On Non-simple blowup solutions of Liouville equation

TL;DR

This paper proves that non-simple blowup solutions do not occur for Liouville equations with quantized singular sources on bounded domains, confirming a long-standing conjecture and extending the result beyond the unit ball.

Contribution

It completely settles the conjecture that non-simple blowup does not occur on bounded domains and extends the result beyond the unit ball, impacting solution analysis and degree counting.

Findings

Non-simple blowup phenomenon is ruled out on bounded domains.

The result confirms the conjecture for the unit ball and extends it.

Implications for uniqueness, degree counting, and solution construction.

Abstract

For Liouville equation with quantized singular sources, the non-simple blowup phenomenon has been a major difficulty for years. It was conjectured by the first two authors that the non-simple blowup phenomenon does not occur if the equation is defined on the unit ball with Dirichlet boundary condition. In this article we not only completely settle this conjecture in its entirety, but also extend our result to cover any bounded domain. Since the main theorem in this article rules out the non-simple phenomenon in commonly observed applications, it may pave the way for advances in degree counting programs, uniqueness of blowup solutions and construction of solutions, etc.