  In this paper, we study Hardy's inequality in a limiting case: $$
  \int_{\Omega} |\nabla u |^N dx \ge C_N(\Omega) \int_{\Omega}
\frac{|u(x)|^N}{|x|^N \left(\log \frac{R}{|x|} \right)^N} dx $$ for functions
$u \in W^{1,N}_0(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^N$
with $R = \sup_{x \in \Omega} |x|$. We study the (non-)attainability of the
best constant $C_N(\Omega)$ in several cases. We provide sufficient conditions
that assure $C_N(\Omega) > C_N(B_R)$ and $C_N(\Omega)$ is attained, here $B_R$
is the $N$-dimensional ball with center the origin and radius $R$. Also we
provide an example of $\Omega \subset \mathbb{R}^2$ such that $C_2(\Omega) >
C_2(B_R) = 1/4$ and $C_2(\Omega)$ is not attained.
