A Weighted Trudinger-Moser Inequality and Its Extremal Functions in Dimension Two  

作　　者：ZHAO Juan YU Pengxiu 

机构地区：[1]School of Mathematics,Renmin University of China,Beijing 100872,China

出　　处：《Journal of Partial Differential Equations》2024年第4期402-416,共15页偏微分方程(英文版)

基　　金：supported by the Outstanding Innovative Talents Cultivation Funded Programs 2023 of Renmin University of China。

摘　　要：Let Ω be a smooth bounded domian in R^(2),H_(0)^(1)(Ω)be the standard Sobolev space,and λ_(f)(Ω)be the first weighted eigenvalue of the Laplacian,namely,λ_(f)(Ω)=inf_(u∈H_(0)^(1)(Ω),∫Ωu^(2)dx=1)∫_(Ω)|∇u|^(2)fdx,where f is a smooth positive function onΩ.In this paper,using blow-up analysis,we prove sup_(u∈H_(0)^(1)(Ω),∫^(Ω)|∇u|^(2)fdx)≤1∫_(Ω)e^(4πfu^(2)(1+α||u||_(2)^(2)))dx<+∞ for any 0≤α<λ_(f)(Ω).Furthermore,extremal functions for the above inequality exist when α>0 is chosen sufficiently small.

关 键 词：Trudinger-Moser inequality extremal functions blow-up analysis 

分 类 号：O175[理学—数学]