Existence and stability of standing waves for a weakly coupled nonlinear fractional Schrödinger system

Abstract

We study the system of two weakly coupled nonlinear fractional Schrödinger equations{(−Δ)su+ωu=|u|2p−2u+β|u|p−2u|v|pin RN,(−Δ)sv+ωv=|v|2p−2v+β|u|p|v|p−2vin RN, where N≥2,s∈(0,1],2<2p<2s⁎=2N/(N−2s),ω>0 and β>0. For s∈(0,1] and 1<p<2, by finding a new kind of solution to the above system and comparing its energy level with those of other kinds of solutions, we show that each component of the least energy solution is nontrivial for any β>0, which forms a striking contrast with case p≥2. We emphasize that this result can be viewed as a complement to that of Guo and He (2016) [13] and Maia et al. (2006) [28]. When s∈(0,1),1<p<1+2s/N and ω>0 is fixed, we obtain the orbital stability of standing waves associated with the synchronized solution to the above system. To this end, we introduce an auxiliary mass constraint problem and establish the connection between the orbital stability of standing waves and the stability of ground state set to the auxiliary problem.