Ground state energy threshold and blow-up for NLS with competing nonlinearities

Abstract

We consider the nonlinear Schrödinger equation with combined nonlinearities, where the leading term is an intracritical focusing power-type nonlinearity, and the perturbation is given by a power-type defocusing one. We completely answer the question wether the ground state energy, which is a threshold between global existence and formation of singularities, is achieved. For any prescribed mass, for mass-supercritical or mass-critical defocusing perturbations, the ground state energy is achieved by a radially symmetric and decreasing solution to the associated stationary equation. For mass-subcritical perturbations, we show the existence of a critical prescribed mass, precisely the mass of the unique, static, positive solution to the associated elliptic equation, such that the ground state energy is achieved for any mass equal or smaller than the critical one. Moreover, the ground state energy is not achieved for mass larger than the critical one. As a byproduct of the variational characterization of the ground state energy, we prove the existence of blowing-up solutions in finite time, for any energy below the ground state energy threshold.

Analysis of stability and instability for standing waves of the double power one dimensional nonlinear Schr{ö}dinger equation

“…The function F (defined in ( 8)) can be written, substituting Φ p and Φ q by their expressions (7), as…”

“…Proof. Using the formula (8) of F (φ 0 ) and replacing in the numerator of the integrand Φ p and Φ q by their expressions (7), we obtain…”

“…Uniqueness and nondegeneracy was considered in [28]. Existence or non-existence of minimizers of the energy at fixed mass was obtained in [7]. Let us also mention in dimension 1 the work [19], which is devoted to the stability of standing waves for cubic-quintic nonlinearities in the presence of a δ potential (see [4,5] for further developments).…”

Dynamics of solutions to the Gross-Pitaevskii equation describing dipolar Bose-Einstein condensates

“…the solution is in Σ 3 . See also [34] for an early work on NLS in anisotropic spaces, and [4,8,15] for these techniques applied to other dispersive models.…”

Blowup of cylindrically symmetric solutions for biharmonic NLS

“…This is the reason that we need to assume that d ≥ 5. Such restrictions are reminiscent of analogous ones for the study of blowup of solutions to NLS with cylindrically symmetric data, see[4,9,14].Theorem 1.2. (Blowup for Mass-Critical Case) Let d ≥ 4, µ ≥ 0 and s c = 0.…”