Let N≥ 2,αN =NωN 1/(N-1)N-1,where ωN-1 denotes the area of the unit sphere in IRN.In this note,we prove that for any 0 < α < αN and anyβ > 0,the supremum u∈W1,N(IRN),sup||u||W1,N(IRN)≤1∫IRN|u|β(eα|u|N/N-1-N-2Σj=0αj/j!|u|Nj/N-1)dxcan be attained by some function u ∈ W1'N (RN) with || u|| w1,N (IRN) =1.Moreover,when α ≥ αN,the above supremum is infinity.
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机译:让N≥2,αn=nωn1/(n-1)n-1,其中ωn-1表示Irn中的单位球体的区域。本说明,我们证明了任何0 <α<αn和任何β> 0,SupremumU∈w1,n(Irn),sup ||||| w1,n(Irn)≤1∫Rn|β(eα| n / n-1-n-2σj=0αj/ J!| U | NJ / N-1)DXCAN由某种功能U∈W1'N(RN)与||你|| W1,N(IRN)= 1.moreover,当α≥αn时,上述超级是无穷大。
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