Project CEMAPRE internal
| Title | Ground States in DIspersive Systems |
| Participants | Filipe Oliveira (Principal Investigator) |
| Summary | This project concerns the analysis of ground states for linearly and nonlinearly coupled nonlinear Schrödinger systems in Rn. We focus on stationary solutions arising from a two-component NLS system with linear coupling and cubic nonlinearities, which appear naturally in nonlinear optics and Bose–Einstein condensates. The main objective is to establish existence, nonexistence, and qualitative properties of ground states, distinguishing between action ground states and energy ground states. A particular emphasis is placed on low-dimensional cases, especially n=1, where standard compactness arguments based on symmetric decreasing rearrangements fail in general Sobolev spaces. Using variational methods on suitable constraint manifolds (notably the Nehari manifold and fixed-mass manifolds), combined with rearrangement techniques in spaces of radially decreasing functions, we prove the existence of positive action ground states for all dimensions under appropriate assumptions on the coupling and nonlinear coefficients. The compact embedding of Sobolev Spaces in Lp spaces ensured by Strauss-type lemmas, play a crucial role. Furthermore, we investigate the relationship between action and energy minimization problems, showing that in several cases energy ground states coincide with action ground states for suitable frequencies. We also address nonexistence results in the critical dimension n=4, complementing known results in the literature. |