High Order Vanishing Theorems for Nonsimple Blowup Solutions of Singular Liouville Equation

TL;DR

This paper establishes conditions under which non-simple blowup occurs in singular Liouville equations, providing a comprehensive framework to understand and control complex blowup profiles in mathematical physics.

Contribution

It proves that non-simple blowup happens only when derivatives of certain coefficients vanish, unifying previous results and enabling analysis of moving poles.

Findings

Non-simple blowup occurs only when specific derivatives approach zero.

Theorems determine vanishing order for quantized singular sources.

Results help eliminate complex blowup scenarios in applications.

Abstract

For a singular Liouville equation, it is plausible that a non-simple blowup phenomenon occurs around a quantized singular pole. The presence of complex blowup profiles of bubbling solutions presents substantial challenges in applications. In this article, we demonstrate that under natural assumptions, non-simple blowup takes place only when the derivatives of certain coefficient functions approach zero. Our main result encompasses all previous findings and determines the vanishing order for any specific quantized singular source. Our theorems can be utilized not only to eliminate multiple non-simple blowup scenarios in applications but also to investigate blowup solutions with moving poles.