Solitary waves and excited states for Boson stars

We study the nonlinear, nonlocal, time-dependent partial differential equation [Formula: see text] which is known to describe the dynamics of quasi-relativistic boson stars in the mean-field limit. For positive mass parameter [Formula: see text] we establish existence of infinitely many (corresponding to distinct energies [Formula: see text]) traveling solitary waves, [Formula: see text], with speed [Formula: see text], where [Formula: see text] corresponds to the speed of light in our choice of units. These traveling solitary waves cannot be obtained by applying a Lorentz boost to a solitary wave at rest (with [Formula: see text]) because Lorentz covariance fails. Instead, we study a suitable variational problem for which the functions [Formula: see text] arise as solutions (called boosted excited states) to a Choquard-type equation in [Formula: see text], where the negative Laplacian is replaced by the pseudo-differential operator [Formula: see text] and an additional term [Formula: see text] enters. Moreover, we give a new proof for existence of boosted ground states. The results are based on perturbation methods in critical point theory.