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Standing wave and global existence to nonlocal nonlinear Schrödinger equations: the two-dimensional case

In this paper, we consider the standing waves and the global existence for two-dimensional nonlocal nonlinear Schrödinger equations. It is a coupled system which describes the spontaneous generation of a magnetic field in a cold plasma under the static limit. The main difficulty in the proofs lies in exploring the inner structure of the equations due to the fact that the nonlocal terms violate the inner scaling invariance, which may cause the non-zero energy for the ground state. For this reason, we first make a proper use of the inner structure of the equations to establish the existence of standing waves, and then we apply an energy scaling to obtain the instability of standing waves. Finally we show a sharp threshold for the global existence of solutions to the nonlocal nonlinear Schrödinger equations by a variational method, which depends again on the inner structure of the equations under consideration.