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Existence of solutions for a class of noncooperative elliptic systems. (English) Zbl 1196.35085

Summary: We establish the existence of a nontrivial solution for a class of noncooperative elliptic systems with nonlinearities of superlinear growth. All results are obtained by the minimax methods in critical point theory.

MSC:

35J57 Boundary value problems for second-order elliptic systems
35J61 Semilinear elliptic equations
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J50 Variational methods for elliptic systems
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[1] Clément, Ph.; Felmer, P.; Mitidieri, E., Homoclinic orbits for a class of infinite-dimensional Hamiltonian systems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 24, 2, 367-393 (1997) · Zbl 0902.35051
[2] Clément, Ph.; de Figueiredo, D. G.; Mitidieri, E., Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations, 17, 923-940 (1992) · Zbl 0818.35027
[3] Costa, D. G.; Magalhães, C. A., Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal., 23, 1401-1412 (1994) · Zbl 0820.35059
[4] Drábek, P.; Robinson, S. B., Resonance problems for the \(p\)-Laplacian, J. Funct. Anal., 169, 1, 189-200 (1999) · Zbl 0940.35087
[5] Felmer, P.; Martínez, S., Existence and uniqueness of positive solutions to certain differential systems, Adv. Differential Equations, 3, 575-593 (1998) · Zbl 0946.35028
[6] de Figueiredo, D. G.; Felmer, P., On superquadratic elliptic systems, Trans. Amer. Math. Soc., 343, 99-116 (1994) · Zbl 0799.35063
[7] de Figueiredo, D. G.; Miyagaki, O. H.; Ruf, B., Elliptic equations in \(R^2\) with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3, 2, 139-153 (1995) · Zbl 0820.35060
[8] de Figueiredo, D. G.; Ruf, B., Elliptic systems with nonlinearities of arbitrary growth, Mediterr. J. Math., 1, 417-431 (2004) · Zbl 1135.35026
[9] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Classics Math. (2001), Springer-Verlag: Springer-Verlag Berlin, reprint of the 1998 edition · Zbl 0691.35001
[10] Hulshof, J.; van der Vorst, R., Differential systems with strongly indefinite variational structure, J. Funct. Anal., 114, 32-58 (1993) · Zbl 0793.35038
[11] Kichenassamy, S.; Véron, L., Singular solutions of the \(p\)-Laplace equation, Math. Ann., 275, 4, 599-615 (1986) · Zbl 0592.35031
[12] Lazer, A. C.; McKenna, P. J., Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32, 4, 537-578 (1990) · Zbl 0725.73057
[13] Li, G. B.; Zhou, H. S., Asymptotically linear Dirichlet problem for the \(p\)-Laplacian, Nonlinear Anal., 43, 8, 1043-1055 (2001) · Zbl 0983.35046
[14] Mitidieri, E., A Rellich type identity and applications, Comm. Partial Differential Equations, 18, 125-151 (1993) · Zbl 0816.35027
[15] Miyagaki, O. H.; Souto, M. A.S., Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations, 245, 12, 3628-3638 (2008) · Zbl 1158.35400
[16] Ruf, B., Lorentz spaces and nonlinear elliptic systems, (Contributions to Nonlinear Analysis. Contributions to Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., vol. 66 (2006), Birkhäuser: Birkhäuser Basel), 471-489 · Zbl 1274.35106
[17] Salvatore, A., Multiple solutions for elliptic systems with nonlinearities of arbitrary growth, J. Differential Equations, 244, 10, 2529-2544 (2008) · Zbl 1143.35028
[18] dos Santos, E. M., Multiplicity of solutions for a fourth-order quasilinear nonhomogeneous equation, J. Math. Anal. Appl., 342, 1, 277-297 (2008) · Zbl 1139.35042
[19] Schechter, M., Linking Methods in Critical Point Theory (1999), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 0915.35001
[20] Schechter, M.; Tintarev, K., Pairs of critical points produced by linking subsets with applications to semilinear elliptic problems, Bull. Soc. Math. Belg. Ser. B, 44, 3, 249-261 (1992) · Zbl 0785.58017
[21] Talbi, M.; Tsouli, N., Existence and uniqueness of a positive solution for a non homogeneous problem of fourth order with weights, Proceedings of the 2005 Oujda International Conference on Nonlinear Analysis. Proceedings of the 2005 Oujda International Conference on Nonlinear Analysis, Electron. J. Differ. Equ. Conf., 14, 231-240 (2006), (electronic) · Zbl 1129.35026
[22] van der Vorst, R., Variational identities and applications to differential systems, Arch. Ration. Mech. Anal., 116, 375-398 (1991) · Zbl 0796.35059
[23] Wang, W. H.; Zhao, P. H., Nonuniformly nonlinear elliptic equations of \(p\)-biharmonic type, J. Math. Anal. Appl., 348, 2, 730-738 (2008) · Zbl 1156.35045
[24] Zhou, J. W.; Wu, X., Sign-changing solutions for some fourth-order nonlinear elliptic problems, J. Math. Anal. Appl., 342, 1, 542-558 (2008) · Zbl 1138.35335
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