Stability and instability of standing waves to a system of Schrödinger equations with combined power-type nonlinearities

Abstract

We obtain the existence of standing wave solutions to a coupled nonlinear Schrödinger systemwhere 0 < p < 4 (N−2) + , μ, γ and α k , β k , p k , q k are real constants with 1 < p k < p + 1, 1 < q k < p + 1, p k + q k = p + 2 and α k q k = β k p k (k = 1, . . . ,m). We also discuss the stability and instability of the standing wave solutions.

Coupled nonlinear Schrödinger equations with harmonic potential

“…These works have been mainly on 2-systems or with small couplings. Moreover, most works treat the focusing case by considering the stationary associated problem [3][4][5]15,27]. Despite the partial progress made so far, many difficult questions remain open and little is known about m-systems for m ≥ 3.…”

A note on coupled nonlinear Schrödinger equations

“…These works have been mainly on 2-systems or with small couplings. Moreover, most works treat the focusing case by considering the stationary associated problem [2,21,13,3,4]. Despite the partial progress made so far, many difficult questions remain open and little is known about m-systems for m ≥ 3.…”

Invariant sets and the blow up threshold for coupled systems of reaction–diffusion

“…Since the literature is huge, we invite the readers to [16][17][18][19][20][21] for more references.…”