Normalized ground states to a cooperative system of Schrödinger equations with generic $L^2$-subcritical or $L^2$-critical nonlinearity

Abstract

We look for ground state solutions to the Schrödinger-type system\[\begin{cases} -\Delta u_j+\lambda_ju_j=\partial_jF(u)\\ \displaystyle \int_{\mathbb R^N}u_j^2\,dx=a_j^2\\ (\lambda_j,u_j)\in\mathbb R\times H^1 (\mathbb R^N)\end{cases}j\in\{1,\dots,M\}\]with $N,M \ge 1$, where $a=(a_1,\dots,a_M)\in ( 0,\infty ) ^M$ is prescribed and $(\lambda,u)=(\lambda_1,\dots,\lambda_M,u_1,\dots u_M)$ is the unknown. We provide generic assumptions about the nonlinearity $F$ which correspond to the $L^2$-subcritical and $L^2$-critical cases, i.e., when the energy is bounded from below for all or some values of $a$. Making use of a recent idea, we minimize the energy over the constraint $\{\left|u_j\right|_{L^2} \le a_j \text{ for all } j \}$ and then provide further assumptions that ensure $|u_j|_{L^2}=a_j$.