带Caffarelli--Kohn--Nirenberg临界指数的加权椭圆系统归一化基态的存在性与渐近行为
Existence and Asymptotic Behavior of Normalized Ground States for a Weighted Elliptic System with Caffarelli--Kohn--Nirenberg Critical Exponent
AI总结：
本文研究带Caffarelli--Kohn--Nirenberg临界指数的加权椭圆系统，证明β≤0时基态不存在，β>0时在质量次临界、临界和超临界区域存在正归一化基态，并给出超临界区域阈值β₀的条件及β趋于0和无穷时的渐近行为。
AI中文摘要：
本文研究如下耦合奇异加权椭圆系统：$$\left\{ \begin{aligned} -\operatorname{div}(|x|^{-2a}\nabla u)+\lambda_1 \frac{u}{|x|^{2a}}&=\beta p\frac{|v|^q|u|^{p-2}u}{|x|^{b(p+q)}}+\frac{|u|^{2^{\sharp}-2}u}{|x|^{b2^{\sharp}}},\quad\text{in}~\mathbb{R}^N, -\operatorname{div}(|x|^{-2a}\nabla v)+\lambda_2\frac{v}{|x|^{2a}}&=\beta q\frac{|u|^p|v|^{q-2}v}{|x|^{b(p+q)}}+\frac{|v|^{2^\sharp-2}v}{|x|^{b{2^\sharp}}},\quad\text{in}~\mathbb{R}^N, \displaystyle\int_{\mathbb{R}^N}\frac{|u|^2}{|x|^{2a}} dx=\rho_1^2,\quad &\int_{\mathbb{R}^N}\frac{|v|^2}{|x|^{2a}}dx=\rho_2^2. \end{aligned} \right. $$ 其中$N\ge3$，$\beta\in\mathbb{R}$，$\rho_1,\rho_2>0$，$\max\{0,\tfrac{N-4}{2}\}\le a<\tfrac{N-2}{2}$，$a<b<a+1$，$2^{\sharp}:=\frac{2N}{N-2(1+a-b)}$，$p,q>1$且$2<p+q<2^{\sharp}$。这里$2^\sharp$是Caffarelli--Kohn--Nirenberg不等式的临界指数，而$\lambda_1,\lambda_2\in\mathbb{R}$是未知的，因为两个加权质量被指定。我们证明当$\beta\le0$时基态水平无法达到。对于$\beta>0$，在质量次临界、质量临界和质量超临界区域，在$\beta$的显式范围内，我们获得具有正拉格朗日乘子的正归一化基态。在质量超临界区域，出现阈值$\beta_0\ge0$：对于每个$\beta>\beta_0$存在基态，而对于$0<\beta<\beta_0$不存在基态。我们给出$p,q$的条件使得$\beta_0=0$，并证明当$p,q\ge2$时$\beta_0>0$。最后，我们描述当$\beta\to0^+$和$\beta\to+\infty$时基态的渐近行为：经过伸缩后，它们收敛到没有临界项的极限系统的基态，或者一个分量消失而另一个分量集中在Caffarelli--Kohn--Nirenberg不等式的一个极值上。
英文摘要：
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 $N\ge3$, $β\in\mathbb{R}$, $ρ_1,ρ_2>0$, $\max\{0,\tfrac{N-4}{2}\}\le a<\tfrac{N-2}{2}$, $a<b<a+1$, $2^{\sharp}:=\frac{2N}{N-2(1+a-b)}$, $p,q>1$ and $2<p+q<2^{\sharp}$. Here $2^\sharp$ is the critical exponent of the Caffarelli--Kohn--Nirenberg inequality, and $λ_1,λ_2\in\mathbb{R}$ are unknown, since the two weighted masses are prescribed. We show that the ground state level is not attained for $β\le0$. 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 $β_0\ge0$ 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 $p,q\ge2$. Finally, we describe the ground states as $β\to0^+$ and as $β\to+\infty$: 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.
