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 ∂Σ, 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 S = u ∈ W 1,2 (Σ, g) : ∇ g u 2 2 ≤ 1 and Σ u dv g = 0 , where W 1,2 (Σ, g) is the usual Sobolev space, • 2 denotes the standard L 2 -norm and ∇ g represent the gradient. By the method of blow-up analysis, we obtainthe supremum is infinite. Moreover, we prove the supremum is attained by a function u α ∈ C ∞ Σ ∩ S for sufficiently small α > 0. Based on the similar work in the Euclidean space, which was accomplished by Lu-Yang [19], we strengthen the result of Yang [29].