Blow-up solutions for mean field equations with non-quantized singularities on Riemann surfaces with boundary

TL;DR

This paper investigates blow-up solutions for mean field equations with non-quantized singularities on Riemann surfaces with boundary, revealing new phenomena and constructing solutions using Lyapunov-Schmidt reduction.

Contribution

It introduces the first construction of blow-up solutions with non-quantized singularities on Riemann surfaces with boundary, including mixed singular-regular cases, using a Lyapunov-Schmidt reduction approach.

Findings

Constructed blow-up solutions with non-quantized singularities.

Identified parameters approaching resonant values for blow-up.

Extended understanding of singular mean field equations on surfaces with boundary.

Abstract

We study mean field equations with singular sources on a compact Riemann surface with boundary $(\Sigma,g)$, subject to homogeneous Neumann boundary conditions: \[ -\Delta_g v = \rho\left( \frac{V e^{v}}{\int_\Sigma V e^{v}\, d v_g} - \frac{1}{|\Sigma|_g}\right) - \sum_{\xi\in Q} \frac{\varrho(\xi)}{2}\gamma(\xi) \left(\delta_{\xi}- \dfrac{1}{|\Sigma|_g}\right) \text{in }\Sigma; \qquad \partial_{\nu_g} v = 0 \text{ on }\partial\Sigma. \] Here, $V$ is a smooth positive function, $\rho$ is a non-negative parameter, $Q\subset\Sigma$ is a finite set of prescribed singular points, and the singular weights satisfy $\gamma(\xi)\in(-1,+\infty)\setminus(\mathbb{N}\cup\{0\})$. The coefficients are given by $\varrho(\xi)=8\pi$ for $\xi \in\Sigma\setminus\partial\Sigma$ and $\varrho(\xi)=4\pi$ for $\xi \in\partial\Sigma$. We construct blow-up solutions in the non-quantized singular regime, including purely singular and mixed singular-regular blow-up cases, with parameters approaching resonant values. The construction is achieved via a Lyapunov-Schmidt reduction under suitable stability assumptions. Key words: Singular mean field equations, Blow-up phenomena, Lyapunov-Schmidt reduction, Riemann surfaces with boundary