In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to...In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to the concentration level of singular Trudinger-Moser functional on the unit ball.展开更多
Let IB be the unit disc in R^2,H be the completion of C∞0(B)under the norm||u||H=(∫B|▽u|^2dx-∫Bu^2/(1-|x^2|^2dx)^1/2,■u∈C∞0(B).Using blow-up analysis,we prove that for anyγ≤4π,the supremum sup u∈H,||u||1,h...Let IB be the unit disc in R^2,H be the completion of C∞0(B)under the norm||u||H=(∫B|▽u|^2dx-∫Bu^2/(1-|x^2|^2dx)^1/2,■u∈C∞0(B).Using blow-up analysis,we prove that for anyγ≤4π,the supremum sup u∈H,||u||1,h≤1∫Beγu^2dx can be attained by some function u0∈H with||u0||1,h=1,where is a decreasingly nonnegative,radially symmetric function,and satisfies a coercive cond让ion.Namely there exists a constantδ>0 satisfying||u||21,h=||u||2H-∫Bhu^2dx≥δ||u||H^2,■u∈H.This extends earlier results of Wang-Ye[1]and Yang-Zhu[2].展开更多
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(Σ, D) be such a surface∑with divi...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(Σ, D) be such a surface∑with divisor D =Σ_(i=1)~mβ_(ipi), where β_i >-1 and p_i ∈Σ for i = 1,..., m, and g be a metric representing D.Denote b_0 = 4π(1 + min_(1≤i≤mβ_i). Suppose ψ : Σ→ R is a continuous function with ∫_Σψdv_g ≠0 and define■Then for any α∈ [0, λ_1^(**)(Σ, g)), we have■When b > b0, the integrals■are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.展开更多
The classical critical Trudinger-Moser inequality in R^(2)under the constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1 was established through the technique of blow-up analysis or the rearrangement-free argument:for anyτ&g...The classical critical Trudinger-Moser inequality in R^(2)under the constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1 was established through the technique of blow-up analysis or the rearrangement-free argument:for anyτ>0,it holds that sup u∈H^(1)(r^(2)fR^(2)(τ|U|^(2)+|▽u|^(2)dx≤1∫_(R^(2)(E^4π|u|^(2)-1)dx≤C(τ)<+∞)))and 4πis sharp.However,if we consider the less restrictive constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1,where V(x)is nonnegative and vanishes on an open set in R^(2),it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π.The loss of a positive lower bound of the potential V(x)makes this problem become fairly nontrivial.The main purpose of this paper is twofold.We will first establish the Trudinger-Moser inequality sup u∈H^(1)(r^(2)fR^(2)(τ|U|^(2)+|▽u|^(2)dx≤1∫_(R^(2)(E^4π|u|^(2)-1)dx≤C(τ)<+∞)))when V is nonnegative and vanishes on an open set in R^(2).As an application,we also prove the existence of ground state solutions to the following Sciridinger equations with critical exponeitial growth:-Δu+V(x)u=f u)in R^(2),(0.1)where V(x)≥0 and vanishes on an open set of R^(2)and f has critical exponential growth.Having a positive constant lower bound for the potential V(x)(e.g.,the Rabinowitz type potential)has been the standard assumption when one deals with the existence of solutions to the above Schrodinger equations when the nonlinear term has the exponential growth.Our existence result seems to be the first one without this standard assumption.展开更多
In this paper,we are concerned with a sharp fractional Trudinger-Moser type inequality in bounded intervals of IR under the Lorentz-Sobolev norms constraint.For any 1<q<∞andβ≤(√π)q'≡βq'q'=q/q-...In this paper,we are concerned with a sharp fractional Trudinger-Moser type inequality in bounded intervals of IR under the Lorentz-Sobolev norms constraint.For any 1<q<∞andβ≤(√π)q'≡βq'q'=q/q-1,we obtain u∈Н1/2,2(I),^sup||(-△)1/4u||2,q≤1^(∫Ieβ|u(x)|q'dx≤c0|I|),andβq is optimal in the sense that u∈Н1/2,2(I),^sup||(-△)1/4u||2,q≤1^(∫Ieβ|u(x)|q'dx+∞),for anyβ>βq.Furthermore,when q is even,we obtain u∈Н1/2,2(I),^sup||(-△)1/4u||2,q≤1^(∫Ih(u)eβq|u(x)|q')dx≤+∞),for any function h:[0,∞→[0,∞)with lim t→∞h(t)=∞.As for the key tools of proof,we use Green functions for fractional Laplace operators and the rearrangement of a convolution to the rearrangement of the convoluted functions.展开更多
In this paper,we obtained the extremal function for a weighted singular Trudinger-Moser inequality by blow-up analysis in the Euclidean space R^(2).This extends recent results of Hou(J.Inequal.Appl.,2018)and similar r...In this paper,we obtained the extremal function for a weighted singular Trudinger-Moser inequality by blow-up analysis in the Euclidean space R^(2).This extends recent results of Hou(J.Inequal.Appl.,2018)and similar result was proved by Zhu(Sci.China Math.,2021).展开更多
In this note, we give a new proof of subcritical Trudinger-Moser inequality on R^n. All the existing proofs on this inequality are based on the rearrangement ar-gument with respect to functions in the Sobolev space W^...In this note, we give a new proof of subcritical Trudinger-Moser inequality on R^n. All the existing proofs on this inequality are based on the rearrangement ar-gument with respect to functions in the Sobolev space W^1,n （R^n）. Our method avoids this technique and thus can be used in the Riemamlian manifold case and in the entire Heisenberg group.展开更多
The first aim of this article is to study the sharp singular(two-weight)Trudinger-Moser inequalities with Finsler norms on R^(2).The second goal is to propose a different approach to study a vanishing-concentration-co...The first aim of this article is to study the sharp singular(two-weight)Trudinger-Moser inequalities with Finsler norms on R^(2).The second goal is to propose a different approach to study a vanishing-concentration-compactness principle for the Trudinger-Moser inequalities and use this to investigate the existence and the nonexistence of the maximizers for the Trudinger-Moser inequalities in the subcritical regions.Finally,by applying our Finsler Trudinger-Moser inequalities to suitable Finsler norms,we derive the sharp affine Trudinger-Moser inequalities which are essentially stronger than the Trudinger-Moser inequalities with standard energy of the gradient.展开更多
Let B be a unit ball in R^(2),W_(0)^(1,2)(B)be the standard Sobolev space.For anyϵ>0,de Figueiredo,doÓ,dos Santons,Yang and Zhu proved the existence of extremals of a Trudinger-Moser inequality in the unit bal...Let B be a unit ball in R^(2),W_(0)^(1,2)(B)be the standard Sobolev space.For anyϵ>0,de Figueiredo,doÓ,dos Santons,Yang and Zhu proved the existence of extremals of a Trudinger-Moser inequality in the unit ball.Precisely,u∈W_(0)^(1,2)(B)^(sup),∫_(B)|▽u|^(2)dx≤1∫_(B)^(|x|^(2∈)e^(4π(1+∈)u^(2)dx)can be attained by some radially symmetric function u_(∈)∈W_(0)^(1,2)(B)∫_(B)with∫_(B)|▽u|^(2)dx=1.In this note,we concern the compactness of the function family{uϵ}ϵ>0 and prove that up to a subsequence uϵconverges to some function u 0 in C^(1)(B)asϵ→0.Furthermore,u_(0) is an extremal function of the supremum u∈W_(0)^(1,2)(B)^(sup),∫_(B)|▽u|^(2)dx≤1∫_(B)e^(4πu^(2)dx)Let us explain the result in geometry.Denoteω_(0)=dx_(1)^(2)+dx_(2)^(2)be the standard Euclidean metricω_(∈)|x|^(2ϵ)ω_(0)for xϵB..Then the extremal family{u_(∈)}∈>0 of the following Trudinger-Moser functionals∫_(B)e^(4π(1+∈)u^(2))dvω_(∈)under the constraint W_(0)^(1,2)(B)and∫_(B)|▽ω_∈)u|^(2)dvω_∈)≤1 is compact asϵ→0.展开更多
In this paper,the author concerns two trace Trudinger-Moser inequalities and obtains the corresponding extremal functions on a compact Riemann surface(Σ,g)with smooth boundaryθΣ.Explicitly,letλ_(1)(θΣ)=inf_(u∈W...In this paper,the author concerns two trace Trudinger-Moser inequalities and obtains the corresponding extremal functions on a compact Riemann surface(Σ,g)with smooth boundaryθΣ.Explicitly,letλ_(1)(θΣ)=inf_(u∈W^(1,2)(Σ,g),∫_(θΣ)uds_(g)=0,u≠0∫_(Σ)(|▽_(g)u|^(2)+u^(2))dv_(g)/∫_(θΣ)u^(2)ds_(g)and H={u∈W^(1,2)(Σ,g):∫_(Σ)(|▽_(g)u|^(2)+u^(2))dv_(g)-α∫_(θΣ)u^(2)ds_(g)≤1 and∫_(θΣ)uds_(g)=0},where W^(1,2)(Σ,g)denotes the usual Sobolev space and▽g stands for the gradient operator.By the method of blow-up analysis,we obtain sup_(u∈H)∫_(θΣ)e^(πu^(2))ds_(g){<+∞,0≤α﹤λ_(1)(∂Σ),=+∞,α≥λ_(1)(∂Σ)Moreover,the author proves the above supremum is attained by a function u∈H∩C^(∞)(∑)for any 0≤α<λ_(1)(θΣ).Further,he extends the result to the case of higher order eigenvalues.The results generalize those of[Li,Y.and Liu,P.,Moser-Trudinger inequality on the boundary of compact Riemannian surface,Math.Z.,250,2005,363–386],[Yang,Y.,Moser-Trudinger trace inequalities on a compact Riemannian surface with boundary,Pacific J.Math.,227,2006,177–200]and[Yang,Y.,Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two,J.Diff.Eq.,258,2015,3161–3193].展开更多
In this paper,we present the singular supercritical Trudinger-Moser inequalities on the unit ball B in Rn,where n≥2.More precisely,we show that for any given α>0 and 0<t<n,then the following two inequalitie...In this paper,we present the singular supercritical Trudinger-Moser inequalities on the unit ball B in Rn,where n≥2.More precisely,we show that for any given α>0 and 0<t<n,then the following two inequalities hold for ∀u∈W^1,n0,r(B),∫Bsup∣▽u∣^ndx≤1∫Bexp((αn,t+∣x∣^α∣)u∣^n/n-1)/∣x∣^tdx<∞ and ∫Bsup∣▽u∣^ndx≤1∫Bexp(αn,t+∣u∣^n/n-1+∣x∣^α)/∣x∣^tdx<∞.We also consider the problem of the sharpness of the constantαn,t.Furthermore,by employing the method of estimating the lower bound and using the concentration-compactness principle,we establish the existence of extremals.These results extend the known results when t=0 to the singular version for 0<t<n.展开更多
Let(∑,g)be a compact Riemannian surface,pj∈∑,βj>-1,forj=1,..i,m.Denoteβ=min{0,β1……Bm}.Let H∈C^(0)(∑)be a positivefunction and h(x)=H(x)(dg(x,pj))^(2βj),where dg(x,pj)denotes the geodesic distance between...Let(∑,g)be a compact Riemannian surface,pj∈∑,βj>-1,forj=1,..i,m.Denoteβ=min{0,β1……Bm}.Let H∈C^(0)(∑)be a positivefunction and h(x)=H(x)(dg(x,pj))^(2βj),where dg(x,pj)denotes the geodesic distance between x and p;for each j=1,...,m.In this paper,using a method of blow-up analysis,we prove that the functional J(u)=1/2∫∑|ΔgU|^(2)dV_(g)+8π(1+β)1/volg(∑)∫∑udvg-8π(1+β)log∫_(∑)he^(U)dv_(g)is bounded from below on the Sobolev space w^(1,2)(g).展开更多
Let W^(1,n)(R^(n))be the standard Sobolev space.For any T>0 and p>n>2,we denote■Define a norm in W^(1,n)(R^(n))by■where 0≤α<λ_(n,p).Using a rearrangement argument and blow-up analysis,we will prove■c...Let W^(1,n)(R^(n))be the standard Sobolev space.For any T>0 and p>n>2,we denote■Define a norm in W^(1,n)(R^(n))by■where 0≤α<λ_(n,p).Using a rearrangement argument and blow-up analysis,we will prove■can be attained by some function u_(0)∈W^(1,n)(R^(n))∩C^(1)(R^(n))with ||u_(0)||_(n,p)=1,here a_(n)=n■_(n-1)^(1/n-1) and ■_(n-1) is the measure of the unit sphere in R^(n).展开更多
文摘In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to the concentration level of singular Trudinger-Moser functional on the unit ball.
文摘Let IB be the unit disc in R^2,H be the completion of C∞0(B)under the norm||u||H=(∫B|▽u|^2dx-∫Bu^2/(1-|x^2|^2dx)^1/2,■u∈C∞0(B).Using blow-up analysis,we prove that for anyγ≤4π,the supremum sup u∈H,||u||1,h≤1∫Beγu^2dx can be attained by some function u0∈H with||u0||1,h=1,where is a decreasingly nonnegative,radially symmetric function,and satisfies a coercive cond让ion.Namely there exists a constantδ>0 satisfying||u||21,h=||u||2H-∫Bhu^2dx≥δ||u||H^2,■u∈H.This extends earlier results of Wang-Ye[1]and Yang-Zhu[2].
基金supported by National Natural Science Foundation of China (Grant No. 11401575)
文摘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(Σ, D) be such a surface∑with divisor D =Σ_(i=1)~mβ_(ipi), where β_i >-1 and p_i ∈Σ for i = 1,..., m, and g be a metric representing D.Denote b_0 = 4π(1 + min_(1≤i≤mβ_i). Suppose ψ : Σ→ R is a continuous function with ∫_Σψdv_g ≠0 and define■Then for any α∈ [0, λ_1^(**)(Σ, g)), we have■When b > b0, the integrals■are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.
基金supported by National Natural Science Foundation of China(Grant No.11901031)supported by a Simons grant from the Simons Foundation+1 种基金supported by National Natural Science Foundation of China(Grant Nos.12071185 and 12061010)Outstanding Young Foundation of Jiangsu Province(Grant No.BK20200042)。
文摘The classical critical Trudinger-Moser inequality in R^(2)under the constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1 was established through the technique of blow-up analysis or the rearrangement-free argument:for anyτ>0,it holds that sup u∈H^(1)(r^(2)fR^(2)(τ|U|^(2)+|▽u|^(2)dx≤1∫_(R^(2)(E^4π|u|^(2)-1)dx≤C(τ)<+∞)))and 4πis sharp.However,if we consider the less restrictive constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1,where V(x)is nonnegative and vanishes on an open set in R^(2),it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π.The loss of a positive lower bound of the potential V(x)makes this problem become fairly nontrivial.The main purpose of this paper is twofold.We will first establish the Trudinger-Moser inequality sup u∈H^(1)(r^(2)fR^(2)(τ|U|^(2)+|▽u|^(2)dx≤1∫_(R^(2)(E^4π|u|^(2)-1)dx≤C(τ)<+∞)))when V is nonnegative and vanishes on an open set in R^(2).As an application,we also prove the existence of ground state solutions to the following Sciridinger equations with critical exponeitial growth:-Δu+V(x)u=f u)in R^(2),(0.1)where V(x)≥0 and vanishes on an open set of R^(2)and f has critical exponential growth.Having a positive constant lower bound for the potential V(x)(e.g.,the Rabinowitz type potential)has been the standard assumption when one deals with the existence of solutions to the above Schrodinger equations when the nonlinear term has the exponential growth.Our existence result seems to be the first one without this standard assumption.
文摘In this paper,we are concerned with a sharp fractional Trudinger-Moser type inequality in bounded intervals of IR under the Lorentz-Sobolev norms constraint.For any 1<q<∞andβ≤(√π)q'≡βq'q'=q/q-1,we obtain u∈Н1/2,2(I),^sup||(-△)1/4u||2,q≤1^(∫Ieβ|u(x)|q'dx≤c0|I|),andβq is optimal in the sense that u∈Н1/2,2(I),^sup||(-△)1/4u||2,q≤1^(∫Ieβ|u(x)|q'dx+∞),for anyβ>βq.Furthermore,when q is even,we obtain u∈Н1/2,2(I),^sup||(-△)1/4u||2,q≤1^(∫Ih(u)eβq|u(x)|q')dx≤+∞),for any function h:[0,∞→[0,∞)with lim t→∞h(t)=∞.As for the key tools of proof,we use Green functions for fractional Laplace operators and the rearrangement of a convolution to the rearrangement of the convoluted functions.
文摘In this paper,we obtained the extremal function for a weighted singular Trudinger-Moser inequality by blow-up analysis in the Euclidean space R^(2).This extends recent results of Hou(J.Inequal.Appl.,2018)and similar result was proved by Zhu(Sci.China Math.,2021).
文摘In this note, we give a new proof of subcritical Trudinger-Moser inequality on R^n. All the existing proofs on this inequality are based on the rearrangement ar-gument with respect to functions in the Sobolev space W^1,n （R^n）. Our method avoids this technique and thus can be used in the Riemamlian manifold case and in the entire Heisenberg group.
文摘The first aim of this article is to study the sharp singular(two-weight)Trudinger-Moser inequalities with Finsler norms on R^(2).The second goal is to propose a different approach to study a vanishing-concentration-compactness principle for the Trudinger-Moser inequalities and use this to investigate the existence and the nonexistence of the maximizers for the Trudinger-Moser inequalities in the subcritical regions.Finally,by applying our Finsler Trudinger-Moser inequalities to suitable Finsler norms,we derive the sharp affine Trudinger-Moser inequalities which are essentially stronger than the Trudinger-Moser inequalities with standard energy of the gradient.
基金Supported by the National Natural Science Foundation of China(Grant No.12201234)the Natural Science Foundation of Anhui Province of China(Grant No.2008085MA07)。
文摘Let B be a unit ball in R^(2),W_(0)^(1,2)(B)be the standard Sobolev space.For anyϵ>0,de Figueiredo,doÓ,dos Santons,Yang and Zhu proved the existence of extremals of a Trudinger-Moser inequality in the unit ball.Precisely,u∈W_(0)^(1,2)(B)^(sup),∫_(B)|▽u|^(2)dx≤1∫_(B)^(|x|^(2∈)e^(4π(1+∈)u^(2)dx)can be attained by some radially symmetric function u_(∈)∈W_(0)^(1,2)(B)∫_(B)with∫_(B)|▽u|^(2)dx=1.In this note,we concern the compactness of the function family{uϵ}ϵ>0 and prove that up to a subsequence uϵconverges to some function u 0 in C^(1)(B)asϵ→0.Furthermore,u_(0) is an extremal function of the supremum u∈W_(0)^(1,2)(B)^(sup),∫_(B)|▽u|^(2)dx≤1∫_(B)e^(4πu^(2)dx)Let us explain the result in geometry.Denoteω_(0)=dx_(1)^(2)+dx_(2)^(2)be the standard Euclidean metricω_(∈)|x|^(2ϵ)ω_(0)for xϵB..Then the extremal family{u_(∈)}∈>0 of the following Trudinger-Moser functionals∫_(B)e^(4π(1+∈)u^(2))dvω_(∈)under the constraint W_(0)^(1,2)(B)and∫_(B)|▽ω_∈)u|^(2)dvω_∈)≤1 is compact asϵ→0.
基金supported by the Outstanding Innovative Talents Cultivation Funded Programs 2020 of Renmin University of China
文摘In this paper,the author concerns two trace Trudinger-Moser inequalities and obtains the corresponding extremal functions on a compact Riemann surface(Σ,g)with smooth boundaryθΣ.Explicitly,letλ_(1)(θΣ)=inf_(u∈W^(1,2)(Σ,g),∫_(θΣ)uds_(g)=0,u≠0∫_(Σ)(|▽_(g)u|^(2)+u^(2))dv_(g)/∫_(θΣ)u^(2)ds_(g)and H={u∈W^(1,2)(Σ,g):∫_(Σ)(|▽_(g)u|^(2)+u^(2))dv_(g)-α∫_(θΣ)u^(2)ds_(g)≤1 and∫_(θΣ)uds_(g)=0},where W^(1,2)(Σ,g)denotes the usual Sobolev space and▽g stands for the gradient operator.By the method of blow-up analysis,we obtain sup_(u∈H)∫_(θΣ)e^(πu^(2))ds_(g){<+∞,0≤α﹤λ_(1)(∂Σ),=+∞,α≥λ_(1)(∂Σ)Moreover,the author proves the above supremum is attained by a function u∈H∩C^(∞)(∑)for any 0≤α<λ_(1)(θΣ).Further,he extends the result to the case of higher order eigenvalues.The results generalize those of[Li,Y.and Liu,P.,Moser-Trudinger inequality on the boundary of compact Riemannian surface,Math.Z.,250,2005,363–386],[Yang,Y.,Moser-Trudinger trace inequalities on a compact Riemannian surface with boundary,Pacific J.Math.,227,2006,177–200]and[Yang,Y.,Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two,J.Diff.Eq.,258,2015,3161–3193].
基金Supported by NSFC(Grant No.11901031)Beijing Institute of Technology Research Fund Program for Young Scholars(Grant No.3170012221903)。
文摘In this paper,we present the singular supercritical Trudinger-Moser inequalities on the unit ball B in Rn,where n≥2.More precisely,we show that for any given α>0 and 0<t<n,then the following two inequalities hold for ∀u∈W^1,n0,r(B),∫Bsup∣▽u∣^ndx≤1∫Bexp((αn,t+∣x∣^α∣)u∣^n/n-1)/∣x∣^tdx<∞ and ∫Bsup∣▽u∣^ndx≤1∫Bexp(αn,t+∣u∣^n/n-1+∣x∣^α)/∣x∣^tdx<∞.We also consider the problem of the sharpness of the constantαn,t.Furthermore,by employing the method of estimating the lower bound and using the concentration-compactness principle,we establish the existence of extremals.These results extend the known results when t=0 to the singular version for 0<t<n.
基金the National Science Foundation of China(GrantNo.11401575).
文摘Let(∑,g)be a compact Riemannian surface,pj∈∑,βj>-1,forj=1,..i,m.Denoteβ=min{0,β1……Bm}.Let H∈C^(0)(∑)be a positivefunction and h(x)=H(x)(dg(x,pj))^(2βj),where dg(x,pj)denotes the geodesic distance between x and p;for each j=1,...,m.In this paper,using a method of blow-up analysis,we prove that the functional J(u)=1/2∫∑|ΔgU|^(2)dV_(g)+8π(1+β)1/volg(∑)∫∑udvg-8π(1+β)log∫_(∑)he^(U)dv_(g)is bounded from below on the Sobolev space w^(1,2)(g).
基金supported by National Science Foundation of China(Grant No.12201234)Natural Science Foundation of Anhui Province of China(Grant No.2008085MA07)the Natural Science Foundation of the Education Department of Anhui Province(Grant No.KJ2020A1198).
文摘Let W^(1,n)(R^(n))be the standard Sobolev space.For any T>0 and p>n>2,we denote■Define a norm in W^(1,n)(R^(n))by■where 0≤α<λ_(n,p).Using a rearrangement argument and blow-up analysis,we will prove■can be attained by some function u_(0)∈W^(1,n)(R^(n))∩C^(1)(R^(n))with ||u_(0)||_(n,p)=1,here a_(n)=n■_(n-1)^(1/n-1) and ■_(n-1) is the measure of the unit sphere in R^(n).
