Extension of Díaz-Saá's inequality in $\mathbb{R}^N$ and application to a system of $p$-Laplacian

The purpose of this paper is to extend the Díaz-Saá's inequality for the unbounded domains as $ \mathbb{R}^N $:
\begin{multline*}
\int_{\mathbb{R}^N} \left( -\frac{\Delta_p u}{u^{p-1}} + \frac{\Delta_p v}{v^{p-1}} \right) \left( u^p -v^p \right)\,dx \geq 0\\ \text{with } \Delta_p u = \operatorname{div}\left(|\nabla u|^{p-2} \nabla u \right).
\end{multline*}
The proof is based on the Picone's identity which is very useful in problems involving p-Laplacian. In a second part, we study some properties of the first eigenvalue for a system of p-Laplacian. We use Díaz-Saá's inequality to prove uniqueness and Egorov's theorem for the isolation. These results generalize J. Fleckinger, R. F. Manásevich, N. M. Stavrakakis and F. de Thélin's work [9] for the first property and A. Anane's one for the isolation.