Compactness of Extremals for Trudinger-Moser Functionals on the Unit Ball in ℝ2

Let \(\mathbb{B}\) be a unit ball in ℝ², \(W_{0}^{1,2}(\mathbb{B})\) be the standard Sobolev space. For any ϵ > 0, de Figueiredo, do Ó, dos Santons, Yang and Zhu proved the existence of extremals of a Trudinger-Moser inequality in the unit ball. Precisely,

$$\mathop {\sup }\limits_{u \in W_0^{1,2}\left( \mathbb{B} \right),\int_\mathbb{B} {|\nabla u{|^2}dx \le 1} } \int_\mathbb{B} {{{\left| x \right|}^{2\epsilon }}{\rm{e}^{4\pi \left( {1 + \epsilon } \right){u^2}}}} dx$$

can be attained by some radially symmetric function \(u_{\epsilon}\in W_{0}^{1,2}(\mathbb{B})\) with \(\int_{\mathbb{B}}\vert\nabla u_{\epsilon}\vert^{2}dx=1\). In this note, we concern the compactness of the function family {u_ϵ}_ϵ>0 and prove that up to a subsequence u_ϵ converges to some function u ₀ in \(C^{1}(\overline{\mathbb{B}})\) as ϵ → 0. Furthermore, u₀ is an extremal function of the supremum

$$\mathop {\sup }\limits_{u \in W_0^{1,2}\left( \mathbb{B} \right),\int_\mathbb{B} {|\nabla u{|^2}dx \le 1} } \int_{\mathbb{B}}\rm{e}^{4\pi u^{2}}dx.$$

Let us explain the result in geometry. Denote \(\omega_{0}=dx_{1}^{2}+dx_{2}^{2}\) be the standard Euclidean metric. Define a conical metric \(\omega_{\epsilon}=\vert x\vert^{2\epsilon}\omega_{0}\) for \(x\in\mathbb{B}\). Then the extremal family {u _ϵ}_ϵ>0 of the following Trudinger-Moser functionals

$$\int_{\mathbb{B}}\rm{e}^{4\pi(1+\epsilon)u^{2}}dv_{w_{\epsilon}}$$

under the constraint \(u\in W_{0}^{1,2}(\mathbb{B})\) and \(\int_{\mathbb{B}}\vert\nabla_{\omega_{\epsilon}u}\vert^{2}dv_{\omega_{\epsilon}}\leq 1\) is compact as ϵ → 0.