Connecting orbits for a periodically forced singular planar Newtonian system

In this paper we are concerned with the study of the existence and multiplicity of connecting orbits for a singular planar Newtonian system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\ddot{q} + V_q(t, q) = 0}$$\end{document} with a periodic strong force Vq(t, q), an infinitely deep well of Gordon's type at one point and two stationary points at which a potential V (t, q) achieves a strict global maximum. To this end we minimize the corresponding actiön functional over the classes of functions in the Sobolev space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${W^{1, 2}_{\rm loc}(\mathbb{R}, \mathbb{R}^2)}$$\end{document} that turn a given number of times around the singularity.


Introduction
In this paper, we study the existence of connecting orbits for a certain class of planar singular Newtonian systems that are periodically forced. Let us consider a systemq + V q (t, q) = 0, q ∈ R 2 , (1.1) with a potential V satisfying the following assumptions: Without loss of generality, we can assume that τ = 1.
The above lemma is a consequence of inequality (1.3) and Corollary 1.3. In the polar coordinate system with the pole ξ and the polar axis {x ∈ R 2 : x = ξ + s · ξa, s ≥ 0} one has q(t) = (r(t) cos ϕ(t), r(t) sin ϕ(t)) for all q ∈ Λ (polar angles are measured counterclockwise from the axis). Moreover, we can assume that r(t) and ϕ(t) are continuous. Definition 1.5. For each q ∈ Λ such that q(±∞) ∈ M, we define the rotation (winding) number rot(q) as follows. If q(−∞) = b and q(∞) = a, then Otherwise, Here [s] denotes the integral part of s ∈ R.
We denote by F the collection of all subsets Z of Λ that satisfy the following three conditions: where n ∈ N. We see at once that these sets belong to F. Set From (iii) it follows that if Z ∈ F and q is a minimizer of I on Z, then q is a weak solution of (1.1). Analysis similar to that in [8] (see the proof of Proposition 3.18, pp. 339-340) shows that q is a classical solution. Moreover, in the same manner as in [4] (see the proof of Lemma 2.9, pp. 387-388) we can see thatq(±∞) = 0.
In the next section we will prove the following result. JFPTA Theorem 1.6 (Main theorem). Suppose that V satisfies (V 1)-(V 5). Then (1.1) possesses at least two nontrivial connecting orbits.
There is always one heteroclinic solution of (1.1). The second solution may be heteroclinic or homoclinic depending on the geometry of V . In particular, there might be no homoclinics at all.
Our work extends [9, Theorem 2.7] for the case #M = 1 to the case #M = 2 and [7, Conclusion 1.5] for an autonomous system to a periodic one. The proof of Theorem 1.6 is based on minimization of I on appropriate sets in F.
Connecting orbits are global in time. For this reason it is natural to apply global methods, in particular variational ones, to obtain them in a direct way, working with a class of functions possessing the qualitative properties sought.
The study of singular Hamiltonian systems seems to be important, because certain potentials arising in physics possess infinitely deep wells. As pointed out by Gordon in [3], it is a little disappointing that the strong force condition excludes gravitational potentials. However, he also wrote "the definition of the strong force condition is well motivated and leads to the disclosure of certain differences between the behaviour of strong force systems and gravitational (or other weak force) systems".

Technicalities
Before we show Theorem 1.6 we state and prove some technical results.
Proof. We will apply the same arguments as in the proof of [  Vol. 12 (2012) Connecting orbits for a singular planar Newtonian system 63 Therefore {q m } m∈N is bounded in L ∞ (R, R 2 ). Consequently, {q m } m∈N is bounded in E.

From now on,
for T, T 1 , T 2 ∈ R and q ∈ Λ.

Remark 2.2.
It is easily seen that the functionals I T2 T1 , I T −∞ , I ∞ T for all T 1 , T 2 , T ∈ R and the functional I are weakly lower semicontinuous.

Lemma 2.3. For each
The proof is immediate and therefore it is left to the reader. If q ∈ Λ has the endpoints q(±∞) in M, then for a time T ∈ R such that q(T ) ∈ B ε (x), where x ∈ M and ε > 0 small enough, we will denote by rot(q| (−∞,T ] ) and rot(q| [T,∞) ) the rotation numbers of appropriate paths in Λ that arise from q| (−∞,T ] and q| [T,∞) resp. by connecting q(T ) to x by a line segment. The proof is similar to that of [7,Theorem 1.3]. However, for the convenience of the reader we present it.
Proof. There is no loss of generality in assuming that and if t < τ m , then |q m (t) − a| < ε. Without loss of generality, we can assume that τ m ∈ [0, 1). By Proposition 2.1 it follows that there is Q ∈ E such that going to a subsequence if necessary q m converges to Q weakly in E and strongly in L ∞ loc (R, R 2 ). From Remark 2.2 we obtain Conversely, suppose that Q(∞) = a. Let δ > 0 be small enough such that 4δ < ε, and there is t δ > 1 such that Q(t δ ) ∈ ∂B δ (a) and where m ≥ m 0 . We see that each Q m joins a to b, 2α ε 2 , and consequently, To finish the proof we have to show that rot(Q) < 0. On the contrary, suppose that rot(Q) ≥ 0. If γ − < γ + , then I(Q) > γ − , contrary to (2.3).
. We can choose T > 1 such that Q([T, ∞)) ⊂ B ε (b) and moreover, Since q m goes to Q almost uniformly on R, there is m 1 ∈ N such that for all m ≥ m 1 , rot(q m | (−∞,T ] ) = rot(Q), and hence rot(q m | [T,∞) ) < − rot(Q) ≤ 0. From Lemma 1.2 it may be concluded that for m ≥ m 1 . Finally, by Remark 2.2, there exists m 2 ∈ N such that for all m ≥ m 2 , By the above, for m ≥ max{m 1 , m 2 } we obtain contrary to (2.2). Thus Q ∈ Γ − , and, in consequence, I(Q) = γ − . Lemma 2.5. Assume that V satisfies (V 1)-(V 5) and Z ∈ F. If {q m } m∈N ⊂ Z is a minimizing sequence of I in Z and q 0 is its weak limit in E, then q 0 is a connecting orbit of (1.1).
Proof. By assumption, From Remark 2.2 and Corollary 1.3 it follows that I(q 0 ) ≤ z and q 0 (±∞) ∈ M. From Lemma 1.4 we deduce that q 0 (t) = ξ for all t ∈ R. It is sufficient to prove that Fix 0 < ε ≤ R. Without restriction of generality, we can assume that for each m ∈ N there exists σ m ∈ [0, 1) such that p m (σ m ) ∈ ∂B ε (a) and if t < σ m , then |p m (t) − a| < ε. From Proposition 2.1 we deduce that there is P ∈ E such that going to a subsequence if necessary p m goes to P weakly in E and strongly in L ∞ loc (R, R 2 ). From Lemma 1.4 it follows that P ∈ Λ. By Remark 2.2 we get I(P ) ≤ γ + , and by Corollary 1.3, P (±∞) ∈ M. Furthermore, P ((−∞, 0]) ⊂ cl B ε (a), which implies P (−∞) = a. According to Lemma 2.5, P is a connecting orbit of (1.1). Two cases are now possible: P (∞) = a or P (∞) = b. If P (∞) = a, then P is a nontrivial homoclinic (to a) orbit of (1.1). We leave it to the reader to verify that rot(P ) > 0. If P (∞) = b and rot(P ) ≥ 0, then P = Q, where Q is a heteroclinic solution of Theorem 2.4. If P (∞) = b and rot(P ) < 0, then it may happen that P = Q. In this case, to receive the second connecting orbit of (1.1), we need the following lemma. Lemma 2.6. If P ∈ Γ − , then γ + = I(P ) +ω n b , where n = − rot(P ).