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The purpose of this paper is to extend the Díaz-Saá's inequality for the unbounded domains as RN: [fórmula]. 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

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