Search

Let (M,g) be a compact Riemann surface, h be a positive smooth function on M. It is well known that the functional \begin{equation*} J(u)=\frac {1}{2}\int _M|\nabla u|^2dv_g+8\pi \int _M udv_g-8\pi \log \int _Mhe^{u}dv_g \end{equation*} achieves its minimum under Ding-Jost-Li-Wang condition. This result was generalized to nonnegative h by Yang and the author. Later, Sun and Zhu [ Existence of Kazdan-Warner equation with sign-changing prescribed function , arXiv: 2012.12840 , 2020] showed the Ding-Jost-Li-Wang condition is also sufficient when h changes sign, which was reproved later by Wang and Yang [J. Funct. Anal. 282 (2022), Paper No. 109449] and Li and Xu [Calc. Var. Partial Differential Equations 61 (2022), Paper No. 143] respectively using a flow approach. The aim of this note is to give a new proof of Sun and Zhu's result. Our proof is based on the variational method and the maximum principle. [ABSTRACT FROM AUTHOR]

In this paper, using the method of blow-up analysis, we establish a trace Trudinger-Moser inequality involving L^p-norm and obtain the corresponding extremal on a compact Riemann surface with a smooth boundary. The result generalizes those of Li-Liu [Math. Z. 250 (2005), pp. 363–686] and Zhang [Commun. Pure Appl. Anal. 20 (2021), pp. 1721–1735]. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]