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

Abstract

Let (Σ, g) be a compact Riemann surface with smooth boundary ∂E, ∆_g be the Laplace-Beltrami operator, and h be a positive smooth function. Using a min-max scheme introduced by Djadli and Malchiodi (2008) and Djadli (2008), we prove that if Σ is non-contractible, then for any ρ Σ (8kπ, 8(k +1)π) with k Σ ℕ*, the mean field equation

$$\left\{ {\matrix{{{\Delta _g}u = \rho {{h{{\rm{e}}^u}} \over {\int_\Sigma {h{{\rm{e}}^u}d{v_g}} }}} \hfill & {{\rm{in}}\,\,\Sigma ,} \hfill \cr {u = 0} \hfill & {{\rm{on}}\,\,\partial \Sigma } \hfill \cr } } \right.$$

has a solution. This generalizes earlier existence results of Ding et al. (Ann Inst H Poincaré Anal Non Linéaire, 1999) and Chen and 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 ρ ∈ (4kπ, 4(k + 1)π) with k ∈ ℕ*, the mean field equation

$$\left\{ {\matrix{{{\Delta _g}u = \rho \left( {{{h{{\rm{e}}^u}} \over {\int_\Sigma {h{{\rm{e}}^u}d{v_g}} }} - {1 \over {\left| \Sigma \right|}}} \right)} \hfill & {{\rm{in}}\,\,\Sigma ,} \hfill \cr {\partial u/\partial v = 0} \hfill & {{\rm{on}}\,\,\partial \Sigma } \hfill \cr } } \right.$$

has a solution, where v denotes the unit normal outward vector on ∂Σ. Note that in this case we do not require the surface to be non-contractible.