Least energy solutions to a cooperative system of Schrödinger equations with prescribed $L^2$-bounds: at least $L^2$-critical growth

Abstract

We look for least energy solutions to the cooperative systems of coupled Schrödinger equations \begin{equation*} \begin{cases} -Δu_i + λ_i u_i = \partial_iG(u)\quad \mathrm{in} \ \mathbb{R}^N, \ N \geq 3, u_i \in H^1(\mathbb{R}^N), \int_{\mathbb{R}^N} |u_i|^2 \, dx \leq ρ_i^2 \end{cases} i\in\{1,\dots,K\} \end{equation*} with $G\ge 0$, where ${\rho }_i>0$ is prescribed and $({\lambda }_i,u_i)\in \mathbb{R}\times H^1({\mathbb{R}}^N)$ is to be determined, $i\in \{1,\dots ,K\}$. Our approach is based on the minimization of the energy functional over a linear combination of the Nehari and Pohožaev constraints intersected with the product of the closed balls in $L^2({\mathbb{R}}^N)$ of radii ${\rho }_i$, which allows to provide general growth assumptions about $G$ and to know in advance the sign of the corresponding Lagrange multipliers. We assume that $G$ has at least $L^2$-critical growth at $0$ and admits Sobolev critical growth. The more assumptions we make about $G$, $N$, and $K$, the more can be said about the minimizers of the corresponding energy functional. In particular, if $K=2$, $N\in \{3,4\}$, and $G$ satisfies further assumptions, then $u=(u_1,u_2)$ is normalized, i.e., ${\int }_{{\mathbb{R}}^N}|u_i{|}^{2}dx={\rho }_i^2$ for $i\in \{1,2\}$.

To appear in Calc. Var. Partial Differential Equations