Homoclinic solutions for a second-order nonperiodic asymptotically linear Hamiltonian systems. (English) Zbl 1268.37080

Zheng, Qiang

Abstr. Appl. Anal. 2013, Article ID 417020, 7 p. (2013).

The second-order Hamiltonian system \[ \ddot{u}(t)+\nabla V(t,u(t))=0 \] is studied, where \(V(t,u)=-K(t,u)+W(t,u)\) and \(K,W:\mathbb R \times \mathbb R ^N\to \mathbb R\) are \(C^1\)-maps. The main result is the following theorem:
Let \(A:=\sup \{ K(t,u): t\in \mathbb R, |u|\leq 1\} <+\infty\). Moreover, assume the following:
1) \(K(t,0)\equiv 0\), and there is a constant \(a>0\) such that \[ K(t,u)\geq a|u|^2, \quad \forall (t,u)\in \mathbb R \times \mathbb R^N; \]
2) there exists \(\beta \in (1,2]\) such that \[ K(t,u)\leq (u,\nabla K(t,u))\leq \beta K(t,u),\quad \forall (t,u)\in \mathbb R \times \mathbb R^N; \]
3) \(W(t,0)\equiv 0\), and \(\nabla W(t,u)=o(|u|)\) as \(u\to 0\) uniformly in \(t\), and there exists \(M_0>0\) such that \[ \frac{|\nabla W(t,u)|}{|u|}\leq M_0, \quad\forall (t,u)\in \mathbb R \times \mathbb R^N; \] 4) \(W(t,u)-w(t)|u|^2=o(|u|^2)\) as \(|u|\to \infty\) uniformly in \(t\), where \(w\in L^\infty(\mathbb R, \mathbb R)\) with \(w_\infty :=\inf_{t\in \mathbb R}w(t)>2A;\)
5) \(\widetilde{W}(t,u):=1/2(\nabla W(t,u),u)-W(t,u)\to +\infty\) as \(|u|\to +\infty\), and \[ \inf \left\{ \frac{\widetilde{W}(t,u)}{|u|^2}: t\in \mathbb R , \quad c\leq |u|<d \right\} , \] for any \(c,d>0\).
Then the above system possesses a nontrivial homoclinic solution \(u\in W^{1,2}(\mathbb R, \mathbb R^N)\) such that \(\dot{u}(t)\to 0\) as \(t\to \pm \infty\).
The author was motivated by a recent result of A. Daouas [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 11, 3407–3418 (2011; Zbl 1221.37115)] dealing with superquadratic \(W\). Results of M. Izydorek and J. Janczewska [J. Differ. Equations 219, No. 2, 375–389 (2005; Zbl 1080.37067)] are used in the proof.

Reviewer: Zdzisław Dzedzej (Gdansk)

MSC:

37J45	Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34C37	Homoclinic and heteroclinic solutions to ordinary differential equations
37C29	Homoclinic and heteroclinic orbits for dynamical systems

Keywords:

second order Hamiltonian systems; homoclinic solutions

Citations:

Zbl 1221.37115; Zbl 1080.37067

References:

[1]	Mawhin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems, 74 (1989), New York, NY, USA: Springer, New York, NY, USA · Zbl 0676.58017
[2]	Rabinowitz, P. H., Variational methods for Hamiltonian systems, Handbook of Dynamical Systems, 1, part 1, 1091-1127 (2002) · Zbl 1048.37055
[3]	Daouas, A., Homoclinic orbits for superquadratic Hamiltonian systems without a periodicity assumption, Nonlinear Analysis: Theory, Methods & Applications, 74, 11, 3407-3418 (2011) · Zbl 1221.37115
[4]	Ding, Y., Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms, Communications in Contemporary Mathematics, 8, 4, 453-480 (2006) · Zbl 1104.70013
[5]	Ding, Y.; Lee, C., Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system, Journal of Differential Equations, 246, 7, 2829-2848 (2009) · Zbl 1162.70014
[6]	Ding, Y.; Jeanjean, L., Homoclinic orbits for a nonperiodic Hamiltonian system, Journal of Differential Equations, 237, 2, 473-490 (2007) · Zbl 1117.37032
[7]	Sun, J.; Chen, H.; Nieto, J. J., Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero, Journal of Mathematical Analysis and Applications, 378, 1, 117-127 (2011) · Zbl 1218.37081
[8]	Ding, Y. H., Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Analysis: Theory, Methods & Applications, 25, 11, 1095-1113 (1995) · Zbl 0840.34044
[9]	Izydorek, M.; Janczewska, J., Homoclinic solutions for a class of the second order Hamiltonian systems, Journal of Differential Equations, 219, 2, 375-389 (2005) · Zbl 1080.37067
[10]	Izydorek, M.; Janczewska, J., Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential, Journal of Mathematical Analysis and Applications, 335, 2, 1119-1127 (2007) · Zbl 1118.37032
[11]	Lv, X.; Lu, S.; Yan, P., Homoclinic solutions for nonautonomous second-order Hamiltonian systems with a coercive potential, Nonlinear Analysis: Theory, Methods & Applications, 72, 7-8, 3484-3490 (2010) · Zbl 1203.34067
[12]	Rabinowitz, P. H., Homoclinic orbits for a class of Hamiltonian systems, Proceedings of the Royal Society of Edinburgh A, 114, 1-2, 33-38 (1990) · Zbl 0705.34054
[13]	Rabinowitz, P. H.; Tanaka, K., Some results on connecting orbits for a class of Hamiltonian systems, Mathematische Zeitschrift, 206, 3, 473-499 (1991) · Zbl 0707.58022
[14]	Sun, J.; Chen, H.; Nieto, J. J., Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems, Journal of Mathematical Analysis and Applications, 373, 1, 20-29 (2011) · Zbl 1230.37079
[15]	Tang, X. H.; Lin, X., Homoclinic solutions for a class of second-order Hamiltonian systems, Journal of Mathematical Analysis and Applications, 354, 2, 539-549 (2009) · Zbl 1179.37082
[16]	Tang, X. H.; Xiao, L., Homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Analysis: Theory, Methods & Applications, 71, 3-4, 1140-1152 (2009) · Zbl 1185.34056
[17]	Wan, L.-L.; Tang, C.-L., Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition, Discrete and Continuous Dynamical Systems B, 15, 1, 255-271 (2011) · Zbl 1216.34033
[18]	Zhang, Z.; Yuan, R., Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems, Nonlinear Analysis: Theory, Methods & Applications, 71, 9, 4125-4130 (2009) · Zbl 1173.34330
[19]	Zhang, Z.; Yuan, R., Homoclinic solutions for some second order non-autonomous Hamiltonian systems with the globally superquadratic condition, Nonlinear Analysis: Theory, Methods & Applications, 72, 3-4, 1809-1819 (2010) · Zbl 1193.34093
[20]	Korman, P.; Lazer, A. C., Homoclinic orbits for a class of symmetric Hamiltonian systems, Electronic Journal of Differential Equations, 1994, 1-10 (1994)
[21]	Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations. Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS, 65 (1986) · Zbl 0609.58002
[22]	Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Analysis: Theory, Methods & Applications, 7, 9, 981-1012 (1983) · Zbl 0522.58012