CONCENTRATION OF SOLUTIONS FOR THE MEAN CURVATURE PROBLEM

doi:10.1016/S0252-9602(13)60026-0

摘要/Abstract

摘要：

We consider the problem of conformal metrics equivalent to the Euclidean met-ric, with zero scalar curvature and prescribed mean curvature on the boundary of the ball B_n , n≥4. By variational methods, we prove the existence of two peak solutions that concen-trate around a strict local maximum points of the mean curvature under certain conditions.

关键词: Boundary mean curvature, critical Sobolev exponent, critical points at infinity

Abstract:

Key words: Boundary mean curvature, critical Sobolev exponent, critical points at infinity

中图分类号:

35J20

Wael ABDELHEDI. CONCENTRATION OF SOLUTIONS FOR THE MEAN CURVATURE PROBLEM[J]. 数学物理学报(英文版), 2013, 33(3): 631-642.

Wael ABDELHEDI. CONCENTRATION OF SOLUTIONS FOR THE MEAN CURVATURE PROBLEM[J]. Acta mathematica scientia,Series B, 2013, 33(3): 631-642.

参考文献

[1] Abdelhedi W, Chtioui H. The Prescribed boundary mean curvature problem on standard n-dimensional ball. Nonlinear Anal TMA, 2007, 67: 668–686

[2] Abdelhedi W, Chtioui H, Ould Ahmedou M. A Morse theoretical approach for boundary mean Curvature Problem on B4. Journal of Functional Analysis, 2008: 254(5): 1307–1341

[3] Abdelhedi W, Chtioui H. Prescribing mean curvature on Bn. International Journal of Math, 2010, 21: 1157–1187

[4] Bahri A. Critical point at infinity in some variational problems//Pitman Res Notes Math, Ser 182. Longman Sci Tech Harlow, 1989

[5] Bahri A, Coron J M. On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of topology of the domain. Comm Pure Appli Math, 1988, 41: 255–294

[6] Bahri A, Li Y Y, Rey O. On a variational problem with lack of compactness: the topological effect of the critical points at infinity. Calc Var and Part Diff Equ, 1995, 3: 67–94

[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. Probl`emes de Neumann non lin´eaires sur les vari´et´es Riemanniennes. J Funct Anal, 1984, 57: 154–207

[9] Escobar J F. Conformal metric with prescribed mean curvature on the boundary. Calc Var and PDE’s, 1996, 4: 559–592

[10] 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

[11] Djadli Z, Malchiodi A, Ould Ahmedou M. The prescribed boundary mean curvature problems on B4. J Differential Equations, 2004, 206: 373–398

[12] Li Y Y. The Nirenberg problem in a domain with boundary. Top Meth Nonlin Anal, 1995, 6: 309–329

[13] Li Y Y, Zhu M. Uniqueness theorems through the method of moving spheres. Duke Math J, 1995, 80: 383–417

[14] Rey O. The role of Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent. J Funct Anal, 1990, 89: 1–52