Title:
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On a nonlinear elliptic system: resonance and bifurcation cases (English) |
Author:
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Zuluaga, Mario |
Language:
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English |
Journal:
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Commentationes Mathematicae Universitatis Carolinae |
ISSN:
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0010-2628 (print) |
ISSN:
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1213-7243 (online) |
Volume:
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40 |
Issue:
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4 |
Year:
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1999 |
Pages:
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701-711 |
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Category:
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math |
. |
Summary:
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In this paper we consider an elliptic system at resonance and bifurcation type with zero Dirichlet condition. We use a Lyapunov-Schmidt approach and we will give applications to Biharmonic Equations. (English) |
Keyword:
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elliptic system at resonance |
Keyword:
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bifurcation points |
Keyword:
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Lyapunov-Schmidt met\-hod |
MSC:
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35B34 |
MSC:
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35J55 |
MSC:
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35J65 |
MSC:
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47J15 |
MSC:
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58J55 |
idZBL:
|
Zbl 1064.35052 |
idMR:
|
MR1756546 |
. |
Date available:
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2009-01-08T18:56:43Z |
Last updated:
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2012-04-30 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/119124 |
. |
Reference:
|
[1] Ahmad S., Lazer A., Paul J.: Elementary critical point theory and perturbation of elliptic boundary value problems at resonance.Indiana Univ. Math. J. 25 (1976), 933-944. MR 0427825 |
Reference:
|
[2] Ambrosetti A., Mancini G.: Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance. The case of the simple eigenvalue.J. Diff. Equations 28 (1978), 220-245. Zbl 0393.35032, MR 0492839 |
Reference:
|
[3] Anane A.: Etude des valeurs propers et la resonance pour le operateur P-Laplacian.Ph.D. Thesis, Univ. Bruxelles, 1988. |
Reference:
|
[4] Bartolo P., Benci V., Fortunato D.: Abstract critical point Theorems and applications to some nonlinear problems with strong resonance.Nonlinear Analysis T.M.A. 7 9 (1983), 981-1012. Zbl 0522.58012, MR 0713209 |
Reference:
|
[5] Berestycki H., Defigueiredo D.: Double resonance in semilinear elliptic problems.Comm. Partial Diff. Equations 6 91-120 (1981). MR 0597753 |
Reference:
|
[6] Brown K.J.: Spatially inhomogeneous steady-state solutions for systems of equations describing interacting populations.J. Math. Anal. Appl. 95 (1983), 251-264. Zbl 0518.92017, MR 0710432 |
Reference:
|
[7] Capozzi A., Lupo D., Solimini S.: On the existence of a nontrivial solution to nonlinear problem at resonance.Nonlinear Analysis T.M.A. 13 2 151-163 (1989). MR 0979038 |
Reference:
|
[8] Cesari L., Kannan R.: Qualitative study of a class of nonlinear boundary value problems at resonance.J. Diff. Equations 56 63-81 (1985). Zbl 0554.34009, MR 0772121 |
Reference:
|
[9] Chiappinelli R., Mawhin J., Nugari R.: Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities.Nonlinear Analysis T.M.A, in press. Zbl 0780.35038 |
Reference:
|
[10] Chow S., Hale J.: Methods of Bifurcation Theory.Springer Verlag (1982). Zbl 0487.47039, MR 0660633 |
Reference:
|
[11] Costa D., Magalh aes: A variational approach to subquadratic perturbations of elliptic systems.J. Diff. Equations 111 1 103-122 (1994). MR 1280617 |
Reference:
|
[12] De Figueiredo D., Chiappinelli R.: Bifurcation from infinity and multiple solutions for an elliptic system.Differential and Integral Equations 6 4 (1993), 757-771. Zbl 0784.35008, MR 1222299 |
Reference:
|
[13] De Figueiredo D., Gossez J.: Resonance below the first eigenvalue for a semilinear elliptic problem.Math. Ann. 281 589-610 (1988). MR 0958261 |
Reference:
|
[14] De Figueiredo D., Mitidieri E.: A maximum principle for an elliptic system and applications to semilinear problems.Siam. J. Math. Anal. 17 (1986), 836-849. Zbl 0608.35022, MR 0846392 |
Reference:
|
[15] Gossez J.: Some nonlinear differential equations at resonance at first eigenvalue.Conf. Sem. Mat. Univ Bari 167 355-389 (1979). |
Reference:
|
[16] Hernández J.: Maximum Principles and Decoupling for Positive Solutions of Reaction-Diffusion Systems.Oxford University Press, K.J. Brown and A. Lacey eds, 1990, pp.199-224. MR 1086647 |
Reference:
|
[17] Iannacci R., Nkashama M.: Nonlinear boundary value problems at resonance.Nonlinear Analysis T.M.A. 11 455-473 (1987). Zbl 0676.35023, MR 0887655 |
Reference:
|
[18] Iannacci R., Nkashama M.: Nonlinear second order elliptic partial differential equations at resonance.Report 87-12, Memphis State University, 1987. Zbl 0686.35045 |
Reference:
|
[19] Krasnosels'kii M., Zabreico F.: Geometrical Methods of Nonlinear Analysis.Springer Verlag (1984). MR 0736839 |
Reference:
|
[20] Landesman E., Lazer A.: Nonlinear perturbation of elliptic boundary value problems at resonance.J. Math. Mech. 19 609-623 (1970). MR 0267269 |
Reference:
|
[21] Lazer A., McKenna P.J.: On steady-state solutions of a system of reaction-diffusion equations from biology.Nonlinear Analysis T.M.A. 6 (1982), 523-530. Zbl 0488.35039, MR 0664014 |
Reference:
|
[22] Lupo D., Solimini S.: A note on a resonance problem.Proc. Royal Soc. Edinburgh 102 A 1-7 (1986). Zbl 0593.35036, MR 0837156 |
Reference:
|
[23] Mawhin J.: Bifurcation from infinity and nonlinear boundary value problems, in ordinary and partial differential equations.vol. II, Sleeman and Jarvis eds, Longman, Ifarlow, 1989, pp.119-129. MR 1031727 |
Reference:
|
[24] Mawhin J., Schmitt K.: Landesman-Lazer type problems at an eigenvalue of odd multiplicity.Results in Math. 14 (1988), 138-146. Zbl 0780.35043, MR 0956010 |
Reference:
|
[25] Mawhin J., Schmitt K.: Nonlinear eigenvalue problems with the parameter near resonance.Ann. Polon. Math. 51 (1990), 241-248. Zbl 0724.34025, MR 1093994 |
Reference:
|
[26] Omari P., Zanolin F.: A note on nonlinear oscillations at resonance.Acta Math. Sinica 3 351-361 (1987). Zbl 0648.34040, MR 0930765 |
Reference:
|
[27] Rabinowitz P.: Minimax methods in critical point theory with applications to differential equations.CBMS 65 Regional Conference Series in Math, A.M.S., 1986. Zbl 0609.58002, MR 0845785 |
Reference:
|
[28] Rothe F.: Global existence of branches of stationary solutions for a system of reaction-diffusion equations from biology.Nonlinear Analysis T.M.A. 5 (1981), 487-498. Zbl 0471.35031, MR 0613057 |
Reference:
|
[29] Schechter M.: Nonlinear elliptic boundary value problems at resonance.Nonlinear Analysis T.M.A. 14 10 889-903 (1990). Zbl 0708.35033, MR 1055536 |
Reference:
|
[30] Smoller J.: Shock Waves and Reaction-Diffusion Equations.Springer-Verlag (1983). Zbl 0508.35002, MR 0688146 |
Reference:
|
[31] Solimini S.: On the solvability of some elliptic partial differential equations with the linear part at resonance.J. Math. Anal. Appl. 117 138-152 (1986). Zbl 0634.35030, MR 0843010 |
Reference:
|
[32] Vargas C., Zuluaga M.: On a nonlinear Dirichlet problem type at resonance and bifurcation.Partial Differential Equations Pitmann Research Notes in Math. Series 273, pp.248-252 (1992). Zbl 0795.35035 |
Reference:
|
[33] Vargas C., Zuluaga M.: A nonlinear elliptic problem at resonance with a nonsimple eigenvalue.Nonlinear Analysis T.M.A. (1996), 711-721. Zbl 0861.35030, MR 1399070 |
Reference:
|
[34] Zuluaga M.: A nonlinear elliptic system at resonance.Dynamic Systems and Applications 3 4 501-510 (1994). Zbl 0811.35037, MR 1304129 |
Reference:
|
[35] Zuluaga M.: Nonzero solutions of a nonlinear elliptic system at resonance.Nonlinear Analysis T.M.A 31 3/4 445-454 (1996). MR 1487555 |
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