Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

An improved Hardy inequality for a nonlocal operator

Pages: 1143 - 1157, Volume 36, Issue 3, March 2016      doi:10.3934/dcds.2016.36.1143

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Boumediene Abdellaoui - Laboratoire D'Analyse Nonlinéaire et Mathématiques Appliquées, Département de Mathématiques, Faculté des sciences, Université About Baker Belkad, Tlemcen 13000, Algeria (email)
Fethi Mahmoudi - Centro de Modelamiento Matemático (CMM), Universidad de Chile, Beauchef 851, Santiago, Chile (email)

Abstract: Let $0 < s < 1$ and $1< p < 2$ be such that $ps < N$ and let $\Omega$ be a bounded domain containing the origin. In this paper we prove the following improved Hardy inequality:
    Given $1 \le q < p$, there exists a positive constant $C\equiv C(\Omega, q, N, s)$ such that $$ \int\limits_{\mathbb{R}^N}\int\limits_{\mathbb{R}^N} \, \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy - \Lambda_{N,p,s} \int\limits_{\mathbb{R}^N} \frac{|u(x)|^p}{|x|^{ps}}\,dx$$$$\geq C \int\limits_{\Omega}\int\limits_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{N+qs}}dxdy $$ for all $u \in \mathcal{C}_0^\infty({\Omega})$. Here $\Lambda_{N,p,s}$ is the optimal constant in the Hardy inequality (1.1).

Keywords:  Fractional Sobolev spaces, weighted Hardy inequality, nonlocal problems.
Mathematics Subject Classification:  49J35, 35A15, 35S15.

Received: January 2015;      Revised: June 2015;      Available Online: August 2015.