International Journal of Differential Equations
Volume 2011 (2011), Article ID 358087, 26 pages
http://dx.doi.org/10.1155/2011/358087
Research Article

Ground State for the Schrödinger Operator with the Weighted Hardy Potential

J. Chabrowski1 and K. Tintarev2

1Department of Mathematics, The University of Queensland, St. Lucia 4072, QLD, Australia
2Department of Mathematics, Uppsala University, P.O. Box 480, 751 06 Uppsala, Sweden

Received 6 May 2011; Accepted 17 July 2011

Academic Editor: Kanishka Perera

Copyright © 2011 J. Chabrowski and K. Tintarev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We establish the existence of ground states on ℝ 𝑁 for the Laplace operator involving the Hardy-type potential. This gives rise to the existence of the principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We also obtain a higher integrability property for the principal eigenfunction. This is used to examine the behaviour of the principal eigenfunction around 0.