On a singular minimizing problem

We study a minimizing problem associated with the singular problem \left\{ \begin{array} [c]{ll} -\operatorname{div}\left( \left\vert \nabla u\right\vert ^{p-2}\nabla u\right) =\lambda u^{-1} & \mathrm{in\ }\Omega\\ u>0 & \mathrm{in\ }\Omega\\ u=0 & \mathrm{on\ }\partial\Omega, \end{array} \right. where $p>1$, $\lambda>0$ and $\Omega$ is a bounded and smooth domain of $\mathbb{R}^{N}$, $N\geq2.$ A new log-Sobolev type inequality is proved and the corresponding best constant is identifyied.