Non-degeneracy of the bubble in a fractional and singular 1D Liouville equation

TL;DR

This work establishes the non-degeneracy of bubbles for the one-dimensional fractional singular Liouville equation $(-\Delta)^{1/2}u=|x|^{\alpha-1}e^u$ on $\mathbb{R}$ for $0<\alpha<2$. By a sequence of conformal changes, the authors reduce the nonlocal linearized problem around the bubble to a Steklov eigenvalue problem on a bounded planar domain determined by $\alpha$—the domain being the intersection or union of two unit disks depending on whether $\alpha\in(0,1)$ or $\alpha\in(1,2)$. They prove that the relevant eigenvalue $\mu_\alpha=1/\sqrt{1+\tau_\alpha^2}$ is simple, with $\tau_\alpha=(1+\cos(\pi\alpha))/\sin(\pi\alpha)$, which yields non-degeneracy and a Morse index change at $\alpha=1$. The analysis combines harmonic extension, cone and bounded-domain reductions, and a detailed Steklov eigenvalue study on symmetric two-disk domains, employing Rellich-Pohozaev identities and Weinstock’s inequality to exclude multiple eigenvalues. The results provide a precise spectral framework for nonlocal 1D Liouville problems and underpin concentration phenomena in related nonlinear PDEs.

Abstract

We prove the non-degeneracy of solutions to a fractional and singular Liouville equation defined on the whole real line in presence of a singular term. We use conformal transformations to rewrite the linearized equation as a Steklov eigenvalue problem posed in a bounded domain, which is defined either by an intersection or a union of two disks. We conclude by proving the simplicity of the corresponding eigenvalue.

Figures (2)

• Figure 1:
• Figure 2:

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Lemma 2.4
• proof
• Proposition 2.5
• proof
• Proposition 2.6