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

Let $(Σ,g)$ be a compact Riemann surface with smooth boundary $\partialΣ$, $Δ_g$ 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 $Σ$ is non-contractible, then for any $ρ\in(8kπ,8(k+1)π)$ with $k\in\mathbb{N}^\ast$, the mean field equation $$\left\{\begin{array}{lll} Δ_g u=ρ\frac{he^u}{\int_Σhe^udv_g}&{\rm in}&Σ\\[1.5ex] u=0&{\rm on}&\partialΣ\end{array}\right.$$ 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 $ρ\in(4kπ,4(k+1)π)$ with $k\in\mathbb{N}^\ast$, the mean field equation $$\left\{\begin{array}{lll} Δ_g u=ρ\left(\frac{he^u}{\int_Σhe^udv_g}-\frac{1}{|Σ|}\right)&{\rm in}&Σ\\[1.5ex] \partial u/\partial{\mathbf{v}}=0&{\rm on}&\partialΣ\end{array}\right.$$ has a solution, where $\mathbf{v}$ denotes the unit normal outward vector on $\partialΣ$. Note that in this case we do not require the surface to be non-contractible.