Liouville-Type Theorems for Fractional and Higher-Order Henon-Hardy Type Equations via the Method of Scaling Spheres

doi:10.1093/imrn/rnac079

CSpace

	Liouville-Type Theorems for Fractional and Higher-Order Henon-Hardy Type Equations via the Method of Scaling Spheres
	Dai, Wei 1,2; Qin, Guolin 3,4
	2022-04-11
发表期刊	INTERNATIONAL MATHEMATICS RESEARCH NOTICES
ISSN	1073-7928
页码	70
摘要	In this paper, we aim to develop the (direct) method of scaling spheres, its integral forms, and the method of scaling spheres in a local way. As applications, we investigate Liouville properties of nonnegative solutions to fractional and higher-order Henon-Hardy type equations (-Delta)(alpha/2) u(x) = f (x, u(x)) in R-n, R-+(n) or B-R(0) with n > alpha, 0 < alpha < 2 or alpha = 2m with 1 <= m < n/2. We first consider the typical case f (x, u) =vertical bar x vertical bar(a)u(p) with a is an element of (-alpha,infinity) and 0 < p < p(c)(a) := n+alpha+2a/n-alpha. By using the method of scaling spheres, we prove Liouville theorems for the above Henon-Hardy equations and equivalent integral equations (IEs). In R-n, our results improve the known Liouville theorems for some especially admissible subranges of a and 1 < p < min {n+alpha+a/n-alpha, p(c)(a)} to the full range alpha is an element of (-alpha,infinity) and p is an element of (0, p(c)(a)). In particular, when a > 0, we covered the gap p is an element of [n+alpha+a/n-alpha, p(c)(a)). For bounded domains (i.e., balls), we also apply the method of scaling spheres to derive Liouville theorems for super-critical problems. Extensions to PDEs and IEs with general nonlinearities f (x, u) are also included (Theorem 1.31). In addition to improving most of known Liouville type results to the sharp exponents in a unified way, we believe the method of scaling spheres developed here can be applied conveniently to various fractional or higher order problems with singularities or without translation invariance or in the cases the method of moving planes in conjunction with Kelvin transforms do not work.
DOI	10.1093/imrn/rnac079
收录类别	SCI
语种	英语
资助项目	National Natural Science Foundation of China[11971049] ; National Natural Science Foundation of China[11501021] ; Fundamental Research Funds for the Central Universities ; State Scholarship Fund of China[201806025011]
WOS研究方向	Mathematics
WOS类目	Mathematics
WOS记录号	WOS:000784154500001
出版者	OXFORD UNIV PRESS
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/60296
专题	中国科学院数学与系统科学研究院
通讯作者	Dai, Wei
作者单位	1.Beihang Univ, Sch Math Sci, Beijing 100191, Peoples R China 2.Univ Sorbonne Paris Nord, Inst Galilee, LAGA, UMR 7539, F-93430 Villetaneuse, France 3.Chinese Acad Sci, Inst Appl Math, Beijing 100190, Peoples R China 4.Univ Chinese Acad Sci, Beijing 100049, Peoples R China
推荐引用方式 GB/T 7714	Dai, Wei,Qin, Guolin. Liouville-Type Theorems for Fractional and Higher-Order Henon-Hardy Type Equations via the Method of Scaling Spheres[J]. INTERNATIONAL MATHEMATICS RESEARCH NOTICES,2022:70.
APA	Dai, Wei,&Qin, Guolin.(2022).Liouville-Type Theorems for Fractional and Higher-Order Henon-Hardy Type Equations via the Method of Scaling Spheres.INTERNATIONAL MATHEMATICS RESEARCH NOTICES,70.
MLA	Dai, Wei,et al."Liouville-Type Theorems for Fractional and Higher-Order Henon-Hardy Type Equations via the Method of Scaling Spheres".INTERNATIONAL MATHEMATICS RESEARCH NOTICES (2022):70.