Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS

Abstract

International audienc

Classification of minimal mass blow-up solutions for an $${L^{2}}$$ L 2 critical inhomogeneous NLS

“…Assuming that g is sufficiently flat at the origin, they proved the existence of critical mass blow-up solutions by adapting a fixed point argument developed by Bourgain and Wang [3] in the classic case of (1.1) with b = 0. It is worth noting here that problem (1.1) does not fall within the scope of [2,11,13] due to the singularity at x = 0. Moreover, our approach strongly benefits from the scaling properties of (1.1)-notably the pseudo-conformal invariance, which is not present in [2,11,13].…”

“…It is worth noting here that problem (1.1) does not fall within the scope of [2,11,13] due to the singularity at x = 0. Moreover, our approach strongly benefits from the scaling properties of (1.1)-notably the pseudo-conformal invariance, which is not present in [2,11,13].…”

“…was considered by Merle [11], and later by Raphaël and Szeftel [13] (in the case N = 2), where the inhomogeneity k is supposed to be smooth, positive and bounded.…”

“…Raphaël and Szeftel [13] proved the existence and the classification of critical mass blow-up solutions for (1.7), provided k attains its maximum in R N . Banica, Carles and Duyckaerts [2] studied the problem…”

Blow up for the critical gKdV equation. II: Minimal mass dynamics

“…i∂ t u + ∆u + k(x)|u| 2 u = 0, while Merle [38] derived sufficient conditions on k(x) to ensure the nonexistence of minimal elements, Raphaël and Szeftel [53] introduced a more dynamical approach to existence and uniqueness under a necessary and sufficient condition on k(x). A robust energy method is implemented to completely classify the minimal mass blow up, in regimes such that the inhomogeneity k influences dramatically the bubble of concentration (1.6) -in contrast with direct perturbative methods developed in [3], [4], [1], see also [20] for existence in the one dimensional half wave problem.…”