Boundary value problem for the mean field equation on a compact Riemann surface

Abstract

Let $(\hat{I}\pounds ,g)$ be a compact Riemann surface with smooth boundary $\partial \hat{I}\pounds$, $\hat{I}{”}_g$ be the Laplace-Beltrami operator, and $h$ be a positive smooth function. Using a min-max scheme introduced by Djadli-Malchiodi (2006) and Djadli (2008), we prove that if $\hat{I}\pounds$ is non-contractible, then for any $\ddot{I}\in (8k\ddot{I}€,8(k+1)\ddot{I}€)$ with $k\in {\mathbb{N}}^{\ast }$, the mean field equation $$\{\begin{array}{lll}\hat{I}{”}_{g}u=\ddot{I}\frac{h{e}^u}{{\int }_{\hat{I}}\pounds h{e}^{u}d{v}_g}& \mathrm{in}& \hat{I}\pounds \\ u=0& \mathrm{on}& \partial \hat{I}\pounds \end{array}$$ has a solution. This generalizes earlier existence results of Ding-Jost-Li-Wang (1999) and Chen-Lin (2003) in the Euclidean domain. Also we consider the corresponding Neumann boundary value problem. If $h$ is a positive smooth function, then for any $\ddot{I}\in (4k\ddot{I}€,4(k+1)\ddot{I}€)$ with $k\in {\mathbb{N}}^{\ast }$, the mean field equation $$\{\begin{array}{lll}\hat{I}{”}_{g}u=\ddot{I}\left(\frac{h{e}^u}{{\int }_{\hat{I}}\pounds h{e}^{u}d{v}_g}-\frac{1}{|\hat{I}\pounds |}\right)& \mathrm{in}& \hat{I}\pounds \\ \partial u/\partial \mathbf{v}=0& \mathrm{on}& \partial \hat{I}\pounds \end{array}$$ has a solution, where $\mathbf{v}$ denotes the unit normal outward vector on $\partial \hat{I}\pounds$. Note that in this case we do not require the surface to be non-contractible.