In this paper,using the method of blow-up analysis,we obtained a Trudinger–Moser inequality involving L^(p)-norm on a closed Riemann surface and proved the existence of an extremal function for the corresponding Trud...In this paper,using the method of blow-up analysis,we obtained a Trudinger–Moser inequality involving L^(p)-norm on a closed Riemann surface and proved the existence of an extremal function for the corresponding Trudinger–Moser functional.展开更多
Let x=(x',x'')with x'∈Rk and x''∈R^N-k andΩbe a x'-symmetric and bounded domain in R^N(N≥2).We show that if 0≤a≤k-2,then there exists a positive constant C>0 such that for all x...Let x=(x',x'')with x'∈Rk and x''∈R^N-k andΩbe a x'-symmetric and bounded domain in R^N(N≥2).We show that if 0≤a≤k-2,then there exists a positive constant C>0 such that for all x'-symmetric function u∈C0^∞(Ω)with∫Ω|■u(x)|^N-a|x'|^-adx≤1,the following uniform inequality holds1/∫Ω|x|^-adx∫Ωe^βa|u|N-a/N-a-1|x'|^-adx≤C,whereβa=(N-a)(2πN/2Γ(k-a/2)Γ(k/2)/Γ(k/2)r(N-a/2))1/N-a-1.Furthermore,βa can not be replaced by any greater number.As the application,we obtain some weighted Trudinger–Moser inequalities for x-symmetric function on Grushin space.展开更多
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...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).展开更多
基金Supported by the Outstanding Innovative Talents Cultivation Funded Programs 2020 of Renmin University of China。
文摘In this paper,using the method of blow-up analysis,we obtained a Trudinger–Moser inequality involving L^(p)-norm on a closed Riemann surface and proved the existence of an extremal function for the corresponding Trudinger–Moser functional.
基金Supported by the National Natural Science Foundation of China(Grant No.11201346)。
文摘Let x=(x',x'')with x'∈Rk and x''∈R^N-k andΩbe a x'-symmetric and bounded domain in R^N(N≥2).We show that if 0≤a≤k-2,then there exists a positive constant C>0 such that for all x'-symmetric function u∈C0^∞(Ω)with∫Ω|■u(x)|^N-a|x'|^-adx≤1,the following uniform inequality holds1/∫Ω|x|^-adx∫Ωe^βa|u|N-a/N-a-1|x'|^-adx≤C,whereβa=(N-a)(2πN/2Γ(k-a/2)Γ(k/2)/Γ(k/2)r(N-a/2))1/N-a-1.Furthermore,βa can not be replaced by any greater number.As the application,we obtain some weighted Trudinger–Moser inequalities for x-symmetric function on Grushin space.
文摘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).
