Existence and Asymptotic Behavior of Normalized Ground States for a Weighted Elliptic System with…

Abstract

In this paper, we study the coupled singular weighted elliptic system $$\left\{ \begin{aligned} -\operatorname{div}(|x|^{-2a}\nabla u)+λ_1 \frac{u}{|x|^{2a}}&=βp\frac{|v|^q|u|^{p-2}u}{|x|^{b(p+q)}}+\frac{|u|^{2^{\sharp}-2}u}{|x|^{b2^{\sharp}}},\quad\text{in}~\mathbb{R}^N, -\operatorname{div}(|x|^{-2a}\nabla v)+λ_2\frac{v}{|x|^{2a}}&=βq\frac{|u|^p|v|^{q-2}v}{|x|^{b(p+q)}}+\frac{|v|^{2^\sharp-2}v}{|x|^{b{2^\sharp}}},\quad\text{in}~\mathbb{R}^N, \displaystyle\int_{\mathbb{R}^N}\frac{|u|^2}{|x|^{2a}} dx=ρ_1^2,\quad &\int_{\mathbb{R}^N}\frac{|v|^2}{|x|^{2a}}dx=ρ_2^2. \end{aligned} \right. $$ where N≥3, β∈ℝ, ρ₁,ρ₂>0, max{0,(N−4)/2}≤ a<(N−2)/2, a<b<a+1, 2^♯:=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 λ₁,λ₂∈ℝ are unknown, since the two weighted masses are prescribed. We show that the ground state level is not attained for β≤0. For β>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 β. In the mass supercritical regime, a threshold β₀≥0 appears: a ground state exists for every β>β₀ and does not exist for 0<β<β₀. We give conditions on p,q under which β₀=0, and we prove that β₀>0 when p,q≥2. Finally, we describe the ground states as β→0⁺ and as β→+∞: 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.

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BibTeX

@misc{rai2026existence,
  title = {{Existence and Asymptotic Behavior of Normalized Ground States for a Weighted Elliptic System with Caffarelli--Kohn--Nirenberg Critical Exponent}},
  author = {Asmita Rai and Divya Goel},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11494},
  eprint = {2609.11494},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11494},
}

RIS

TY  - UNPB
TI  - Existence and Asymptotic Behavior of Normalized Ground States for a Weighted Elliptic System with Caffarelli--Kohn--Nirenberg Critical Exponent
AU  - Asmita Rai
AU  - Divya Goel
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11494
UR  - https://arxiv.org/abs/2609.11494
AB  - In this paper, we study the coupled singular weighted elliptic system $$\left\{ \begin{aligned} -\operatorname{div}(|x|^{-2a}\nabla u)+λ_1 \frac{u}{|x|^{2a}}&=βp\frac{|v|^q|u|^{p-2}u}{|x|^{b(p+q)}}+\frac{|u|^{2^{\sharp}-2}u}{|x|^{b2^{\sharp}}},\quad\text{in}~\mathbb{R}^N, -\operatorname{div}(|x|^{-2a}\nabla v)+λ_2\frac{v}{|x|^{2a}}&=βq\frac{|u|^p|v|^{q-2}v}{|x|^{b(p+q)}}+\frac{|v|^{2^\sharp-2}v}{|x|^{b{2^\sharp}}},\quad\text{in}~\mathbb{R}^N, \displaystyle\int_{\mathbb{R}^N}\frac{|u|^2}{|x|^{2a}} dx=ρ_1^2,\quad &\int_{\mathbb{R}^N}\frac{|v|^2}{|x|^{2a}}dx=ρ_2^2. \end{aligned} \right. $$ where N≥3, β∈ℝ, ρ_1,ρ_2>0, max{0,(N−4)/2}≤ a<(N−2)/2, a<b<a+1, 2^♯:=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 λ_1,λ_2∈ℝ are unknown, since the two weighted masses are prescribed. We show that the ground state level is not attained for β≤0. For β>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 β. In the mass supercritical regime, a threshold β_0≥0 appears: a ground state exists for every β>β_0 and does not exist for 0<β<β_0. We give conditions on p,q under which β_0=0, and we prove that β_0>0 when p,q≥2. Finally, we describe the ground states as β→0^+ and as β→+∞: 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.
ER  -

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