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: $$\{\begin{array}{ll}-\Delta u+{\lambda }_{1}u={\mu }_1|u{|}^{p-2}u+\beta r_1|u{|}^{r_1-2}|v{|}^{r_2}u& in{\mathbb{R}}^N,\\ -\Delta v+{\lambda }_{2}v={\mu }_2|v{|}^{q-2}v+\beta r_2|u{|}^{r_1}|v{|}^{r_2-2}v& in{\mathbb{R}}^N,\\ {\int }_{{\mathbb{R}}^N}{u}^2=a_1^{2}and{\int }_{{\mathbb{R}}^N}{v}^2=a_2^2,& \end{array}$$ where $p,q,r_1+r_2$ can be Sobolev critical. To this purpose, we study the geometry of the Pohozaev manifold and the associated minimizition problem. Under some assumption on $a_1,a_2$ and $\beta$, we obtain the existence of the positive normalized ground state solution to the above system. We have solved some unsolved open problems in this area.