an: Zbl 1196.26021
au: Kufner, A.
ti: Weighted inequalities and spectral problems.
la: EN
so: Banach J. Math. Anal. 4, No. 1, 116-121, electronic only (2010).
py: 2010
dt: J
cc: *26D10 34L05 47E05
ut: Hardy inequality; nonlinear Sturm-Liouville problem
ab: 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.
rv: James Adedayo Oguntuase (Abeokuta)