含Hardy型势的临界Grushin算子方程解的存在性和渐近估计

数学物理学报 ›› 2021, Vol. 41 ›› Issue (4): 997-1012.

含Hardy型势的临界Grushin算子方程解的存在性和渐近估计

张金国*(),杨登允()

^{江西师范大学数学与统计学院南昌 330022}

收稿日期:2020-05-13 出版日期:2021-08-26 发布日期:2021-08-09
通讯作者: 张金国 E-mail:jgzhang@jxnu.edu.cn;yangdengyun@139.com
作者简介:杨登允, E-mail: yangdengyun@139.com
基金资助:
国家自然科学基金(11761049)

Existence and Asymptotic Behavior of Solution for a Degenerate Elliptic Equation Involving Grushin-Type Operator and Critical Sobolev-Hardy Exponents

Jinguo Zhang*(),Dengyun Yang()

^{School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022}

Received:2020-05-13 Online:2021-08-26 Published:2021-08-09
Contact: Jinguo Zhang E-mail:jgzhang@jxnu.edu.cn;yangdengyun@139.com
Supported by:
the NSFC(11761049)

摘要/Abstract

摘要：

该文研究含Hardy型势和临界指数的退化椭圆方程 $ -（\Delta_{x}+|x|^{2\alpha}\Delta_{y}） u-\mu\frac{\psi^{2}u}{d （z）^{2}}=\frac{\psi^{s}|u|^{2^*（s）-2}u}{d （z）^{s}}，\, \, z=（x，y）\in\mathbb{R}^{m}\times\mathbb{R}^{n}， $ 其中 $-（\Delta_{x}+|x|^{2\alpha}\Delta_{y}）$ 是Grushin型退化算子，$\alpha>0, 2^*（s）=\frac{2（Q-s）}{Q-2}, Q=m+（\alpha+1） n$ 是空间 $\mathbb{R}^{m}\times\mathbb{R}^{n} $ 在伸缩变换 $\delta_{\lambda} $ 下的空间齐次维数.当 $0 \leq\mu <\mu_{G}：=（\frac{Q-2}{2}）^{2} ，0\leq s<2$ 时，该文证明了上述方程非平凡解的存在性；并且给出了方程的解在原点和无穷远点的渐近性质，即当 $d （z）\to 0 $ 时，$ u （z）=O （d （z）^{-（\frac{Q-2}{2}-\sqrt{（\frac{Q-2}{2}）^{2}-\mu}）}）$；当 $ d （z）\to+\infty$ 时，$u （z）=O （d （z）^{-（\frac{Q-2}{2}+\sqrt{（\frac{Q-2}{2}）^{2}-\mu}）}） $.

关键词: Grushin型算子, Moser迭代, 渐近性质, Sobolev-Hardy临界指数

Abstract:

In this paper, we study the existence and asymptotic behavior of solutions for a class of degenerate elliptic equation involving Grushin-type operator and Hardy potentials $ -(\Delta_{x}+|x|^{2\alpha}\Delta_{y}) u-\mu\frac{\psi^{2}u}{d (z)^{2}}=\frac{\psi^{s}|u|^{2^*(s)-2}u}{d (z)^{s}},\, \, z=(x,y)\in\mathbb{R}^{m}\times\mathbb{R}^{n}, $ where $-(\Delta_{x}+|x|^{2\alpha}\Delta_{y}) $ is the Grushin-type operator, $\alpha>0, 2^*(s)=\frac{2(Q-s)}{Q-2} $ is the critical Sobolev-Hardy exponent and $Q=m+(\alpha+1)n $ is the homogenous dimension for Grushin operator. If $0 \leq \mu<(\frac{Q-2}{2})^{2}, 0 < s <2$, we will prove the existence of nontrivial, nonnegative solutions for this degenerate problem, and give the asymptotic behavior of solutions, at the singularity and at infinity.

Key words: Grushin-type operator, Moser iteration, Asymptotic behavior, Critical Sobolev-Hardy exponents

中图分类号:

O175.29

张金国,杨登允. 含Hardy型势的临界Grushin算子方程解的存在性和渐近估计[J]. 数学物理学报, 2021, 41(4): 997-1012.

Jinguo Zhang,Dengyun Yang. Existence and Asymptotic Behavior of Solution for a Degenerate Elliptic Equation Involving Grushin-Type Operator and Critical Sobolev-Hardy Exponents[J]. Acta mathematica scientia,Series A, 2021, 41(4): 997-1012.

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