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

• Published in: Proceedings of the London Mathematical Society 2013-08, Vol.107 (2), p.303-339
• Main Authors: Bellazzini, Jacopo, Jeanjean, Louis, Luo, Tingjian
• Format: Article
• Language: English
• Subjects: Analysis of PDEs; Critical point; Energy levels; Instability; Mathematical analysis; Mathematics; Mountains; Schroedinger equation; Stability; Standing waves
• ISSN: 0024-6115; 1460-244X
• DOI: 10.1112/plms/pds072

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.