Normalized ground states for semilinear elliptic systems with critical and subcritical nonlinearities

Abstract

In the present paper, we study the normalized solutions with least energy to the following system: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda _1u=\mu _1 |u|^{p-2}u+\beta r_1|u|^{r_1-2}|v|^{r_2}u\quad &{}\hbox {in}~{{\mathbb {R}}^N},\\ -\Delta v+\lambda _2v=\mu _2 |v|^{q-2}v+\beta r_2|u|^{r_1}|v|^{r_2-2}v\quad &{}\hbox {in}~{{\mathbb {R}}^N},\\ \int _{{{\mathbb {R}}^N}}u^2=a_1^2\quad \hbox {and}\quad \int _{{{\mathbb {R}}^N}}v^2=a_2^2, \end{array}\right. } \end{aligned}$$where \(p,r_1+r_2<2^*\) and \(q\le 2^*\). To this purpose, we study the geometry of the Pohozaev manifold and the associated minimization problem. Under some assumptions on \(a_1,a_2\) and \(\beta \), we obtain the existence of the positive normalized ground state solution to the above system.