Solutions for periodic Schrِdinger equation with spectrum zero and general superlinear nonlinearities

In this paper we consider the following Schrödinger equation: { − Δ u + V ( x ) u = g ( x , u ) for x ∈ R N , u ( x ) → 0 as | x | → ∞ , where V ( x ) and g ( x , u ) are periodic with respect to x and 0 is a boundary point of the spectrum σ ( − Δ + V ) . Replacing the classical Ambrosetti–Rabinowitz superlinear assumption on g ( x , u ) by a general super-quadratic condition, we are able to obtain the existence of nontrivial solutions.