Non degeneracy of blow-up solutions of non-quantized singular Liouville-type equations and the convexity of the mean field entropy of the Onsager vortex model with singular sources

TL;DR

This paper proves the non-degeneracy of bubbling solutions in singular mean field equations with non-quantized sources, leading to the convexity of the mean field entropy in the Onsager vortex model.

Contribution

It extends previous results to include non-quantized singular sources and demonstrates entropy convexity in the vortex model.

Findings

Non-degeneracy of bubbling solutions with singular sources

Convexity of the mean field entropy in large energy limit

Extension of prior regular blow-up point results

Abstract

We establish the non-degeneracy of bubbling solutions for singular mean field equations when the blow-up points are either regular or involve non-quantized singular sources. This extends the results from Bartolucci-Jevnikar-Lee-Yang \cite{bart-5}, which focused on regular blow-up points. As a consequence, we establish the strict convexity of the Entropy in the large energy limit for a specific class of two-dimensional domains in the Onsager mean field vortex model with singular sources.