Ground state for a system of nonlinear Schrödinger equations with three waves interaction and critical nonlinearities

In this paper, we consider a system of nonlinear Schrödinger equations with three waves interaction and critical exponents. We discuss the existence of a nontrivial ground state solution. This problem has been studied by several researchers, for example Pomponio (2010) [7] and Kurata and Osada (2021) [2] in the case where all the exponents of the nonlinearities are subcritical. In this paper, we will demonstrate that even when some of or all of the exponents of the nonlinearities 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 $(u_1,u_2,u_3)$ that satisfies $u_i\ne 0$ for all $i=1,2,3$. Our method is to consider a minimization problem on the Nehari manifold.