Ground State for the Schrödinger Operator with the Weighted Hardy Potential

Abstract

We establish the existence of ground states on ℝ𝑁 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.