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Zbl 1196.26021
Kufner, A.
Weighted inequalities and spectral problems. (English)
[J] Banach J. Math. Anal. 4, No. 1, 116-121, electronic only (2010). ISSN 1735-8787/e

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.
[James Adedayo Oguntuase (Abeokuta)]

MSC 2000:: *26D10 Inequalities involving derivatives, diff. and integral operators
34L05 General spectral theory for ODE
47E05 Ordinary differential operators

Keywords: Hardy inequality; nonlinear Sturm-Liouville problem

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