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.
@article{119124,
author = {Mario Zuluaga Uribe},
title = {On a nonlinear elliptic system: resonance and bifurcation cases},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {40},
year = {1999},
pages = {701-711},
zbl = {1064.35052},
mrnumber = {1756546},
language = {en},
url = {http://dml.mathdoc.fr/item/119124}
}
Zuluaga Uribe, Mario. On a nonlinear elliptic system: resonance and bifurcation cases. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) pp. 701-711. http://gdmltest.u-ga.fr/item/119124/
Elementary critical point theory and perturbation of elliptic boundary value problems at resonance, Indiana Univ. Math. J. 25 (1976), 933-944. (1976) | MR 0427825
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. (1978) | MR 0492839 | Zbl 0393.35032
Etude des valeurs propers et la resonance pour le operateur P-Laplacian, Ph.D. Thesis, Univ. Bruxelles, 1988.
Abstract critical point Theorems and applications to some nonlinear problems with strong resonance, Nonlinear Analysis T.M.A. 7 9 (1983), 981-1012. (1983) | MR 0713209 | Zbl 0522.58012
Double resonance in semilinear elliptic problems, Comm. Partial Diff. Equations 6 91-120 (1981). (1981) | MR 0597753
Spatially inhomogeneous steady-state solutions for systems of equations describing interacting populations, J. Math. Anal. Appl. 95 (1983), 251-264. (1983) | MR 0710432 | Zbl 0518.92017
On the existence of a nontrivial solution to nonlinear problem at resonance, Nonlinear Analysis T.M.A. 13 2 151-163 (1989). (1989) | MR 0979038
Qualitative study of a class of nonlinear boundary value problems at resonance, J. Diff. Equations 56 63-81 (1985). (1985) | MR 0772121 | Zbl 0554.34009
Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities, Nonlinear Analysis T.M.A, in press. | Zbl 0780.35038
Methods of Bifurcation Theory, Springer Verlag (1982). (1982) | MR 0660633 | Zbl 0487.47039
A variational approach to subquadratic perturbations of elliptic systems, J. Diff. Equations 111 1 103-122 (1994). (1994) | MR 1280617
Bifurcation from infinity and multiple solutions for an elliptic system, Differential and Integral Equations 6 4 (1993), 757-771. (1993) | MR 1222299 | Zbl 0784.35008
Resonance below the first eigenvalue for a semilinear elliptic problem, Math. Ann. 281 589-610 (1988). (1988) | MR 0958261
A maximum principle for an elliptic system and applications to semilinear problems, Siam. J. Math. Anal. 17 (1986), 836-849. (1986) | MR 0846392 | Zbl 0608.35022
Some nonlinear differential equations at resonance at first eigenvalue, Conf. Sem. Mat. Univ Bari 167 355-389 (1979). (1979)
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
Nonlinear boundary value problems at resonance, Nonlinear Analysis T.M.A. 11 455-473 (1987). (1987) | MR 0887655 | Zbl 0676.35023
Nonlinear second order elliptic partial differential equations at resonance, Report 87-12, Memphis State University, 1987. | Zbl 0686.35045
Geometrical Methods of Nonlinear Analysis, Springer Verlag (1984). (1984) | MR 0736839
Nonlinear perturbation of elliptic boundary value problems at resonance, J. Math. Mech. 19 609-623 (1970). (1970) | MR 0267269
On steady-state solutions of a system of reaction-diffusion equations from biology, Nonlinear Analysis T.M.A. 6 (1982), 523-530. (1982) | MR 0664014 | Zbl 0488.35039
A note on a resonance problem, Proc. Royal Soc. Edinburgh 102 A 1-7 (1986). (1986) | MR 0837156 | Zbl 0593.35036
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
Landesman-Lazer type problems at an eigenvalue of odd multiplicity, Results in Math. 14 (1988), 138-146. (1988) | MR 0956010 | Zbl 0780.35043
Nonlinear eigenvalue problems with the parameter near resonance, Ann. Polon. Math. 51 (1990), 241-248. (1990) | MR 1093994 | Zbl 0724.34025
A note on nonlinear oscillations at resonance, Acta Math. Sinica 3 351-361 (1987). (1987) | MR 0930765 | Zbl 0648.34040
Minimax methods in critical point theory with applications to differential equations, CBMS 65 Regional Conference Series in Math, A.M.S., 1986. | MR 0845785 | Zbl 0609.58002
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. (1981) | MR 0613057 | Zbl 0471.35031
Nonlinear elliptic boundary value problems at resonance, Nonlinear Analysis T.M.A. 14 10 889-903 (1990). (1990) | MR 1055536 | Zbl 0708.35033
Shock Waves and Reaction-Diffusion Equations, Springer-Verlag (1983). (1983) | MR 0688146 | Zbl 0508.35002
On the solvability of some elliptic partial differential equations with the linear part at resonance, J. Math. Anal. Appl. 117 138-152 (1986). (1986) | MR 0843010 | Zbl 0634.35030
On a nonlinear Dirichlet problem type at resonance and bifurcation, Partial Differential Equations Pitmann Research Notes in Math. Series 273, pp.248-252 (1992). (1992) | Zbl 0795.35035
A nonlinear elliptic problem at resonance with a nonsimple eigenvalue, Nonlinear Analysis T.M.A. (1996), 711-721. (1996) | MR 1399070 | Zbl 0861.35030
A nonlinear elliptic system at resonance, Dynamic Systems and Applications 3 4 501-510 (1994). (1994) | MR 1304129 | Zbl 0811.35037
Nonzero solutions of a nonlinear elliptic system at resonance, Nonlinear Analysis T.M.A 31 3/4 445-454 (1996). (1996) | MR 1487555