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

In this paper, we make a comprehensive study for the orbital stability of standing waves for the fractional Schrodinger equation with combined power-type nonlinearities partial derivative(t)psi - (-Delta)(s)psi + a vertical bar psi vertical bar(p1) psi + vertical bar psi vertical bar(p)(2)psi = 0. We prove that when p(2) = 4s/N and a(p(1) - 4s/N) < 0, there exist the standing waves of (FNLS), which are orbitally stable. When a = 0 and 4s/N < p(2) < 4s/N-2s we present a new, simpler method to study the strong instability of standing waves. When a = - 1, 0 < p(1) < p(2) and 4s/N <= p(2) < 4s/N - 2s, or a = 1 and 4s/N <= p(1) < p(2) < 4s/N-2s or a = 1, 0 < p(1) < 4s/N < p(2) < 4s/N - 2s and partial derivative(2)(lambda) S-omega (u(omega)(lambda))vertical bar(lambda=1 )<= 0, we deduce that the ground state standing waves of (FNLS) are strongly unstable by blow-up. (C) 2021 Elsevier Inc. All rights reserved.