Blow-up solutions for mean field equations with Neumann boundary conditions on Riemann surfaces
Zhengni Hu, Thomas Bartsch, Mohameden Ahmedou
TL;DR
This work addresses blow-up phenomena for the mean field equation with Neumann boundary conditions on a compact Riemann surface by developing a Lyapunov–Schmidt reduction blended with variational reduction. The authors construct bubbling solutions for parameters $\lambda$ near the critical values $\lambda_{k,m}=4\pi(m+k)$, with exactly $k$ interior and $m-k$ boundary blow-up points, governed by a finite-dimensional reduced energy $\mathcal{F}_{k,m}$ and auxiliary quantities $\mathcal{A}_1$, $\mathcal{A}_2$, $\mathcal{B}$. The analysis hinges on refined isothermal coordinates, Green's function theory, and a careful asymptotic expansion of the energy and its derivatives, yielding conditions under which blow-up occurs on specific configurations, including left or right neighborhoods of the critical values. The results extend the bubbling theory to Riemann surfaces with boundary, providing a rigorous framework for prescribed interior and boundary singularities and their dependence on geometric data such as Gaussian curvature and geodesic curvature, with potential implications for related geometric-analytic problems and models in mathematical physics.
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 $\partial Σ$.