A radial symmetry and Liouville theorem for systems involving fractional Laplacian

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• 作者：Dongsheng Li ; Zhenjie Li
• 关键词：Fractional Laplacian ; method of moving planes ; Kelvin transform ; Liouville theorem
• 刊名：Frontiers of Mathematics in China
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：12
• 期：2
• 页码：389-402
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Higher Education Press
• ISSN：1673-3576
• 卷排序：12

文摘

We investigate the nonnegative solutions of the system involving the fractional Laplacian: $$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {( - \Delta )^\alpha u_i (x) = f_i (u),} & {x \in \mathbb{R}^n , i = 1,2, \ldots ,m,} \\ \end{array} } \\ {u(x) = (u_1 (x),u_2 (x), \ldots ,u_m (x)),} \\ \end{array} } \right.$$ where 0 < α < 1, n > 2, f_i(u), 1 ≤ i ≤ m, are real-valued nonnegative functions of homogeneous degree p_i ≥ 0 and nondecreasing with respect to the independent variables u₁, u₂,..., u_m. By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point x₀ if p_i = (n + 2α)/(n − 2α) for each 1 ≤ i ≤ m; and the only nonnegative solution of this system is u ≡ 0 if 1 < p_i < (n + 2α)/(n − 2α) for all 1 ≤ i ≤ m.KeywordsFractional Laplacianmethod of moving planesKelvin transformLiouville theoremMSC35A0535J4535S0547G3058J70References1.Applebaum D. Lévy Processes and Stochastic Calculus. 2nd ed. Cambridge Studies in Advanced Mathematics, Vol 116. Cambridge: Cambridge University Press, 2009CrossRefMATHGoogle Scholar2.Barrios B, Colorado E, Pablo A, Sánchez U. On some critical problems for the fractional Laplacian operator. J Differential Equations, 2012, 252: 6133–6162MathSciNetCrossRefMATHGoogle Scholar3.Bogdan K, Zak T. On Kelvin transformation. J Theoret Probab, 2006, 19: 89–120MathSciNetCrossRefMATHGoogle Scholar4.Bouchard J P, Georges A. Anomalous diffusion in disordered media, statistical mechanics, models and physical applications. Phys Rep, 1990, 195: 127–293MathSciNetCrossRefGoogle Scholar5.Brändle C, Colorado E, Pablo A, Sánchez U. A concave convex elliptic problem involving the fractional Laplacian. Proc Roy Soc Edinburgh, 2013, 143: 39–71MathSciNetCrossRefMATHGoogle Scholar6.Cabré X, Tan Jinggang. Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv Math, 2010, 224: 2052–2093MathSciNetCrossRefMATHGoogle Scholar7.Caffarelli L, Gidas B, Spruck J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm Pure Appl Math, 1989, 42: 271–297MathSciNetCrossRefMATHGoogle Scholar8.Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Comm Partial Differential Equations, 2007, 32: 1245–1260MathSciNetCrossRefMATHGoogle Scholar9.Caffarelli L, Vasseur L. Drift diffusion equations with fractional diffusion and the quasigeostrophic equation. Ann Math, 2010, 171(3): 1903–1930MathSciNetCrossRefMATHGoogle Scholar10.Chen Wenxiong, Fang Yanqin, Yang R. Semilinear equations involving the fractional Laplacian on domains. 2013, arXiv: 1309.7499Google Scholar11.Chen Wenxiong, Li Congming. Classification of solutions of some nonlinear elliptic equations. Duke Math J, 1991, 63: 615–622MathSciNetCrossRefMATHGoogle Scholar12.Chen Wenxiong, Li Congming. Classification of positive solutions for nonlinear differential and integral systems with critical exponents. Acta Math Sci Ser B Engl Ed, 2009, 29(4): 949–960MathSciNetCrossRefMATHGoogle Scholar13.Chen Wenxiong, Li Congming, Li Y. A direct method of moving planes for the fractional Laplacian. 2014, arXiv: 1411.1697Google Scholar14.Chen Wenxiong, Li Congming, Ou B. Classification of solutions for a system of integral equations. Comm Partial Differential Equations, 2005, 30: 59–65MathSciNetCrossRefMATHGoogle Scholar15.Chen Wenxiong, Li Congming, Ou B. classification of solutions for an integral equation. Comm Pure Appl Math, 2006, 59: 330–343MathSciNetCrossRefMATHGoogle Scholar16.Chen Wenxiong, Zhu Jiuyi. Indefinite fractional elliptic problem and Liouville theorems. J Differential Equations, 2015, 260(5): 4758–4785MathSciNetCrossRefMATHGoogle Scholar17.Constantin P. Euler equations, Navier-Stokes equations and turbulence. In: Constantin P, Gallavotti G, Kazhikhov A V, Meyer Y, Ukai S, eds. Mathematical Foundation of Turbulent Viscous Flows. Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 1–5, 2003. Lecture Notes in Math, Vol 1871. Berlin: Springer, 2006, 1–43Google Scholar18.Damascelli L, Gladiali F. Some nonexistence results for positive solutions of elliptic equations in unbounded domains. Rev Mat Iberoam, 2004, 20: 67–86MathSciNetMATHGoogle Scholar19.Dipierro S, Pinamonti A. A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian. J Differential Equations, 2013, 255: 85–119MathSciNetCrossRefMATHGoogle Scholar20.Dipierro S, Pinamonti A. Symmetry results for stable and monotone solutions to fibered systems of PDEs. Commun Contemp Math, 2015, 17(4): 1450035MathSciNetCrossRefMATHGoogle Scholar21.Fall M M, Jarohs S. Overdetermined problems with fractional Laplacian. ESAIM Control Optim Calc Var, 2015, 21(4): 924–938MathSciNetCrossRefMATHGoogle Scholar22.Felmer P, Wang Y. Radial symmetry of positive solutions to equations involving the fractional Laplacian. Commun Contemp Math, 2014, 16(1): 1350023MathSciNetCrossRefMATHGoogle Scholar23.Figueiredo D G, Felmer P L. A Liouville-type theorem for elliptic systems. Ann Sc Norm Super Pisa, 1994, 21: 387–397MathSciNetMATHGoogle Scholar24.Fowler P H. Further studies of Emden’s and similar differential equations. Quart J Math (Oxford), 1931, 2: 259–288CrossRefMATHGoogle Scholar25.Gidas B, Ni Weiming, Nirenberg L. Symmetry and related properties via the maximum principle. Comm Math Phys, 1979, 68: 209–243MathSciNetCrossRefMATHGoogle Scholar26.Guo Yuxia, Liu Jiaquan. Liouville type theorems for positive solutions of elliptic system in ℝ^{n. Comm Partial Differential Equations, 2008, 33: 263–284MathSciNetCrossRefMATHGoogle Scholar27.Jarohs S, Weth T. Symmetry via antisymmetric maximum principles in nonlocal problems of variable order. Ann Mat Pura Appl (4), 2014, DOI: 10.1007/s10231-014-0462-yGoogle Scholar28.Landkof N S. Foundations of Modern Potential Theory. New York: Springer-Verlag, 1972CrossRefMATHGoogle Scholar29.Lin Changshou. A classification of solutions of a conformally invariant fourth order equation in ℝn. Comment Math Helv, 1998, 73: 206–231MathSciNetCrossRefMATHGoogle Scholar30.Mitidieri E. Non-existence of positive solutions of semilinear systems in ℝn. Differential Integral Equations, 1996, 9: 465–479MathSciNetMATHGoogle Scholar31.Nezza E, Palatucci G, Valdinoci E. Hitchhiker’s guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136(5): 521–573MathSciNetCrossRefMATHGoogle Scholar32.Reichel W. Radial symmetry for elliptic boundary value problems on exterior domains. Arch Ration Meth Anal, 1997, 137: 381–394MathSciNetCrossRefMATHGoogle Scholar33.Serrin J. A symmetry problem in potential theory. Arch Ration Meth Anal, 1971, 43: 304–318MathSciNetMATHGoogle Scholar34.Serrin J, Zou Henghui. Non-existence of positive solutions of Lane-Emden system. Differential Integral Equations, 1996, 9: 635–653MathSciNetMATHGoogle Scholar35.Souplet P. The proof of the Lane-Emden conjecture in four space dimensions. Adv Math, 2009, 221: 1409–1427MathSciNetCrossRefMATHGoogle ScholarCopyright information© Higher Education Press and Springer-Verlag Berlin Heidelberg 2017Authors and AffiliationsDongsheng Li1Zhenjie Li1Email author1.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anChina About this article CrossMark Publisher Name Higher Education Press Print ISSN 1673-3452 Online ISSN 1673-3576 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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