Standing waves for Schrödinger equations with Kato–Rellich potentials

Abstract

We show the existence of standing waves for the nonlinear Schrödinger equation with Kato-Rellich type potential. We consider both resonant with the nonlinearity satisfying one of Landesman–Lazer type or sign conditions and non-resonant case where the linearization at infinity has zero kernel. The approach relies on the geometric and topological analysis of the parabolic semiflow associated to the involved elliptic problem. Tail estimates techniques and spectral theory of unbounded linear operators are used to exploit subtle compactness properties necessary for use of the Conley index theory due to Rybakowski. The obtained results extend those by Prizzi (2003) for the non-resonant case, resolve the existence problem at resonance and complete those from A. Ćwiszewski and W. Kryszewski (2019) where the bifurcation of stationary solutions from infinity was studied.