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

We study mean field equations with singular sources on a compact Riemann surface with boundary $(Σ,g)$, 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, $ρ$ is a non-negative parameter, $Q\subsetΣ$ is a finite set of prescribed singular points, and the singular weights satisfy $γ(ξ)\in(-1,+\infty)\setminus(\mathbb{N}\cup{0})$. The coefficients are given by $\varrho(ξ)=8π$ for $ξ\inΣ\setminus\partialΣ$ and $\varrho(ξ)=4π$ for $ξ\in\partialΣ$. 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

💡 Research Summary

The paper investigates blow‑up phenomena for mean‑field equations with singular sources on a compact Riemann surface Σ equipped with a smooth boundary ∂Σ. The equation under homogeneous Neumann boundary conditions is
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