A note about the generalized Hardy-Sobolev inequality with potential in L-p,L-d (R-n)

摘要

We present a generalized version of the Hardy-Sobolev inequality, in which the homogeneous potential vertical bar x vertical bar(-alpha) is replaced by any potential V belonging to the Lorentz space L-n/alpha,L-infinity (R-n). We show that the best constant in these inequalities is achieved provided that L-n/alpha,L-d(R-n) where 1 <= d < infinity. We also analyze the limit case d = infinity. Finally an application to a non-linear eigenvalues problem with rough potentials is presented.