| We have studied the Moser-Trudinger inequalities with singularity, by ap-plying the properties of blow-up analysis theory of semi-linear elliptic equation, and Green’s function on the compact Riemann surface.In this paper, we derive the following Moser-Trudinger inequalities by the analysis of blow-up and the classification of singularities:Further more, we prove that functional attains its minimal on H1, under the assumption: where H1={u∈H1,2(M)|∫M g(x)|x-p0|2αeu= 1}, C’=-2α∫M log|x-p0|。 M is a compact Riemann surface， g(x) is a positive smooth function on M. |
