Existence and Asymptotic Behavior of Normalized Ground States for a Weighted Elliptic System with Caffarelli--Kohn--Nirenberg Critical Exponent

In this paper, we study the coupled singular weighted elliptic system $$\{\begin{array}{rlrl}{\displaystyle -\mathrm{div}(\mathrm{\mid }x{\mathrm{\mid }}^{-2a}\mathrm{\nabla }u)+{\lambda }_1\frac{u}{\mathrm{\mid }x{\mathrm{\mid }}^{2a}}}& {\displaystyle =\beta p\frac{\mathrm{\mid }v{\mathrm{\mid }}^q\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u}{\mathrm{\mid }x{\mathrm{\mid }}^{b(p+q)}}+\frac{\mathrm{\mid }u{\mathrm{\mid }}^{2^{\mathrm{♯}}-2}u}{\mathrm{\mid }x{\mathrm{\mid }}^{b{2}^{\mathrm{♯}}}},\text{in }{\mathbb{R}}^N,-\mathrm{div}(\mathrm{\mid }x{\mathrm{\mid }}^{-2a}\mathrm{\nabla }v)+{\lambda }_2\frac{v}{\mathrm{\mid }x{\mathrm{\mid }}^{2a}}}& {\displaystyle =\beta q\frac{\mathrm{\mid }u{\mathrm{\mid }}^p\mathrm{\mid }v{\mathrm{\mid }}^{q-2}v}{\mathrm{\mid }x{\mathrm{\mid }}^{b(p+q)}}+\frac{\mathrm{\mid }v{\mathrm{\mid }}^{2^{\mathrm{♯}}-2}v}{\mathrm{\mid }x{\mathrm{\mid }}^{b{2}^{\mathrm{♯}}}},\text{in }{\mathbb{R}}^N,{\displaystyle {\int }_{{\mathbb{R}}^N}\frac{\mathrm{\mid }u{\mathrm{\mid }}^2}{\mathrm{\mid }x{\mathrm{\mid }}^{2a}}dx={\rho }_1^2,}}& {\displaystyle {\int }_{{\mathbb{R}}^N}\frac{\mathrm{\mid }v{\mathrm{\mid }}^2}{\mathrm{\mid }x{\mathrm{\mid }}^{2a}}dx={\rho }_2^2\mathrm{.}}\end{array}$$ where $N\ge 3$, $\beta \in \mathbb{R}$, ${\rho }_1,{\rho }_2>0$, $\max \{0,\frac{N-4}{2}\}\le a<\frac{N-2}{2}$, $a<b<a+1$, $2^{\mathrm{♯}}:=\frac{2N}{N-2(1+a-b)}$, $p,q>1$ and $2<p+q<2^{\mathrm{♯}}$. Here $2^{\mathrm{♯}}$ is the critical exponent of the Caffarelli–Kohn–Nirenberg inequality, and ${\lambda }_1,{\lambda }_2\in \mathbb{R}$ are unknown, since the two weighted masses are prescribed. We show that the ground state level is not attained for $\beta \le 0$. For $\beta >0$ we obtain positive normalized ground states, with positive Lagrange multipliers, in the mass subcritical, mass critical, and mass supercritical regimes, for explicit ranges of $\beta$. In the mass supercritical regime, a threshold ${\beta }_0\ge 0$ appears: a ground state exists for every $\beta >{\beta }_0$ and does not exist for $0<\beta <{\beta }_0$. We give conditions on $p,q$ under which ${\beta }_0=0$, and we prove that ${\beta }_0>0$ when $p,q\ge 2$. Finally, we describe the ground states as $\beta \to 0^{+}$ and as $\beta \to +\mathrm{\infty }$: after a dilation, they converge to the ground states of the limit system without critical terms, or one component vanishes and the other concentrates on an extremal of the Caffarelli–Kohn–Nirenberg inequality.