Mass-subcritical Half-Wave Equation with mixed nonlinearities: existence and non-existence of ground states
Jacopo Bellazzini, Luigi Forcella
TL;DR
The paper addresses the existence and nonexistence of ground states for the focusing mass-subcritical Half-Wave equation with a defocusing subcritical perturbation under a mass constraint, establishing a sharp critical mass $\rho_0^v$ such that $I_{\rho^2}=0$ for $0<\rho\le\rho_0^v$ and $I_{\rho^2}<0$ for $\rho>\rho_0^v$, with minimizers existing at and above the threshold and none below. The authors develop a variational framework using scaling, subadditivity, and a careful mass-threshold analysis, complemented by a concentration-compactness-type argument to prove existence of constrained minimizers at the critical mass. In 1D, they prove global well-posedness in $H^{1/2}$ and orbital stability of the ground-state set for masses above the threshold, thereby connecting variational structure to long-time dynamics. The work extends ground-state theory to the Half-Wave operator with mixed nonlinearities in the mass-subcritical regime and provides a rigorous dynamical stability picture for traveling/standing waves in one dimension.
Abstract
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. At the dynamical level, in the one dimensional case, we show that the ground states are orbitally stable.