Discrete nonlinear Schrödinger equations with double power nonlinearities

For discrete nonlinear Schrödinger equations (DNLS) with double power nonlinearities we prove the existence of normalized ground state solutions by means of variational methods. Considering the corresponding constrained (with prescribed mass) minimization problem of least energy, respectively action, for DNLS where both nonlinear terms are of focusing type, the existence proof utilizes the Concentration Compactness Principle. For DNLS for which the leading nonlinearity is focusing while the lower order one is defocusing, the method of the Nehari manifold is used. Orbital stability of the ground state solutions is shown. Furthermore, we discuss the existence of excitation thresholds for the creation of ground state solutions focusing on the impact of the lower order nonlinearity on the threshold values of the mass for the excitation of ground state solutions.