Existence of positive solutions to nonlinear elliptic equations involving critical Sobolev exponents

In this paper we extend the results of Brezis and Nirenberg in [4] to the problem $$left{ begin{gathered} Lu = - D_i (a_{ij} (x)D_j u) = b(x)u^p + f(x,u) inOmega , hfill p = (n + 2)/(n - 2) hfill u > 0 inOmega , u = 0 partial Omega , hfill end{gathered} right.$$ whereL is a uniformly elliptic operator,b(x)≥0,f(x,u) is a lower order perturbation ofu ^p at infinity. The existence of solutions to (A) is strongly dependent on the behaviour ofa _ij (x), b(x) andf(x, u). For example, for any bounded smooth domain Ω, we have (a_{ij} left( x right) in Cleft( {bar Omega } right)) such thatLu=u ^p possesses a positive solution inH ₀ ¹ (Ω). We also prove the existence of nonradial solutions to the problem ?Δu=f(|x|, u) in Ω,u>0 in Ωu=0 on ?Ω, Ω=B(0,1). for a class off(r, u).