Mathematische Annalen

, Volume 361, Issue 3–4, pp 787–809 | Cite as

Nonradial maximizers for a Hénon type problem and symmetry breaking bifurcations for a Liouville-Gel’fand problem with a vanishing coefficient

  • Yasuhito MiyamotoEmail author


We study the radiality and nonradiality of the maximizers of
$$\begin{aligned} \sup \left\{ \int _D|x|^{\alpha }(e^u-1)dx;\ u\in H_0^1(D),\ \int _D\left| \nabla u\right| ^2dx=\beta ^2\right\} \end{aligned}$$
from a viewpoint of the bifurcation diagram of the Euler-Lagrange equation
$$\begin{aligned} \Delta u+\beta ^2\frac{|x|^{\alpha }e^u}{\int _D|x|^{\alpha }ue^udx}=0\ \ \hbox {in}\ \ D,\qquad u=0\ \ \hbox {on}\ \ \partial D, \end{aligned}$$
where \(D:=\{x\in {\mathbb {R}}^2;\ |x|<1\}\). We show that the maximizers are nonradial if
$$\begin{aligned} \beta >\sqrt{8\pi \left\{ (\alpha +2)\log \frac{2\alpha +4}{\alpha } -\frac{\alpha +4}{2}\right\} } \end{aligned}$$
and that the maximizers are radial if \(\alpha >0\) or \(\beta >0\) is small. We also give the bifurcation diagrams of the Liouville-Gel’fand problem with a vanishing coefficient
$$\begin{aligned} \Delta u+\lambda |x|^{\alpha }e^u=0\ \ \hbox {in}\ \ D,\qquad u=0\ \ \hbox {on}\ \ \partial D \end{aligned}$$
and of the associated mean field equation. The analysis is based on exact solutions and explicit calculations.

Mathematics Subject Classification

Primary 35B32 35J20 Secondary 35J61 35B09 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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