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 $(\mathrm{\Sigma },g)$, subject to homogeneous Neumann boundary conditions: $$-{\mathrm{\Delta }}_{g}v=\rho (\frac{V{e}^v}{{\int }_{\mathrm{\Sigma }}V{e}^{v}d{v}_g}-\frac{1}{\mathrm{\mid }\mathrm{\Sigma }{\mathrm{\mid }}_g})-\sum _{\xi \in Q}\frac{\varrho (\xi )}{2}\gamma (\xi )({\delta }_{\xi }-{\displaystyle \frac{1}{\mathrm{\mid }\mathrm{\Sigma }{\mathrm{\mid }}_g}})\text{in }\mathrm{\Sigma };{\mathrm{\partial }}_{{\nu }_g}v=0\text{ on }\mathrm{\partial }\mathrm{\Sigma }\mathrm{.}$$ Here, $V$ is a smooth positive function, $\rho$ is a non-negative parameter, $Q\subset \mathrm{\Sigma }$ is a finite set of prescribed singular points, and the singular weights satisfy $\gamma (\xi )\in (-1,+\mathrm{\infty })\setminus (\mathbb{N}\cup \{0\})$. The coefficients are given by $\varrho (\xi )=8\pi$ for $\xi \in \mathrm{\Sigma }\setminus \mathrm{\partial }\mathrm{\Sigma }$ and $\varrho (\xi )=4\pi$ for $\xi \in \mathrm{\partial }\mathrm{\Sigma }$. 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