A Note on Nonlinear Schr\"odinger Equations: Unveiling the Relation Between Spectral Gaps and the Nonlinearity Behavior

We establish the existence of a nontrivial weak solution to strongly indefinite asymptotically linear and superlinear Schr"odinger equations. The novelty is to identify the essential relation between the spectrum of the operator and the behavior of the nonlinear term, in order to weaken the necessary assumptions to obtain a linking structure to the problem, for instance to allow zero being in the spectrum or the nonlinearity being sign-changing. Our main difficulty is to overcome the lack of monotonicity on the nonlinear term, as well, as the lack of compactness since the domain is unbounded. With this purpose, we require periodicity on $V$.