EXISTENCE AND REGULARITY OF POSITIVE SOLUTIONS OF THE ELLIPTIC EQUATIONS INVOLVING CRITICAL NONLINEAR AND SUBLINEAR

0 IntroductionLet Ωbe a smooth bounded domain in Rn with n>3, consider the problem of finding a function u（x）which satisfieswhere 0<α<1, 1<P=2n-1, 2n=2n/n-2 is the critical exponent for Sobolev embedding H01→L2 .Indeed solutions of （0 ,1） correspond to critical points of the functionalBrezis and Nirenberg considered the case α≥1 and p= 2* -1, and obtainedTheorem （B , N） Let n≥4, when α=1 and 0 <λ< λ1,there is a solution of （0. 1） ; when α> 1 there is asolution of （0. 1） for any λ>0.The principal tecncihique of argument in [1] is using positive radial solution （0. 3） of equation -△u=u2n-1 in R*.