Stability and instability of standing waves for the fractional nonlinear Schrödinger equations

Abstract

In this paper, we make a comprehensive study for the orbital stability of standing waves for the fractional Schrödinger equation with combined power-type nonlinearities ( FNLS ) i ∂ t ψ − ( − Δ ) s ψ + a | ψ | p 1 ψ + | ψ | p 2 ψ = 0 . We prove that when p 2 = 4 s N and a ( p 1 − 4 s N ) < 0 , there exist the standing waves of (FNLS), which are orbitally stable. When a = 0 and 4 s N < p 2 < 4 s N − 2 s , we present a new, simpler method to study the strong instability of standing waves. When a = − 1 , 0 < p 1 < p 2 and 4 s N ≤ p 2 < 4 s N − 2 s , or a = 1 and 4 s N ≤ p 1 < p 2 < 4 s N − 2 s , or a = 1 , 0 < p 1 < 4 s N < p 2 < 4 s N − 2 s and ∂ λ 2 S ω ( u ω λ ) | λ = 1 ≤ 0 , we deduce that the ground state standing waves of (FNLS) are strongly unstable by blow-up.