Weighted inequalities and spectral problems (Q967167)

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.