A Trudinger-Moser inequality with mean value zero on a compact Riemann surface with boundary

Abstract

In this paper, on a compact Riemann surface $(\Sigma ,g)$ with smooth boundary $\partial \Sigma$, we concern a Trudinger-Moser inequality with mean value zero. To be exact, let ${\lambda }_1(\Sigma )$ denotes the first eigenvalue of the Laplace-Beltrami operator with respect to the zero mean value condition and $\mathcal{S}=\{u\in W^{1,2}(\Sigma ,g):\Vert {\nabla }_{g}u{\Vert }_2^2\le 1$ and ${\int }_{\Sigma }ud{v}_g=0\},$ where $W^{1,2}(\Sigma ,g)$ is the usual Sobolev space, $\Vert \cdot {\Vert }_2$ denotes the standard $L^2$-norm and ${\nabla }_g$ represent the gradient. By the method of blow-up analysis, we obtain \begin{eqnarray*} \sup_{u \in \mathcal{S}} \int_Σ e^{ 2πu^{2} \left(1+α\|u\|_2^{2}\right) }d v_{g} <+\infty, \ \forall \ 0 \leqα<λ_1(Σ); \end{eqnarray*} when $\alpha \ge {\lambda }_1(\Sigma )$, the supremum is infinite. Moreover, we prove the supremum is attained by a function $u_{\alpha }\in C^{\infty }\left(\overline{\Sigma }\right)\cap \mathcal{S}$ for sufficiently small $\alpha >0$. Based on the similar work in the Euclidean space, which was accomplished by Lu-Yang \cite{Lu-Yang}, we strengthen the result of Yang \cite{Yang2006IJM}.

arXiv admin note: substantial text overlap with arXiv:1911.07299