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 Liouville-Type Theorems for Fractional and Higher-Order Henon-Hardy Type Equations via the Method of Scaling Spheres Dai, Wei1,2; Qin, Guolin3,4 2022-04-11 Source Publication INTERNATIONAL MATHEMATICS RESEARCH NOTICES ISSN 1073-7928 Pages 70 Abstract 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 Indexed By SCI Language 英语 Funding Project 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 Research Area Mathematics WOS Subject Mathematics WOS ID WOS:000784154500001 Publisher OXFORD UNIV PRESS Citation statistics Document Type 期刊论文 Identifier http://ir.amss.ac.cn/handle/2S8OKBNM/60296 Collection 中国科学院数学与系统科学研究院 Corresponding Author Dai, Wei Affiliation 1.Beihang Univ, Sch Math Sci, Beijing 100191, Peoples R China2.Univ Sorbonne Paris Nord, Inst Galilee, LAGA, UMR 7539, F-93430 Villetaneuse, France3.Chinese Acad Sci, Inst Appl Math, Beijing 100190, Peoples R China4.Univ Chinese Acad Sci, Beijing 100049, Peoples R China Recommended CitationGB/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.
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