中文摘要：

给出了具临界指数的Baouendi—Grushi。方程Pu=-u^Q＋2/Q-2的显式解为u=c[（ε^2｜z｜^2）^2＋4｜t｜^2]^-Q-2/4，其中P=△x＋｜z｜^2△t为α=1时的广义Baouendi—Grushin算子，z∈R^n,t∈R^m,Q=n＋2m为齐次维数，c=[（Q-2）nε^2]^Q-2/4,ε〉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=2m/Qπ-n＋m/2（n＋2m）/{n[n＋2（m-1）]}×[Г（n＋m）/Г（n＋m）/2]^1/n＋2m,极值函数为[（1＋｜z｜^2）^2＋4｜t｜^2]^-1/4.当n=m=1时，本文的结论与Beckner的结果一致．

英文摘要：

In this paper, the explicit entire solution of the Baouendi - Grushin equation with critical exponent Pu=-u^Q＋2/Q-2 is given by u=c[（ε^2｜z｜^2）^2＋4｜t｜^2]^-Q-2/4,where P=△x＋｜z｜^2△t 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）nε^2]^Q-2/4 and ε〉0.From this, the embedding constant and the extremal function in the Sobolev~ inequal-ty associated with P are presented by S=2m/Qπ-n＋m/2（n＋2m）/{n[n＋2（m-1）]}×[Г（n＋m）/Г（n＋m）/2]^1/n＋2m and [（1＋｜z｜^2）^2＋4｜t｜^2]^-1/4, [4] in respectively. When n = m = 1, the conclusion is the same as Beckner obained.