Hardy's inequality in a limiting case on general bounded domains

In this paper, we study Hardy’s inequality in a limiting case: $${\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{N}dx\ge C_N(\mathrm{\Omega }){\int }_{\mathrm{\Omega }}\frac{\mathrm{\mid }u(x){\mathrm{\mid }}^N}{\mathrm{\mid }x{\mathrm{\mid }}^N{(\log \frac{R}{\mathrm{\mid }x\mathrm{\mid }})}^N}dx$$ for functions $u\in W_0^{1,N}(\mathrm{\Omega })$, where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$ with $R={\sup }_{x\in \mathrm{\Omega }}\mathrm{\mid }x\mathrm{\mid }$. We study the (non-)attainability of the best constant $C_N(\mathrm{\Omega })$ in several cases. We provide sufficient conditions that assure $C_N(\mathrm{\Omega })>C_N(B_R)$ and $C_N(\mathrm{\Omega })$ is attained, here $B_R$ is the $N$-dimensional ball with center the origin and radius $R$. Also we provide an example of $\mathrm{\Omega }\subset {\mathbb{R}}^2$ such that $C_2(\Omega) > C_2(B_R) = 1/4$and$C_2(\Omega)$ is not attained.