@article{1196.26021,
author="Kufner, A.",
title="{Weighted inequalities and spectral problems.}",
language="English",
journal="Banach J. Math. Anal. ",
volume="4",
number="1",
pages="116-121",
year="2010",
abstract="{The author studies the mutual connection between the $n$-dimensional
    Hardy inequality 
 $$\bigg(\int_\Omega |f|^q u\,dx\bigg)^{\frac1q}\le
    C\bigg(\int_\Omega |\nabla f|^p v\,dx\bigg)^{\frac1{p}}, \quad f\in
    C^\infty_0$$ 
 and the spectral problem 
 $$-\text{div}\big(v|\nabla
    f|^{p-2}|\nabla f|\big)= \lambda u|f|^{q-2}f \quad \text {in }\Omega, \qquad
    u= 0 \quad \text {on }\partial \Omega,$$ 
 where $\Omega$ is a domain in
    $\Bbb R^n$ with boundary $\partial\Omega$, $p,q$ are real parameters,
    $1<p,q<\infty$, and $u,v$ are weight functions on $\Omega$. The author
    establishes that the conditions for the validity of the Hardy inequality
    coincide with the conditions on the spectrum of some (nonlinear)
    differential operators to be bounded from below and discrete. Furthermore,
    examples are given to illustrate this mutual connection.}",
reviewer="{James Adedayo Oguntuase (Abeokuta)}",
keywords="{Hardy inequality; nonlinear Sturm-Liouville problem}",
classmath="{*26D10 (Inequalities involving derivatives, diff. and integral
    operators)
34L05 (General spectral theory for ODE)
47E05 (Ordinary differential operators)
}",
}