Extension of Díaz-Saá's inequality in \(\mathbb{R}^N\) and application to a system of \(p\)-Laplacian.
The goal of this paper is to extend the Díaz-Saá inequality for the unbounded domain as \(\mathbb R^N\): \[ \int_{\mathbb R^N} \bigg( \frac{-\Delta_pu}{u^{p-1}}+ \frac{\Delta_pv}{v^{p-1}}\bigg) (u^p-v^p)\,dx\geq 0, \] with \(\Delta_pu= \operatorname{div}(|\nabla u|^{p-2}\nabla u)\). To this end the author uses so-called Picone's identity which is very useful in problems involving \(p\)-Laplacian. Moreover, the author studies some properties of the first eigenvalue for a system of \(p\)-Laplacian. Finally, the author uses Díaz-Saá's inequality to prove uniqueness and Egorov's theorem for the isolation.
scientific article; zbMATH DE number 1876881