  The paper studies the existence of minimizers for Rayleigh quotients
$\mu_{\Omega}=\inf\frac{\int_\Omega|\nabla u|^2}{\int_\Omega V{|u|^2}} $, where
$\Omega$ is a domain in $\mathbb{R}^N$, and $V$ is a nonzero nonnegative
function that may have singularities on $\partial\Omega$. As a model for our
results one can take $\Omega$ to be a Lipschitz cone and $V$ to be the Hardy
potential $V(x)=\frac{1}{|x|^2} $.
