Schrödinger operators with negative potentials and Lane-Emden densities

We consider the Schrödinger operator −Δ+V for negative potentials V, on open sets with positive first eigenvalue of the Dirichlet-Laplacian. We show that the spectrum of −Δ+V is positive, provided that V is greater than a negative multiple of the logarithmic gradient of the solution to the Lane-Emden equation −Δu=uq−1 (for some 1≤q<2). In this case, the ground state energy of −Δ+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.