Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials.

We study the existence of solutions of the following nonlinear Schrödinger equation − Δ u + ( V ( x ) − μ | x | 2 ) u = f ( x , u ) for x ∈ R N ∖ { 0 } , where V : R N → R and f : R N × R → R are periodic in x ∈ R N . We assume that 0 does not lie in the spectrum of − Δ + V and μ < ( N − 2 ) 2 4 , N ≥ 3 . The superlinear and subcritical term f satisfies a weak monotonicity condition. For sufficiently small μ ≥ 0 we find a ground state solution as a minimizer of the energy functional on a natural constraint. If μ < 0 and 0 lies below the spectrum of − Δ + V , then ground state solutions do not exist. [ABSTRACT FROM AUTHOR]