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

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