Nonlinear Differential Equations, Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 223–235 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

Multichain-type solutions for Hamiltonian systems ^{∗}

Paul H. Rabinowitz & Vittorio Coti Zelati

Dedicated to Alan Lazer on his 60th birthday

Abstract

The existence of basic and more complicated multichain heteroclinic solutions is established for a class of forced slowly oscillating Hamiltonian systems. Constrained minimization arguments are the key tool in obtaining the results.

1 Introduction

Consider the Hamiltonian system

q̈ + V_{q}(t, q) = 0 ,  (1.1)

where q = (q_{1}, …, q_{n}) ∈ ℝ^{n} and V satisfies

(V_{1}) V ∈ C^{2}(ℝ × ℝ^{n}, ℝ), is 1-periodic in t and 1-periodic in q_{i}, 1 ≤ i ≤ n;

(V_{2}) V (t, 0) = 0 > V (t, x) with x ∈ ℝ^{n}\ℤ^{n}.

This system was studied by Strobel [18] who proved the following:

(a) For each ξ ∈ ℤ^{n}, there is an η ∈ ℤ^{n}\{ξ} and a solution Q of (1.1) heteroclinic from ξ to η, i.e. Q(−∞) = ξ and Q(∞) = η

(b) For each ξ ≠ η ∈ ℤ^{n}, there is a heteroclinic chain of solutions of (1.1) joining ξ and η, i.e. there exist ξ_{0} = ξ, ξ_{1}, …, ξ_{k} = η and solutions Q_{i} of (1.1) heteroclinic from ξ_{i−1} to ξ_{i}, 1 ≤ i ≤ k.

∗Mathematics Subject Classifications: 34C37, 37J45, 37J50, 58E99. Key words: multichain solutions, Hamiltonian systems, minimization methods. \begin{matrix}c \\ ◯ \end{matrix}2000 Southwest Texas State University. Published October 25, 2000. (V.C.Z.) was supported by MURST, Programma di richerca scientifica di interesse nazionale “Methode Variazionali e Equazioni Differenziali Nonlineari” (P.H.R.) was supported by the NSF under grant MCS8110556

Earlier versions of (a) and (b) when V = V (x) were obtained in [14, 16, 8]. More recently, Bertotti and Montecchiari [7] have treated (1.1) where V (t, x) = a(t)W (x) with W satisfying (V_{1})–(V_{2}) and a almost periodic in t. They also find infinitely many heteroclinic solutions of (1.1) but without a nondegeneracy condition as in [18], they cannot make as precise existence statements as [18].

In his setting, under a further nondegeneracy condition involving the functions Q_{i} in (b), Strobel proved that in fact there exist infinitely many solutions of (1.1) heteroclinic from ξ to η which are near the chain Q_{1}, …, Q_{k} and are distinguished by the amount of time they spend near Q_{1}(∞), …, Q_{k−1}(∞).

In this paper, results related to [18] will be proved for two classes of potentials that are of a more restricted form than V (t, x), namely a(t)W (x). However a(t) is not necessarily periodic in t and unlike [18], no nondegeneracy conditions will be required. The function W satisfies the time independent version of (V_{1})– (V_{2}):

(W_{1}) W ∈ C^{2}(ℝ^{n}, ℝ) is 1-periodic in q_{i}, and 1 ≤ i ≤ n;

(W_{2}) W (0) = 0 > W (x) with x ∈ ℝ^{n}\ℤ^{n}.

For the first class of potentials, roughly speaking, a(t) is nearly constant near a sequence of its local maxima and minima which are sufficiently far apart. This will be made precise in §2. For example if a(t) is 1-periodic, continuous, positive, and non-constant, for all small ϵ > 0, a(ϵt)W (x) will be an allowable potential. A second class of potentials are of the form (α_{1}(ϵt) + α_{2}(t))W (x) where α_{1}, α_{2} are e.g. each like the a just described.

Bolotin and MacKay [10] have recently studied multichain type solutions for a class of slowly oscillating problems in a setting that is more general than ours in some ways but less general in particular in t dependence. Their approach involves a mixture of analytical and minimization arguments. In very recent work, Alessio, Bertotti, and Montecchiari [4] studied a generalization of [7] and also showed that by perturbing such a situation by a term of the form α(ϵt)W (x) with α almost periodic and ϵ small, they get solutions of multichain type. Although there is some intersection with this paper, the point of view taken here is quite different from that of [4]. For other related results in a small perturbation setting, see Ambrosetti and Badiale [1], Ambrosetti and Berti [3], Berti [5], Berti and Bolle [6].

It is also worth noting that there has been a considerable amount of work in a PDE setting on standing wave solutions for nonlinear Schrödinger equations which have slowly oscillating spacially dependent potentials. See e.g. Floer and Weinstein [12], Oh [13], Thandi [19], del Pino and Felmer [11], and Ambrosetti, Badiale, and Cingolani [2] to mention a few.

In §2, the existence of basic heteroclinic solutions will be established. The existence of heteroclinics near finite chains of basic solutions will be given in §3. Some simple observations then yield the case of solutions of infinite chain type. The proofs involve elementary minimization and comparison arguments.

2 Basic heteroclinic solutions

In this section, the existence of basic heteroclinic orbits will be established. To begin, let

A = \{a ∈ C(ℝ, ℝ) | 0 < \underline{a} ≤ a(t) ≤ \overline{a} < ∞\}

where \underline{a} < \overline{a}. Our first goal is to find a solution of (1.1) heteroclinic from 0 to some ξ ∈ ℝ^{n}\{0}. Choose r > 0 which is small compared to 1 ≡ inf{|ξ_{i} − ξ_{j}| | ξ_{i} ≠ ξ_{j} ∈ ℤ^{n}}, i.e. r ≪ 1. A further condition will be imposed on r later. Let B_{r}(z) denote an open ball of radius r about z ∈ ℝ^{n}. Let b_{1} < b_{2} − 1. A heteroclinic solution of (1.1) will be obtained such that the transition between the end states occurs mainly in [b_{1}, b_{2}]. Define

Γ = Γ(b_{1}, b_{2}) = \{q ∈ W_{loc}^{1,2}(ℝ, ℝ^{n}) : q(t) ∈ \overline{B}_{r}(0), t ≤ b_{1},
and q(t) ∈ \overline{B}_{r}(ξ) for some ξ ∈ ℤ^{n}\\{0\}, t ≥ b_{2}\}.

Set

L(q) = \frac{1}{2}|q̇|^{2} − a(t)W (q),

the Lagrangian for (1.1), and define the associated functional

I(q) = ∫_{ℝ} L(q)dt.

Finally define

c = c(b_{1}, b_{2}) = inf_{q∈Γ} I(q) .  (2.1)

Proposition 2.1 If a ∈ A and W satisfies (W_{1})–(W_{2}), there is a Q ∈ Γ(b_{1}, b_{2}) such that I(q) = c(b_{1}, b_{2}).

Proof: Let (q_{m}) be a minimizing sequence for (2.1). Then the form of I and Γ imply (q_{m}) is bounded in W_{loc}^{1,2} and converges weakly in W_{loc}^{1,2} and strongly in L_{loc}^{∞} to Q ∈ Γ(b_{1}, b_{2}). Moreover standard weak lower semicontinuity arguments imply I(Q) = c(b_{1}, b_{2}).

Remark 2.2 (i) As in [14], Q(−∞) = 0 and Q(∞) ∈ ℤ^{n}\{0}.

(ii) Standard regularity arguments show Q is a solution of (1.1) for t ∈ (b_{1}, b_{2}) and also for those values of t ≤ b_{1}, t ≥ b_{2} when Q(t) ∉ ∂B_{r}(0), Q(t) ∉ ∂B_{r}(Q(∞)) respectively.

It remains to choose a subfamily A^{∗} ⊂ A for which a ∈ A^{∗} implies Q is a solution of (1.1). First a few observations about Q are necessary. Suppose Q(∞) = ξ.

Lemma 2.3 For 0 < ρ < r, there is an ω = ω(ρ) > 0 and t_{1} = t_{1}(ρ) ∈ [b_{1} − ω, b_{1}] such that Q(t_{1}) ∈ \overline{B}_{ρ}(0). Moreover ω can be chosen independently of a ∈ A.

Proof: Since Q(−∞) = 0, Q(t) ∈ B_{ρ}(0) for t near −∞. The point is to find ω independently of a ∈ A. Let η ∈ ℤ^{n}\{0} and define

R(t) = \{ \begin{matrix}0 & if t ≤ b_{1}, \\ (t − b_{1})η & if b_{1} ≤ t ≤ b_{1} + 1, \\ η & if t ≥ b_{1} + 1. \end{matrix}  (2.2)

Then R belongs to Γ, so

c = I(Q) ≤ I(R) ≡ M .  (2.3)

Set

β(ρ) = inf_{|x−ℤ^{n}|≥ρ} −W (x) .  (2.4)

By (W_{1})–(W_{2}), β(ρ) > 0. If |Q(t)| > ρ in [b_{1} − ω, b_{1}], by (2.3)–(2.4),

M ≥ I(φ) ≥ ∫_{b_{1}−ω}^{b_{1}} −a(t)W (Q)dt ≥ \underline{a}β(ρ)ω .

Thus the Lemma holds for any ω > M(\underline{a}β(ρ))^{−1}.

Corollary 2.4 There is a t_{2} = t_{2}(ρ) ∈ [b_{2}, b_{2} + ω] such that Q(b_{2}) ∈ \overline{B}_{ρ}(ξ).

Proof: As in Lemma 2.3. After obtaining t_{1}, we define

P (t) = \{ \begin{matrix}0 & if t ≤ t_{1} − 1, \\ (t − (t_{1} − 1))Q(t_{1}) & if t_{1} − 1 ≤ t ≤ t_{1}, \\ Q(t) & if t ≥ t_{1}. \end{matrix}  (2.5)

Then P ∈ Γ(b_{1}, b_{2}) so I(Q) ≤ I(P) and in particular by (2.5),

∫_{−∞}^{t_{1}} L(Q)dt ≤ ∫_{−∞}^{t_{1}} L(P)dt = ∫_{t_{1}−1}^{t_{1}} L(P)dt ≡ φ(ρ)  (2.6)

and the definition of φ(ρ) shows φ(ρ) → 0 as ρ → 0. Similarly, it can be assumed that

∫_{t_{2}}^{∞} L(Q)dt ≤ φ(ρ).

Lemma 2.5 For ρ ≪ r, Q(t) ∈ B_{r/2}(0) for t ≤ t_{1} and Q(t) ∈ B_{r/2}(ξ) for t ≥ t_{2}.

Proof: The first assertion will be proved. If it is not valid, Q(s) ∈ ∂B_{r/2}(0) for some s < t_{1}. By Lemma 2.3, Q(t_{1}) ∈ \overline{B}_{ρ}(0). For ρ ≪ r, the cost of Q going from ∂B_{r/2}(0) to ∂B_{ρ}(0), as measured by I, exceeds that of going from 0 to ∂B_{ρ}(0) as in (2.5)–(2.6). Since Q minimizes I in Γ, the Lemma follows.

Lemma 2.6 There is an s_{1} ∈ [b_{1}, b_{1} + ω] and s_{2} ∈ [b_{2} − ω, b_{2}] such that Q(s_{1}), Q(s_{2}) ∈ \overline{B}_{ρ}(0) ∪ \overline{B}_{ρ}(ξ).

Proof: The proof of Lemma 2.3 shows there exists s_{i} with Q(s_{i}) ∈ \overline{B}_{ρ}(x_{i}) for some x_{i} ∈ ℤ^{n}, i = 1, 2. Thus the possibility that x_{i} ∉ {0, ξ} must be excluded. Since Q(−∞) = 0, Q(b_{2}) ∈ \overline{B}_{r}(ξ), r ≪ 1, and Q minimizes I in Γ_{1}, simple comparison arguments in the spirit of Lemma 2.5 show x_{i} = 0 or ξ, i = 1, 2.

To show that Q is a solution of (1.1), further conditions will have to be imposed on a ∈ A:

(a_{1}) there is a T > 0 and a sequence of points (m_{i})_{i∈ℤ} ⊂ ℝ such that m_{i+1} − m_{i} ≥ T

(a_{2}) there is a γ > 0 and θ_{i} ∈ (2ω, m_{i} − m_{i−1} − 2ω), such that for all i ∈ ℤ, where

(i) a(t) − a(s) ≥ γ, t ∈ [m_{i} − ω, m_{i} + ω], s ∈ [m_{i} − θ_{i} − ω, m_{i} − θ_{i} + ω].

(ii) a(t)−a(s) ≥ γ, t ∈ [m_{i} −ω, m_{i} +ω], s ∈ [m_{i} +θ_{i+1} −ω, m_{i} +θ_{i+1} +ω].

Define

A^{∗} = {a ∈ A : (a_{1}) and (a_{2}) hold} .

Conditions (a_{1})–(a_{2}) are satisfied if e.g. a is T periodic in t, with T appropriately large, m_{i+1} = m_{i} + T, a(m_{i}) = max a, θ_{i+1} = θ_{i} + T, a(m_{i} + θ_{i}) = min a, γ = \frac{1}{2}(a(m_{i}) − a(m_{i} + θ_{i})) and a oscillates slowly so (a_{2}) holds. More generally, it suffices that a remains near its maximum and minimum on a large time interval. In particular, as mentioned in the Introduction, these conditions will be satisfied if a(t) = b(ϵt) with b positive, continuous, 1-periodic in t, and ≢ constant, and ϵ sufficiently small. Suppose further

φ(ρ) < \frac{γ}{32M} (\underline{a}/\overline{a})β(r) .  (2.7)

Choosing (b_{1}, b_{2}) = (m_{i}, m_{i+1}), we have

Theorem 2.7 Suppose (W_{1})–(W_{2}) hold, ρ and r satisfy ρ ≪ r ≪ 1 and (2.7), and a ∈ A^{∗}. Then for each i ∈ ℤ, (1.1) has a solution Q = Q_{i} ∈ Γ(m_{i}, m_{i+1}) with I(Q_{i}) = c(m_{i}, m_{i+1}).

Proof: Since it does not effect the argument, for notational simplicity, we set i = 1. By Remark 2.2 and Lemma 2.5, Q is a solution of (1.1) except possibly for t ∈ (t_{1}, m_{1}] ∪ [m_{2}, t_{2}). Suppose e.g. Q(t) ∈ ∂B_{r}(0) for some t ∈ (t_{1}, m_{1}]. Then the cost analysis of Lemma 2.5 shows, Q(s_{1}) ∈ \overline{B}_{ρ}(ξ), Q(t) ∈ B\frac{r}{2} (ξ) for t ≥ s_{1}, and

∫_{s_{1}}^{∞} L(Q)dt ≤ φ(ρ) .

Therefore, Q^{∗}(t) = Q(t−τ) ∈ Γ for any τ ∈ [0, m_{2}−s_{1}]. Since θ_{2} < m_{2}−m_{1}−2ω and s_{1} < m_{1} + ω, θ_{2} < m_{2} − s_{1} − ω < m_{2} − s_{1} so taking τ = θ_{2} shows

0 ≥ I(Q) − I(Q^{∗}) = − ∫_{ℝ}(a(t) − a(t + θ_{2}))W (Q) dt .  (2.8)

Now

| ∫_{−∞}^{t_{1}}(a(t) − a(t + θ_{2}))W (Q)dt| ≤ 2\frac{\overline{a}}{\underline{a}} ∫_{−∞}^{t_{1}} L(Q)dt ≤ 2\frac{\overline{a}}{\underline{a}}φ(ρ)  (2.9)

and similarly

| ∫_{s_{1}}^{∞}(a(t) − a(t + θ_{2}))W (Q)dt| ≤ 2\frac{\overline{a}}{\underline{a}} ∫_{s_{1}}^{∞} L(Q)dt ≤ 2\frac{\overline{a}}{\underline{a}}φ(ρ)  (2.10)

while by (a_{2})(ii),

− ∫_{t_{1}}^{s_{1}}(a(t) − a(t + θ_{2}))W (Q)dt ≥ γ ∫_{t_{1}}^{s_{1}} W (Q) dt .  (2.11)

In the interval [t_{1}, s_{1}], Q goes from ∂B_{ρ}(0) to ∂B_{ρ}(ξ). In particular, since Q minimizes (2.1), there is a subinterval [σ, s] of [t_{1}, s_{1}] in which Q lies in ℝ^{n}\B_{r}(ℤ^{n}) and joins ∂B_{r}(0) to ∂B_{r}(ξ). Hence by the definition of M,

\frac{1}{2} ≤ |Q(s) − Q(σ)| = | ∫_{σ}^{s} Q̇(t)dt|
≤ (s − σ)^{1/2}( ∫_{σ}^{s} |Q̇|^{2}dt)^{1/2}
≤ (s − σ)^{1/2}(2I(Q))^{1/2}
≤ (2M(s − σ))^{1/2}

so that s − σ ≥ 1/(8M), and

− ∫_{t_{1}}^{s_{1}} W (Q)dt ≥ − ∫_{σ}^{s} W (Q)dt ≥ \frac{1}{8M} β(r) .

Combining (2.9)–(2.11) and the above equation yields

0 ≥ γ \frac{1}{8M} β(r) − 4\frac{\overline{a}}{\underline{a}}φ(ρ)

contrary to (2.7). Hence it is not possible that Q(t) ∈ ∂B_{r}(0) for t ∈ (t_{1}, m_{1}]. Similarly using (a_{2}) (i), Q(t) ∈ ∂B_{r}(ξ) for t ∈ [m_{2}, t_{2}) cannot occur. Thus Q is a solution of (1.1) and Theorem 2.7 is proved.

Remark 2.8 (i) Replacing Q_{i} + j for j ∈ ℤ^{n} gives a solution of (1.1) in Γ(m_{i}, m_{i+1}) heteroclinic from j to j + ξ.

(ii) Possibly Q_{i}(∞) ≠ Q_{i−1}(∞).

(iii) Although Q_{i} need not be unique, when a ∈ A^{∗} is T -periodic, one choice for Q_{i−1}(t) is Q_{i}(t − T).

Remark 2.9 Modifying slightly arguments as in [8, 15, 9] gives at least n + 1 distinct points ξ_{0} ≡ ξ, ξ_{1}, …, ξ_{n} ∈ ℤ^{n}\{0} and corresponding heteroclinic solutions Q_{i}^{0} ≡ Q_{i}, Q_{i}^{1}, …, Q_{i}^{n} ∈ Γ(m_{i}, m_{i+1}) provided that (2.7) is strengthened. E.g. once Q_{i}^{0}, …, Q_{i}^{ℓ−1} have been found, Q_{i} is the minimizer of the variational problem

inf_{q∈Γ_{ℓ}(m_{i},m_{i+1})} I(q)

where

Γ_{ℓ}(m_{i}, m_{i+1}) = {q ∈ Γ(m_{i}, m_{i+1}) | q(∞) ∉ span_{ℕ}{ξ_{0}, …, ξ_{ℓ−1}}

and span_{ℕ}X denotes the span with coefficients in ℕ ∪ {0} of elements in X. Moreover Q_{i}^{ℓ} is a solution of (1.1), 0 ≤ i ≤ n as in Theorem 2.7 provided that (2.7) is replaced by

φ(ρ) < \frac{γ}{32M^{∗}} (\underline{a}/\overline{a})β(r) ,  (2.12)

where M^{∗} is defined as follows. Let e.g. e_{1}, …, e_{n} be the usual basis in ℝ^{n}, i.e. e_{1} = (1, 0, …, 0), etc. Set e_{n+1} = (−1, …, −1). Replace η in (2.2) by e_{i}, calling the resulting function R_{i}. Then at least one of R_{1}(1), …, R_{ℓ+1}(1) ∉ span_{ℕ}{ξ_{0}, …, ξ_{ℓ−1}}. Set

M^{∗} = max_{1≤i≤n+1} I(R_{i}) ..

Remark 2.10 As mentioned in the Introduction, the conclusions of Theorem 2.7 hold for a more general class of a’s than A^{∗}. Rather than formalizing such a result, we just give an example of this type. Suppose a = α_{1} + α_{2} where α_{1} ∈ A^{∗} and α_{2} ≥ 0 is continuous and periodic with period p ≤ 1 which for convenience will be taken to be 1. (Some small modifications in the argument that follows are needed if p < 1.) It can be assumed that ω >> 1. Let μ denote the greatest integer in θ_{2}, μ = [θ_{2}], so 0 ≤ θ_{2} − μ < 1 ≪ ω. Now in the proof of Theorem 2.7, choose τ = μ. Since μ ≤ θ_{2} < m_{2} − s_{1} − ω + 1 < m_{2} − s_{1}, Q^{∗} ∈ Γ as earlier so (2.8) becomes

0 ≥ − ∫_{ℝ}(α_{1}(t) − α_{1}(t + μ))W (Q) dt − ∫_{ℝ}(α_{2}(t) − α_{2}(t + μ))W (Q) dt .

By the 1-periodicity of α_{2}, the second integral on the right vanishes so the earlier argument can be used again to get existence here. The multiplicity results of Remark 2.9 are also valid for this more general class of a’s.

3 Solutions of multichain type

Consider a heteroclinic ℓ-chain constructed by gluing together ℓ basic heteroclinics or their translates as obtained in Remarks 2.9 and 2.8(i). Suppose the chain begins at ξ_{0} and ends at ξ_{ℓ}. The goal of this section is to show there are infinitely many heteroclinic solutions of (1.1) that spend as much time as desired near ξ_{1}, …, ξ_{ℓ−1}. To be more precise, let r, ρ, and A^{∗} be as in §2. Let k ∈ ℤ^{2ℓ} where k = (k_{1}, …, k_{2ℓ}), k_{j} < k_{j+1}, and k_{j} = m_{i_{j}} for some i_{j} where (m_{i}) is as in (a_{1}). Define

Γ_{k} = \{q ∈ W_{loc}^{1,2} | q(t) ∈ \overline{B}_{r}(ξ_{0}), t ≤ k_{1}, q(t) ∈ \overline{B}_{r}(ξ_{j}), t ∈ [k_{2j}, k_{2j+1}],
1 ≤ j ≤ ℓ − 1, and q(t) ∈ \overline{B}_{r}(ξ_{ℓ}), t ≥ k_{2ℓ}\}

Set

c_{k} = inf_{q∈Γ_{k}} I(q) .  (3.1)

Repeating arguments from §2 gives

Proposition 3.1 1. There exists Q = Q_{k} ∈ Γ_{k} such that I(Q_{k}) = c_{k}.

2. There are numbers t_{1} ∈ [k_{1} − ω, k_{1}], t_{2j} ∈ [k_{2j}, k_{2j} + ω], t_{2j+1} ∈ [k_{2j+1} − ω, k_{2j+1}], 1 ≤ j ≤ ℓ − 1, t_{2ℓ} ∈ [k_{2ℓ}, k_{2ℓ} + ω] such that Q(t_{1}) ∈ \overline{B}_{ρ}(ξ_{0}), Q(t_{2j}), Q(t_{2j+1}) ∈ \overline{B}_{ρ}(ξ_{j}), Q(t_{2ℓ}) ∈ \overline{B}_{ρ}(ξ_{ℓ}).

3. There is a φ(ρ) → 0 as ρ → 0 such that

∫_{−∞}^{t_{1}} L(Q)dt, ∫_{t_{2ℓ}}^{∞} L(Q)dt ≤ φ(ρ)

and similarly,

∫_{t_{2j}}^{t_{2j+1}} L(Q)dt ≤ φ(ρ), 1 ≤ j ≤ ℓ − 1

4. Q(t) ∈ B_{r/2}(ξ_{j}), t ≤ t_{1}, Q(t) ∈ B_{r/2}(ξ_{j}), t ∈ [t_{2j}, t_{2j+1}], Q(t) ∈ B_{r/2}(ξ_{ℓ}), t ≥ t_{2ℓ}.

5. There is an s_{1} ∈ [k_{1}, k_{1} + ω], s_{2j} ∈ [k_{2j} − ω, k], s_{2j+1} ∈ [k_{2j+1}, k_{2j+1} + ω], 1 ≤ j ≤ ℓ − 1, s_{2ℓ} ∈ [k_{2ℓ} − ω, k_{2ℓ}] such that Q(s_{1}), Q(s_{2}) ∈ \overline{B}_{ρ}(ξ_{0}) ∪ B_{ρ}(ξ_{1}), ⋯ Q(s_{2ℓ−1}), Q(s_{2ℓ}) ∈ \overline{B}_{ρ}(ξ_{ℓ−1}) ∪ \overline{B}_{ρ}(ξ_{ℓ}).

The characterization of Q_{k} as a minimum in (3.1) implies there is an \overline{M} > 0 and independent of ℓ such that

∫_{k_{2j−1}}^{k_{2j}} L(Q_{k})dt ≤ \overline{M}, 1 ≤ j ≤ ℓ  (3.2)

Replacing (2.12) by

φ(ρ) < \frac{γ}{80\overline{M}} β(r),  (3.3)

we have

Theorem 3.2 If (W_{1})–(W_{2}) hold, ρ ≪ r ≪ 1, (3.3) is satisfied, and a ∈ A^{∗}, then (1.1) has a solution, Q_{k} ∈ Γ_{k} with I(Q_{k}) = c_{k}.

Proof: As earlier, it suffices to show Q(t) = Q_{k}(t) ∉ ∂B_{r}(ξ_{0}), t ≤ k_{1}; Q(t) ∉ ∂B_{r}(ξ_{j}), t ∈ [k_{2j}, k_{2j+1}], 1 ≤ j ≤ ℓ − 1; Q(t) ∉ ∂B_{r}(ξ_{ℓ}), t ≥ k_{2ℓ}. The idea is to show if one of these conditions is violated, it is possible to construct an appropriate Q^{∗} ∈ Γ_{k} and obtain a contradiction as in §2. There are basically two cases to consider.

Suppose first that Q(t) ∈ ∂B_{r}(ξ_{0}) for some t ∈ (t_{1}, k_{1}]. Then as in the proof of Theorem 2.7, Q(s_{1}) ∈ \overline{B}_{ρ}(ξ_{i}), Q(t) ∈ B_{r/2}(ξ_{1}) for t ∈ [s_{1}, t_{2}], and

∫_{s_{1}}^{t_{2}} L(Q)dt ≤ φ(ρ) .  (3.4)

Set

Q^{∗}(t) = \{ \begin{matrix}Q(t − k_{1})  if t ≤ s_{1} + θ_{k_{1}+1}, \\ (s_{1} + θ_{k_{1}+1} + 1 − t)Q(s_{1}) + (t − (s_{1} + θ_{k_{1}+1}))ξ_{1}, \\ if s_{1} + θ_{k_{1}+1} ≤ t ≤ s_{1} + θ_{k_{1}+1} + 1 \\ (s_{1} + θ_{k_{1}+1} + 2 − t)ξ_{1} + (t − (s_{1} + θ_{k_{1}+1} + 1))Q(s_{1} + θ_{k_{1}+1} + 2), \\ if s_{1} + θ_{k_{1}+1} + 1 ≤ t ≤ s_{1} + θ_{k_{1}+1} + 2 \\ Q(t),  if t ≥ s_{1} + θ_{k_{1}+1} + 2. \end{matrix}

(3.5)

Then Q^{∗} ∈ Γ_{k} and

0 ≥ I(Q) − I(Q^{∗}) = ∫_{−∞}^{s_{1}} L(Q)dt − ∫_{−∞}^{s_{1}+θ_{k_{1}+1}+1} L(Q^{∗}) dt
+ ∫_{s_{1}}^{s_{1}+θ_{k_{1}+1}+2} L(Q)dt − ∫_{s_{1}+θ_{k_{1}+1}}^{s_{1}+θ_{k_{1}+1}+2} L(Q^{∗}) dt .

By (3.4), each of the last two terms in this inequality is less than or equal to φ(ρ). Therefore,

0 ≥ − ∫_{−∞}^{s_{1}}(a(t) − a(t + θ_{k_{1}+1}))W (Q)dt − 2φ(ρ).

As in §2, this leads to

0 ≥ γ \frac{1}{8\overline{M}} β(r) − 3φ(ρ)

contrary to (3.3).

Using (a_{2})(i), a similar argument holds if Q(t) ∈ ∂B_{r}(ξ_{2ℓ}) for some t ∈ [k_{2ℓ}, t_{2ℓ}). If Q(t) ∈ ∂B_{r}(ξ_{j}) for some t ∈ [k_{2j}, k_{2j+1}], then by Proposition 3.1, either t ∈ [k_{2j}, t_{2j}) or t ∈ (t_{2j+1}, k_{2j+1}]. The argument is similar in either event, so suppose t ∈ [k_{2j}, t_{2j}). Then Q(s_{2j}) ∈ \overline{B}_{ρ}(ξ_{j−1}) and Q(t) ∈ B_{r/2}(ξ_{j−1}) for t ∈ [t_{2j−1}, s_{2j}]. It is now convenient to use two comparison functions. Define

Q̃(t) = \{ \begin{matrix}Q(t) & if t ≤ t_{2j−1} \\ ξ_{j−1} & if t_{2j−1} + 1 ≤ t ≤ s_{2j} − 1 \\ Q(t) & if s_{2j} ≤ t ≤ t_{2j} \\ ξ_{j} & if t_{2j+1} ≤ t ≤ t_{2j+1} − 1 \\ Q(t) & if t ≥ t_{2j} \end{matrix}

with a linear interpolant, as in (3.5) for the four intermediate intervals. Then Q̃ ∈ Γ_{k} and

0 ≤ I(Q̃) − I(Q)
= ∫_{t_{2j−1}}^{t_{2j−1}+1} L(Q̃)dt + ∫_{s_{2j−1}}^{s_{2j}} L(Q̃) dt + ∫_{t_{2j}}^{t_{2j+1}} L(Q̃) dt

+ ∫_{t_{2j+1}−1}^{t_{2j+1}} L(Q̃) dt − ∫_{t_{2j−1}}^{s_{2j}} L(Q) dt − ∫_{t_{2j}}^{t_{2j+1}} L(Q) dt .

Each of the terms on the right-hand side of this inequality is less than or equal to φ(ρ) so

0 ≤ I(Q̃) − I(Q) ≤ 6φ(ρ) .  (3.6)

Now define

Q^{∗}(t) = \{ \begin{matrix}Q̃(t) & if t ≤ t_{2j} + 1 − θ_{j} \\ Q̃(t + θ_{j}) & if t_{2j+1} + 1 − θ_{j} ≤ t ≤ t_{2j} + 1 \\ Q̃(t) & if t ≥ t_{2j} + 1 . \end{matrix}

Again Q^{∗} ∈ Γ_{k} and

0 ≤ I(Q^{∗}) − I(Q) = I(Q^{∗}) − I(Q̃) + I(Q̃) − I(Q) .  (3.7)

Hence by (3.13),

I(Q̃) − I(Q^{∗}) ≤ I(Q̃) − I(Q) ≤ 6φ(ρ) .  (3.8)

But by the definition of Q^{∗} and Q̃,

I(Q̃) − I(Q^{∗}) = ∫_{t_{2j−1}+1}^{t_{2j}+1}(L(Q̃) − L(Q^{∗}))dt

= − ∫_{s_{2j−1}}^{t_{2j+1}}(a(t) − a(t − θ_{j}))W (Q)dt  (3.9)

≥ − ∫_{s_{2j}}^{t_{2j}}(a(t) − a(t − θ_{j}))W (Q)dt − 4φ(ρ)
≥ \frac{γ}{8\overline{M}} β(r) − 4φ(ρ).

Combining (3.7)–(3.9) shows

\frac{γβ(r)}{8\overline{M}} ≤ 10φ(ρ)

contrary to (3.3). The proof is complete.

Remark 3.3 By choosing k appropriately, the solution, Q_{k}, of (1.1) is near each of the equilibrium points ξ_{1}, …, ξ_{ℓ−1} for as long a time interval as desired.

However Q_{k} need not be near the original heteroclinic chain joining 0 and ξ_{ℓ}, i.e. Q_{k}|_{k_{2j−1}}^{k_{2j}} is not necessarily near any basic heteroclinic joining ξ_{j−1} and ξ_{j}.

Nevertheless, a Q_{k}|_{k_{2j−1}}^{k_{2j}} near such P_{j} can be constructed by taking k_{2j} − k_{2j−1}

sufficiently large as in [17]. Indeed (3.2) implies an L^{∞} upper bound for Q_{k}|_{k_{2j−1}}^{k_{2j}} independent of k_{2j} − k_{2j−1} and (1.1) then yields such a bound in C^{2}. As k_{2j} − k_{2j−1} → ∞, by standard arguments as in [17], Q_{k}|_{k_{2j−1}}^{k_{2j}} approaches a chain of heteroclinic H, …, H_{s} joining ξ_{j−1} and ξ_{j} with

∑_{1}^{s} I(H_{i}) = I(P_{j}).

The construction of P_{j} as indicated in Remark 2.9 implies s = 1. Hence for k_{2j} − k_{2j−1} large, Q_{k}|_{k_{2j−1}}^{k_{2j}} will be near a basic heteroclinic P_{j} joining ξ_{j−1} and ξ_{j}.

A standard consequence of Theorem 3.2 is the existence of solutions of infinite chain type of (1.1). Consider any formal doubly infinite heteroclinic chain made up of the basic heteroclinics of Remark 2.9. The endpoints of the chain form a sequence Ξ = (ξ_{i})_{i∈ℤ}, ξ_{i} ∈ ℤ^{n}. Let k = (k_{i})_{i∈ℤ} with k_{i} < k_{i+1} and each k_{i} = m_{i_{j}} for some j. Now set

Γ_{k} = \{q ∈ W_{loc}^{1,2} | q(t) ∈ \overline{B}_{r}(ξ_{j}), t ∈ [k_{2j}, k_{2j+1}], j ∈ ℤ\}.

Then we have

Theorem 3.4 Under the hypothesis of Theorem 3.2, for each Ξ, k as above, there is a solution, Q_{k} ∈ Γ_{k}, of (1.1).

Proof: Note that the construction of Theorem 3.2 is independent of ℓ, the number of basic homoclinics. For Ξ and k as above, let Ξ_{ℓ} = (ξ_{−ℓ}, …, ξ_{ℓ}) and K_{ℓ} = (k_{−2ℓ}, …, k_{2ℓ}) ∈ ℤ^{4ℓ}. Then by Theorem 3.2, there is a solution Q_{ℓ} of (1.1) in Γ_{K_{ℓ}}, heteroclinic from ξ_{−ℓ} to ξ_{ℓ}. Since Q_{ℓ} is a solution of (1.1), for each j ∈ ℤ, the form of Γ_{K_{ℓ}} yields C^{2}([k_{2j}, k_{2j+1}], ℝ^{n}) bounds for Q_{ℓ} (independent of ℓ). Moreover in the intervals [k_{2j−1}, k_{2j}], the bound (3.2) holds with Q = Q_{ℓ}. As in Remark 3.3, this gives bounds in C^{2}([k_{2j−1}, k_{2j}], ℝ^{n}) for Q_{ℓ} independent of ℓ. The Arzela-Ascoli Theorem then yields the desired solution Q_{k} of (1.1).

Acknowledgement Coti Zelati’s research was done while he was visiting the University of Wisconsin–Madison. He thanks the faculty and staff of the Mathematics Department for their kind hospitality.

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Paul H. Rabinowitz Department of Mathematics University of Wisconsin-Madison Madison, WI 53706, USA e-mail: rabinowi@math.wisc.edu

Vittorio Coti Zelati Departimento di Matematica Universita di Napoli Napoli, Italy e-mail: zelati@unina.it