On a class of Kazdan–Warner equations

Abstract

Let (Σ, g) be a compact Riemannian surface without boundary and W 1,2 (Σ) be the usual Sobolev space. For any real number p > 1 and α ∈ R , we define a functional Jα,8π(u) = 1 2 (∫ Σ |∇gu| 2 dvg − α(∫ Σ |u| p dvg) 2/p) − 8π log ∫ Σ he u dvg on a function space H = { u ∈ W 1,2 (Σ) : ∫ Σ udvg = 0 } , where h is a positive smooth function on Σ. Denote λ1,p(Σ) = inf u∈H, ∫ Σ |u| p dvg =1 ∫ Σ |∇gu| 2 dvg. If α < λ1,p(Σ) and Jα,8π has no minimizer on H , then we obtain the exact value of infH Jα,8π by using a method of blow-up analysis. Hence, if infH Jα,8π is not equal to that value, then Jα,8π|H has a critical point that satisfies a Kazdan-Warner equation. This recovers a recent result of Yang and Zhu

A Trudinger-Moser inequality involving L^p -norm on a closed Riemann surface

“…Moreover, for sufficiently small α > 0, the supremum is attained. Analogs of (3) are naturally expected for the cases of do Ó-de Souza [6,8], Nguyen [19,20], Li [12], Li-Yang [13], Zhu [33], Fang-Zhang [9] and Yang-Zhu [29,30].…”

Extremal functions for a class of trace Trudinger-Moser inequalities

“…Theorem 2. Let (Σ, g) be a compact Riemannian surface with smooth boundary ∂Σ, ℓ be an positive integer and λ ℓ+1 (∂Σ) be defined by (10). For any 0 ≤ α < λ ℓ+1 (∂Σ), we let…”

“…For theirs proofs, we employ the method of blow-up analysis, which was originally used by Carleson-Chang [5], Ding-Jost-Li-Wang [8], Adimurthi-Struwe [2], Li [15], Liu [20], Li-Liu [17], and Yang [31,32]. This method is now standard, for related works, we refer the reader to Adimurthi-Druet [1], do Ó-de Souza [7,9], Nguyen [22,23], Li [13], Li-Yang [14], Zhu [38], Fang-Zhang [10] and Yang-Zhu [34,35].…”

A Trudinger-Moser inequality with mean value zero on a compact Riemann surface with boundary

“…For the proof, we employ the method of blow-up analysis, which was originally used by Carleson-Chang [4], Ding-Jost-Li-Wang [9], Adimurthi-Struwe [2], Li [14], and Yang [31,33]. For related works, we refer the reader to Adimurthi-Druet [1], do Ó-de Souza [8,10], Nguyen [21,22], Li-Yang [16], Zhu [39], Fang-Zhang [11], Yang-Zhu [35,36] and Csató-Nguyen-Roy [7]. We should point out that the blow-up occurs on the boundary ∂Σ in our case.…”