Improved Trudinger–Moser inequality involving Lp-norm on a closed Riemann surface with isometric group actions

Abstract

In this paper, using the method of blow-up analysis, the authors obtain an improved Trudinger–Moser inequality involving [Formula: see text]-norm ([Formula: see text]) and prove the existence of its extremal function on a closed Riemann surface [Formula: see text] with the action of a finite isometric group [Formula: see text]. To be exact, let [Formula: see text] be the usual Sobolev space, a function space [Formula: see text] and [Formula: see text] [Formula: see text] and [Formula: see text] where [Formula: see text] stands for the number of all distinct points in the set [Formula: see text]. Define [Formula: see text], where [Formula: see text] is the standard [Formula: see text]-norm on [Formula: see text]. Using blow-up analysis, we prove that if [Formula: see text], the supremum [Formula: see text] and this supremum can be attained; if [Formula: see text], the above supremum is infinite. This kind of inequality will play an important role in the study of prescribing Gaussian curvature problem and mean field equations. In particular, their result generalizes those of Chen [A Trudinger inequality on surfaces with conical singularities, Proc. Amer. Math. Soc. 108 (1990) 821–832], Yang [Extremal functions for Trudinger–Moser inequalities of Adimurthi–Druet type in dimension two, J. Differential Equations 258 (2015) 3161–3193] and Fang–Yang [Trudinger–Moser inequalities on a closed Riemannian surface with the action of a finite isometric group, Ann. Sc. Norm. Super. Pisa Cl. Sci. 20 (2020) 1295–1324].