EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A HARMONIC EQUATION WITH CRITICAL NONLINEARITY

doi:10.1016/S0252-9602(16)30070-4

数学物理学报(英文版) ›› 2016, Vol. 36 ›› Issue (5): 1305-1316.doi: 10.1016/S0252-9602(16)30070-4

EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A HARMONIC EQUATION WITH CRITICAL NONLINEARITY

Kamal OULD BOUH

Department of Mathematics, College of Sciences, Taibah University, P. O. Box 30097, Almadinah Almunawwarah, KSA

收稿日期:2015-06-23 修回日期:2015-11-17 出版日期:2016-10-25 发布日期:2016-10-25
作者简介:Kamal OULD BOUH,kamalbouh@yahoo.fr,hbouh@taibahu.edu.sa

EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A HARMONIC EQUATION WITH CRITICAL NONLINEARITY

Kamal OULD BOUH

Department of Mathematics, College of Sciences, Taibah University, P. O. Box 30097, Almadinah Almunawwarah, KSA

Received:2015-06-23 Revised:2015-11-17 Online:2016-10-25 Published:2016-10-25

摘要/Abstract

摘要：

This paper is concerned with the harmonic equation (P_?ε):△u=0, u>0 in Bⁿ and (∂u)/(∂ν)+(n-2)/(2)u=(n-2)/(2)Ku^(n)/(n-2)?ε on S^n-1 where Bⁿ is the unit ball in Rⁿ, n≥4 with Euclidean metric g0, ∂Bⁿ=S^n-1 is its boundary, K is a function on S^n-1 and ε is a small positive parameter. We construct solutions of the subcritical equation (P_-ε) which blow up at one critical point of K. We give also a sufficient condition on the function K to ensure the nonexistence of solutions for (P_-ε) which blow up at one point. Finally, we prove a nonexistence result of single peaked solutions for the supercritical equation (P_+ε).

关键词: variational problem, critical points, harmonic equation, mean curvature, critical exponent

Abstract:

Key words: variational problem, critical points, harmonic equation, mean curvature, critical exponent

中图分类号:

35J20

Kamal OULD BOUH. EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A HARMONIC EQUATION WITH CRITICAL NONLINEARITY[J]. 数学物理学报(英文版), 2016, 36(5): 1305-1316.

Kamal OULD BOUH. EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A HARMONIC EQUATION WITH CRITICAL NONLINEARITY[J]. Acta mathematica scientia,Series B, 2016, 36(5): 1305-1316.

参考文献

[1] Abdelhedi W, Chtioui H, Ould Ahmedou M. A Morse theoretical approach for boundary mean Curvature Problem on B⁴. J Funct Anal, 2008, 254(5):1307-1341
[2] Bahri A. Critical Point at Infinity in Some Variational Problems. Pitman Res Notes Math, Ser 182. Harlow:Longman Sci Tech, 1989
[3] Bahri A, Li Y Y, Rey O. On a variational problem with lack of compactness:the topological effect of the critical points at infinity. Cal Var Partial Diff Equa, 1995, 3:67-94
[4] Ben Ayed M, El Mehdi K, Grossi M, Rey O. A Nonexistence result of single peaked solutions to a supercritical nonlinear problem. Comm Contenporary Math, 2003, 2:179-195
[5] Ben Ayed M, Ghoudi R, Ould Bouh K. Existence of conformal metrics with prescribed scalar curvature on the four dimensional half sphere. Nonlinear Diff Equ Appl, 2012, 19:629-662
[6] Ben Ayed M, Ould Bouh K. Nonexistence results of sign-changing solution to a supercritical nonlinear problem. Comm Pure Appl Anal, 2008, 7(5):1057-1075
[7] Chang S A, Xu X, Yang P C. A perturbation result for prescribing mean curvature. Math Ann, 1998, 310(3):473-496
[8] Cherrier P. Problèmes de Neumann non linéaires sur les variétés Riemaniennes. J Funct Anal, 1984, 57:154-207
[9] Djadli Z, Malchiodi A, Ould Ahmedou M. The prescribed boundary mean curvature problems on B4. J Differ Equ, 2004, 206:373-398
[10] El Mehdi K, Hammami M. Blowing up solutions for a biharmonic equation with critical nonlinearity. Asym Analm 2005, 45(3/4):191-225
[11] Escobar J. Conformal deformation of Riemannian metric to scalar flat metric with constant mean curvature on the boundary. Ann of Math, 1992, 136:1-50
[12] Escobar J. Conformal metrics with prescribed mean curvature on the boundary. Cal Var, 1996, 4:559-592
[13] Escobar J, Schoen R. Conformal metrics with prescribed scalar curvature. Invent Math, 1986, 86:243-254
[14] Escobar J F, Garcia G. Conformal metric on the ball with zero scalare and prescribed mean curvature on the boundary. J Funct Anal, 2004, 211(1):71-152
[15] Li Y Y. The Niremberg problem in a domain with boundary. Top Meth Nonli Anal, 1995, 6:309-329
[16] Ould Bouh K. Nonexistence result of sign-changing solutions for a supercritical problem of the scalar curvature type. Adv Nonlinear Stud, 2012, 12:149-171
[17] Ould Bouh K. Sign-changing solutions of a fourth-order elliptic equation with supercritical exponent. Electron J Diff Equ, 2014, 2014(77):1-13
[18] Rey O. The topological impact of critical points at infinity in a variational problem with lack of compactness:the dimension 3. Adv Differ Equ, 1999, 4:581-616

EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A HARMONIC EQUATION WITH CRITICAL NONLINEARITY

EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A HARMONIC EQUATION WITH CRITICAL NONLINEARITY

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	马如云, 高红亮. MULTIPLE POSITIVE SOLUTIONS FOR A CLASS OF SEMIPOSITONE NEUMANN PROBLEMS WITH SINGULAR φ-LAPLACIAN[J]. 数学物理学报(英文版), 2017, 37(5): 1472-1482.
[2]	何春蕾, 黄守军, 邢晓敏. SELF-SIMILAR SOLUTIONS TO THE HYPERBOLIC MEAN CURVATURE FLOW[J]. 数学物理学报(英文版), 2017, 37(3): 657-667.
[3]	杨瑞瑞, 张薇, 刘祥清. SIGN-CHANGING SOLUTIONS FOR p-BIHARMONIC EQUATIONS WITH HARDY POTENTIAL IN R^N[J]. 数学物理学报(英文版), 2017, 37(3): 593-606.
[4]	韩欢, 周焕松. A VARIATIONAL PROBLEM ARISING IN REGISTRATION OF DIFFUSION TENSOR IMAGES[J]. 数学物理学报(英文版), 2017, 37(2): 539-554.
[5]	Mohamed BEN AYED, Habib FOURTI. SCALAR CURVATURE TYPE PROBLEM ON THE THREE DIMENSIONAL BOUNDED DOMAIN[J]. 数学物理学报(英文版), 2017, 37(1): 139-173.
[6]	王静, 张银山. RIGIDITY OF COMPACT SURFACES IN HOMOGENEOUS 3-MANIFOLDS WITH CONSTANT MEAN CURVATURE[J]. 数学物理学报(英文版), 2016, 36(6): 1609-1618.
[7]	Batirkhan TURMETOV. ON SOME BOUNDARY VALUE PROBLEMS FOR NONHOMOGENOUS POLYHARMONIC EQUATION WITH BOUNDARY OPERATORS OF FRACTIONAL ORDER[J]. 数学物理学报(英文版), 2016, 36(3): 831-846.
[8]	张靖, 马世旺. INFINITELY MANY SIGN-CHANGING SOLUTIONS FOR THE BRÉZIS-NIRENBERG PROBLEM INVOLVING HARDY POTENTIAL[J]. 数学物理学报(英文版), 2016, 36(2): 527-536.
[9]	郑攀, 穆春来, 胡学刚, 张付臣. SECONDARY CRITICAL EXPONENT AND LIFE SPAN FOR A DOUBLY SINGULAR PARABOLIC EQUATION WITH A WEIGHTED SOURCE[J]. 数学物理学报(英文版), 2016, 36(1): 244-256.
[10]	李海霞, 韩玉柱, 高文杰. CRITICAL EXTINCTION EXPONENTS FOR POLYTROPIC FILTRATION EQUATIONS WITH NONLOCAL SOURCE AND ABSORPTION[J]. 数学物理学报(英文版), 2015, 35(2): 366-374.
[11]	曾小雨\|张贻民\|周焕松. EXISTENCE AND STABILITY OF STANDING WAVES FOR A COUPLED NONLINEAR SCHR¨\|ODINGER SYSTEM[J]. 数学物理学报(英文版), 2015, 35(1): 45-70.
[12]	A.S. BERDYSHEV, A. CABADA, B.Kh. TURMETOV. ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR A NONHOMOGENEOUS BIHARMONIC EQUATION WITH A BOUNDARY#br# OPERATOR OF A FRACTIONAL ORDER[J]. 数学物理学报(英文版), 2014, 34(6): 1695-1706.
[13]	Zagharide Zine EL ABIDINE. COMBINED EFFECTS IN A SEMILINEAR POLYHARMONIC PROBLEM IN THE UNIT BALL[J]. 数学物理学报(英文版), 2014, 34(5): 1404-1416.
[14]	尚月赟, 王莉. MULTIPLE NONTRIVIAL SOLUTIONS FOR A CLASS OF SEMILINEAR POLYHARMONIC EQUATIONS[J]. 数学物理学报(英文版), 2014, 34(5): 1495-1509.
[15]	王莉, 汪继秀. CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS[J]. 数学物理学报(英文版), 2014, 34(5): 1603-1618.