Mass-subcritical Half-Wave equation with mixed nonlinearities: Existence and non-existence of ground states and traveling waves

We consider the problem of existence of constrained minimizers for the focusing mass-subcritical Half-Wave equation with a defocusing mass-subcritical perturbation. We show the existence of a critical mass such that minimizers do exist for any mass larger than or equal to the critical one, and do not exist below it. As a byproduct, the same existence and non-existence results hold for traveling waves. At the dynamical level, in the one dimensional case, we show that the ground states are orbitally stable.