MULTIPLICITY OF SOLUTIONS FOR BIHARMONIC ELLIPTIC SYSTEMS INVOLVING CRITICAL NONLINEARITY

Lu, Dengfeng; Xiao, Jianhai

doi:10.4134/BKMS.2013.50.5.1693

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MULTIPLICITY OF SOLUTIONS FOR BIHARMONIC ELLIPTIC SYSTEMS INVOLVING CRITICAL NONLINEARITY

Journal title : Bulletin of the Korean Mathematical Society
Volume 50, Issue 5, 2013, pp.1693-1710
Publisher : The Korean Mathematical Society
DOI : 10.4134/BKMS.2013.50.5.1693

Title & Authors

MULTIPLICITY OF SOLUTIONS FOR BIHARMONIC ELLIPTIC SYSTEMS INVOLVING CRITICAL NONLINEARITY
Lu, Dengfeng; Xiao, Jianhai;

Abstract

In this paper, we consider the biharmonic elliptic systems of the form $$\{{\Delta}^2u

Keywords

biharmonic elliptic system;critical Sobolev exponent;variational method;multiple solutions;

Language

English

Cited by

1time(s) in CrossRef

ON A CLASSIFICATION OF WARPED PRODUCT SPACES WITH GRADIENT RICCI SOLITONS, Korean Journal of Mathematics, 2016, 24, 4, 627

References

C. O. Alves and Y. H. Ding, Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity, J. Math. Anal. Appl. 279 (2003), no. 2, 508-521.

T. Bartsch and Y. Guo, Existence and nonexistence results for critical growth polyhar- monic elliptic systems, J. Differential Equations 220 (2006), no. 2, 531-543.

F. Bernis, J. Garcia-Azorero, and I. Peral, Existence and multiplicity of nontrivial so- lutions in semilinear critical problems of fourth order, Adv. Differential Equations 1 (1996), no. 2, 219-240.

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and con- vergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486-490.

J. Chabrowski and J. Marcos do O, On some fourth-order semilinear elliptic problems in RN, Nonlinear Anal. 49 (2002), no. 6, 861-884.

L. Ding and S. W. Xiao, Multiple positive solutions for a critical quasilinear elliptic system, Nonlinear Anal. 72 (2010), no. 5, 2592-2607.

D. E. Edmunds, D. Fortunato, and E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Arch. Rational Mech. Anal. 112 (1990), no. 3, 269-289.

D. C. de Morais Filho and M. A. S. Souto, Systems of p-Laplacian equations involv- ing homogeneous nonlinearities with critical Sobolev exponent degrees, Comm. Partial Differential Equations 24 (1999), no. 7-8, 1537-1553.

F. Gazzola, H.C. Grunau, and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations 18 (2003), no. 2, 117-143.

10.

Y. X. Ge, J. C. Wei, and F. Zhou, A critical elliptic problem for polyharmonic operators, J. Funct. Anal. 260 (2011), no. 8, 2247-2282.

11.

H. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents, Calc. Var. Partial Differential Equations 3 (1995), no. 2, 243-252.

12.

P. G. Han, The effect of the domain topology on the number of positive solutions of an elliptic systems involving critical Sobolev exponents, Houston J. Math. 32 (2006), no. 4, 1241-1257.

13.

T. S. Hsu and H. L. Lin, Multiple positive solutions for a critical elliptic system with concave-convex nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), no. 6, 1163-1177.

14.

D. S. Kang and S. J. Peng, Existence and asymptotic properties of solutions to elliptic systems involving multiple critical exponents, Sci.China Math. 54 (2011), no. 2, 243-256.

15.

D. F. Lu, Multiple solutions for a class of biharmonic elliptic systems with Sobolev critical exponent, Nonlinear Anal. 74 (2011), no. 17, 6371-6382.

16.

D. F. Lu and J. H. Xiao, Multiple solutions for weighted nonlinear elliptic system in-volving critical exponents, Math. Comput. Modelling 55 (2012), no. 3-4, 816-827.

17.

M. Montenegro, On nontrivial solutions of critical polyharmonic elliptic systems, J. Differential Equations 247 (2009), no. 3, 906-916.

18.

E. S. Noussair, C. A. Swanson, and J. Yang, Critical semilinear biharmonic equations in $R^N$, Proc. Roy. Soc. Edinburgh Sect. A 121 (1992), no. 1-2, 139-148.

19.

O. Rey, A multiplicity results for a variational problem with lack of compactness, Nonlinear Anal. 13 (10) (1989), no. 10, 1241-1249.

20.

Y. Shen and J. H. Zhang, Multiplicity of positive solutions for a semilinear p-Laplacian system with Sobolev critical exponent, Nonlinear Anal. 74 (2011), no. 4, 1019-1030.

21.

Y. J. Wang and Y. T. Shen, Multiple and sign-changing solutions for a class of semi-linear biharmonic equation, J. Differential Equations 246 (2009), no. 8, 3109-3125.

22.

M. Willem, Minimax Theorems, Birkhauser, Boston, 1996.

23.

Y. J. Zhang, Positive solutions of semilinear biharmonic equations with critical Sobolev exponents, Nonlinear Anal. 75 (2012), no. 1, 55-67.