An Indefinite Elliptic Problem on RN Autonomous at Infinity: The Crossing Effect of the Spectrum and the Nonlinearity

We present a new approach to solve a Schr\"odinger Equation autonomous at infinity, by identifying the relation between the arrangement of the spectrum of the concerned operator and the behavior of the nonlinearity at zero and at infinity. In order to apply variational methods, we set up a suitable linking structure depending on the growth of the nonlinear term and making use of information about the autonomous problem at infinity. Our method allows us to circumvent the lack of compactness. The main novelty is that none monotonicity assumption is required on the nonlinearity, which may be sign-changing as well as the potential. Furthermore, depending on the nonlinearity, the limit of the potential at infinity may be non-positive, so that zero may be an interior point in the essential spectrum of the Schr\"odinger operator.