Trudinger–Moser type inequalities with a symmetric conical metric and a symmetric potential

Abstract

Let (M,g) be a closed Riemann surface, where the metric g has certain conical singularities at finite points. Suppose Γ is a group with elements of isometries acting on (M,g). In this paper, Trudinger–Moser inequalities involving Γ and the operador Δg+V are established, where Δg denotes the Laplace–Beltrami operator associated to g and the potential V:M→(0,∞) belongs to a class of symmetric and continuous functions. Moreover, via the method of blow-up analysis, the corresponding extremals are also obtained.