A non-linear problem involving a critical Sobolev exponent

We study a non-linear minimization problem on H 0 1 ( Ω ) ⊂ L q with q = 2 n n − 2 : inf ‖ u ‖ L q = 1 ∫ Ω ( 1 + | x | β | u | k ) | ∇ u | 2 . We show that minimizers exist only for the range β < k n / q which corresponds to a dominant non-linear term. However, for β ≥ k n / q the influence of linearity prevents their existence.