Ground state for the Schr\"odinger operater with the weighted Hardy   potential
  We establish the existence of ground states on Euclidean space 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. 
Jan Chabrowski, Kyril Tintarev
math.AP