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

TL;DR

This paper investigates the existence of least energy solutions to a system of coupled Schrödinger equations with prescribed $L^2$ bounds, considering critical growth conditions and employing a variational approach with combined constraints.

Contribution

It introduces a novel variational method using combined Nehari and Pohožaev constraints to find solutions with prescribed $L^2$ bounds under critical growth conditions.

Findings

Established existence of solutions with prescribed $L^2$ norms.

Provided conditions under which solutions are normalized.

Extended analysis to cases with Sobolev critical growth.

Abstract

We look for least energy solutions to the cooperative systems of coupled Schr\"odinger equations \begin{equation*} \begin{cases} -\Delta u_i + \lambda_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 \rho_i^2 \end{cases} i\in\{1,\dots,K\} \end{equation*} with $G\geq 0$, 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 Poho\v{z}aev 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 $i\in\{1,2\}$.