Weighted polyharmonic equation with Navier boundary conditions in a half space

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• 作者：Ran Zhuo
• 关键词：Navier boundary conditions ; half space ; super polyharmonic ; equivalence ; integral equation ; rotational symmetry ; non ; existence
• 刊名：Science China Mathematics
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：60
• 期：3
• 页码：491-510
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Applications of Mathematics;
• 出版者：Science China Press
• ISSN：1869-1862
• 卷排序：60

文摘

We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space: $$\left\{ {\begin{array}{*{20}c}{( - \Delta )^m u(x) = \frac{{u^p (x)}}{{\left| x \right|^s }},} & {in \mathbb{R}_ + ^n ,} \\{u(x) = - (\Delta )u(x) = \cdots = ( - \Delta )^{m - 1} u(x) = 0,} & {on \partial \mathbb{R}_ + ^n ,} \\\end{array} } \right.$$ (0.1) where m is any positive integer satisfying 0 < 2m < n. We first prove that the positive solutions of (0.1) are super polyharmonic, i.e., $${\left( { - \Delta } \right)^i}u > 0,\;i = 0,1, \ldots ,m - 1.$$ (0.2) For α = 2m, applying this important property, we establish the equivalence between (0.1) and the integral $$u \left( x \right) = {c_n}\int_{\mathbb{R}_ + ^n} {\left( {\frac{1}{{{{\left| {x - y} \right|}^{n - \alpha }}}} - \frac{1}{{{{\left| {{x^*} - y} \right|}^{n - \alpha }}}}} \right)} \frac{{{u ^p}\left( y \right)}}{{{{\left| y \right|}^s}}}dy,$$ (0.3) where x* = (x₁,..., x_n−1, −x_n) is the reflection of the point x about the plane Rn−1. Then, we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of (0.3), in which α can be any real number between 0 and n. By some Pohozaev type identities in integral forms, we prove a Liouville type theorem—the non-existence of positive solutions for (0.1).KeywordsNavier boundary conditionshalf spacesuper polyharmonicequivalenceintegral equationrotational symmetrynon-existenceMSC(2010)31B0531B1035B0635B53References1.Aubin T. Problems isopérimétriques et espaces de Sobolev. J Differential Geom, 1976, 11: 573–598MATHGoogle Scholar2.Badiale M, Tarantello G. A Hardy-Sobolev inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch Ration Mech Anal, 2002, 163: 259–293MathSciNetCrossRefMATHGoogle Scholar3.Caffarelli L, Kohn R, Nirenberg L. First order interpolation inequalities with weights. Compos Math, 1984, 53: 259–275MathSciNetMATHGoogle Scholar4.Cao D M, Li Y Y. Results on positive solutions of elliptic equations with a critical Hardy-Sobolev operator. Methods Appl Anal, 2008, 15: 81–96MathSciNetMATHGoogle Scholar5.Cao L F, Chen W X. 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J Funct Anal, 2005, 221: 482–492MathSciNetCrossRefMATHGoogle ScholarCopyright information© Science China Press and Springer-Verlag Berlin Heidelberg 2016Authors and AffiliationsRan Zhuo12Email author1.Department of Mathematical SciencesHuanghuai UniversityZhumadianChina2.Department of Mathematical SciencesYeshiva UniversityNew YorkUSA About this article CrossMark Publisher Name Science China Press Print ISSN 1674-7283 Online ISSN 1869-1862 About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; border-bottom: 0; padding: 8px 12px; position: relative; background-color: #f2f2f2; } .buybox__header .buybox__login { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; 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