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

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.

### 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
Full Text:

### References:

  Mawhin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems, 74 (1989), New York, NY, USA: Springer, New York, NY, USA · Zbl 0676.58017  Rabinowitz, P. H., Variational methods for Hamiltonian systems, Handbook of Dynamical Systems, 1, part 1, 1091-1127 (2002) · Zbl 1048.37055  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  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  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  Ding, Y.; Jeanjean, L., Homoclinic orbits for a nonperiodic Hamiltonian system, Journal of Differential Equations, 237, 2, 473-490 (2007) · Zbl 1117.37032  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  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  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  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  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  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  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  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  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  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  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  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  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  Korman, P.; Lazer, A. C., Homoclinic orbits for a class of symmetric Hamiltonian systems, Electronic Journal of Differential Equations, 1994, 1-10 (1994)  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  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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.