Normalized solutions with positive energies for a coercive problem and application to the cubic-quintic nonlinear Schrödinger equation

Abstract

In any dimension N ≥ 1, for given mass m > 0 and when the C 1 energy functionalis coercive on the mass constraintwe are interested in searching for constrained critical points at positive energy levels. Under general conditions on F ∈ C 1 (R, R) and for suitable ranges of the mass, we manage to construct such critical points which appear as a local minimizer or correspond to a mountain pass or a symmetric mountain pass level. In particular, our results shed some light on the cubic-quintic nonlinear Schrödinger equation in R 3 .

Self-Bound vortex states in nonlinear Schrödinger equations with LHY correction

“…For the case with competing power-law nonlinearities, it has recently been shown that there exist least action solutions to (3.2), which are not equal to constrained energy minimizers, see [9,20,21,24] for a broader discussion on all this.…”