Quantization analysis of Moser-Trudinger equations in the Poincaré disk and applications

TL;DR

This work addresses the quantization of the Moser-Trudinger equation on the Poincaré disk, establishing sharp blow-up descriptions and energy concentration phenomena under the hyperbolic geometry setting. The authors exploit hyperbolic symmetry via a moving-planes approach in integral form and perform a detailed blow-up analysis to obtain energy quantization at $4\pi$, along with convergence/divergence behavior as the parameter $\lambda$ varies. They prove the existence of positive critical points for the Moser-Trudinger functional under a Dirichlet-energy constraint slightly above the threshold $4\pi$, and they derive precise $\lambda$-$c_{\lambda}$ asymptotics, including a high-order expansion for $c_{\lambda}$ as $\lambda\to0$. A final uniqueness result shows that, for small $\lambda$, positive solutions are unique up to Möbius transformations, significantly extending the understanding of Moser-Trudinger phenomena from hyperbolic balls to the full Poincaré disk.

Abstract

In this paper, we first establish the quantitative properties for positive solutions to the Moser-Trudinger equations in the two-dimensional Poincaré disk $\mathbb{B}^2$: \begin{equation*}\label{mt1} \left\{ \begin{aligned} &-Δ_{\mathbb{B}^2}u=λue^{u^2},\ x\in\mathbb{B}^2, &u\to0,\ \text{when}\ ρ(x)\to\infty, &||\nabla_{\mathbb{B}^2} u||_{L^2(\mathbb{B}^2)}^2\leq M_0, \end{aligned} \right. \end{equation*} where $0<λ<\frac{1}{4}=\inf\limits_{u\in W^{1,2}(\mathbb{B}^2)\backslash\{0\}}\frac{\|\nabla_{\mathbb{B}^2}u\|_{L^2(\mathbb{B}^2)}^2}{\|u\|_{L^2(\mathbb{B}^2)}^2}$, $ρ(x)$ denotes the geodesic distance between $x$ and the origin and $M_0$ is a fixed large positive constant (see Theorem 1.1). Furthermore, by doing a delicate expansion for Dirichlet energy $\|\nabla_{\mathbb{B}^2}u\|_{L^2(\mathbb{B}^2)}^2$ when $λ$ approaches to $0,$ we prove that there exists $Λ^\ast>4π$ such that the Moser-Trudinger functional $F(u)=\int_{\mathbb{B}^2}\left(e^{u^2}-1\right) dV_{\mathbb{B}^2}$ under the constraint $\int_{\mathbb{B}^2}|\nabla_{\mathbb{B}^2}u|^2 dV_{\mathbb{B}^2}=Λ$ has at least one positive critical point for $Λ\in(4π,Λ^{\ast})$ up to some Möbius transformation. Finally, when $λ\rightarrow 0$, by doing a more accurate expansion for $u$ near the origin and away from the origin, applying a local Pohozaev identity around the origin and the uniqueness of the Cauchy initial value problem for ODE,Cauchy-initial uniqueness for ODE, we prove that the Moser-Trudinger equation only has one positive solution when $λ$ is close to $0.$ During the process of the proofs, we overcome some new difficulties which involves the decay properties of the positive solutions, as well as some precise expansions for the solutions both near the origin and away from the origin.

Theorems & Definitions (79)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Theorem 1.7
• Proposition 2.1
• proof
• Proposition 3.1