Nonlinear bound states with prescribed angular momentum in the mass supercritical regime
Tianxiang Gou, Xiaoan Shen
TL;DR
This work investigates bound states of the focusing nonlinear Schrödinger equation with super-quadratic confinement in $d=2,3$ under the double mass and angular momentum constraints, connecting to rotating Bose–Einstein condensates. A constrained variational framework with energy functionals $E$ and $E_\Omega$ on $S(m,l)$ yields a two-solution picture for small mass: a local minimizer $u_{m,l}$ and a mountain-pass solution $v_{m,l}$, with $E(u_{m,l})<E(v_{m,l})$; $u_{m,l}$ is orbitally stable while $v_{m,l}$ is strongly unstable. The analysis employs Pohozaev identities $P(u)$, virial-type arguments, and a Jeanjean-type reduction to obtain constrained Palais–Smale sequences, and it establishes regularity of minimizers under mild arithmetic conditions on $l/m$, with cylindrical reductions clarifying the eigenfunction cases. Together, the results extend prior subcritical (NSS) findings to the mass-supercritical regime and illuminate the structure and stability of rotating bound states under double constraints in the Gross–Pitaevskii setting.
Abstract
In this paper, we consider the existence, orbital stability/instability and regularity of bound state solutions to nonlinear Schrödinger equations with super-quadratic confinement in two and three spatial dimensions for the mass supercritical case. Such solutions, which are given by time-dependent rotations of a non-radially symmetric spatial profile, correspond to critical points of the underlying energy function restricted on the double constraints consisting of the mass and the angular momentum. The study exhibits new pictures for rotating Bose-Einstein condensates within the framework of Gross-Pitaevskii theory. It is proved that there exist two non-radially symmetric solutions, one of which is local minimizer and the other is mountain pass type critical point of the underlying energy function restricted on the constraints. Moreover, we derive conditions that guarantee that local minimizers are regular, the set of those is orbitally stable and mountain pass type solutions are strongly unstable. The results extend and complement the recent ones in \cite{NSS}, where the consideration is undertaken in the mass subcritical case.