Finding critical points of the Trudinger-Moser functional through the heat flow

Abstract:

Let $\Omega \subset \mathbb {R}^2$ be a smooth bounded domain and $W_0^{1,2}(\Omega )$ be the standard Sobolev space. In this paper, using the heat flow of Lamm-Robert-Struwe [J. Funct. Anal. 257 (2009), pp. 2951–2998] and blow-up analysis, we study the critical points of the Trudinger-Moser functional \begin{equation*} J(u)=\int _\Omega \exp (4\pi u^2)dx \end{equation*} under the constraint \begin{equation*} {E}(1)=\left \{u\in W_0^{1,2}(\Omega ):\int _\Omega |\nabla u|^2dx=1\right \}. \end{equation*} Precisely, for certain initial data $u_0$, we obtain that up to a subsequence, the flow converges to a critical point of $J$ under the constraint $E(1)$. This complements the results of Lamm-Robert-Struwe.