D. Bartolucci - A. Jevnikar - R. Wu
On the global bifurcation diagram of the equation $-\Delta u=\mu|x|^{2\alpha}e^u$ in dimension two
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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