Abstract
In this paper we establish the existence of multiple solutions for a class of hamiltonian system with subcritical growth. Here, we use variational method to get multiplicity of solutions using the Lusternik–Schnirelmann category of \({\overline{\Omega }}\) in itself.
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Alves, C.O., Carrião, P.C., Miyagaki, O.H.: Nonlinear perturbation of a periodic elliptic problem with critical growth. J. Math. Anal. Appl. 260, 133–146 (2001)
Alves, C.O., Soares, S.H.M., Jianfu, Y.: On existence and concentration of solutions for a class of Hamiltonian systems in \({\mathbb{R}}^N\). Adv. Nonlinear Stud. 3, 161–180 (2003)
Brezis, H.: Functional Analysis, Sobolev spaces and Partial Differential Equations. Springer, Berlin (2010)
Benci, V., Cerami, G.: The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems. Arch. Ration. Mech. Anal. 114, 79–83 (1991)
Benci, V., Cerami, G.: Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. Calc. Var. Partial Differ. Equ. 02, 29–48 (1994)
Clement, P., de Figueiredo, D.G., Mitidieri, E.: Positive solutions of semilinear elliptic systems. Commun. Partial Differ. Equ. 17, 923–940 (1992)
Costa, D.G., Magalhães, C.A.: A variational approach to noncooperative elliptic systems. Nonlinear Anal. 25, 699–715 (1995)
de Figueiredo, D.G., Felmer, P.: On superquadratic elliptic systems. Trans. Am. Math. Soc. 343, 99–116 (1994)
de Figueiredo, D.G., Magalhães, C.A.: On nonquadratic Hamiltonian elliptic systems. Adv. Differ. Equ. 1, 881–898 (1996)
de Figueiredo, D.G., Yang, J.: Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Anal. 33, 211–234 (1998)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations. Springer, Berlin (1977)
Hulshof, J., van der Vorst, R.C.A.M.: Differential systems with strongly indefinite variational structure. J. Funct. Anal. 114, 32–58 (1993)
Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Springer, Berlin (2013)
Palais, R.S.: The principle of symmetric criticality. Commun. Math. Phys. 69, 19–30 (1979)
Ramos, M., Tavares, H.: Solutions with multiple spike patterns for an elliptic system. Calc. Var. Partial Differ. Equ. 31, 1–25 (2008)
Serrin, J., Zou, H.: Existence of positive entire solutions of elliptic Hamiltonian systems. Commun. Partial Differ. Equ. 23, 577–599 (1998)
Sirakov, B.: On the existence of solutions of elliptic systems in \({\mathbb{R}}^n\). Adv. Differ. Equ. 5, 1445–1464 (2000)
Yang, J.: Nontrivial solutions of semilinear elliptic systems in \({\mathbb{R}}^n\). Electron. J. Differ. Equ. Conf. 6, 343–357 (2001)
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Communicated by Ansgar Jüngel.
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C. O. Alves was partially supported by CNPq/Brazil 304804/2017-7.
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Alves, C.O., Mukherjee, T. Existence and multiplicity of solutions for a class of Hamiltonian systems. Monatsh Math 192, 269–289 (2020). https://doi.org/10.1007/s00605-020-01379-7
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DOI: https://doi.org/10.1007/s00605-020-01379-7