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arXiv · 2408.16917

Blow-up solutions for mean field equations with Neumann boundary conditions on Riemann surfaces

Zhengni Hu·Thomas Bartsch·Mohameden Ahmedou
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Abstract

On a compact Riemann surface $(Σ, g)$ with a smooth boundary $\partial Σ$, we consider the following mean field equations with Neumann boundary conditions: $$ -Δ_g u = λ\left(\frac{Ve^u}{\int_Σ Ve^u \, dv_g} - \frac{1}{|Σ|_g}\right) \text{ in } Σ\text{ with } \partial_{ν_g} u = 0 \text{ on } \partial Σ, $$ We find conditions on the potential function $V: Σ\to \mathbb{R}^+$ such that solutions exist for the parameter $λ$ when it is in a small right (or left) neighborhood of a critical value $4π(m+k)$ for $k \leq m \in \mathbb{N}_+$ and blow up as $λ$ approaches the critical parameter. The blow-up occurs exactly at $k$ points in the interior of $Σ$ and $(m-k)$ points on the boundary $\partial Σ$.

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Zhengni Hu, Thomas Bartsch, Mohameden Ahmedou. 2025-01-04. Blow-up solutions for mean field equations with Neumann boundary conditions on Riemann surfaces. https://arxiv.org/abs/2408.16917

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