TITLE

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

AUTHOR(S)
Visciglia, Nicola
PUB. DATE
October 2005
SOURCE
Calculus of Variations & Partial Differential Equations;Oct2005, Vol. 24 Issue 2, p167
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
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.
ACCESSION #
18140573

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