A note about the generalized Hardy-Sobolev inequality with potential in $ L^{{p,d}} {\left( {\mathbb{R}^{n} } \right)}$

Visciglia, Nicola
October 2005
Calculus of Variations & Partial Differential Equations;Oct2005, Vol. 24 Issue 2, p167
Academic Journal
We present a generalized version of the Hardy-Sobolev inequality, in which the homogeneous potential $\vert x\vert^{-\alpha}$ is replaced by any potential V belonging to the Lorentz space $ L^{{\frac{n} {\alpha },\infty }} {\left( {\mathbb{R}^{n} } \right)} $. We show that the best constant in these inequalities is achieved provided that $ V \in L^{{\frac{n}% {\alpha },d}} {\left( {\mathbb{R}^{n} } \right)} $ where $ 1 \leqslant d < \infty $. We also analyze the limit case $d = \infty$. Finally an application to a non-linear eigenvalues problem with rough potentials is presented.


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