Calculus of Variations and Geometric Measure Theory

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

created by jevnikar on 29 Jun 2023
modified on 13 Sep 2023


Accepted Paper

Inserted: 29 jun 2023
Last Updated: 13 sep 2023

Journal: Differential and Integral Equations
Pages: 15
Year: 2023


The aim of this note is to present the first qualitative global bifurcation diagram of the equation $-\Delta u=\mu
^{2\alpha}e^u$. 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