Minimization problems related to generalized Hardy's inequalities. (Q1867294)

Let \(\Omega\) be a domain in \(\mathbb R^N\) (bounded or unbounded) with a compact boundary of class \(C^2\). Denote by \(D^{1,2}_{\varepsilon}(\Omega)\) the closure of \(D(\Omega)\) with respect to the inner product \((u,v):=\int_{\Omega} (\delta(x))^{\varepsilon}\, \nabla u \cdot \nabla v \, dx\), where \(0\leq \varepsilon < 1\) and \(\delta (x) : = \text{dist}\,(x,\partial\Omega)\), \(x\in \mathbb R^N\). The main aim of the paper is to study the connection between the quantity \[ J_{\lambda,\varepsilon} := \inf_{u\in D^{1,2}_{\varepsilon}(\Omega)} \Big[ \int_{\Omega}| \nabla u| ^2 \delta^{\varepsilon} \,dx - \lambda \int_{\Omega} p\,u^2 \delta^{\varepsilon} \,dx\Big] \Big/ \int_{\Omega} u^2\delta^{\varepsilon-2}\,dx \leqno{(*)} \] (where \(p\in L^{\infty}(\Omega)\) is positive on \(\Omega\) and such that \(\,\) \(p\,\delta^{\varepsilon} \in L^{N/2} (\Omega)\)) and the existence of a minimizer for \((*)\).