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 Maz'ya in [V.G. Maz'ja, 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⩽N1⩽k⩽N denote points ξ∈RN=Rk×RN−kξ∈RN=Rk×RN−k as pairs (x,y)(x,y). Let p∈(1,N)p∈(1,N), q∈(p,p∗]q∈(p,p∗] and assume ba:=N−qN−p+ap0c>0 such thatc(∫RN|x|−ba|u|qdξ)p/q⩽∫RN|x|a|∇u|pdξ,∀u∈Cc∞(RN). We prove that the best constant is achieved for any a,p,ka,p,k, provided that q