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

Abstract

In this paper, on a compact Riemann surface $(Σ, g)$ with smooth boundary $\partialΣ$, we concern a Trudinger-Moser inequality with mean value zero. To be exact, let $λ_1(Σ)$ denotes the first eigenvalue of the Laplace-Beltrami operator with respect to the zero mean value condition and $\mathcal{ S }= \left\{ u \in W^{1,2} (Σ, g) : \|\nabla_g u\|_2^2 \leq 1\right.$ and $\left.\int_Σu \,dv_g = 0 \right \},$ where $W^{1,2}(Σ, g)$ is the usual Sobolev space, $\|\cdot\|_2$ 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 $α\geqλ_1(Σ)$, the supremum is infinite. Moreover, we prove the supremum is attained by a function $u_α \in C^\infty\left(\overlineΣ\right)\cap \mathcal {S}$ for sufficiently small $α> 0$. 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}.