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arXiv · 2306.16990

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

Daniele Bartolucci·Aleks Jevnikar·Ruijun Wu
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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 $\mu\to-\infty$.

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Daniele Bartolucci, Aleks Jevnikar, Ruijun Wu. 2023-06-29. On the global bifurcation diagram of the equation $-\Delta u=\mu|x|^{2\alpha}e^u$ in dimension two. https://doi.org/10.57262/die037-0708-425

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