Liouville type theorem for higher order Henon equations on a half space
Dai, Wei1,2; Qin, Guolin3,4; Zhang, Yang3,4
AbstractIn this paper, we are concerned with the higher order Henon equations with Navier boundary condition on a half space R-+(n): { (-Delta)(m)u(x) = vertical bar x vertical bar(a)u(p)(x), u(x) >= 0, x is an element of R-+(n), (0.1) u= (-Delta)u = . . . = (-Delta)(m-1)u = 0, x is an element of partial derivative R-+(n), where u is an element of C-2m (R-+(n)) boolean AND C2m-2 (<(R-+(n))over bar>), a >= 0, n >= 3, 1 <= m < n/2 and 1 < p < n+2m+2a/n-2m. We first prove the super poly-harmonic properties and establish the equivalence between (0.1) and the corresponding integral equation. Then, we consider the equivalent integral equation of generalized form, that is, ( ) u(x) = integral(R+n) G(x,y)vertical bar y vertical bar(a)u(p)(y)dy (0.2) where G(x, y) denotes the Green's function for (-Delta)(m) on R-+(n) with Navier or Dirichlet boundary conditions. We establish Liouville theorem for (0.2) via "the method of scaling spheres" in integral forms developed initially in [14] (2018) by Dai and Qin. As a consequence, we obtain the Liouville theorem for (0.1). Extensions to IEs and PDEs with general nonlinearities are also included. (C) 2019 Elsevier Ltd. All rights reserved.
KeywordThe method of scaling spheres in integral forms Henon equations Liouville theorems Nonnegative solutions Navier problems
Funding ProjectNNSF of China[11501021] ; State Scholarship Fund of China[201806025011]
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000462288100013
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Document Type期刊论文
Corresponding AuthorQin, Guolin
Affiliation1.Beihang Univ BUAA, Sch Math & Syst Sci, Beijing 100083, Peoples R China
2.Univ Paris 13, LAGA, UMR 7539, Paris, France
3.Chinese Acad Sci, Inst Appl Math, Beijing 100190, Peoples R China
4.Univ Chinese Acad Sci, Beijing 100049, Peoples R China
Recommended Citation
GB/T 7714
Dai, Wei,Qin, Guolin,Zhang, Yang. Liouville type theorem for higher order Henon equations on a half space[J]. NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS,2019,183:284-302.
APA Dai, Wei,Qin, Guolin,&Zhang, Yang.(2019).Liouville type theorem for higher order Henon equations on a half space.NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS,183,284-302.
MLA Dai, Wei,et al."Liouville type theorem for higher order Henon equations on a half space".NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 183(2019):284-302.
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