Existence and instability of standing waves with prescribed norm for a class of Schrödinger–Poisson equations

By Jacopo Bellazzini, Louis Jeanjean, Tingjian Luo. 2011

284 citations recorded by the source corpus.

In this paper, we study the existence and the instability of standing waves with prescribed L2‐norm for a class of Schrödinger–Poisson–Slater equations in ℝ3 (*) iψt+Δψ−(| x |−1*| ψ |2)ψ+| ψ |p−2ψ=0 when p∈(103,6) . To obtain such solutions, we look into critical points of the energy functional F(u)=12‖ ∇u ‖L2(R3)2+14∫R3 ∫R3 | u(x) |2| u(y) |2| x−y |dx dy−1p∫R3 | u |pdx, on the constraints given by S(c)={ u∈H1(R)3:‖ u ‖L2(R3)2=c,c>0 }. For the values p∈(103,6) considered, the functional F is unbounded from below on S(c) and the existence of critical points is obtained by a mountain‐pass argument developed on S(c). We show that critical points exist provided that c > 0 is sufficiently small and that when c > 0 is not small a nonexistence result is expected. Regarding the dynamics, we show for initial condition u0∈H1(ℝ3) of the associated Cauchy problem with ‖ u0 ‖22=c that the mountain‐pass energy level γ(c) gives a threshold for global existence. Also, the strong instability of standing waves at the mountain pass energy level is proved. Finally, we draw a comparison between the Schrödinger–Poisson–Slater equation and the classical nonlinear Schrödinger equation.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a5186ffacece443c0b031d36bb6d27f6f110c3b5

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Citations in the corpus

Normalized solutions for a Schrödinger equation with critical growth in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{documen (2021, unknown): “For the reader interested in normalized solutions for the Schrödinger equations, we would also like to refer [9,11,12,17,25,29,31,34,38,39,41] and references therein.”
Normalized ground states for the NLS equation with combined nonlinearities (2018, unknown): “More recently, however, problems of this type received much attention, see [1, 10–12, 15, 16, 32] for normalized solutions to scalar equations in the whole space R , [6–9, 28] for normalized solutions to systems”
The existence of normalized solutions for L2-critical constrained problems related to Kirchhoff equations (2014, methodology): “As far as we know, there is no paper to study L(2)critical constrained problems in this way, which was used in [3,10,26] to deal with L(2) subcritical cases.”
A mass supercritical problem revisited (2020, background): “To overcome this difficulty, a by now standard strategy is the one initially proposed in [12].”
Normalized solutions for the Chern–Simons–Schrödinger equation in R^2 (2017, methodology): “8) S(c) = {u ∈ H(1)(R) : ‖u‖2L2(RN ) = c}, c > 0, [6] proved that for p ∈ ((10) 3 , 6), there exists c0 > 0 such that for any c ∈ (0, c0) there exists a couple (uc, λc) ∈ H1(RN)×R− of weak solution to (1.”
Orbital stability of ground states for a Sobolev critical Schrödinger equation (2020, background): “This approach, that we shall follow here, is relevant from the physical point of view, in particular, since the L2 norm is a preserved quantity of the evolution and since the variational characterization of such solutions is often a strong help to analyze their orbital stability, see for example, [6”
Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case (2019, unknown): “More recently, this kind of idea has been exploited and further developed in other contexts, and more in general the search for normalized solutions in cases when the energy is unbounded from below on the L-constraint attracted much attention: we refer to [1, 7, 12, 13, 16, 20, 34] for normalized so”