ABSTRACT

In this paper, we are concerned with the Hénon–Hardy type systems on ${\mathbb{R}}^n$: $\begin{array}{rl}(-\mathrm{\Delta }{)}^{\frac{\alpha }{2}}u(x)=|x{|}^a{v}^p(x),& u\left(x\right)\ge 0,x\in {\mathbb{R}}^n,\\ (-\mathrm{\Delta }{)}^{\frac{\alpha }{2}}v(x)=|x{|}^b{u}^q(x),& v\left(x\right)\ge 0,x\in {\mathbb{R}}^n,\end{array}$ where $n\ge 2$, $n>\alpha$, $0<\alpha \le 2$ or $\alpha =2m$. We prove Liouville theorems (i.e. non-existence of nontrivial nonnegative solutions) for the above Hénon–Hardy systems. The arguments used in our proof is the method of scaling spheres developed in [Dai W, Qin GLiouville type theorems for fractional and higher-order Hénon–Hardy type equations via the method of scaling spheres. preprint, submitted for publication, arXiv: 1810.02752.]. Our results generalize the Liouville theorems for single Hénon–Hardy equation on ${\mathbb{R}}^n$ in Bidaut-Véron and Pohozaev [Nonexistence results and estimates for some nonlinear elliptic problems. J Anal Math. 2001;84:1.49], Chen et al. [Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy–Hénon equations in ${\mathbb{R}}^N$. preprint, submitted, arXiv: 1808.06609], Dai et al. [Liouville type theorems, a priori estimates and existence of solutions for non-critical higher-order Lane–Emden–Hardy equations. preprint, submitted for publication, arXiv: 1808–10771], Dai and Qin [Liouville type theorems for Hardy–Hénon equations with concave nonlinearities. Math Nachrichten. 2020;293(6):1084–1093. https://doi.org/10.1002/mana.201800532; Liouville type theorems for fractional and higher-order Hénon–Hardy type equations via the method of scaling spheres. preprint, submitted for publication, arXiv: 1810.02752], Guo and Liu [Liouville-type theorems for polyharmonic equations in ${\mathbb{R}}^N$ and in Liouville-type theorems for. Proc Roy Soc Edinburgh Sect A. 2008;138(2):339–359], and Phan and Souplet [Liouville-type theorems and bounds of solutions of Hardy–Hénon equations. J Diff Equ. 2012;252:2544–2562] to systems.

摘要

在本文中，我们关注的是 Hénon-Hardy 类型系统 ${\mathbb{电阻}}^n$： $\begin{array}{rl}(-\mathrm{\Delta }{)}^{\frac{\alpha }{2}}你(X)=|X{|}^{\mathrm{一种}}{v}^{磷}(X),& 你\left(X\right)\ge 0,X\in {\mathbb{电阻}}^n,\\ (-\mathrm{\Delta }{)}^{\frac{\alpha }{2}}v(X)=|X{|}^{乙}{你}^q(X),& v\left(X\right)\ge 0,X\in {\mathbb{电阻}}^n,\end{array}$ 在哪里 $n\ge 2$, $n>\alpha$, $0<\alpha \le 2$ 或者 $\alpha =2米$. 我们证明了上述 Hénon-Hardy 系统的 Liouville 定理（即不存在非平凡非负解）。在我们的证明中使用的参数是在 [Dai W, Qin GLiouville 型定理中通过缩放球体的方法为分数和高阶 Hénon-Hardy 型方程开发的缩放球体的方法。预印本，提交出版，arXiv：1810.02752。]。我们的结果概括了单个 Hénon-Hardy 方程的 Liouville 定理${\mathbb{电阻}}^n$在 Bidaut-Véron 和 Pohozaev [一些非线性椭圆问题的不存在结果和估计。J肛门数学。2001;84:1.49]，陈等人。[Liouville 型定理、临界阶 Hardy-Hénon 方程的先验估计和解的存在性${\mathbb{电阻}}^N$. 预印本，提交，arXiv：1808.06609]，戴等人。[Liouville 型定理、非临界高阶 Lane-Emden-Hardy 方程的先验估计和解的存在。预印本，提交出版，arXiv：1808-10771]，戴和秦 [具有凹非线性的 Hardy-Hénon 方程的刘维尔型定理。数学新闻。2020;293(6):1084–1093。https://doi.org/10.1002/mana.201800532；分数阶和高阶 Hénon-Hardy 型方程的 Liouville 型定理，通过缩放球体的方法。预印本，提交出版，arXiv：1810.02752]，Guo 和 Liu [Liouville-type theorems for polyharmonic equations in${\mathbb{电阻}}^N$并在 Liouville 型定理中。Proc Roy Soc Edinburgh Sect A. 2008;138(2):339-359]，以及 Phan 和 Souplet [Liouville 型定理和 Hardy-Hénon 方程解的界限。J Diff Equ。2012;252:2544-2562] 到系统。