We present a generalized version of the Hardy-Sobolev inequality, in which the homogeneous potential |x|^−α is replaced by any potential V belonging to the Lorentz space L^n/α,∞(Rn). We show that the best constant in these inequalities is achieved provided that V ∈ Lnα,d(Rn) where 1 ≤ d < ∞. We also analyze the limit case d = ∞. Finally an application to a non-linear eigenvalues problem with rough potentials is presented
A note about the generalized Hardy-Sobolev inequality with potential in L^{p, d}
VISCIGLIA, NICOLA
2005-01-01
Abstract
We present a generalized version of the Hardy-Sobolev inequality, in which the homogeneous potential |x|^−α is replaced by any potential V belonging to the Lorentz space L^n/α,∞(Rn). We show that the best constant in these inequalities is achieved provided that V ∈ Lnα,d(Rn) where 1 ≤ d < ∞. We also analyze the limit case d = ∞. Finally an application to a non-linear eigenvalues problem with rough potentials is presentedFile in questo prodotto:
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