Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-24T00:10:04.515Z Has data issue: false hasContentIssue false
  • English
  • Français

MULTIPLE SOLUTIONS FOR RESONANT ELLIPTIC SYSTEMS VIA REDUCTION METHOD

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

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)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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