Moser–Trudinger inequalities of vector bundle over a compact Riemannian manifold of dimension 2

Abstract

Abstract. Let (M, g) be a 2-dimensional compact Riemannian manifold. In this paper, we use the method of blowing up analysis to prove several Moser-Trdinger type inequalities for vector bundle over (M, g). We also derive an upper bound of such inequalities under the assumption that blowing up occur.

Concentration compactness of Moser functionals on manifolds

“…1.5 with a k = 0 and f k = 1. However, in Li (2001Li ( , 2005, we proved the blow up (i.e. c k → +∞) will never happen, while in this paper, we need the function sequence u k to blow up and to concentrate at p. For this sake, we will chose suitable a k so that the blow up happen, and suitable f k so that |∇ g u k | n dV g δ p .…”

“…We mainly follows the ideas in Li 2001Li , 2005 in which we considered Eq. 1.5 with a k = 0 and f k = 1.…”

On Trudinger–Moser type inequalities involving Sobolev–Lorentz spaces

“…The existence of extremals in the TM-inequality was obtained by Carleson and Chang [10] for Ω = B 1 (0) ⊂ R 2 , by Flucher [23] for arbitrary bounded domains in R 2 , and by Lin [33] for bounded domains in R N ; de Figueiredo et al [19] gave an alternative proof and some generalization, using an optimal normalized concentrating sequence. For extensions of the TM-inequality to manifolds, see Cherrier [12], Fontana [25], Li [31,32], Yang [44]. Related elliptic equations with "critical" TM growth were considered by Adimurthi [3] and de Figueiredo et al [18], giving sufficient conditions on the lower order terms for the existence of solutions; in de Figueiredo and Ruf [21] the non-existence of radial solutions was proved for equations with critical TM-growth whose lower order term does not satisfy the above existence conditions.…”

A Moser-Trudinger inequality on the boundary of a compact Riemann surface

“…Similar to section 2 in [20], for a constant δ ∈ (0, µ(∂M)) we define Proof. The proof will be accomplished through the following four steps.…”