Existence and characterization of ground states for fourth order nonlinear Schrödinger systems

The purpose of this paper is to establish the existence of ground states without restricting ourselves to the space of radial functions for a class of fourth order nonlinear Schrödinger systems of the form:$ \left\{\begin{array}{ll} \Delta^2 u_1 - \alpha_1 \Delta u_1 + \lambda_1 u_1 = \mu_1 |u_1|^{2}u_1 + \beta |u_2|^{2}u_1 \\ \Delta^2 u_2 - \alpha_2 \Delta u_2 + \lambda_2 u_2 = \mu_2 |u_2|^{2}u_2 + \beta |u_1|^{2}u_2 \end{array} \right. \quad \hbox{in} \ \mathbb{R}^N. $For our particular problem, neither the maximum principle nor the Schwarz symmetric rearrangement can be applied, as is usually performed for second order problems. Consequently, in order to arrive at the results obtained here, we remove the radial symmetry by applying the Fourier rearrangement, following a totally different approach from the usual one used for these types of problems. A classification of whether the ground states are semi-trivial or fully-nontrivial is also presented, obtaining a precise analysis depending on the parameters involved in the system. Our results complement those given by Álvarez-Caudevilla, Colorado and Galaktionov in [1] for the case $ \alpha_j = 0 $ $ (j = 1, 2) $.