We consider a system of nonlinear Schr\{o}dinger equations with three wave interactions and critical exponents and concern the existence of a nontrivial ground state solution. This problem has been studied by several researchers, for example Pomponio(2010) and Osada (2021, 2022, 2024) in the case where all the exponents of the nonlinearlities are subcritical.

In this talk, we will demonstrate that even when the exponents of the nonlinearity admit the Sobolev critical exponent, a nontrivial ground state solution can still be obtained if the coupling constant is sufficiently large. Additionally, we show that when the coupling constant is large enough, the ground state solution is a vector solution, namely, a solution $(u_1,u_2,u_3)$ which satisfies $u_i \neq 0$ for all $i=1,2,3$. Our method is to consider a minimization problem on the Nehari manifold.