Existence and multiplicity of solutions for a class of Hamiltonian systems

Khaled Khachnaoui

doi:10.3842/umzh.v76i5.7497

Khaled Khachnaoui Department of Mathematics, University of Kairouan, Preparatory Institute for Engineering Studies, Tunisia

DOI: https://doi.org/10.3842/umzh.v76i5.7497

Анотація

УДК 517.9

Існування та множинність розв’язків для одного класу гамільтонових систем

Mи досліджуємо клас гамільтонoвих систем \begin{gather} {-q''}(t)+(L(t)-\xi)q(t)= a(t)|q(t)|^{p-2}q(t)+\eta f(t), \\ q\in H^1(\mathbb{R},\mathbb{R}^N),\end{gather} де $(t,q)\in \mathbb{R}\times \mathbb{R}^N$, $p>2,$ $a\inC(\mathbb{R},(0,+\infty)),$ $f\in C(\mathbb{R},\mathbb{R}^N),$ $\xi$ та $\eta$ --- дійсні параметри і$L\in C(\mathbb{R},\mathbb{R}^{N^2})$ — додатно визначена симетрична матриця для всіх $t\in \mathbb{R}.$ Основний технічний підхід базується на методі многовиду Нехарі спільно з варіаційними і топологічними методами. Отримані результати розширюють і доповнюють результати, доступні в літературі.

Посилання

A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349–381 (1973). DOI: https://doi.org/10.1016/0022-1236(73)90051-7

A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations II: a generalized Nehari manifold approach, Discrete and Contin. Dyn. Syst., 19, № 2, 419–430 (2007). DOI: https://doi.org/10.3934/dcds.2007.19.419

A. Benhassine, Multiple of homoclinic solutions for a perturbed dynamical systems with combined nonlinearities, Mediterr. J. Math., 14, 132 (2017). DOI: https://doi.org/10.1007/s00009-017-0930-x

C. O. Alves, P. C. Carrião, O. H. Miyagaki, Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation, Appl. Math. Lett., 16, 639–642 (2003). DOI: https://doi.org/10.1016/S0893-9659(03)00059-4

H. Berestycki, I. Capuzzo Dolcetta, L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, Nonlinear Different. Equat. and Appl., 2, 553–572 (1995). DOI: https://doi.org/10.1007/BF01210623

G. Liu, Periodic solutions of second order Hamiltonian systems with nonlinearity of general linear growth, Electron. J. Qual. Theory Different. Equat., 27, 1–19 (2021). DOI: https://doi.org/10.14232/ejqtde.2021.1.27

I. Ekeland, On the variational principle, Math. Anal. and Appl., 17, 324–353 (1974). DOI: https://doi.org/10.1016/0022-247X(74)90025-0

J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, Springer-Verlag, New York (1989). DOI: https://doi.org/10.1007/978-1-4757-2061-7

K. C. Chang, Methods in nonlinear analysis, Springer Monogr. Math., Springer, Berlin (2005).

K. Khachnaoui, Homoclinic orbits for damped vibration systems with small forcing terms, Nonlinear Dyn. and Syst. Theory, 18, № 1, 80–91 (2018).

K. J. Brown, The Nehari manifold for a semilinear elliptic equation involving a sublinear term, Calc. Var., 22, 483–494 (2005). DOI: https://doi.org/10.1007/s00526-004-0289-2

K. J. Brown, Y. Zhang, The Nehari manifold for a semilinear elliptic problem with a sign changing weight function, J. Different. Equat., 193, 481–499 (2003). DOI: https://doi.org/10.1016/S0022-0396(03)00121-9

K. Khachnaoui, New results on periodic solutions for second order damped vibration systems, Ric. Mat. (2021).

M. Izydorek, J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Different. Equat., 219, № 2, 375–389 (2005). DOI: https://doi.org/10.1016/j.jde.2005.06.029

M. Izydorek, J. Janczewska, Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential, J. Math. Anal. and Appl., 335, 1119–1127 (2007). DOI: https://doi.org/10.1016/j.jmaa.2007.02.038

M. Willem, Minimax theorems, Birkhäuser, Boston (1996). DOI: https://doi.org/10.1007/978-1-4612-4146-1

M. R. Heidari Tavani, Existence results for a class of $p$-Hamiltonian systems, J. Math. Ext., 15, № 2 (4), 1–15 (2021).

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence (1986). DOI: https://doi.org/10.1090/cbms/065

T. F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. and Appl., 318, 253–270 (2006). DOI: https://doi.org/10.1016/j.jmaa.2005.05.057

T. F. Wu, Multiplicity results for a semilinear elliptic equation involving sign-changing weight function, Rocky Mountain J. Math., 39 (2009). DOI: https://doi.org/10.1216/RMJ-2009-39-3-995

V. Coti Zelati, P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing super-quadratic potentials, J. Amer. Math. Soc., 4, № 4, 693–727 (1991). DOI: https://doi.org/10.1090/S0894-0347-1991-1119200-3

Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95, 101–123 (1960). DOI: https://doi.org/10.1090/S0002-9947-1960-0111898-8

Z. Liu, S. Guo, Z. Zhang, Homoclinic orbits for the second-order Hamiltonian systems, Nonlinear Anal., Real World Appl., 36, 116–138 (2017). DOI: https://doi.org/10.1016/j.nonrwa.2016.12.006