Normalized solutions to Schrödinger systems with critical nonlinearities

Abstract

We consider a system of coupled Schrödinger equations involving critical exponent given by $\left(\begin{array}{cc}-\Delta u+{\lambda }_{1}u=\mu {\left|u\right|}^{q-2}u+\frac{2\alpha }{\alpha +\beta }{\left|u\right|}^{\alpha -2}u{\left|v\right|}^{\beta }& \text{in}{R}^N,\\ -\Delta v+{\lambda }_{2}v=\mu {\left|v\right|}^{q-2}v+\frac{2\beta }{\alpha +\beta }{\left|u\right|}^{\alpha }{\left|v\right|}^{\beta -2}v& \text{in}{R}^N.\end{array}\right)$We study the existence of positive ground state solutions having prescribed mass ${\int }_{R^N}{\left|u\right|}^{2}dx=a_1^2\text{and}{\int }_{R^N}{\left|v\right|}^{2}dx=a_2^2,$where $N=3,4$, $a_1,a_2>0$, $q\in (2,2^{\ast })$, $\alpha ,\beta >1$ with $\alpha +\beta =2^{\ast }=\frac{2N}{N-2}$, the Sobolev critical exponent, ${\lambda }_1,{\lambda }_2\in R$ are parameters to be specified and will appear as Lagrange multipliers, and $\mu >0$ is a parameter. Under some $L^2$-subcritical, $L^2$-critical and $L^2$-supercritical perturbations $\mu {\left|u\right|}^{q-2}u$ and $\mu {\left|v\right|}^{q-2}v$, respectively, we prove several existence results by using variational methods, which can be considered as a counterpart of the Brézis-Nirenberg problem in the context of normalized solutions for coupled Schrödinger equations. Our results extend and improve the existing literature in several directions.