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

  title={Least energy solutions to a cooperative system of Schr{\"o}dinger equations with prescribed \$L^2\$-bounds: at least \$L^2\$-critical growth},
  author={Jarosław Mederski and Jacopo Schino},
  journal={arXiv: Analysis of PDEs},
We look for least energy solutions to the cooperative systems of coupled Schrodinger equations $$\left\{\begin{array}{l} -\Delta u_i+\lambda_i u_i = \partial_i G(u) \quad\text{in }\mathbb{R}^N,\ N\ge3, \\ u_i\in H^1(\mathbb{R}^N), & & \\ \textstyle\int_{\mathbb{R}^N}|u_i|^2\,dx\le \rho_i^2 \end{array} i\in\{1,\ldots,K\}\right.$$ with $G\ge0$, where $\rho_i>0$ is prescribed and $(\lambda_i,u_i)\in\mathbb{R}\times H^1(\mathbb{R}^N)$ is to be determined, $i\in\{1,\ldots,K\}$. Our approach is… 

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