Blow-up solutions for mean field equations with Neumann boundary conditions on Riemann surfaces

Abstract

On a compact Riemann surface $(\Sigma ,g)$ with a smooth boundary $\partial \Sigma$, we consider the following mean field equations with Neumann boundary conditions: $$-{\Delta }_{g}u=\lambda \left(\frac{V{e}^u}{{\int }_{\Sigma }V{e}^{u}d{v}_g}-\frac{1}{|\Sigma {|}_g}\right)\text{ in }\Sigma \text{ with }{\partial }_{{\nu }_g}u=0\text{ on }\partial \Sigma ,$$ We find conditions on the potential function $V:\Sigma \to {\mathbb{R}}^{+}$ such that solutions exist for the parameter $\lambda$ when it is in a small right (or left) neighborhood of a critical value $4\pi (m+k)$ for $k\le m\in {\mathbb{N}}_{+}$ and blow up as $\lambda$ approaches the critical parameter. The blow-up occurs exactly at $k$ points in the interior of $\Sigma$ and $(m-k)$ points on the boundary $\partial \Sigma$.