Existence and orbital stability of the ground states with prescribed mass for the $L^2$-critical and supercritical NLS on bounded domains

By Benedetta Noris, Hugo Tavares, Gianmaria Verzini. 2013

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Given > 0, we study the elliptic problem , N and p) are provided for the existence of solutions. Moreover, we show that standing waves associ- ated to least energy solutions are orbitally stable for every (in the exis- tence range) when p is L 2 -critical and subcritical, i.e. 1 < p 1 + 4=N, while they are stable for almost every in the L 2 -supercritical regime 1 + 4=N < p < 2 1. The proofs are obtained in connection with the study of a variational problem with two constraints, of independent inter- est: to maximize the L p+1 -norm among functions having prescribed L 2 and H 1 0 -norm.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c161ab4a0e797ea473cc391a516a87b8d37e2ab0

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Citations in the corpus

Stationary focusing mean-field games (2016, background): “e(ϕ) is not bounded from below; if 2β<2N/(N− 2), one expects the existence of a critical point under the additional condition that the L2 constraint is small, or equivalently that Cf is small (see [33], where the setting of a bounded domain Ω and homogeneous Dirichlet boundary data is considered)”
Multiple positive normalized solutions for nonlinear Schrödinger systems (2017, background): “Let us also mention the recent papers [28, 29, 34] dealing with situations where the equation is set on a bounded domain or when a trapping potential is acting.”
Normalized ground states for the NLS equation with combined nonlinearities (2018, unknown): “in R , and [23, 41–44] for normalized solutions to equations or systems in bounded domains.”
Normalized solutions to mass supercritical Schrödinger equations with negative potential (2021, unknown): “potentials as in (Pρ) satisfying lim |x|→+∞ −V (x) = +∞ (or even −V ≡ +∞ outside some bounded Ω), have been considered in [28, 29, 33, 30] (see also [11] for the case of partial confinement).”
Normalized solutions of mass supercritical Schrödinger equations with potential (2020, background): “On the other hand, when dealing on bounded domains instead of R , it is known that solutions having bounded Morse index cannot exist for arbitrarily large q, see [7,8].”