On ground states for the 2D Schrödinger equation with combined nonlinearities and harmonic potential

Abstract

We consider the nonlinear Schrödinger equation with a harmonic potential in the presence of two combined energy-subcritical power nonlinearities. We assume that the larger power is defocusing, and the smaller power is focusing. Such a framework includes physical models, and ensures that finite energy solutions are global in time. We address the questions of the existence and the orbital stability of the set of standing waves. Given the mathematical features of the equation (external potential and inhomogeneous nonlinearity), the set of parameters for which standing waves exist in unclear. In the twodimensional case, we adapt the method of fundamental frequency solutions, introduced by the second author in the higher dimensional case without potential. This makes it possible to describe accurately the set of fundamental frequency standing waves and ground states, and to prove its orbital stability.

Critical points with prescribed energy for a class of functionals depending on a parameter: existence, multiplicity and bifurcation results

“…To this end, we shall follow the nonlinear generalized Rayleigh quotient method, introduced by Y Il'yasov [23] and used successfully also in other papers, as e.g. [12,24,[26][27][28].…”

Existence and Stability of Standing Waves for the Nonlinear Schrödinger Equation with Combined Nonlinearities and a Partial Harmonic Potential

“…Schrödinger equation with harmonic limiting potential and W indicates that the external potential is uniformly distributed in all directions of space in a harmonic form. This type of equations has been studied in [4] [5] [6]. In fact, when ( )…”

“…In [4], Carles and Il'yasov considered the nonlinear Schrödinger equation with a harmonic potential in the presence of two combined energy-subcritical power nonlinearities. They address the equations of the existence and the orbital stability of the set of standing waves by the method of fundamental frequency solutions.…”

Prescribed energy saddle-point solutions of nonlinear indefinite problems

“…The prescribed energy solutions of nonlinear problems was studied recently in [7,18,19] by using the nonlinear Rayleigh quotients [17]. The nonlinear Rayleigh quotients have the remarkable property that the critical points of these functionals correspond to the solutions of the equations while having a simpler structure than the corresponding energy functionals (see, e.g., [17]).…”

“…The nonlinear Rayleigh quotients have the remarkable property that the critical points of these functionals correspond to the solutions of the equations while having a simpler structure than the corresponding energy functionals (see, e.g., [17]). They were particularly useful (see, e.g., [17,19]) for finding nonnegative solutions to zero-mass problems [6] and detecting S-shaped bifurcations of nonlinear partial differential equations [7]. The nonlinear Rayleigh quotients method and solutions with prescribed energies were used to introduce a generalization of the Poincaré and Courant-Fischer-Weil minimization principles to nonlinear problems [18], as well as to study the orbital stability for ground states of the NLS equations [7].…”