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Advances in Nonlinear Analysis

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Homoclinics for singular strong force Lagrangian systems

Marek Izydorek
  • Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Narutowicza 11/12, 80-233, Gdańsk, Poland
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/ Joanna Janczewska
  • Corresponding author
  • Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Narutowicza 11/12, 80-233, Gdańsk, Poland
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/ Jean Mawhin
  • Département de mathématique, Université Catholique de Louvain, chemin du cyclotron, 2, B-1348, Louvain-la-Neuve, Belgium
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Published Online: 2019-06-29 | DOI: https://doi.org/10.1515/anona-2020-0018

Abstract

We study the existence of homoclinic solutions for a class of Lagrangian systems ddt(∇Φ((t))) + ∇uV(t, u(t)) = 0, where t ∈ ℝ, Φ : ℝ2 → [0, ∞) is a G-function in the sense of Trudinger, V : ℝ × (ℝ2 ∖ {ξ}) → ℝ is a C1-smooth potential with a single well of infinite depth at a point ξ ∈ ℝ2 ∖ {0} and a unique strict global maximum 0 at the origin. Under a strong force condition around the singular point ξ, via minimization of an action integral, we will prove the existence of at least two geometrically distinct homoclinic solutions u± : ℝ → ℝ2 ∖ {ξ}.

Keywords: homoclinic solution; homotopy class; Lagrangian system; strong force; rotation number (winding number)

MSC 2010: Primary: 37J45 46E30; Secondary: 34C37 70H05

1 Introduction

In this work we will be concerned with the problem of existence of solutions for a class of Lagrangian systems

ddtΦ(u˙(t))+uV(t,u(t))=0,limt±u(t)=limt±u˙(t)=0,(LS)

where t ∈ ℝ, Φ : ℝn → [0, ∞) is a G-function in the sense of Trudinger, and V : ℝ × (ℝn ∖ {ξ}) → ℝ is a C1-smooth potential possessing a single well of infinite depth at a point ξ ∈ ℝn ∖ {0} and a strict global maximum 0 at the origin.

We begin with the notion of G-function. Let a C1-function Φ : ℝn → ℝ satisfy the following conditions:

  • (G1)

    Φ(0) = 0,

  • (G2)

    Φ is coercive, i.e. lim|x|Φ(x)|x|=,

  • (G3)

    Φ is convex, i.e. Φ(ax + (1 − a)y) ≤ (x) + (1 − a)Φ(y) for each a ∈ [0, 1] and all x, y ∈ ℝn,

  • (G4)

    Φ is symmetric, i.e. Φ(x) = Φ(−x) for all x ∈ ℝn,

  • (G5)

    ΦC1(ℝn ∖ {0}, ℝn).

In particular, Φ is a G-function in the sense of Trudinger (compare [1]). Let us recall that the Fenchel transform Φ* of a G-function Φ is the function Φ* : ℝn → ℝ defined by

Φ(y)=supxRn(x,y)Φ(x),

where (⋅, ⋅) : ℝn × ℝn → ℝ is the standard inner product in ℝn (c.f. [2, 3]). It is well known that Φ* is continuous and satisfies (G1)−(G4) (c.f. [4]). Furthermore, Φ** = Φ (c.f. [5]).

Troughout the paper we will assume that Φ and Φ* are globally Δ2-regular [6], i.e. there is a constant L > 0 such that for each x ∈ ℝn,

Φ(2x)LΦ(x)12Φ(Lx).(Δ2)

Given a function Φ we define ϕ : ℝ → ℝ by

ϕ(r)=min{Φ(x);|x|=r}

and ϕ(−r) = ϕ(r). Here ∣ ⋅ ∣ : ℝn → [0, ∞) is the standard norm. Let us recall that the epigraph of a function f : ℝn → ℝ is the set

epi f={(x,t)Rn×R;f(x)t}

(c.f. [2]). We define the supporting function φ : ℝ → ℝ for Φ by the formula:

φ=conv ϕ,

which means that epi φ = conv (epi ϕ). Obviously,

Φ(x)φ(|x|)forxRn.(1)

One can easily check that

  • φ is continuous and satisfies (G1)−(G4), i.e. φ is a G-function;

  • φ satisfies the (Δ2)-condition, i.e. φ and φ* are globally Δ2-regular.

    Our intention is to generalize the following result by Paul H. Rabinowitz from [7] to the Lagrangian systems (LS).

Theorem 1.1

Assume that

  • (V1)

    V : ℝ × (ℝ2 ∖ {ξ}) → ℝ, where ξ ∈ ℝ2 ∖ {0}, is a C1-smooth potential, 1-periodic in t ∈ ℝ and

    limxξV(t,x)=

    uniformly in the time variable t,

  • (V2)

    for all t ∈ ℝ, x ∈ ℝ2 ∖ {0}, V(t, x) ≤ 0 and V(t, x) = 0 iff x = 0,

  • (V3)

    there is a negative constant V0 such that for all t ∈ ℝ,

    lim sup|x|V(t,x)V0,

  • (V4)

    there are a neighbourhood 𝓝 ⊂ℝ2 of the singular point ξ and a function UC1(𝓝 ∖ {ξ}, ℝ) such thatU(x)∣ → ∞ as xξ, and for all x ∈ 𝓝 ∖ {ξ} and t ∈ ℝ,

    |U(x)|2V(t,x).

    Then the problem

    u¨(t)+uV(t,u(t))=0,limt±u(t)=limt±u˙(t)=0(HS)

    has at least two solutions u± : ℝ → ℝ2 ∖ {ξ}, which wind around ξ in opposite directions.

    The proof of Theorem 1.1 in [7] is of variational nature. The basic idea is to take the Lagrangian action corresponding to the problem (HS), defined on the subset of all the functions of the Sobolev space W1,2(ℝ, ℝn) omitting the singularity at a finite time and to minimize this functional both over the subset of functions with a positive winding number around ξ and the subset of functions possessing a negative rotation.

    We are thus led to the following strengthening of Theorem 1.1.

Theorem 1.2

Let Φ : ℝ2 → [0, ∞) satisfy (G1)−(G5) and (Δ2). Assume also that the potential V : ℝ × (ℝ2 ∖ {ξ}) → ℝ satisfies (V1)−(V3), and moreover,

  • (V4′)$\begin{array}{} (V_{4}') \end{array} $

    there are a neighbourhood 𝓝 ⊂ℝ2 of the point ξ and a function UC1(𝓝 ∖ {ξ}, ℝ) such thatU(x)∣ → ∞ as xξ, and for all x ∈ 𝓝 ∖ {ξ} and t ∈ ℝ,

    φ(|U(x)|)V(t,x).

    Then there exist at least two classical solutions u± : ℝ → ℝ2 ∖ {ξ} of the problem (LS) winding around ξ in opposite directions.

    Let us remark that if we substitute Φ(x) = 12x2, x ∈ ℝ2, into (LS) then we obtain (HS). What is more, for Φ(x) = 1pxp, x ∈ ℝ2, p > 1, we have

    ddtΦ(u˙(t))=ddt|u˙(t)|p2u˙(t),

    i.e. the p-Laplacian, and for Φ(x) = χ(∣x∣), where χ : ℝ → ℝ is a so-called N-function (a G-function of one variable with extra growth conditions, c.f. [8]) we obtain a χ-Laplacian. Let us note that φ* in the condition (V4) is the Fenchel transform of the supporting function φ for Φ. Thus φ* depends on Φ. Let us briefly discuss now our assumptions in Theorem 1.2.

    Condition (V4) was introduced by W.B. Gordon in [9] and in the literature it is known as the strong force condition or Gordon’s condition. It governs the rate at which V(x) → −∞ as xξ and holds, for example, if α ≥ 2 for V(x) = −∣xξα nearby ξ. Gordon’s condition excludes the gravitational case and leads to the disclosure between the behaviour of strong force systems and gravitational ones. Condition (V4) is an extension of (V4) to the Lagrangian system (LS). Following Gordon, if V : ℝ × (ℝ2 ∖ {ξ}) → ℝ satisfies (V4) then ∇uV : ℝ × (ℝ2 ∖ {ξ}) → ℝ2 will be called a strong force. Moreover, (LS) is said to be a strong force Lagrangian system. (V4) implies that the system (LS) does not possess solutions in the Orlicz-Sobolev space associated with φ, entering the singular point ξ in a finite time. Condition (V3) can be replaced by a somewhat weaker assumption, namely,

  • (V3′)$\begin{array}{} (V_{3}') \end{array} $

    lim|x||x|2V(x)=.

    During the past thirty years, there has been made a great deal of progress in the use of variational methods to investigate homoclinic solutions for Lagrangian systems. Some basic material on variational methods can be found in [2, 10, 11, 12, 13]. Since homoclinics are global in time, it is natural to use global methods to study their existence. Both minimization and minimax arguments have been employed to obtain homoclinic solutions (see [7, 14, 15, 16, 17, 18]. The variational formulation for Lagrangian systems leads to action functionals. Although there may be a natural class of curves or functions to work with, there is not always an easy choice of an associated norm or metric. Choosing a good setting in which to formulate the variational problem is often a great difficulty.

    To study homoclinic solutions of the problem (LS), in Section 2 a technical framework will be introduced to treat a corresponding action functional in an appropriate Sobolev-Orlicz space. Section 3 contains the proof of our main result. The basic idea of the proof of Theorem 1.2 is to find two minimizers of the action functional winding around the singularity in opposite directions.

2 Preliminaries

From now on, we assume that Φ : ℝn → [0, ∞) satisfy (G1)−(G5) and (Δ2).

Let Ω ⊂ℝ be a domain. Following Trudinger [1] we define the space

LΦ(Ω)=u:ΩRn:uis Lebesgue measurableandΩΦ(u)dt<.

This space equipped with the Luxemburg norm

uΦ=infν>0:ΩΦuνdt1(2)

is a Banach space. Since Φ is Δ2-regular, LΦ(Ω) is also a separable space (c.f. Rem. 8.22 in [8]). Furthermore, LΦ(Ω) is reflexive if and only if (Δ2) is satisfied (c.f. Thm. 8.20 in [8]).

Set ψ = φ∘∣ ⋅ ∣, i.e. ψ(x) = φ(∣x∣) for each x ∈ ℝn. As a consequence of (1), the space LΦ(Ω) is continuously imbedded in Lψ(Ω) (c.f. Thm. 8.12 in [8]),

LΦ(Ω)Lψ(Ω).

Note that ∥uψ = ∥∣u∣∥φ.

For simplicity of notation, we write LΦ instead of LΦ(ℝ). Although the norm formula (2) depends on the domain Ω, we use the same notation ∥ ⋅ ∥Φ for different subsets of ℝ. It will be clear from the context what Ω is.

Let ACloc(ℝ, ℝn) be the space of locally absolutely continuous functions on ℝ with values in ℝn. Finally, let E denote the Orlicz-Sobolev space

E=uACloc(R,Rn):u˙LΦ(R,Rn)

with the norm

u=u˙Φ+|u(0)|.

We note for later reference that E is a separable reflexive Banach space (see [19]).

For every T > 0 we define the Banach space ET consisting of restrictions of uE to the interval [0, T] with the induced norm,

uET=|u(0)|+u˙Φ.

Let C([0, T], ℝn) denote the space of continuous functions from [0, T] into ℝn with the standard norm.

Proposition 2.1

The inclusion map ETC([0, T], ℝn) is continuous, i.e. there is CT > 0 such that for each uET one has

maxt[0,T]|u(t)|CTuET.

Proof

One has

|u(t)|=u(0)+0tu˙(s)ds|u(0)|+0t|u˙(s)|ds|u(0)|+0T|u˙(s)|ds|u(0)|+21φ|u˙|φ(1+21φ)|u(0)|+|u˙|φCT|u(0)|+u˙Φ=CTuET.

Proposition 2.2

If a sequence {uk}k∈ℕET converges weakly to u0ET then it converges uniformly to u0 in C([0, T], ℝn).

Proof

Since {uk}k∈ℕ converges to u0 weakly in ET then, by Proposition 2.1, it also converges to u0 weakly in C([0, T], ℝn). Furthermore, ∥ukETM for some M > 0 and every k ∈ ℕ.

Let 0 ≤ stT. Then

|uk(t)uk(s)|=stu˙k(τ)dτst|u˙k(τ)|dτ21φ|u˙k|φ21φukET2M(φ)11ts1.

Thus {uk}k∈ℕ is a sequence of equicontinuous functions. By the Arzela-Ascoli Theorem, every sequence {uki}i∈ℕ contains a subsequence converging to a certain û in C([0, T], ℝn). By the uniqueness of the weak limit, û = u0, which completes the proof.□

In what follows, Φ : ℝ2 → ℝ and V : ℝ × (ℝ2 ∖ {ξ}) → ℝ satisfy the assumptions of Theorem 1.2.

For each uE, we define a functional I by setting

I(u)=Φ(u˙(t))V(t,u(t))dt.(3)

Let

αε=inf{V(t,x):xBε(0)},(4)

where 0 < ε12ξ∣ and Bε(0) denotes the ball of radius ε centered at the origin. By (V1)−(V3) we have αε > 0.

Lemma 2.3

Suppose that uE and u(t) ∉ Bε(0) for each t ∈ [a, b]. Then, there is C > 0 such that

(I(u)+1)2Clength(u|[a,b])C|u(b)u(a)|.(5)

Proof

One has

|u(b)u(a)|=abu˙(t)dtab|u˙(t)|dt2|u˙|φ1φ.

The last estimation follows from Hölder’s inequality in Orlicz spaces (c.f. [5], Par. 8.11). Directly from the definition, one has

1φ=(φ)11ba1.

Set δ = length (u∣[a, b]) and τ = ba. Then

|u˙|φ12δ1φ1=12δ(φ)11τ.

Consequently,

I(u)abΦ(u˙(t))V(t,u(t))dt=abΦ(u˙(t))dt+abV(t,u(t))dtabφ(|u˙(t)|)dt+αετ|u˙|φ1+αετ12δ(φ)11τ1+αετ.(6)

Hence

I(u)+112δ(φ)11τ+αετ12δτk(φ)1(k)+αετ,

where the natural number k satisfies τ k ≥ 1 and the last inequality follows from the fact that (φ*)−1 is concave. We choose the smallest k with the property τ k ≥ 1. In particular, we set k = 1 if τ ≥ 1. Now, if τ ≥ 1 then

f(τ)=12δτ(φ)1(1)+αετ

achieves its minimum at the point

τmin=δ(φ)1(1)2αε12,

which is equal to fmin=(2δαε(φ)1(1))12. If τ < 1 then

12δτk(φ)1(k)+αετ14δ(φ)1(k)+αετ14δ(φ)1(1).

Finally, set

C=min2αε(φ)1(1),14(φ)1(1).

Remark 2.4

In the above lemma the interval [a, b] can be replaced by a finite sum of disjoint intervals.

We will denote by L(ℝ, ℝ2) the space of Lebesgue measurable essentially bounded functions from ℝ into ℝ2 with the norm

u=ess sup|u(t)|.

Corollary 2.5

If uE and I(u) < ∞ then uL(ℝ, ℝ2).

Proof

Assume that uL(ℝ, ℝ2). Then for every n ∈ ℕ there exists tn ∈ ℝ such that ∣u(tn)∣ > n. Consequently, by Lemma 2.3 we get

(I(u)+1)2C|u(tn)u(t1)|C(|u(tn)||u(t1)|)C(n|u(t1)|)

for n ∈ ℕ, contrary to I(u) < ∞.□

Lemma 2.6

If uE and I(u) < ∞ then limt± u(t) = 0.

Lemma 2.6 is analogous to Proposition 3.11 of [20] and Lemma 2.4 of [21]. In spite of different assumptions on the potential V, the claims are similar.

Proof

Let A(u) denote the set of limit points of u(t), as t → −∞. From Corollary 2.5 we conclude that A(u) ≠ ∅. Assume that there are ε > 0 and ρ ∈ ℝ such that if t < ρ then u(t) ∉ Bε(0). By (4) we obtain,

I(u)ρV(t,u(t))dt=,

a contradiction. Thus A(u) contains 0. It is sufficient to note that A(u) consists of a point. If not, there is ε > 0 such that u(t) intersects Bε2(0) and Bε(0) infinitely many times. Let τ0 ≥ 0 be the smallest number such that

I(u)+112ε2(φ)11τ0+αε2τ0.

Since limτ→∞(φ*)−1(τ) = ∞, one has τ0 > 0. By Remark 2.4, we obtain

I(u)+1nαε2τ0

for each n ∈ ℕ, and hence I(u) = ∞, a contradiction.

In the same manner we can see that limt u(t) = 0.□

Lemma 2.7

If [a, b] is an interval such that u([a, b]) ⊂𝓝 ∖ {ξ} then it holds

|U(u(b))||U(u(a))|2(I(u)+1)2.(7)

Proof

We first note that

|U(u(b))||U(u(a))|+abddtU(u(t))dt|U(u(a))|+abU(u(t)),u˙(t)dt|U(u(a))|+ab|U(u(t))||u˙(t)|dt|U(u(a))|+2|U(u)|φ|u˙|φ

Since

|U(u)|φ1+abφ(|U(u(t))|)dt1+abV(t,u(t))dt

and

|u˙|φ1+abφ(|u˙(t)|)dt

we obtain

|U(u(b))||U(u(a))|+2(I(u)+1)2.

As an immediate consequence of (7) one has that u(t) ≠ ξ for t ∈ ℝ provided that I(u) < ∞ (c.f. [7], Eq. (2.21)). In fact, we obtain the following

Corollary 2.8

(c.f. [17]) If the action functional I is bounded on some set WE, say I(W) ⊂[0, β] then there is ρ > 0 depending on β such that for every uW and t ∈ ℝ one hasu(t) − ξ∣ ≥ ρ.

Set

Λ=uE:limt±u(t)=0,u(R)R2{ξ}.

If I(u) < ∞ then uΛ. Consequently, u describes a closed curve in ℝ2 ∖ {ξ} that starts and ends at 0. Hence its homotopy class [u] represents an element of the fundamental group π1(ℝ2 ∖ {ξ}).

Let us remind that two functions u0, u1Λ are homotopic if and only if there exists a continuous map h : [0, 1] → Λ such that h(0) = u0 and h(1) = u1. The rotation number (or winding number) rotξ(u) of u around ξ is constant on every connected component of Λ and induces an isomorphism rot* : π1(ℝ2 ∖ {ξ}) → ℤ,

rot([u])=rotξ(u).

Equivalently, Λ is a sum of its path connected components labeled by the integers.

Similarily to [17] one can prove the following result.

Proposition 2.9

Let WΛ be a set such that the functional I restricted to W is bounded. Then there exists D ∈ ℕ such thatrotξ(u)∣ ≤ D for all uW.

Let

Λ±={uΛ:±rotξ(u)>0},

and

λ±=infuΛ±I(u).(8)

Our main result is an immediate consequence of the following.

Theorem 2.10

If the assumptions of Theorem 1.2 are satisfied then there exists u±Λ± such that I(u±) = λ± > 0. Moreover, u± is a classical homoclinic solution of (LS).

3 Proof of Theorem 2.10

The proof will be carried out for the “+” case. The proof for the “-” case is similar. We set λ = λ+. Let {un}n=1 be a minimizing sequence for (8). With no loss of generality we assume that for every n ∈ ℕ,

λI(un)λ+1,

and by Proposition 2.9, for some d ∈ ℕ,

rotξ(un)=d.

Since d > 0, there are νn and θn > 1 such that un(νn) = θnξ. In particular, by Corollary 2.8

un>|ξ|.

Furthermore, there are σn, μn and τn ∈ [σn, μn] such that:

  1. un([σn, μn]) ⊂ R2B|ξ|2(0),

  2. un(σn)∣ = ∣un(μn)∣ = 12ξ∣,

  3. un(τn)∣ = ∥un

Hence, by Lemma 2.3,

(λ+2)2(I(un)+1)2Clength (un|[σn,μn])>C(2un|ξ|)>Cun,

and thus the sequence {∥un}n∈ℕ is bounded. Furthermore, since by (6)

λ+2I(un)+112δ(φ)11τ+αετ

with δ ≥ ∣ξ∣, there are M > m > 0 such that m < τ < M. In particular, μnσn > m for each n ∈ ℕ. Consequently, λ = inf{I(un) ; n ∈ ℕ} ≥ αεm > 0. From (G3) we obtain

RΦ(Aω(t))dtARΦ(ω(t))dt

for 0 ≤ A ≤ 1 and ωLΦ. If we let A = (λ +1)−1 then

RΦ((λ+1)1u˙n(t))dt(λ+1)1RΦ(u˙n(t))dt(λ+1)1I(un)1,

which implies that ∥nΦλ +1. In consequence, {un}n=1 is bounded in E.

Now, let C0(ℝ, ℝ2) denote the space of smooth functions from ℝ into ℝ2 with compact supports.

We say that a set ZΛ has the perturbation property and write Z ∈ 𝓟 if for each uZ and for each vC0(ℝ, ℝ2) there exists δ > 0 such that if s ∈ (−δ, δ) then u+sv ∈ Z.

Let us remark that if u is a minimizer of I on a set Z ∈ 𝓟 then

ddsI(u+sv)|s=0=0=((Φ(u˙(t)),v˙(t))(V(t,u(t)),v(t)))dt,

and consequently, u is a weak solution of (LS). A similar argument as in the proof of Proposition 3.18 in [20] shows that u is a classical solution of (LS). Finally, using (LS), (V1) and (V2) as in [18] gives (±∞) = 0.

Of course Λ± ∈ 𝓟. We expect that minimizing I over Λ+ and Λ gives two solutions.

Let Lloc(ℝ, ℝ2) be the space of Lebesgue measurable functions from ℝ into ℝ2 that are essentially bounded on each compact subset of ℝ.

Since E is reflexive, the sequence {un}n=1 converges along a subsequence to QE weakly in E and, by Proposition 2.2, strongly in Lloc(ℝ, ℝ2). It follows from Fatou’s Lemma that I(Q) ≤ λ. Thus QΛ. Finally, we apply the following version of the shadowing chain lemma

Lemma 3.1

Let Z ∈ 𝓟 be an arbitrary set all of whose elements have the same rotation number d ∈ ℤ. Set

z=inf{I(q):qZ}.

Under the conditions of Thm.1.2, there are a finite number of homoclinic solutions: Q1, Q2, …, QlΛ of (LS) such that

z=I(Q1)+I(Q2)++I(Ql)

and

d=rotξ(Q1)+rotξ(Q2)++rotξ(Ql).

The proof is analogous to that of Lemma 3.2 in [17].

Since d > 0 there is at least one Qi with rotξ(Qi) > 0. In fact, this nontrivial solution is unique. If Qj is another nontrivial solution then I(Qj) > 0. Thus I(Qi) < λ, which is a contradiction.

Acknowledgement

The first two authors are supported by Grant BEETHOVEN2 of the National Science Centre, Poland, no. 2016/23/G/ST1/04081.

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About the article

Received: 2018-10-06

Accepted: 2019-01-18

Published Online: 2019-06-29

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 644–653, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0018.

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© 2020 Marek Izydorek et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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