Extremal functions for Trudinger–Moser inequalities of Adimurthi–Druet type in dimension two

Abstract

Combining Carleson-Chang's result [9] with blow-up analysis, we prove existence of extremal functions for certain Trudinger-Moser inequalities in dimension two.

Existence of solutions to a class of Kazdan-Warner equations on compact Riemannian surface

“…It follows from the Poincaré inequality that if α < λ 1 (Σ), then u 1,α = Σ (|∇ g u| 2 − αu 2 )dv g 1/2 defines a Sobolev norm on H. In a previous work [15], using the method of blow-up analysis, we proved the following: for any α < λ 1 (Σ), there holds sup u∈H, u 1,α ≤1 Σ e 4πu 2 dv g < ∞ (6) and the supremum is attained. As a consequence of (6), there exists some constant C depending only on (Σ, g) and α < λ 1 (Σ) such that for all u ∈ H,…”

A gradient flow for the prescribed Gaussian curvature problem on a closed Riemann surface with conical singularity

“…This together with (30), (31) and (38) leads to (34). In view of (22), ∇L(u(t)) = (∆ g + I) −1 (Ke 2u(t) ).…”

“…We remark that (47) is a weak version of Trudinger-Moser inequality. For related strong versions, we refer the reader to recent works [1,26,38,39,40] and the references therein.…”

A Trudinger–Moser inequality for a conical metric in the unit ball

“…In this section, we use the standard blow-up analysis to prove Theorem 2. This method was originally introduced by Ding-Jost-Li-Wang [9] and Li [17,18], and extensively employed by Yang [34,35,36,37], Lu-Yang [23], Li-Ruf [19], Zhu [41], doÓ-de Souza [10,11], Li-Yang [16], Li [15], Nguyen [25,26] and others. Comparing with the case p ≤ N [41,26], we need more analysis to deal with the general case p > 1.…”