Non-degeneracy and local uniqueness of bubbling solutions for a singular Liouville equation

Abstract

We consider the following singular Liouville equation $-\Delta v=\lambda V(x)|x|^2e^v \quad \text{in } B_1, \quad v=0 \quad \text{on } \partial B_1, $ where $B_1\subset\mathbb R^2$ is the unit disk, $\lambda>0$ is a small parameter, and $V$ is a positive smooth function. We first prove a non-degeneracy result for the bubbling solutions constructed in \cite{D-W-Z2026} by the local Pohozaev identities. Then we use this non-degeneracy result to establish the local uniqueness of bubbling solutions whose concentration parameters are sufficiently close.