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 , where is a bounded domain with smooth boundary , is the biharmonic operator, , < , denotes the critical Sobolev exponent, is homogeneous function of degree . By using the variational methods and the Ljusternik-Schnirelmann theory, we obtain multiplicity result of nontrivial solutions under certain hypotheses on and .
ON A CLASSIFICATION OF WARPED PRODUCT SPACES WITH GRADIENT RICCI SOLITONS, Korean Journal of Mathematics, 2016, 24, 4, 627
References
1.
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.
2.
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.
3.
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.
4.
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.
5.
J. Chabrowski and J. Marcos do O, On some fourth-order semilinear elliptic problems in RN, Nonlinear Anal. 49 (2002), no. 6, 861-884.
6.
L. Ding and S. W. Xiao, Multiple positive solutions for a critical quasilinear elliptic system, Nonlinear Anal. 72 (2010), no. 5, 2592-2607.
7.
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.
8.
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.
9.
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.