Hardy's inequality in unbounded domains

Abstract

The aim of this paper is to consider Hardy's inequality with weight on unbounded domains. In particular, using a decomposition lemma, we study the existence of a minimizer for $$ S_\varepsilon(\Omega):= \inf_{u \in D_{\varepsilon}^{1,2}(\Omega)} \frac {\int_{\Omega}{\vert\nabla u\vert}^2{\delta^{\varepsilon}}dx} {\int_{\Omega}{\vert u\vert}^2\delta^{\varepsilon - 2}dx}. $$