LOCATION OF THE BLOW UP POINT FOR POSITIVE SOLUTIONS OF A BIHARMONIC EQUATION INVOLVING NEARLY CRITICAL EXPONENT

Abstract

Abstract:

In this paper a semilinear biharmonic problem involving nearly critical growth
with Navier boundary condition is considered on an any bounded smooth domain. It is
proved that positive solutions concentrate on a point in the domain, which is also a critical
point of the Robin’s function corresponding to the Green’s function of biharmonic operator
with the same boundary condition. Similar conclusion has been obtained in [6] under the
condition that the domain is strictly convex

Key words: Biharmonic operator；Navier boundary conditions, asymptotic behavior;
critical exponents, Green’s function

CLC Number:

35B50

GENG Di. LOCATION OF THE BLOW UP POINT FOR POSITIVE SOLUTIONS OF A BIHARMONIC EQUATION INVOLVING NEARLY CRITICAL EXPONENT[J].Acta mathematica scientia,Series B, 2005, 25(2): 283-295.

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References

[1] Atkinson F V, Peletier L A. Elliptic equations with near critical growth. J Diff Eq’ns, 1987, 70: 349-365

[2] Azorero J G, Alonso J P. On limits of solutions of elliptic problems with nearly critical exponent. Comm
Partial Diff Eqn’s, 1992, 17: 2113-2126

[3] Bahri A, Li Y, Rey O. On a variational problem with lack of compactness: the topological effect of the
critical points at infinity. Calc Var, 1995, 3: 67-93

[4] Brezis H, Coron J M. On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect
of the topology of the domain. Comm Pure Appl Math, 1988, 41: 253-290

[5] Brezis H, Peletier L A. Asymptotics for elliptic equations involving critical growth. In: Colombini F et al,
eds. Partial Differential Equations and Calculus of Variations, Vol 1. Basel: Birkh¨auser, 1989. 149-192

[6] Chou K S, Geng D. Asymptotics of positive solutions for a biharmonic equation involving critical exponent.
Differential and Integral Equations, 2000, 13(7-9): 921-940

[7] Edmunds D E, Fortunato D, Jannelli E. Critical exponents, critical dimensions and the biharmonic operator.
Arch Rational Mech Anal, 1990, 112: 269-289

[8] de Figueiredo D G, Lions P L, Nussbaum R D. A priori estimates and existence of positive solutions of
semilinear equations. J Math Pure Appl, 1982, 61: 41-63

[9] Han Z. Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev
exponent. Ann Inst H Poincar´e, 1991, 8: 159-174

[10] Lewandowski R. Little holes and convergence of solutions of −u = u(N+2)/(N−2). Nonlinear Analysis,
TMA, 1990, 14(10): 873-888

[11] Lions P L. The concentration compactness principle in the calculus of variations. C r Acad Sci Paris, 1983,
296: 645-648

[12] Lin C S. Classification of solutions of fourth order conformally invariant equation in Rn. Comment, Math
Helv, 1998, 73: 206-231

[13] Lu Guozhen, Wei Juncheng. On a Sobolev inequality with remainder terms. Proc Amer Math Soc, 2000,
128: 75-84

[14] Rey O. Proof of two conjectures of H. Brezis and L. A. Peletier. Manus Math, 1989, 65: 19-37

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

[16] Wei Juncheng. Asymptotic behavior of a nonlinear fourth order eigenvalue problem. Comm Partial Diff
Eqn’s, 1996, 21: 1451-1467