Fast homoclinic orbits for a class of damped vibration systems
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Fast homoclinic orbits for a class of damped vibration systems Wafa Selmi1 · Mohsen Timoumi1 Received: 24 June 2020 / Accepted: 3 September 2020 © Università degli Studi di Napoli "Federico II" 2020
Abstract We study the existence of fast homoclinic orbits for the following damped vibration system u(t) ¨ + q(t)u(t) ˙ + ∇V (t, u(t)) = 0; where q ∈ C(R, R) and V ∈ C 1 (R × N R , R) is of the type V(t,x)=-K(t,x)+W(t,x). A map K is not a quadratic form in x and W (t, x) is superquadratic in x. Keywords Critical point · Damped vibration system · Homoclinic orbit Mathematics Subject Classification 34C37 · 35J61 · 58E30
1 Introduction In this paper, we shall study the existence of fast homoclinic solutions for the following damped vibration system u(t) ¨ + q(t)u(t) ˙ + ∇V (t, u(t)) = 0, t ∈ R;
(1)
where V : R × R N → R is a continuous function, differentiable with respect to the second variable with continuous derivative ∇V (t, x) and q : R → R is a continuous t
function such that Q(t) =
q(s)ds → ∞, as |t| → +∞.
0
When q = 0, system (1) reduces to the following second-order Hamiltonian system u(t) ¨ + ∇V (t, u(t)) = 0, t ∈ R.
B
(2)
Wafa Selmi [email protected] Mohsen Timoumi [email protected]
1
Department of Mathematics, Faculty of Sciences of Monastir, Monastir, Tunisia
123
W. Selmi, M. Timoumi
As usual, we say that a solution u of (2) is homoclinic (to 0) if u(t) → 0, u(t) ˙ →0 as |t| → ∞. In addition if u ∓ 0, then u is called nontrivial homoclinic solution. The existence of homoclinic solutions for the Hamiltonian system (2) is one of the most important problems in the history of Hamiltonian systems which has been intensively studied by many mathematicians. If the potential V (t, x) is of the form 1 V (t, x) = − L(t)x.x + W (t, x), 2
(3)
and assuming that L(t) and W (t, x) are either independent of t or T-periodic in t, several authors have studied the existence of homoclinic solutions for Hamiltonian system (2) via critical point theory and variational methods, see [1–5] and the references therein. If L(t) and W (t, x) are neither autonomous nor periodic in t, this problem is quite different from the periodic case because of the lack of compactness of the Sobolev embedding, see [6–16] and the references therein. In [3], Isydorek and Janczewska have studied the existence of homoclinic solutions for (2) when the potential V is of the form V (t, x) = −K (t, x) + W (t, x) − f (t).x,
(4)
where K , W and f are periodic in t. After this work, several authors obtained some interesting results on the existence of homoclinic orbits for system (2) under various suitable assumptions on K , W and f , see [17–21] and the references therein. Let us note that in all these last papers, K , W and f are assumed to be periodic in the first variable. Moreover, they are obtained the existence of such solutions as limit of 2kT -periodic solutions of (2). If q ∓ 0, there exist only a few results concerning the existence of fast homoclinic solutions for (1) (see Definition 2.1) when V is of the type (4), see for
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