一类具临界指数椭圆方程的非平凡解存在性

Existence of Non-Trivial Solutions for a Class of Elliptic Equations with the Critical Sobolev Exponents

当N 4时,Capozzi A(1985),Ambrosetti A(1986)给出了具临界指数2*的椭圆型方程-Δku+|u|2*-2u,inΩRN;u=0,onΩ(*)非平凡解的存在性结论,其中λk是算子-Δ的第k个特征值。然而N=3是问题(*)的临界维数,在适当添加一个次临界扰动项后,利用P.L.Lions集中紧性原理获得了一对非平凡解的存在性结论。

It is well known that Capozzi A（ 1985 ） and Ambrosetti A（1986） have got existence theorems of the fol- lowing elliptic equation with critical Sobolev exponent if N ≥ 4,△Ku＋｜u｜^2＊-2u,inΩ∈R^N;u=O,on ЭΩ where λk is the kth eigen - value of - △. However,N = 3 is the critical dimension of the problem（ ＊ ）. Adding a subcritical perturbation,the authors have given existence theorems by ways of the concentration - compactness principle of P. L. Lions.