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

Mohameden Ahmedou, Zhengni Hu, Miaomiao Zhu

TL;DR

The paper addresses blow-up phenomena for mean field equations with non-quantized singularities on Riemann surfaces with boundary under Neumann conditions. It develops a Lyapunov–Schmidt reduction using projected bubbles to construct blow-up solutions as the parameter approaches resonant values, characterizing both purely singular and mixed singular-regular blow-ups. A central outcome is the reduced finite-dimensional functional $oldsymbol{F}^{K,Q_1}_{p,q}$ whose $C^1$-stable critical points yield blow-up configurations with precise mass concentration and energy asymptotics; the analysis requires careful handling of boundary effects, Green functions, and singular weights with $oldsymbol{ u}$-type constraints, notably $oldsymbol{ ho}_*<1$. The results extend the understanding of non-quantized singular blow-up in a geometric setting and establish a framework for constructing such solutions via a rigorous finite-dimensional reduction.

Abstract

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

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

TL;DR

The paper addresses blow-up phenomena for mean field equations with non-quantized singularities on Riemann surfaces with boundary under Neumann conditions. It develops a Lyapunov–Schmidt reduction using projected bubbles to construct blow-up solutions as the parameter approaches resonant values, characterizing both purely singular and mixed singular-regular blow-ups. A central outcome is the reduced finite-dimensional functional whose -stable critical points yield blow-up configurations with precise mass concentration and energy asymptotics; the analysis requires careful handling of boundary effects, Green functions, and singular weights with -type constraints, notably . The results extend the understanding of non-quantized singular blow-up in a geometric setting and establish a framework for constructing such solutions via a rigorous finite-dimensional reduction.

Abstract

We study mean field equations with singular sources on a compact Riemann surface with boundary , subject to homogeneous Neumann boundary conditions: Here, is a smooth positive function, is a non-negative parameter, is a finite set of prescribed singular points, and the singular weights satisfy . The coefficients are given by for and for . 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
Paper Structure (13 sections, 20 theorems, 167 equations)

This paper contains 13 sections, 20 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Isothermal coordinates and Green functions
  4. Projected bubbles
  5. The Moser-Trudinger inequality
  6. Approximate solutions
  7. Finite-dimensional reduction
  8. Linearized problem
  9. The non-linear problem
  10. Reduced functional and its expansion
  11. Proof of main results
  12. The linearized equations
  13. Asymptotic behavior of the projected bubbles

Key Result

Theorem 1.1

Let $Q_1\subset Q$ be a nonempty subset, and set $N:=|Q_1|>0$. Then there exists a family of blow-up solutions $v_\varepsilon$ to eq:s_mf with such that $v_\varepsilon$ blows up precisely at the points in $Q_1$. Moreover, up to a subsequence, the following estimates hold: as $\varepsilon\to 0$ and where $u_\varepsilon(x)= v_\varepsilon(x) + h_Q(x)$ and $K(x) = V(x) e^{-h_Q(x)}$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 26 more