具临界指数的Baouendi-Grushin方程显式整解及Sobolev嵌入常数

给出了具临界指数的Baouendi-Grushin方程Pu=-uQQ+-22的显式解为u=c[(2|z|2)2+4|t|2]-Q4-2,其中P=Δz+|z|2Δt为α=1时的广义Baouendi-Grushin算子,z∈Rn,t∈Rm,Q=n+2m为齐次维数,c=[(Q-2)n2]Q4-2,>0.本文还由此导出算子P的精确Sobolev不等式中的嵌入常数为S=2Qmπ-2(nn++2mm){n[n+2(m-1)]}21×Γ(n+m)Γ(n+2m)1n+2m,极值函数为[(1+|z|2)2+4|t|2]-41.当n=m=1时,本文的结论与Beckner[4]的结果一致.

The explicit entire solution of the Baouendi-Grushin equation with critical exponent and the Sobolev embedding constant

In this paper,the explicit entire solution of the Baouendi-Grushin equation with critical exponent Pu=-u~(Q+2Q-2) is given by u=c[(~2｜z｜~2)~2+4｜t｜~2]~(-Q-24),where P=Δ_z+｜z｜~2Δt is the generalized Baouendi-Grushin operator when α=1,z∈R~n,t∈R~m,Q=n+2m is the homogeneous dimension,c=[(Q-2)n~2]~(Q-24) and >0.From this,the embedding constant and the extremal function in the Sobolev's inequality associated with P are presented by S=2~(mQ)π-n+m2(n+2m)｛n[n+2(m-1)]｝~(12)×Γ(n+m)Γ(n+m2)~(1n+2m) and [(1+｜z｜~2)~2+(4｜t｜~2]~(-14)),respectively.When n=m=1,the conclusion is the same as Beckner~()obained.