THE DIFFERENTIAL EQUATION Δu = 8π − 8πhe^u ON A COMPACT RIEMANN SURFACE

Abstract

Let M be a compact Riemann surface, h(x) a positive smooth function on M . In this paper, we consider the functionalWe give a sufficient condition under which J achieves its minimum. * dg-ga/9710005

Trudinger-Moser inequalities on a closed Riemannian surface with the action of a finite isometric group

“…In the case G = {Id}, where Id : Σ → Σ is the identity map, Theorems 1-3 are reduced to that of Ding-Jost-Li-Wang [6]. Though we are in the spirit of Ding-Jost-Li-Wang [6] for the proof of the general case of Theorems 1-3, many technical difficulties need to be smoothed. The first issue is to construct Green functions having many singular points by using elliptic estimates and the symmetric properties of (Σ, g).…”

“…This improved the result in [6]. By assuming Σ to be a flat torus and h a positive smooth function with certain symmetrization, Wang [18] obtained an analog of [6] for ρ = 16π.…”

“…Our aim is to extend Ding-Jost-Li-Wang's result [6] to closed Riemannian surface with an isometric group action. More precisely, let (Σ, g), G and ℓ be defined as in (4).…”

Extremal functions for a singular Hardy-Moser-Trudinger inequality

“…The existence of extremal functions for Moser-Trudinger inequality originated in [7]. This result was generalized by M. Struwe [19], F. Flucher [12], K. Lin [16], W. Ding, J. Jost, J. Li and G. Wang [11], Adimurthi and M. Struwe [3], Y. Li [15], Adimurthi and O. Druet [1], and so on. Compared with [26], there are difficulties caused by the term |x| −2β in the process of blow-up analysis.…”

Min–max schemes for SU(3) Toda systems

“…which gives coercivity of Iρ forρ < 4π (see also [18,40] for the borderline case ρ = 4π). Concerning compactness for (5), a useful result was proved in [10,30], giving an alternative behaviour on solutions.…”