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

Abstract

In this paper, we study the coupled singular weighted elliptic system $$\{\begin{array}{rlrl}-\mathrm{div}(|x{|}^{-2a}\nabla u)+{\lambda }_1\frac{u}{|x{|}^{2a}}& =\beta p\frac{|v{|}^q|u{|}^{p-2}u}{|x{|}^{b(p+q)}}+\frac{|u{|}^{2^{♯}-2}u}{|x{|}^{b{2}^{♯}}},\text{in}{\mathbb{R}}^N,-\mathrm{div}(|x{|}^{-2a}\nabla v)+{\lambda }_2\frac{v}{|x{|}^{2a}}& =\beta q\frac{|u{|}^p|v{|}^{q-2}v}{|x{|}^{b(p+q)}}+\frac{|v{|}^{2^{♯}-2}v}{|x{|}^{b{2}^{♯}}},\text{in}{\mathbb{R}}^N,{\int }_{{\mathbb{R}}^N}\frac{|u{|}^2}{|x{|}^{2a}}dx={\rho }_1^2,& {\int }_{{\mathbb{R}}^N}\frac{|v{|}^2}{|x{|}^{2a}}dx={\rho }_2^2.\end{array}$$ where $N\ge 3$, $\beta \in \mathbb{R}$, ${\rho }_1,{\rho }_2>0$, $\max \{0,\frac{N-4}{2}\}\le a<\frac{N-2}{2}$, $a<b<a+1$, $2^{♯}:=\frac{2N}{N-2(1+a-b)}$, $p,q>1$ and $2<p+q<2^{♯}$. Here $2^{♯}$ 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 $\beta \to +\infty$: 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.