|u|^{2^*_s-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} where $a,b>0$, $\frac{N}{4}<s<1$, $2^*_s=\frac{2N}{N-2s}$ and $(-Δ)^s$ is the fractional Laplacian. We prove the uniqueness and nondegeneracy of positive solutions to the problem, which can be used to study the singular perturbation problems concerning fractional Kirchhoff equations.

Critical fractional Kirchhoff problems: Uniqueness and Nondegeneracy

TL;DR

This work studies a critical fractional Kirchhoff equation on with a nonlocal coefficient depending on the gradient energy. It proves the uniqueness of the positive ground state up to translation and a full nondegeneracy property of the linearized operator, by relating the Kirchhoff problem to the standard fractional critical problem through a scaling parameter and exploiting the bubble profile for . The key contributions are a scaling-based uniqueness reduction and a complete angular-momentum–decomposed nondegeneracy analysis, which underpin singular perturbation methods for fractional Kirchhoff models and extend known results to the critical, high-dimensional fractional setting. These results enhance understanding of the qualitative behavior and stability of ground states in nonlocal nonlinear PDEs and provide a framework for further perturbation studies.

Abstract

In this paper, we consider the following critical fractional Kirchhoff equation \begin{equation*} \Big(a+b{\int_{\mathbb{R}^{N}}}|(-Δ)^{\frac{s}{2}}u|^2dx\Big)(-Δ)^su=|u|^{2^*_s-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} where , , and is the fractional Laplacian. We prove the uniqueness and nondegeneracy of positive solutions to the problem, which can be used to study the singular perturbation problems concerning fractional Kirchhoff equations.

Paper Structure

This paper contains 4 sections, 2 theorems, 38 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['Thm1.1']}
  3. Proof of Uniqueness
  4. Proof of nondegeneracy

Key Result

Theorem 1.1

Let $a,b>0$ and $\frac{N}{4}<s<1$. Then equation eq1.2 has a unique ground state solution $U\in D$ up to translation. Moreover, $U$ is nondegenerate in the sense that there holds where $\mathcal{L}_{+}$ is defined as acting on $L^2(\mathbb{R}^N)$ with domain $D$.

Theorems & Definitions (2)

  • Theorem 1.1
  • Lemma 2.1