On the global bifurcation diagram of the equation $-\Delta u=\mu|x|^{2\alpha}e^u$ in dimension two

TL;DR

This paper provides the first qualitative global bifurcation diagram for a singular mean field equation involving a weighted exponential term, analyzing solution branches in various domain types and asymptotic behaviors.

Contribution

It introduces the concept of domains of first/second kind for singular equations and extends bifurcation analysis to non-radial and non-symmetric solutions.

Findings

Solution branches resemble the regular radial case on the disk.

The shape of the bifurcation diagram persists in non-symmetric domains.

Asymptotic profiles are characterized as - for   -.

Abstract

The aim of this note is to present the first qualitative global bifurcation diagram of the equation $-\Delta u=\mu|x|^{2\alpha}e^u$. To this end, we introduce the notion of domains of first/second kind for singular mean field equations and base our approach on a suitable spectral analysis. In particular, we treat also non-radial solutions and non-symmetric domains and show that the shape of the branch of solutions still resembles the well-known one of the model regular radial case on the disk. Some work is devoted also to the asymptotic profile for $\mu\to-\infty$.