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

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