May  2007, 19(2): 387-410. doi: 10.3934/dcds.2007.19.387

An expository survey on the recent development of mean field equations

1. 

Department of Mathematics, Taida Institute of Mathematical Sciences, National Taiwan University, Taipei, 10617, Taiwan

Received  September 2006 Revised  June 2007 Published  July 2007

We consider mean field equations: $ \Delta u+\rho(\frac{he^u}{\int_Mhe^u}-1)=0\, on M, $ where $M$ is a compact Riemann surface with area 1, $h$ is a positive continuous function and $\rho$ is a constant, or

$\Delta u+\rho\frac{he^u}{\int_\Omega he^u}=0$ in $\Omega, $
$ u=0 on \partial\Omega, $

where $\Omega$ is a bounded $\mathcal{C}^1$ domain of $\mathbb{R}^2$. In this paper, we give a short survey on the uniqueness problem, find sharper estimates of bubbling solutions and count the topological degree of solutions.

Citation: Chang-Shou Lin. An expository survey on the recent development of mean field equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 387-410. doi: 10.3934/dcds.2007.19.387
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