Kufner, A. Weighted inequalities and spectral problems. (English) Zbl 1196.26021 Banach J. Math. Anal. 4, No. 1, 116-121 (2010). The author studies the mutual connection between the \(n\)-dimensional Hardy inequality \[ \bigg(\int_\Omega |f|^q u\,dx\bigg)^{\frac1q}\leq C\bigg(\int_\Omega |\nabla f|^p v\,dx\bigg)^{\frac1{p}}, \quad f\in C^\infty_0 \]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 \(\mathbb R^n\) 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) Cited in 1 Document MSC: 26D10 Inequalities involving derivatives and differential and integral operators 34L05 General spectral theory of ordinary differential operators 47E05 General theory of ordinary differential operators Keywords:Hardy inequality; nonlinear Sturm-Liouville problem PDFBibTeX XMLCite \textit{A. Kufner}, Banach J. Math. Anal. 4, No. 1, 116--121 (2010; Zbl 1196.26021) Full Text: DOI EuDML EMIS