Schrodinger operators with negative potentials and Lane-Emden densities

We consider the Schrodinger operator -Delta+V for negative potentials V, on open sets with positive first eigenvalue of the Dirichlet-Laplacian. We show that the spectrum of -Delta+V is positive, provided that V is greater than a negative multiple of the logarithmic gradient of the solution to the Lane-Emden equation -Delta u = u(q-1) (for some 1 <= q < 2). In this case, the ground state energy of -Delta+V is greater than the first eigenvalue of the Dirichlet-Laplacian, up to an explicit multiplicative factor. This is achieved by means of suitable Hardy-type inequalities, that we prove in this paper. (C) 2017 Elsevier Inc. All rights reserved.