Orbital stability of semitrivial standing waves for the Klein–Gordon–Schrödinger system

In the present paper, we study the orbital stability and instability of standing waves of the Klein–Gordon–Schrödinger system. Especially, we are interested in a standing wave which is expressed by the unique positive solution w1 to a certain scalar field equation. By utilizing the property of the positive solution w1, we can apply the general theory of Grillakis, Shatah and Strauss (1987) [11] and show the stability and instability of the standing wave.