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

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 $\mu \to - \infty.$