Blow-up solutions for mean field equations with non-quantized singularities on Riemann surfaces with boundary
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| author | Ahmedou, Mohameden Hu, Zhengni Zhu, Miaomiao |
| author_facet | Ahmedou, Mohameden Hu, Zhengni Zhu, Miaomiao |
| contents | 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_04790 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Blow-up solutions for mean field equations with non-quantized singularities on Riemann surfaces with boundary Ahmedou, Mohameden Hu, Zhengni Zhu, Miaomiao Analysis of PDEs Primary: 35J25, Secondary: 35B40, 35B44, 58J05 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 |
| title | Blow-up solutions for mean field equations with non-quantized singularities on Riemann surfaces with boundary |
| topic | Analysis of PDEs Primary: 35J25, Secondary: 35B40, 35B44, 58J05 |
| url | https://arxiv.org/abs/2602.04790 |