On the capacity approach to non-attainability of Hardy's inequality in $\mathbb{R}^N$

Daniele Cassani; Bernhard Ruf; Cristina Tarsi

doi:10.3934/dcdss.2019017

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2019, Volume 12, Issue 2: 245-250. Doi: 10.3934/dcdss.2019017

This issue Previous Article Fourth-order problems with Leray-Lions type operators in variable exponent spaces Next Article Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence

On the capacity approach to non-attainability of Hardy's inequality in $\mathbb{R}^N$

1.
Dip. di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria
2.
RISM-Riemann International School of Mathematics, via G.B. Vico 46, 21100 - Varese, Italy
3.
Dip. di Matematica, Università degli Studi di Milano, via C. Saldini 50, 20133 - Milano, Italy

* Corresponding author: Daniele Cassani
* Corresponding author: Daniele Cassani

In honor of Vicentiu Rădulescu on the occasion of his 60^th birthday, with friendship and admiration

Received: July 31, 2017

Revised: November 30, 2017

Published: April 2019

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Abstract

In this note we exploit nonlinear capacity estimates in the spirit of Mitidieri-Pohozaev [15] in the context of Lorentz spaces. This from one side yields a simple proof, though non-optimal, of non-attainability of Hardy's inequality in $\mathbb{R}^N$, on the other side gives a partial positive answer to a conjecture raised in [15].

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Mathematics Subject Classification: Primary: 35A23; Secondary: 46E30, 35B53.

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