Existence of normalized solutions for the coupled elliptic system with quadratic nonlinearity

In the present paper, we study the existence of the normalized solutions for the following coupled elliptic system with quadratic nonlinearity − Δ u − λ 1 u = μ 1 ∣ u ∣ u + β u v in R N , − Δ v − λ 2 v = μ 2 ∣ v ∣ v + β 2 u 2 in R N , \left\{\begin{array}{ll}-\Delta u-{\lambda }_{1}u={\mu }_{1}| u| u+\beta uv\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ -\Delta v-{\lambda }_{2}v={\mu }_{2}| v| v+\frac{\beta }{2}{u}^{2}\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\end{array}\right. where u , v u,v satisfying the additional condition ∫ R N u 2 d x = a 1 , ∫ R N v 2 d x = a 2 . \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={a}_{1},\hspace{1em}\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{v}^{2}{\rm{d}}x={a}_{2}. On the one hand, we prove the existence of minimizer for the system with L 2 {L}^{2} -subcritical growth ( N ≤ 3 N\le 3 ). On the other hand, we prove the existence results for different ranges of the coupling parameter β > 0 \beta \gt 0 with L 2 {L}^{2} -supercritical growth ( N = 5 N=5 ). Our argument is based on the rearrangement techniques and the minimax construction.