A TRUDINGER-MOSER INEQUALITY WITH MEAN VALUE ZERO ON A COMPACT RIEMANN SURFACE WITH BOUNDARY

In this paper, on a compact Riemann surface (Sigma, g) with smooth boundary partial derivative 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 S = {u is an element of W-1,W-2(Sigma, g) : parallel to del(g)u parallel to(2)(2) <= 1 and integral(Sigma)udv(g) = 0}, where W-1,W-2(Sigma, g) is the usual Sobolev space, parallel to . parallel to(2) denotes the standard L-2-norm and del(g) represent the gradient. By the method of blow-up analysis, we obtain sup(u subset of S) integral(e pi u2(1+ alpha parallel to u parallel to 22))(Sigma)dv(g) < + infinity, for all 0 <= alpha < lambda(1)(Sigma); when alpha >= lambda(1)(Sigma), the supremum is infinite. Moreover, we prove the supremum is attained by a function u(alpha) is an element of C-infinity (Sigma) boolean AND S for sufficiently small alpha > 0. Based on the similar work in the Euclidean space, which was accomplished by Lu-Yang [19], we strengthen the result of Yang [29].