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

Abstract

We study mean field equations with singular sources on a compact Riemann surface with boundary $(\hat{I}\pounds ,g)$, subject to homogeneous Neumann boundary conditions: \[ -Δ_g v = ρ\left( \frac{V e^{v}}{\int_ΣV e^{v}\, d v_g} - \frac{1}{|Σ|_g}\right) - \sum_{ξ\in Q} \frac{\varrho(ξ)}{2}γ(ξ) \left(δ_ξ- \dfrac{1}{|Σ|_g}\right) \text{in }Σ; \qquad \partial_{ν_g} v = 0 \text{ on }\partialΣ. \] Here, $V$ is a smooth positive function, $\ddot{I}$ is a non-negative parameter, $Q\subset \hat{I}\pounds$ is a finite set of prescribed singular points, and the singular weights satisfy ${\hat{I}}^3(\hat{I}¾)\in (-1,+\infty )\setminus (\mathbb{N}\cup \{0\})$. The coefficients are given by $\varrho (\hat{I}¾)=8\ddot{I}€$ for $\hat{I}¾\in \hat{I}\pounds \setminus \partial \hat{I}\pounds$ and $\varrho (\hat{I}¾)=4\ddot{I}€$ for $\hat{I}¾\in \partial \hat{I}\pounds$. 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