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

TL;DR

This paper investigates the relationship between spectral gaps and nonlinearity in Schr"odinger equations, establishing existence results for solutions under weaker assumptions by linking spectral properties to nonlinear behavior.

Contribution

It introduces a novel approach connecting the spectrum of the operator with the nonlinear term, allowing zero in the spectrum and sign-changing nonlinearities, overcoming previous limitations.

Findings

Existence of nontrivial solutions under weaker spectral conditions

Linking spectrum to nonlinear behavior broadens applicability

Overcomes lack of monotonicity and compactness issues

Abstract

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 $V$.