A generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2019,（4）
• 页码范围: 699-718
• 页数: 20页
• 语种: 英文

摘要

In this paper,using the method of blow-up analysis,we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities.Precisely,let (∑,D) be such a surface with divisor D =∑mi=1 βiPi,where βi ＞-1 and pi ∈ ∑ for i =1 m,and g be a metric representing D.Denote b0 =4π(1 + min1≤i≤m βi).Suppose ψ:∑→ R is a continuous function with ∫∑ ψdvg ≠ 0 and define λ** (∑,g) =u∈H1 (∑,g),f∑inf ψudvg=0,f∑ u2dvg=1∫∑|▽gu|2dvg.Then for any α ∈ [0,λ**1(∑,g)),we haveu∈H1 (∑,g),∫∑ ψu=0,f∑sup |▽gu|2dvg-α ∫∑ u2dvg≤1∫∑eb0u2dvg＜+∞.When b ＞ b0,the integrals ∫∑ ebu2dvg are still finite,but the supremum is infinity.Moreover,we prove that the extremal function for the above inequality exists.