In this paper we extend the results of Brezis and Nirenberg in [4] to the problem
(A)
where
L is a uniformly elliptic operator,
b(x)≥0,
f(x,u) is a lower order perturbation of
up at infinity.The existence of solutions to (A) is strongly dependent on the behaviour of
aij (x),b(x) and
f(x,u).For example,for any bounded smooth domain Ω,we have
such that
Lu=up possesses a positive solution in
H01(Ω).
We also prove the existence of nonradial solutions to the problem -Δu=f(|x|,u) in Ω,u>0 in Ω u=0 on ∂Ω,Ω=B(0,1).for a class of f(r,u).