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MULTIPLE SOLUTIONS FOR RESONANT ELLIPTIC SYSTEMS VIA REDUCTION METHOD

Part of: Variational problems in infinite-dimensional spaces Elliptic equations and systems

Published online by Cambridge University Press:  22 April 2010

MARCELO F. FURTADO  and
FRANCISCO O. V. DE PAIVA
MARCELO F. FURTADO*
Affiliation:
Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília - DF, Brazil (email: mfurtado@unb.br)
FRANCISCO O. V. DE PAIVA
Affiliation:
IMECC-UNICAMP, Caixa Postal 6065, 13081-970, Campinas-SP, Brazil (email: odair@ime.unicamp.br)
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Abstract

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We establish the existence of two nontrivial solution for some elliptic systems. In the proofs we apply variational methods and Morse theory.

Keywords

resonant elliptic systemsvariational methodsMorse theory

MSC classification

Secondary: 35J20: Variational methods for second-order elliptic equations 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 2 , October 2010 , pp. 211 - 220
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Bartsch, T. and Li, S. J., ‘Critical point theory for asymptotically quadratic functionals and applications to problems with resonance’, Nonlinear Anal. 28 (1997), 419–441.CrossRefGoogle Scholar
[2]Castro, A., ‘Reduction methods via minimax’, in: Differential Equations (São Paulo, 1981), Lecture Notes in Mathematics, 957 (Springer, Berlin, 1982), pp. 1–20.Google Scholar
[3]Castro, A. and Cossio, J., ‘Multiple solutions for a nonlinear Dirichlet problems’, SIAM J. Math. Anal. 25 (1994), 1554–1561.CrossRefGoogle Scholar
[4]Chang, K. C., Infinite Dimensional Morse Theory and Multiple Solution Problems (Birkhäuser, Boston, MA, 1993).CrossRefGoogle Scholar
[5]Chang, K. C., ‘An extension of the Hess–Kato theorem to elliptic systems and its applications to multiple solutions problems’, Acta Math. Sin. 15 (1999), 439–454.CrossRefGoogle Scholar
[6]Chang, K. C., ‘Principal eigenvalue for weigth in elliptic systems’, Nonlinear Anal. 46 (2001), 419–433.CrossRefGoogle Scholar
[7]Costa, D. G. and Magalhães, C. A., ‘Variational elliptic problems which are nonquadratic at infinity’, Nonlinear Anal. 23 (1994), 1401–1412.CrossRefGoogle Scholar
[8]Costa, D. G. and Magalhães, C. A., ‘A unified approach to a class of strongly indefinite functionals’, J. Differential Equations 125 (1996), 521–547.CrossRefGoogle Scholar
[9]deFigueiredo, D. G., ‘Positive solutions of semilinear elliptic problems’, in: Differential Equations (São Paulo, 1981), Lecture Notes in Mathematics, 957 (Springer, Berlin, 1982), pp. 34–87.CrossRefGoogle Scholar
[10]Furtado, M. F. and de Paiva, F. O. V., ‘Multiplicity of solutions for resonant elliptic systems’, J. Math. Anal. Appl. 319 (2006), 435–449.CrossRefGoogle Scholar
[11]Furtado, M. F. and Silva, E. A. B., ‘Double resonant problems which are locally nonquadratic at infinity’, Electron. J. Differ. Equ., Conf. 6 (2001), 155–171.Google Scholar
[12]Li, S. J. and Liu, S., ‘Critical groups at infinity, saddle point reduction and elliptic resonant problems’, Commun. Comtem. Math. 5 (2003), 761–773.CrossRefGoogle Scholar
[13]Liu, S., ‘Remarks on multiple solutions for elliptic resonant problems’, J. Math. Anal. Appl. 336 (2007), 498–505.CrossRefGoogle Scholar
[14]Lopes, O., ‘Radial symmetry of minimizers for some translation and rotation invariant functionals’, J. Differential Equations 124 (1996), 378–388.CrossRefGoogle Scholar