A Hardy–Trudinger–Moser Inequality Involving LpNorm in the Hyperbolic Space

Let B be the unit disc in R2,H be the completion of C0∞(B)under the norm||u||=(∫B|▽U|^2dx-∫Bu^2/(1-|x|^2)^2dx)^1/2,U∈C^∞0(B).By the method of blow-up analysis and an argument of rearrangement with respect to the standard hyperbolic metric dv=dx/(1-|x|^2)^2,we prove that,for any fixedα,0≤α<λp(B)-infu∈■,u≠0||U||^2■/||U||^2p,the supremum u∈■、||su||■≤1∫B^e^4π(1+α||su||^2p)U^2dx<+∞,■p>1.This is an analog of early results of Lu–Yang(Discrete Contin.Dyn.Syst.,2009)and Yang(Trans.Amer.Math.Soc.,2007),and extends those of Wang–Ye(Adv.Math.,2012)and Yang–Zhu(Ann.Global Anal.Geom.,2016).