[1]邹文明.变分方法及其在非线性偏微分方程应用方面的进展和未决问题[J].江西师范大学学报(自然科学版),2018,(02):111-129207.[doi:10.16357/j.cnki.issn1000-5862.2018.02.01]
 ZOU Wenming.Results and Open Problems of the Variational Method with Applications to Nonlinear Partial Differential Equations[J].Journal of Jiangxi Normal University:Natural Science Edition,2018,(02):111-129207.[doi:10.16357/j.cnki.issn1000-5862.2018.02.01]
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变分方法及其在非线性偏微分方程应用方面的进展和未决问题()
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《江西师范大学学报》(自然科学版)[ISSN:1006-6977/CN:61-1281/TN]

卷:
期数:
2018年02期
页码:
111-129207
栏目:
特约稿
出版日期:
2018-04-20

文章信息/Info

Title:
Results and Open Problems of the Variational Method with Applications to Nonlinear Partial Differential Equations
文章编号:
1000-5862(2018)02-0111-19
作者:
邹文明
清华大学数学科学系,北京 100084
Author(s):
ZOU Wenming
Department of Mathematical Sciences,Tsinghua University,Beijing 100084,China
关键词:
变分法 非线性偏微分方程 环绕理论 临界指数 变号临界点理论 薛定谔方程
Keywords:
variational method nonlinear partial differential equations linking theory critical exponent critical point theory Schrödinger equations
分类号:
O 176; O 175.29
DOI:
10.16357/j.cnki.issn1000-5862.2018.02.01
文献标志码:
A
摘要:
先介绍变分法发展的简单历史以及将来的发展趋势. 然后综述变分法应用于非线性偏微分方程的基本思想和最新成果. 通俗介绍环绕理论、变号临界点理论及应用,其中包括对称扰动方程和Rabinowitz公开问题、Brezis-Nirenberg临界指数方程、Li-Lin公开问题、Bose-Einstein凝聚、Berestycki-Caffarelli-Nirenberg猜测和Lane-Emden方程及猜想.
Abstract:
The brief history and the development trend of the variational method are introduced.Then the fundamental ideas and the latest achievements of the variational method with applications to nonlinear partial differential equations are summarized.The critical point theory and its applications are briefly reviewed,including the perturbed equation from symmetry,Rabinowitz's open problem,Brezis-Nirenberg's critical exponent equation,Li-Lin's open problem,Bose-Einstein condensation,Berestycki-Caffarelli-Nirenberg's conjecture and Lane-Emden's equation and conjecture.

参考文献/References:

[1] 吴文俊.世界著名数学家传记:上下集[M].北京:科学出版社出版,1997.
[2] Ambrosotti A,Rabinowitz P H.Dual variational methods in critical point theory and applications[J].J Func Anal,1973,14(4):349-381.
[3] “1 000个数学难题”数学编委会.10 000个科学难题:数学卷[M].北京:科学出版社,2009.
[4] Zou Wenming,Schechter M.Critical point theory and its application[M].New York:Springer,2006.
[5] Schechter M,Zou Wenming.Semi-classical bound states of Schrödinger equations[J].Math Proc Cambridge Philos Soc,2014,156(1):167-181.
[6] Schechter M,Zou Wenming.Double linking theorem and multiple solutions[J].J Functional Analysis,2003,205(1):37-61.
[7] Szulkin A,Zou Wenming.Homoclinic orbit for asymptotically linear Hamiltonian systems[J].J Function Analysis,2001,187(1):25-41.
[8] Rabinowitz P H.Minimax methods in critical point theory with applications to differential equations[M].Rhode Island:America Mathematical Society,1986.
[9] Rabinowitz P H.Minimax methods and their application to partial differential equations[M]//Berkeley Calif.Seminar on nonlinear partial differential equations.New York:Springer,1984:307-320.
[10] Struwe M.Variational methods:applications to nonlinear partial differential equations and Hamiltonian systems[M].Berlin:Springer-Verlag,1996.
[11] Benci V,Rabinowitz P H.Critical point theorems for indefinite functionals[J].Invent Math,1979,52(3):241-273.
[12] Lazer A C,Solimini S.Nontrivial solutions of operator equations and Morse indices of critical points of min-max type[J].Nonlinear Anal TMA,1988,12(8):761-775.
[13] Silva E A B.Subharmonic solutions for subquadratic Hamiltonian systems[J].J Differential Equations,1995,115(1):120-145.
[14] Furtado M F,Maia L A,Silva E A B.On a double resonant problem in RN[J].Differential Integral Equations,2002,15(11):1335-1344.
[15] Furtado M F,Maia L A,Silva E A B.Solutions for a resonant elliptic system with coupling in RN[J].Comm Partial Differential Equations,2002,27(7/8):1515-1536.
[16] Zou Wenming.Sign-Changing critical point theory[M].New York:Springer,2008:278.
[17] Zou Wenming.On finding Sign-Changing solutions[J].J Functional Analysis,2006,234(2):364-419.
[18] Zou Wenming.Sign-Changing saddle point[J].J Functional Analysis,2005,219(2):433-468.
[19] Zou Wenming,Li Shujie.On Schechter's linking theorems[J].J Functional Analysis,2010,258(10):3347-3361.
[20] Bahri A,Lions P L.Morse index of some min-max critical points I:application to multiplicity results[J].Comm Pure Appl Math,1988,41(8):1027-1037.
[21] Ramos M,Tavares H,Zou Wenming.A Bahri-Lions theorem revisted[J].Advances in Mathematics,2009,222(6):2173-2195.
[22] Hirano N,Zou Wenming.A purerbation method for multiple Sign-Changing solutions[J].Calc Var PDE,2010,37(1/2):87-98.
[23] Yamabe H.On a deformation of Riemannian structures on compact manifolds[J].Osaka Math J,1960,12(1):21-37.
[24] Trudinger N.Remarks concerning the conformal deformation of Riemannian structures on compact manifolds[J].Ann Scuola Norm Sup Pisa,1968,22(3):265-274.
[25] Aubin T.Equations differentielles non lineaires et probleme de Yamabe concernant la courbure scalaire[J].J Math Pures Appl,1976,55(3):269-296.
[26] Schoen R.Conformal deformation of a Riemannian metric to constant scalar curvature[J].J Differential Geom,1984,20(2):479-495.
[27] Brezis H,Nirenberg L.Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[J].Comm Pure Appl Math,1983,36(4):437-477.
[28] Capozzi A,Fortunato D,Palmieri G.An existence result for nonlinear elliptic problems involving critical Sobolev exponent[J].Ann Inst H Poincaré Anal Non Linéaire,1985,2(6):463-470.
[29] Zhang Dong.On multiple solutions of Δu+λu+|u|4/(n-2)u=0[J].Nonlinear Anal,1989,13(4):353-372.
[30] Fortunato D,Jannelli E.Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains[J].Proc Roy Soc Edinburgh Sect A,1987,105(1):205-213.
[31] Cerami G,Solimini S,Struwe M.Some existence results for superlinear elliptic boundary value problems involving critical exponents[J].J Funct Anal,1986,69(3):289-306.
[32] Lazzo M.Solutions positives multiples pour unequation elliptique nonlineaire avec lexposant critique de Sobolev[J].C R Acad Sci Paris Sér I Math,1992,314(1):61-64.
[33] Devillanova G,Solimini S.Concentration estimates and multiple solutions to elliptic problems at critical growth[J].Adv Differential Equations,2002,7(10):1257-1280.
[34] Cao Daomin,Peng Shuangjie,Yan Shusen.Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth[J].J Funct Anal,2012,262(6):2861-2902.
[35] Schechter M,Zou Wenming.On a Brézis-Nirenberg theorem[J].Archive for Rational Mechanics and Analysis,2010,197(1):337-356.
[36] Devillanova G,Solimini S.A multiplicity result for elliptic equations at critical growth in low dimension[J].Commun Contemp Math,2003,5(2):171-177.
[37] Clapp M,Weth T.Multiple solutions for the Brezis-Nirenberg problem[J].Adv Differential Equations,2005,10(4):463-480.
[38] Chen Zhijie,Shioji N,Zou Wenming.Ground state and multiple solutions for a critical exponent problem[J].No DEA Nonlinear Differential Equations Appl,2012,19(3):253-277.
[39] Berestycki H,Lions P L.Nonlinear scalar field equations(I):existence of a ground state[J].Arch Ration Mech Anal,1983,82(4):313-346.
[40] Berestycki H,Lions P L.Nonlinear scalar field equations(II):existence of infinitely many solutions[J].Arch Ration Mech Anal,1983,82(4):347-375.
[41] Berestycki H,Gallouёt T,Kavian O.Equations de champs scalaires euclidiens non linéaire dans le plan[J].C R Acad Sci Paris:Paris Ser I Math,1983,297(2):307-310.
[42] Brezis H,Lieb E H.Minimum action solutions of some vector field equations[J].Commun Math Phys,1984,96(1):97-113.
[43] Hirata J,Ikoma N,Tanaka K.Nonlinear scalar field equations in RN:mountain pass and symmetric mountain pass approaches[J].Topological Methods in Nonlinear Analysis,2010,35(2):253-276.
[44] Zhang Jianjun,Zou Wenming.A Berestycki-Lions theorem revisited[J].Comm Contemp Math,2012,14(5):1250033.
[45] Pino M D,Felmer P L.Spike-layered solutions of singularly perturbedelliptic problems in a degenerate setting[J].Indiana Univ Math J,1999,48(3):883-898.
[46] Gardner R,Peletier L A.The set of positive solutions of semilinear equations in large balls[J].Proc Roy Soc Edinburgh Sect A,1986,104(1/2):53-72.
[47] Jang J D.On spike solutions of singularly perturbed semilinear Dirichlet problems[J].J Differential Equations,1994,114(2):370-395.
[48] Li Yanyan,Nirenberg L.The Dirichlet problem for singularly perturbed elliptic equations[J].Comm Pure Appl Math,1998,51(11/12):1445-1490.
[49] Ni Weiming,Wei Juncheng.On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems[J].Commun Pure Appl Math,1995,48(7):731-768.
[50] Ni Weiming,Takagi I,Wei Juncheng.On the location and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problem:intermediate solutions[J].Duke Math J,1998,94(3):597-618.
[51] Noussair E,Yan Shusen.The effect of the domain geometry in singular perturbation problems[J].Proc London Math Soc,1998,76(2):427-452.
[52] Byeon J.Mountain pass solutions for singularly perturbed nonlinear Dirichlet problems[J].J Differential Equations,2005,217(2):257-281.
[53] Byeon J.Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity[J].Trans Amer Math Soc,2010,362(4):1981-2001.
[54] Pohozaev S I.On the eigenfunctions of the equation Δu+λf(u)=0[J].Dokl Akad Nauk SSSR,1965,165(6):36-39.
[55] Alves C O,Souto Marco A S,Montenegro Marcelo.Existence of a ground state solution for a nonlinear scalar field equation with critical growth[J].Calculus of Variations and PDE,2012,43(3/4):537-554.
[56] Byeon J,Zhang Jianjun,Zou Wenming.Singularly perturbed nonlinear Dirichlet problems involving critical growth[J].Calc Var PDE,2013,47(1/2):65-85.
[57] Alves C O,Marcos do Ó J,Souto M A S.Local mountain-pass for a class of elliptic problems in RN involving critical growth[J].Nonlinear Anal,2001,46(4):495-510.
[58] Berestycki H,Wei Juncheng.On the least energy solutions to a semilinear elliptic equation in a strip[J].Disc Conti Dyn Sys,2010,28(3):1083-1099.
[59] Ruiz D,Willem M.Elliptic problems with critical exponents and Hardy potentials[J].J Differential Equations,2003,190(2):524-538.
[60] Caffarelli L,Kohn R,Nirenberg L.First order interpolation inequalities with weights[J].Compositio Mathematica,1984,53(3):259-275.
[61] Lin Changshou.Interpolation inequalities with weights[J].Comm Partial Differential Equations,1986,11(14):1515-1538.
[62] Ghoussoub N,Robert F.The effect of curvature on the best constant in the Hardy-Sobolev inequalities[J].Geom Funct Anal,2006,16(6):1201-1245.
[63] Li Yanyan,Lin Changshou.A nonlinear elliptic pde with two Sobolev-Hardy critical exponents[J].Arch Ration Mech Anal,2012,203(3):943-968.
[64] Cerami G,Zhong Xuexiu,Zou Wenming.On some nonlinear elliptic PDEs with Sobolev-Hardy critical exponents and a Li-Lin open problem[J].Calc Var Partial Differential Equations,2015,54(2):1793-1829.
[65] Frantzeskakis D J.Dark solitons in atomic Bose-Einstein condensates:from theory to experiments[J].J Phys A:Math Theor,2010,43(21):213001.
[66] Kivshar Y S,Luther-Davies B.Dark optical solitons:physics and applications[J].Physics Reports,1998,298(2/3):81-197.
[67] Esry B,Greene C,Burke J,et al.Hartree-Fock theory for double condesates[J].Phys Rev Lett,1997,78(19):3594-3597.
[68] Ambrosetti A,Colorado E.Bound and ground states of coupled nonlinear Schrödinger equations[J].C R Math Acad Sci Paris,2006,342(7):453-458.
[69] Ambrosetti A,Colorado E.Standing waves of some coupled nonlinear Schröinger equations[J].J London Math Soc,2007,75(1):67-82.
[70] Bartsch T,Wang Zhiqiang,Wei Juncheng.Bound states for a coupled Schrödinger system[J].J Fixed Point Th Appl,2007,2(2):353-367.
[71] Lin Taichia,Wei Juncheng.Ground state of coupled nonlinear Schrödinger equations in Rn,n≤3[J].Commun Math Phys,2005,255(3):629-653.
[72] Maia L,Montefusco E,Pellacci B.Positive solutions for a weakly coupled nonlinear Schrödinger systems[J].J Differ Eqs,2006,229(2):743-767.
[73] Sirakov B.Least energy solitary waves for a system of nonlinear Schrödinger equations in Rn[J].Commun Math Phys,2007,271(1):199-221.
[74] Lin Taichia,Wei Juncheng.Spikes in two coupled nonlinear Schrödinger equations[J].Ann Inst H Poincaré AN,2005,22(4):403-439.
[75] Lin Taichia,Wei Juncheng.Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials[J].J Differ Eqs,2006,229(2):538-569.
[76] Maia L,Pellacci B,Squassina M.Semiclassical states for weakly coupled nonlinear Schrödinger systems[J].J Eur Math Soc,2007,10(1):47-71.
[77] Pomponio A.Coupled nonlinear Schrödinger systems with potentials[J].J Differ Eqs,2006,227(1):258-281.
[78] Bartsch T,Dancer N,Wang Zhiqiang.A Liouville theorem,a priori bounds,and bifurcating branches of positive solutions for a nonlinear elliptic system[J].Calc Var PDE, 2010,37(3/4):345-361.
[79] Dancer N,Wei Juncheng,Weth T.A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger systems[J].Ann Inst H Poincaré AN,2010,27(3):953-969.
[80] Liu Zhaoli,Wang Zhiqiang.Multiple bound states of nonlinear Schrödinger systems[J].Commun Math Phys,2008,282(3):721-731.
[81] Wei Juncheng,Weth T.Nonradial symmetric bound states for a system of two coupled Schrödinger equations[J].Rend Lincei Mat Appl,2007,18(1):279-293.
[82] Wei Juncheng,Weth T.Radial solutions and phase separation in a system of two coupled Schrödinger equations[J].Arch Ration Meth Anal,2008,190(1):83-106.
[83] Wu Yuanze,Wu Tsungfang,Zou Wenming.On a two-component Bose-Einstein condensate with steep potential wells[J].Ann Mat Pura Appl,2017,196(5):1695-1737.
[84] Conti M,Teraccini S,Verzini G.Nehari's problem and competing species systems[J].Ann Inst H Poincaré Anal Non Linéaire,2002,19(6):871-888.
[85] Conti M,Teraccini S,Verzini G.An optimal partition problem related to nonlinear eigenvalues[J].J Funct Anal,2003,198(1):160-196.
[86] de Figueiredo D G,Lopes O.Solitary waves for some nonlinear Schrödinger systems[J].Ann Inst H Poincaré Anal Non Linéaire,2008,25(1):149-161.
[87] Ikoma N,Tanaka K.A local mountain pass type result for a system of nonlinear Schrödinger equations[J].Calc Var PDE,2011,40(3/4):449-480.
[88] Wei Juncheng,Yao Wei.Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations[J].Comm Pure Appl Anal,2012,11(3):1003-1011.
[89] Chen Zhijie,Zou Wenming.Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent[J].Arch Ration Mech Anal,2012,205(2):515-551.
[90] Chen Zhijie,Zou Wenming.An optimal constant for the existence of least energy solutions of a coupled Schrödinger system[J].Calc Var PDE,2013,48(3/4):695-711.
[91] Wei Juncheng,Weth T.Asymptotic behaviour of solutions of planar elliptic systems with strong competition[J].Nonlinearity,2008,21(2):305-317.
[92] Noris B,Tavares H,Terracini S,et al.Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition[J].Comm Pure Appl Math,2010,63(3):267-302.
[93] Caffarelli L A,Lin Fanghua.Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries[J].J Amer Math Soc,2008,21(3):847-862.
[94] Caffarelli L A,Roquejoffre J M.Uniform Höder estimates in a class of elliptic systems and applications to singular limits in models for diffusion flames[J].Arch Ration Mech Anal,2007,183(3):457-487.
[95] Conti M,Terracini S,Verzini G.Asymptotic estimates for the spatial segregation of competitive systems[J].Adv Math,2005,195(2):524-560.
[96] Aubin T.Problemes isoperimetriques et espaces de Sobolev[J].J Diff Geom,1976,11(4):573-598.
[97] Talenti G.Best constant in Sobolev inequality[J].Ann Mat Pure Appl,1976,110(1):352-372.
[98] Chen Zhijie,Zou Wenming.Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case[J].Calc Var Partial Differential Equations,2015,52(1/2):423-467.
[99] Chen Zhijie,Zou Wenming.Existence and symmetry of positive ground states for a doubly critical Schrödinger system[J].Trans Amer Math Soc,2015,367(5):3599-3646.
[100] Chen Zhijie,Zou Wenming.Standing waves for a coupled system of nonlinear Schrödinger equations[J].Ann Mat Pura Appl,2015,194(1):183-220.
[101] Chen Zhijie,Lin Changshou,Zou Wenming.Sign-changing solutions and phase separation for an elliptic system with critical exponent[J].Comm Partial Differential Equations,2014,39(10):1827-1859.
[102] Chen Zhijie,Zou Wenming.A remark on doubly critical elliptic systems[J].Calc Var Partial Differential Equations,2014,50(3/4):939-965.
[103] Chen Zhijie,Zou Wenming.Standing waves for linearly coupled Schrödinger equations with critical exponent[J].Ann Inst H Poincare Anal Non Lineaire,2014,31(3):429-447.
[104] Chen Zhijie,Zou Wenming.On linearly coupled Schrödinger systems[J].Proc Amer Math Soc,2014,142(1):323-333.
[105] Chen Zhijie,Zou Wenming.Standing waves for coupled nonlinear Schrodinger equations with decaying potentials[J].J Math Phys,2013,54(11):453-458.
[106] Chen Zhijie,Zou Wenming.On the Brezis-Nirenberg problem in a ball[J].Differential Integral Equations,2012,25(5/6):527-542.
[107] Chen Zhijie,Zou Wenming.Ground states for a system of Schrödinger equations with critical exponent[J].J Funct Anal,2012,262(7):3091-3107.
[108] Chen Zhijie,Lin Changshou,Zou Wenming.Infinitely many sign-changing and semi-nodal solutions for a nonlinear Schrödinger system[J].Annali della Scuola Normale Superiore di Pisa,Classe di Scienze,2016,15(5):859-897.
[109] Chen Zhijie,Lin Changshou,Zou Wenming.Monotonicity and nonexistence results to cooperative systems in the half space[J].J Funct Anal,2014,266(2):1088-1105.
[110] Long Wei,Peng Shuangjie.Segregated vector solutions for a class of Bose-Einstein systems[J].J Differential Equations,2014,257(1):207-230.
[111] Long Wei,Wang Qingfang.Segregated and synchronized vector solutions to linearly coupled systems of Schrödinger equations[J].J Math Phys,2015,56(9):091507.
[112] Peng Shuangjie,Wei Shuai,Wang Qingfang.Multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent[J].J Differential Equations,2017,263(1):709-731.
[113] He Qihan,Peng Shuangjie.Synchronized vector solutions to an elliptic system[J].Proc Amer Math Soc,2016,144(9):4055-4063.
[114] Yang Jianfu,Yu Xiaohui.Existence of a cooperative elliptic system involving Pucci operator[J].Acta Math Sci Ser B Engl Ed,2010,30(1):137-147.
[115] Berestycki H,Caffarelli L,Nirenberg L.Symmetry for elliptic equations in a half space[M]//Lions J L,Baiocchi C.Boundary value problems for partial differential equations and applications.Paris:Masson,1993:27-42.
[116] Berestycki H,Caffarelli L,Nirenberg L.Inequalities for second order elliptic equations with applications to unbounded domains(I)[J].Duke Math J,1996,81(2):467-494.
[117] Berestycki H,Caffarelli L,Nirenberg L.Monotonicity for elliptic equations in an unbounded Lipschitz domain[J].Comm Pure Appl Math,1997,50(11):1089-1112.
[118] Berestycki H,Caffarelli L,Nirenberg L.Further qualitative properties for elliptic equations in unbounded domains[J].Ann Sc Norm Super Pisa Cl Sci,1997,25(1):69-94.
[119] Farina A,Valdinoci E.Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems[J].Arch Ration Mech Anal,2010,195(3):1025-1058.
[120] Cowan C.Liouville theorems for stable Lane-Emden systems and biharmonic problems[J].Nonlinearity,2013,26(8):2357-2371.
[121] Quittner P,Souplet Ph.Superlinear parabolic problems:blow-up,global existence and steady states[M].Berlin:Springer,2007.
[122] Gidas B,Spruck J.Global and local behavior of positive solutions of nonlinear elliptic equations[J].Comm Pure Appl Math,1981,34(4):525-598.
[123] Mitidieri E.A Rellich type identity and applications[J].Comm Partial Differential Equations,1993,18(1/2):125-151.
[124] Mitidieri E.Nonexistence of positive solutions of semilinear elliptic systems in RN[J].Differential Integral Equations,1996,9(3):465-479.
[125] Birindelli I,Mitidieri E.Liouville theorems for elliptic inequalities and applications[J].Proc Roy Soc Edinburgh Sect A,1998,128(6):1217-1247.
[126] Busca J,Manasevich R.A Liouville-type theorem for Lane-Emden system[J].Indiana Univ Math J,2002,51(1):37-51.
[127] Clement Ph,de Figueiredo D G,Mitidieri E.Positive solutions of semilinear elliptic systems[J].Comm Partial Differential Equations,1992,17(5/6):923-940.
[128] de Figueiredo D G.Semilinear elliptic systems[M]∥Ambrosetti A,Chang K C,Ekeland I.Nonlinear Functional Analysis and Applications to Differential Equations.Nanjing:World Sci Publishing,1998:122-152.
[129] de Figueiredo D G,Felmer P.A Liouville-type theorem for elliptic systems[J].Ann Sc Norm Super Pisa Cl Sci,1994,21(4):387-397.
[130] Polacik P,Quittner P,Souplet Ph.Singularity and decay estimates in superlinear problems via Liouville-type theorems Part I:Elliptic equations and systems[J].Duke Math J,2007,139(3):555-579.
[131] Reichel W,Zou Henghui.Non-existence results for semilinear cooperative elliptic systems via moving spheres[J].J Differential Equations,2000,161(1):219-243.
[132] Serrin J,Zou Henghui.Non-existence of positive solutions of Lane-Emden systems[J].Differential Integral Equations,1996,9(4):635-653.
[133] Serrin J,Zou Henghui.Existence of positive solutions of the Lane-Emden system[J].Atti Semin Mat Fis Univ Modena,1998,46(suppl):369-380.
[134] Serrin J,Zou Henghui.Non-existence of positive solutions of semilinear elliptic systems[J].Discourses Math Appl,1994,3:55-68.
[135] de Figueiredo D G,Yang Jianfu.A priori bounds for positive solutions of a non-variational elliptic system[J].Comm Partial Differential Equations,2001,26(11/12):2305-2321.
[136] Souto M A S.A priori estimates and existence of positive solutions of non-linear cooperative elliptic systems[J].Differential Integral Equations,1995,8(4):1245-1258.
[137] Souplet P.The proof of the Lane-Emden conjecture in four space dimensions[J].Adv Math,2009,221(5):1409-1427.
[138] Lin Changshou.A classification of solutions of a conformally invariant fourth order equation in Rn[J].Comment Math Helv,1998,73(2):206-231.

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备注/Memo

备注/Memo:
收稿日期:2018-01-20
基金项目:国家自然科学基金(11771234)资助项目.
作者简介:邹文明(1966-), 男,江西宁都人,教授, 博士生导师,国家杰出青年基金获得者,主要从事变分法和非线性微分方程的研究.E-mails:zou-wm@mail.tsinghua.edu.cn
更新日期/Last Update: 2018-04-20