*Accepted Paper*

**Inserted:** 29 jun 2023

**Last Updated:** 13 sep 2023

**Journal:** Differential and Integral Equations

**Pages:** 15

**Year:** 2023

**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 $\mu\to-\infty$.

**Keywords:**
bifurcation analysis, singular Gelfand problem

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