Existence and nonexistence results for a weighted elliptic equation in exterior domains

We consider positive solutions to the weighted elliptic problem -div(|x|θ∇u)=|x|ℓupinRN\B¯,u=0on∂B,where B is the standard unit ball of R^N. We give a complete answer for the existence question for N^′: = N+ θ> 2 and p> 0. In particular, for N^′> 2 and τ: = ℓ- θ> - 2 , it is shown that for 0<p≤ps:=N′+2+2τN′-2, the only nonnegative solution to the problem is u≡ 0. This nonexistence result is new, even for the classical case θ= ℓ= 0 and NN-2<p≤N+2N-2, N≥ 3. The interesting feature here is that we do not require any behavior at infinity or any symmetry assumption.