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Multiple of Homoclinic Solutions for a Perturbed Dynamical Systems with Combined Nonlinearities

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Abstract

In this paper, we study the existence of multiple and infinite homoclinic solutions for the following perturbed dynamical systems

$$\begin{aligned} \ddot{x}+A\cdot \dot{x}-L(t)\cdot x+\nabla W(t,x)=f(t), \end{aligned}$$

where \(t\in {\mathbb R}, x\in {\mathbb R}^N,\) A is an antisymmetric constant matrix, the matrix L(t) is not necessary positive definite for all \(t\in {\mathbb R}\) nor coercive, the nonlinearity \(W \in C^{1}(\mathbb {R}\times \mathbb {R}^{N},\mathbb {R})\) involves a combination of superquadratic and subquadratic terms and is allowed to be sign-changing and \(f\in C({\mathbb R},{\mathbb R}^{N})\cap L^{2}({\mathbb R},{\mathbb R}^{N}).\) Recent results in the literature are generalized and significantly improved and some examples are also given to illustrate our main theoretical results.

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References

  1. Agarwal, R.P.: Difference Equations and Inequalities: Theory, Methods, and Applications, 2nd edn. Marcel Dekker Inc, New York (2000)

    MATH  Google Scholar 

  2. Agarwal, R.P., Perera, K., O’Regan, D.: Multiple Positive Solutions of Singular Discrete p-Laplacian Problems via Variational Methods. Adv. Differ. Equ. 2005, 93–99 (2005)

  3. Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal. 7, 981–1012 (1983)

  4. Chen, C., Tzeng, S.: Existence and Multiplicity Results for Homoclinic Orbits of Hamiltonian Systems. Electron. J. Differ. Equations 1997, 1–19 (1997)

  5. Zelati, V.C., Rabinowitz, P.H.: Homoclinic Orbits for Second Order Hamiltonian Systems Possessing Superquadratic Potentials. J. Am. Math. Soc. 4, 693–727 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. de Figueiredo, D.G., Gossez, J.P., Ubilla, P.: Local Superlinearity and Sublinearity for Indefinite Semilinear Elliptic Problems. J. Funct. Anal. 199(2), 452–467 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ding, Y.H.: Existence and Multiplicity Results for Homoclinic Solutions to a Class of Hamiltonian Systems. Nonlinear Anal. 25(11), 1095–1113 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ekeland, I.: Nonconvex Minimization Problems. Bull. Am. Math. Soc. 1, 443–474 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  9. Guo, C., O’Regan, D., Wang, C., Agarwal, R.P.: Existence of Homoclinic Orbits of Superquadratic Second-Order Hamiltonian Systems. J. Anal. Appl. 34, 27–41 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Izydorek, M., Janczewska, J.: Homoclinic Solutions for a Class of the Second Order Hamiltonian Systems. J. Differ. Equations 219(2), 375–389 (2005)

    Article  MATH  Google Scholar 

  11. Izydorek, M., Janczewska, J.: Homoclinic Solutions for Nonautonomous Second Order Hamiltonian Systems with a Coercive Potential. J. Math. Anal. Appl. 335, 1119–1127 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Korman, P., Lazer, A.C.: Homoclinic Orbits for a Class of Symmetric Hamiltonian Systems. Electron. J. Differ. Equations 1, 1–10 (1994)

  13. Li, Y.: Existence of Multiple Solutions of Semilinear Elliptic Equations in \(\mathbb{R}^{N}\), Variational Methods, Paris, 1988. Prog. nonlinear Differ. Equations Appl. 4, 133–159 (1990)

  14. Lusternik, L., Schnirelmann, L.: Methodes Topologiques dans les Problemes Variationnels. Hermann, Paris (1934)

    MATH  Google Scholar 

  15. Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989)

    Book  MATH  Google Scholar 

  16. Ou, Z.Q., Tang, C.L.: Existence of Homoclinic Orbits for the Second Order Hamiltonian Systems. J. Math. Anal. Appl. 291, 203–213 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Omana, W., Willem, M.: Homoclinic Orbits for a Class of Hamiltonian Systems. Differ. Integr. Equations 5(5), 1115–1120 (1992)

    MathSciNet  MATH  Google Scholar 

  18. Poincaré, H.: Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars, Paris (1897–1899)

  19. Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. In: CBMS Reg. Conf. Ser. in. Math., vol. 65. American Mathematical Society, Provodence, RI (1986)

  20. Rabinowitz, P.H.: Homoclinic Orbits for a Class of Hamiltonian Systems. Proc. R. Soc. Edinb. Sect. A 114, 33–38 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rabinowitz, P.H., Tanaka, K.: Some Results on Connecting Orbits for a Class of Hamiltonian Systems. Math. Z. 206(3), 473–499 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sun, J., Chen, H., Nieto, J.J.: Homoclinic Orbits for a Class of First-Order Nonperiodic Asymptotically Quadratic Hamiltonian Systems With Spectrum Point Zero. J. Math. Anal. Appl. 378, 117–127 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tang, X.H., Xiao, L.: Homoclinic Solutions for a Class of Second-order Hamiltonian Systems. Nonlinear Anal. 71, 1140–1152 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, J., Zhang, F., Xu, J.: Existence and Multiplicity Results of Homoclinic Orbits for the Second Order Hamiltonian Systems. J. Math. Anal. Appl. 366, 569–581 (2010)

  25. Willem, M.: Minimax Theorems. Birkhäuser, Berlin (1996)

    Book  MATH  Google Scholar 

  26. Zhang, Z., Yuan, R.: Homoclinic Solutions of Some Second Order Non-Autonomous Systems. Nonlinear Anal. 71, 5790–5798 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang, L., Tang, X., Chen, Y.: Infinitely Many Homoclinic Solutions for a Class of Indefinite Perturbed Second-Order Hamiltonian Systems. Mediterr. J. Math. doi:10.1007/s00009-016-0708-6

  28. Zhang, Z., Yuan, R.: Homoclinic Solutions for a Class of Non-autonomous Subquadratic Second Order Hamiltonian Systems. Nonlinear Anal. doi:10.1016/j.na.2009.02.071

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Acknowledgements

The author would like to thank the referees for their careful reading, critical comments and helpful suggestions, which helped to improve the quality of the paper.

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  1. Department of Mathematics, Higher Institute of Informatics and Mathematics, 5000, Monastir, Tunisia

    Abderrazek Benhassine

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  1. Abderrazek Benhassine
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Benhassine, A. Multiple of Homoclinic Solutions for a Perturbed Dynamical Systems with Combined Nonlinearities. Mediterr. J. Math. 14, 132 (2017). https://doi.org/10.1007/s00009-017-0930-x

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