Sharp Remainder Terms of the Rellich Inequality and its Application

Alnar Detalla ; Toshio Horiuchi ; Hiroshi Ando

Bulletin of the Malaysian Mathematical Society, Tome 35 (2012) no. 2A

Résumé

In this article we shall study the improvement of the Rellich inequality by adding terms with a singular weight of the type $\left(\log (1/|x|)\right)^{-2}$ in the right hand side. We show that this weight function is optimal in the sense that the improved inequality fails for any other weight more singular than this one. As an application, we use our improved inequality to analyze the behaviour of the first eigenvalue of the weighted eigenvalue problem for the operator $L_\mu u=\Delta\left(|\Delta u|^{p-2}\Delta u \right) - \left(\mu/|x|^{2p}\right)|u|^{p-2}u$ as $\mu$ increases to $\left((n-2p)/p\right)^p \left((np-n)/p\right)^p$ for $1$.

Détail

Classification : Primary: 35J70; Secondary: 35J60.