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

RESUMO

In this paper, we study the coupled singular weighted elliptic system $$\left\{ \begin{aligned} -\operatorname{div}(|x|^{-2a}\nabla u)+\lambda_1 \frac{u}{|x|^{2a}}&=\beta p\frac{|v|^q|u|^{p-2}u}{|x|^{b(p+q)}}+\frac{|u|^{2^{\sharp}-2}u}{|x|^{b2^{\sharp}}},\quad\text{in}~\mathbb{R}^N, -\operatorname{div}(|x|^{-2a}\nabla v)+\lambda_2\frac{v}{|x|^{2a}}&=\beta q\frac{|u|^p|v|^{q-2}v}{|x|^{b(p+q)}}+\frac{|v|^{2^\sharp-2}v}{|x|^{b{2^\sharp}}},\quad\text{in}~\mathbb{R}^N, \displaystyle\int_{\mathbb{R}^N}\frac{|u|^2}{|x|^{2a}} dx=\rho_1^2,\quad &\int_{\mathbb{R}^N}\frac{|v|^2}{|x|^{2a}}dx=\rho_2^2. \end{aligned} \right. $$ where $N\ge3$, $\beta\in\mathbb{R}$, $\rho_1,\rho_2>0$, $\max\{0,\tfrac{N-4}{2}\}\le a<\tfrac{N-2}{2}$, $a<b<a+1$, $2^{\sharp}:=\frac{2N}{N-2(1+a-b)}$, $p,q>1$ and $2<p+q<2^{\sharp}$. Here $2^\sharp$ 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 $\beta\le0$. 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 $\beta_0\ge0$ 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 $p,q\ge2$. Finally, we describe the ground states as $\beta\to0^+$ and as $\beta\to+\infty$: 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.