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

In this paper, using the method of blow-up analysis, the authors obtain an improved audinger-Moser inequality involving L-p-norm (p > 1) and prove the existence of its extremal function on a closed Riemann surface (Sigma, g) with the action of a finite isometric group G = {sigma(1), sigma(2), ..., sigma(N)}. To be exact, let W-1,W-2 (Sigma, g) be the usual Sobolev space, a function space H-G = {u is an element of W-1,W-2 (Sigma, g) : integral(Sigma) ud upsilon(g) = 0 and u(sigma(i)(x)) = u(x), for all x is an element of Sigma, sigma(i) is an element of G} and l = min(x)(is an element of)(Sigma) I(x), where I(x) stands for the number of all distinct points in the set G(x) = {sigma(1)(x), ..., sigma(N) (x)}. Define lambda(G)(p) = inf(u is an element of HG), (u not equivalent to 0) parallel to del(g)u parallel to(2)(2)/parallel to u parallel to(2)(p), where parallel to.parallel to(p) is the standard L-p-norm on (Sigma, g). Using blow-up analysis, we prove that if 0 <= alpha < lambda(G)(p), the supremum sup(u is an element of HG,parallel to del gu parallel to 22-alpha parallel to u parallel to p2 <= 1 )integral(Sigma)e(4 pi lu2)d upsilon(g) < +infinity, and this supremum can be attained; if alpha >= lambda(G)(p), 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. Muth. 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].