Some results for a system of NLS arising in optical materials with χ3 nonlinear response

In this paper, we investigate the nonlinear Schödinger equations with cubic interactions, arising in nonlinear optics. To begin, we prove the existence results for normalized ground state solutions in the subcritical case and L2$$ {L}^2 $$‐supercritical case, respectively. Our proofs relies on the Concentration‐compactness principle, Pohozaev manifold, and rearrangement technique. Then, we establish the nonexistence of normalized ground state solutions in the L2$$ {L}^2 $$‐critical case by finding that there exists a threshold. In addition, based on the existence of the normalized solutions, we also establish the blow‐up results by using localized virial estimates as well as a new blow‐up criterion which is related to normalized solutions.