Liouville type theorem for critical order Henon-Lane-Emden type equations on a half space and its applications

doi:10.1016/j.jfa.2021.109227

	Liouville type theorem for critical order Henon-Lane-Emden type equations on a half space and its applications
	Dai, Wei 2,3,4; Qin, Guolin 1,5
刊名	JOURNAL OF FUNCTIONAL ANALYSIS
	2021-11-15
卷号	281 期号:10 页码:37
关键词	Critical order Henon-Lane-Emden type equations Liouville theorems A priori estimates
ISSN号	0022-1236
DOI	10.1016/j.jfa.2021.109227
英文摘要	In this paper, we are concerned with the critical (i.e., n-th) orderHenon-Lane-Emden type equations with Navier boundary conditions on a half space R-+(n): {-Delta)(n/2) u(x) = f(x,u(x)), u(x) >= 0, x is an element of R-+(n), u(x) = -Delta u(x) = center dot center dot center dot = (-Delta)n/2-1 u(x) = 0, x is an element of partial derivative R-+(n), (0.1) where u is an element of C-n(R-+(n)) boolean AND Cn-2((R) over bar (+n)) and n >= 2 is even. We first consider the typical case f( x, u) = vertical bar x vertical bar(a)u(p) with 0 <= a < infinity and 1 < p < infinity. We prove the super poly-harmonic propertiesand establish the equivalencebetween (0.1) and the corresponding integral equations u(x) = integral(n)(R+) + G(+)( x, y)f( y, u(y)) dy, (0.2) where G(+)( x, y) denotes the Green function for (-Delta)(n/2) on R-+(n) with Navier boundary conditions. Then, we establish Liouville theoremfor (0.2) and hence obtain the Liouville theorem for (0.1) on Rn+. As an application of the Liouville theorem on R-+(n)(Theorem1.6) and Liouville theorems in Rn, we derive a priori estimatesvia blowing-up methodsfor solutions (possibly change signs) to Navier problems involving critical order uniformly elliptic operatorsL. Consequently, by using the Leray-Schauder fixed point theorem, we derive existence of positive solutionsto critical order Lane-Emden equations in bounded domains for all n >= 2 and 1 < p < infinity. In contrast to the subcritical order cases, our results seem to be the firstwork on Navier problems for critical order equations on R-+(n), which is the critical-order counterpartto those results on subcritical order cases in [6,20,21]. Extensions to IEs and PDEs with general nonlinearities f( x, u) are also included. Surprisingly, there are no growth conditions on uand hence f( x, u) can grow exponentially (or even faster) on u. (C) 2021 Elsevier Inc. All rights reserved.
资助项目	NNSF of China[11971049] ; NNSF of China[11501021] ; Fundamental Research Funds for the Central Universities ; State Scholarship Fund of China[201806025011]
WOS研究方向	Mathematics
语种	英语
出版者	ACADEMIC PRESS INC ELSEVIER SCIENCE
WOS记录号	WOS:000696264800011
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/59263]
专题	中国科学院数学与系统科学研究院
通讯作者	Qin, Guolin
作者单位	1.Chinese Acad Sci, Inst Appl Math, Beijing 100190, Peoples R China 2.Beihang Univ BUAA, Sch Math Sci, Beijing 100191, Peoples R China 3.Minist Educ, Key Lab Math Informat & Behav Semant, Beijing 100191, Peoples R China 4.Univ Sorbonne Paris Nord, Inst Galilee, LAGA, UMR 7539, F-93430 Villetaneuse, France 5.Univ Chinese Acad Sci, Beijing 100049, Peoples R China
推荐引用方式 GB/T 7714	Dai, Wei,Qin, Guolin. Liouville type theorem for critical order Henon-Lane-Emden type equations on a half space and its applications[J]. JOURNAL OF FUNCTIONAL ANALYSIS,2021,281(10):37.
APA	Dai, Wei,&Qin, Guolin.(2021).Liouville type theorem for critical order Henon-Lane-Emden type equations on a half space and its applications.JOURNAL OF FUNCTIONAL ANALYSIS,281(10),37.
MLA	Dai, Wei,et al."Liouville type theorem for critical order Henon-Lane-Emden type equations on a half space and its applications".JOURNAL OF FUNCTIONAL ANALYSIS 281.10(2021):37.