As a continuation of our previous work in Forcella et al. Standing waves for a Schrödinger system with three-wave interaction, arXiv:2210.07643, 2022), we investigate the standing wave solutions of a nonlinear Schrödinger system with three-wave interaction on a bounded domain. When the nonlinearities are \(L^2\) -supercritical (energy critical) and the coupling terms are \(L^2\) -subcritical, we obtain that three-wave equations possess at least two solutions, which are characterized by a local minimizer and a Mountain Pass critical point of the corresponding energy functional. Moreover, we also show that the set of local ground states is stable under the three-wave evolution by assuming local well-posedness. Comparing the results in this paper with those in Song and Zou (Two positive normalized solutions and phase Separation for coupled Schrödinger equations on bounded domain with \(L^2\) -supercritical and Sobolev critical or subcritical exponent, arXiv: 2311.16861, 2023), one sees that these results are substantially different from those of pure mass subcritical or mass supercritical.
