Extension of Díaz-Saá's inequality in RN and application to a system of p-Laplacian
- Autores: Karim Chaïb
- Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 46, Nº 2, 2002, págs. 473-488
- Idioma: inglés
- Títulos paralelos:
- Extensión de la desigualdad de Díaz-Saá a RN y aplicación a un sistema con un p-laplaciano
- Enlaces
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Resumen
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.
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