On a Sobolev-type inequality related to the weighted p-Laplace operator

In this paper we deal with some Sobolev-type inequalities with weights that were proved by Mazʹya in [V.G. Mazʹja, Sobolev Spaces, Springer-Verlag, Berlin, 1980] and by Caffarelli, Kohn and Nirenberg in [L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weight, Compos. Math. 53 (1984) 259–275]. For integers 1 ⩽ k ⩽ N denote points ξ ∈ R N = R k × R N − k as pairs ( x , y ) . Let p ∈ ( 1 , N ) , q ∈ ( p , p ∗ ] and assume b a : = N − q N − p + a p < k . Then there exists c > 0 such that c ( ∫ R N | x | − b a | u | q d ξ ) p / q ⩽ ∫ R N | x | a | ∇ u | p d ξ , ∀ u ∈ C c ∞ ( R N ) . We prove that the best constant is achieved for any a , p , k , provided that q < p ∗ or q = p ∗ and a < 0 . Results for weighted Sobolev-type inequalities on cones are also given.