An expository survey on the recent development of mean field equations
Chang‐Shou Lin · Discrete and Continuous Dynamical Systems · 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 theuniqueness problem, find sharper estimates of bubbling solutionsand count the topological degree of solutions.