Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

MULTIPLICITY OF SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS WITH CRITICAL EXPONENTIAL GROWTH

  • Fang, Yanqin (School of Mathematics Sciences Nanjing Normal University) ;
  • Zhang, Jihui (School of Mathematics Sciences Nanjing Normal University)
  • Received : 2010.11.08
  • Accepted : 2011.05.02
  • Published : 2011.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we consider a system of N-Laplacian elliptic equations with critical exponential growth. The existence and multiplicity results of solutions are obtained by a limit index method and Trudinger-Moser inequality.

Keywords

References

  1. C. O. Alves, S. H. Monari, Existence of solution for a class of quasilinear systems, Adv. Nonlinear Stud. 9 (2009), 537-564.
  2. C. O. Alves, G. M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in $R^N$, J. Differential Equations 246 (2009), 1288-1311. https://doi.org/10.1016/j.jde.2008.08.004
  3. L. Boccardo, D. G. DeFIgueiredo, Some remarks on a system of quasilinear elliptic equa- tions, NoDEA Nonlinear Differ. Equ. Appl. 9 (3) (2002), 309-323. https://doi.org/10.1007/s00030-002-8130-0
  4. J. F. Bonder, S. Martinezi, J. D. Rossi, Existence results for Gradient elliptic systems with nonlinear boundary conditions, NoDEA Nonlinear Differ. Equ. Appl. 14 (2007), 153-179. https://doi.org/10.1007/s00030-007-5015-2
  5. V. Benci, On critical point theory for indefinite functional in presence of symmetries, Tran. Amer. Math. Soc. 24 (1982), 533-572.
  6. D. G. Costa, Multiple solutions for a class of stongly indefinite problems, Mat. Contemp. 15 (1998), 87-103.
  7. M. Clapp, Y. H. Ding, S. Hernandez-Linares, Strongly indefinite functional with perturbed symmetries and multiple solutions of nonsymmetric elliptic systems, Electron. J. Differen- tial Equations 100 (2004), 1-18.
  8. Y. Q. Fang, J. H. Zhang, Multiplicity of solutions for a class of elliptic systems with critical Sobolev exponent, accepted by Nonlinear Anal. (2010).
  9. D. G. de Figueiredo, J. M. Bezerra do O, B. Ruf, Critical and subcritical elliptic systems in dimension two, Indiana Univ. Math. J. 53 (2004), no.4, 1037-1054. https://doi.org/10.1512/iumj.2004.53.2402
  10. D. Huang, Y. Li, Multiple solutions for a noncooperative p-Laplacian elliptic system in $R^N$ , J. Differential Equations 215 (2005), 206-223. https://doi.org/10.1016/j.jde.2004.09.001
  11. W. Kryszewski, A. Szulkin, An infinte dimensional Morse theory with applications, Trans. Amer. Math. Soc. 349 (1997), 3181-3234. https://doi.org/10.1090/S0002-9947-97-01963-6
  12. V. Kondvatev, M. Shubin, Discreteness of spectrum for the Schrodinger operators on manifolds of bounded geometry, Oper. Theory Adv. Appl. 110 (1999), 185-226.
  13. F. Lin, Y. Li, Multiplcity of solutions for a noncooperative elliptic system with critical Sobolev exponent, Z. Angew. Math. Phys. 60 (2009), 402-415. https://doi.org/10.1007/s00033-008-7114-2
  14. S. J. Li, M. willem, Applications of local linking to critical point thery, J. Math. Anal. Appl. 189 (1995), 6-32. https://doi.org/10.1006/jmaa.1995.1002
  15. Y. Q. Li, A limit index theory and its application, Nonlinear Anal. 25 (1995), 1371-1389. https://doi.org/10.1016/0362-546X(94)00254-F
  16. J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I. Springer, Berlin (1977).
  17. J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 1077-1092. https://doi.org/10.1512/iumj.1971.20.20101
  18. M. Struwe, Variational Methods, Springer, Berlin 1990.
  19. A. Szulkin, Index theories for indefinite functionals and applications, in: P. Drabek (Ed.), Pitman Research Notes in Mathematics, Series 365, Topological and Variational Methods for Nonlinear Boundary Value Problems, Longman, Harlow 1997, pp. 89-121.
  20. N. S. Trudinger, On the imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-484.
  21. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978).
  22. M. Willem, Minimax Theorems, BirkhÄauser, Boston, 1996.
  23. J. F. Yang, Positive solutions of quasilinear elliptic obstacle problems with critical expo- nents, Nonlinear Anal. 25 (1995), 1283-1306. https://doi.org/10.1016/0362-546X(94)00247-F