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

Abstract

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 ( * ) i ψ t + Δ ψ − ( | x | − 1 * | ψ | 2 ) ψ + | ψ | p − 2 ψ = 0 when p ∈ ( 10 3 , 6 ) . To obtain such solutions, we look into critical points of the energy functional F ( u ) = 1 2 ‖ ∇ u ‖ L 2 ( R 3 ) 2 + 1 4 ∫ R 3 ∫ R 3 | u ( x ) | 2 | u ( y ) | 2 | x − y | d x d y − 1 p ∫ R 3 | u | p d x , on the constraints given by S ( c ) = { u ∈ H 1 ( R ) 3 : ‖ u ‖ L 2 ( R 3 ) 2 = c , c > 0 } . For the values p ∈ ( 10 3 , 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 ‖ u 0 ‖ 2 2 = 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.