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

TL;DR

This paper investigates the blow-up behavior of solutions to mean field equations with Neumann boundary conditions on Riemann surfaces, identifying conditions for existence and describing the precise blow-up points as parameters approach critical values.

Contribution

It establishes conditions on the potential function V for solution existence near critical parameters and characterizes the exact interior and boundary blow-up points.

Findings

Solutions blow up at exactly k interior points as parameter approaches critical value

Solutions blow up at (m-k) boundary points near critical parameter

Conditions on V determine solution existence near critical values

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: $$ -\Delta_g u = \lambda \left(\frac{Ve^u}{\int_{\Sigma} Ve^u \, dv_g} - \frac{1}{|\Sigma|_g}\right) \text{ in } \Sigma \text{ with } \partial_{\nu_g} u = 0 \text{ on } \partial \Sigma, $$ 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 $k \leq m \in \mathbb{N}_+$ 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 $\partial \Sigma$.