Acta mathematica scientia, Series B >
REGULARITY OF THE SOLUTIONS FOR NONLINEAR BIHARMONIC EQUATIONS IN |RN
Received date: 2009-06-29
Revised date: 2007-04-11
Online published: 2009-09-20
Supported by
Research Supported by the National Natural Science Foundation of China (10631030), PHD specialized grant of Ministry of Education of China (20060511001) and supported in part by the Xiao-Xiang Special Fund, Hunan
The purpose of this article is to establish the regularity of the weak solutions for the nonlinear biharmonic equation
{Δ2u+a(x)u=g(x, u),
u∈ H2 (RN), （1.1）
where the condition u ∈ H2(RN) plays the role of a boundary value condition, and as well expresses explicitly that the differential equation is to be satisfied in the weak sense.
Key words： nonlinear biharmonic equation; regularity; fundamental solutions
DENG Yin-Bin , LI Yi . REGULARITY OF THE SOLUTIONS FOR NONLINEAR BIHARMONIC EQUATIONS IN |RN[J]. Acta mathematica scientia, Series B, 2009 , 29(5) : 1469 -1480 . DOI: 10.1016/S0252-9602(09)60119-3
[1] Chou K S, Geng D. Asymptotics of positive solutions for a biharmonic equation involving critical exponent. Differential Integral Equations, 2000, 13(7-9): 921--940
[2] Deng Y B, Li Y. Exponential decay of the solutions for nonlinear biharmonic equations. Communications in Contemporary Mathematics, 2007, 9(5): 753--768
[3] Grunau H C, Sweers G. Sharp estimates for iterated Green functions. Proc Roy Soc Edinburgh Sect A, 2002, 132(1): 91--120
[4] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. 2nd ed. Berlin: Springer-Verlag, 1983
[5] Jannelli E. The role played by space dimension in elliptic critical problems. J Differential Equations, 1999, 156: 407--426
[6] Lebedev N N. Special Functions and Their Applications. Englewood Cliffs N J: Prentice-Hall, 1965
[7] Leis R. Initial-boundary Value Problems in Mathematical Physics. New York: John Wiley & Sons, 1986
[8] Noussair E S, Swanson C A, Yang J F. Transcritical biharmonic equations in RN. Funkcial Ekvac, 1992, 35(3): 533--543
[9] Peletier L A, Van der Vorst R C A M. Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equation. Differential Integral Equations, 1992, 5(4): 747--767
[10] Pucci P, Serrin J. Critical exponents and critical dimensions for polyharmonic operators. J Math Pures Appl, 1990, 69(9): 55--83
[11] Rabier P J, Stuart C A. Fredholm properties of Schr\"odinger operators in Lp(RN). Differential Integral Equations, 2000, 13: 1429--1444
[12] Stuart C A. An introduction to elliptic equations on RN//Ambrosetti A, Chang K C, Ekeland I, eds. Nonlinear Functional Analysis and Applications to Differential Equations (Trieste, 1997). River Edge N J: World Scientific Publishing, 1998: 237--285
[13] Titchmarsh E C. Introduction to the Theory of Fourier Integrals. Oxford: Oxford University Press, 1937
[14] Yao Y, Wang R, Shen Y. Nontrival solution for a class of semilinear biharmonic equation involving crifical exponents. Acta Math Sci, 2007, 27B(3): 509--514
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