Abstract
We study the existence of homoclinic solutions for the second order Hamil- tonian system ¨ u+ Vu(t, u) = f(t). Let V (t, u) = −K(t, u)+ W(t, u) ∈ C 1 (R × R n , R) be T-periodic in t, where K is a quadratic growth function and W may be asymptotically quadratic or super-quadratic at infinity. One homoclinic solution is obtained as a limit of solutions of a sequence of periodic second order differential equations.
Highlights
Introduction and the main resultIn this paper, we consider the existence of homoclinic solutions for the second order Hamiltonian system: u(t) + Vu(t, u(t)) = f (t), (1.1)where f ∈ L2(R, Rn) is continuous and bounded, V (t, u) = −K(t, u) + W (t, u) ∈ C1(R × Rn, R) is T -periodic in t, T > 0.Let us recall that a solution u(t) of (1.1) is homoclinic to 0 if u(t) ≡ 0, u(t) → 0 and u (t) → 0 as t → ±∞.In recent years, the existence of homoclinic solutions for (1.1) has been studied extensively by variational methods(see, for instance, [6,7,11,12,13,15] )
We consider the existence of homoclinic solutions for the second order Hamiltonian system: u(t) + Vu(t, u(t)) = f (t), (1.1)
It is well known that the major difficulty is to check the Palais-Smale (P S) condition when one considers (1.1) on the whole space R via variational methods
Summary
Most of them considered (1.1) with W satisfying the Ambrosetti-Rabinowitz condition, that is, there exists μ > 2 such that 0 < μW (t, u) ≤ (Wu(t, u), u) for all t ∈ R and u ∈ Rn \ {0}. W (u) being independent of t, i.e., the system is autonomous They obtained one homoclinic solution as a limit of solutions of a certain sequence of periodic systems. By this method, [5] considered the case that L(t) and W (t, u) are periodic in t. It assumed that L(t) is positive definite and symmetric, W (t, u) satisfies the Ambrosetti-Rabinowitz growth condition. Given W satisfied the Ambrosetti-Rabinowitz condition, they obtained one homoclinic solution. Vu(t, u) → 0 as |u| → 0 uniformly in t ∈ R; (A2) For all (t, u) ∈ R × Rn, 0 ≤ (u, Ku(t, u)) ≤ 2K(t, u), and there exist 0 < b ≤ b such that b|u|2 ≤ K(t, u) ≤ b|u|2; (A3) There exist constants d1 > 0, r ≥ 2 such that W (t, u) ≤ d1|u|r for all (t, u) ∈ R × Rn; (A4) There exist constants d2 > 0, μ with r ≥ μ > r − 1 and β ∈ L1(R, R+) such that (Wu(t, u), u) − 2W (t, u) ≥ d2|u|μ − β(t) for all (t, u) ∈ R × Rn;
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Electronic Journal of Qualitative Theory of Differential Equations
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
