Existence and uniqueness for Mean Field Equations on multiply connected domains at the critical parameter

Daniele Bartolucci, Chang-Shou Lin

We consider the mean field equation: (1) Δu+ρ\frac{e^u}{\int_Ωe^u}=0 & \hbox{in} \;Ω, u=0 & \hbox{on}\;\partialΩ, where $\Omega \subset {\mathbb{R}}^2$ is an open and bounded domain of cla…