Existence of critical points for a mean field functional on a compact Riemann surface with boundary
Volume 268 / 2023
Abstract
We prove the existence of critical points for a mean field type functional on a compact Riemann surface $( \Sigma ,g)$ with smooth boundary $\partial \Sigma $. More exactly, let $H^1(\Sigma )$ denote the usual Sobolev space, $h:\Sigma \rightarrow \mathbb R$ a smooth positive function and let $\rho \in (2 k \pi , 2(k+1) \pi )$ for some $k \in \mathbb N_+$. We prove that the mean field type functional $$ \tilde {J}_{\rho }(u)=\frac {1}{2} \int _{\Sigma }\,(|\nabla _g u|^{2}+u^2) \,d v_g -\rho \ln \int _{\partial \Sigma } h e^{u}\,d s_g $$ has critical points. The main point here is that we make no assumption on the topology or geometry of the surface. We apply topological methods and min-max schemes, used by Ding–Jost–Li–Wang (1996), Djadli (2008), Djadli–Malchiodi (2008), Malchiodi (2008) and Guo-Liu (2006).
