Multiplicity of Entire Solutions For a Class of Almost Periodic Allen-Cahn Type Equations

Multiplicity of Entire Solutions For a Class of Almost Periodic Allen-Cahn Type Equations

Francesca ^∗ Alessio

Dipartimento di Scienze Matematiche ,

Universit` a Politecnica delle Marche , via Brecce Bianche, I-60131 Ancona e-mail: alessio@dipmat.univpm.it

Piero ^† Montecchiari

Dipartimento di Scienze Matematiche ,

Universit` a Politecnica delle Marche , via Brecce Bianche, I-60131 Ancona e-mail: montecchiari@dipmat.univpm.it

Received 28 August 2005 Communicated by Antonio Ambrosetti

Abstract

We consider a class of semilinear elliptic equations of the form

−∆u(x, y) + a(εx)W

(u(x, y)) = 0, (x, y) ∈ R

(0.1) where ε > 0, a : R → R is an almost periodic, positive function and W : R → R is modeled on the classical two well Ginzburg-Landau potential W (s) = (s

− 1)

. We show via variational methods that if ε is sufficiently small and a is not constant then (0.1) admits infinitely many two dimensional entire solutions verifying the asymptotic conditions u(x, y) → ±1 as x → ±∞ uniformly with respect to y ∈ R.

1991 Mathematics Subject Classification . 35J60, 35B05, 35B40, 35J20, 34C37. .

Key words . Heteroclinic Solutions, Elliptic Equations, Variational Methods.

∗

Supported by MURST Project ‘Metodi Variazionali ed Equazioni Differenziali Non Lineari’

†

Supported by MURST Project ‘Metodi Variazionali ed Equazioni Differenziali Non Lineari’

515

1 Introduction

In this paper, pursuing a study started in [3] and [5], we consider semilinear elliptic equa- tions of the form

−∆u(x, y) + a(εx)W ^� (u(x, y)) = 0, (x, y) ∈ R ² (1.2) where we assume ε > 0 and

(H 1 ) a : R → R is H¨older continuous, almost periodic and satisfies 0 < a ≡ inf

x∈R a(x) < sup

x∈R a(x) ≡ a;

(H 2 ) W ∈ C ² (R) satisfies W (s) ≥ 0 for any s ∈ R, W (s) > 0 for any s ∈ (−1, 1), W ( ±1) = 0 and W ^�� (±1) > 0.

Classical examples of potentials which satisfy the above assumptions are the double- well Ginzburg-Landau potential, W (s) = (s ² − 1) ² , used in the study of phase transitions in binary metallic alloys, and the Sine-Gordon potential, W (s) = 1 + cos(πs), applied to several problems in condensed state Physics. Equation (1.2) with these potentials is respec- tively known as Allen-Cahn and Sine-Gordon stationary equation. The global minima of W represent pure phases of the material and they are energetically favorite.

We look for existence and multiplicity of two phases solutions of (1.2), i.e. solutions of the boundary value problem

� −∆u(x, y) + a(εx)W ^� (u(x, y)) = 0, (x, y) ∈ R ²

x →±∞ lim u(x, y) = ±1, uniformly w.r.t. y ∈ R. (1.3) One interesting feature regarding problem (1.3) is described by N. Ghoussoub and C. Gui in [14]. Indeed, in [14] the authors, dealing with the study of a De Giorgi conjecture (see [11]), proved in particular that in the autonomous case, i.e. when a(x) = a 0 > 0 for any x ∈ R, if u ∈ C ² (R ² ) is a solution of (1.3), then u(x, y) = q(x) for all (x, y) ∈ R ² , where q ∈ C ² (R) is a solution of the ordinary differential equation −¨q(x) + a 0 W ^� (q(x)) = 0, x ∈ R, satisfying lim x→±∞ q(x) = ±1. In other words, in the autonomous case, the problem (1.3) is one dimensional and the set of its solutions consists of one dimensional heteroclinic solutions. We quote also the papers [8], [9], [12], [2], [6] (and the reference therein) where the results in [14] are extended to higher dimensions and to more general settings. We recall moreover that the De Giorgi conjecture has been proven to hold in R ⁿ for any n ≤ 8 in the recent paper by O. Savin, [22].

As shown in [3], the one dimensional symmetry disappears in the non autonomous case, even considering, as in Problem (1.3), potentials depending periodically only on the single variable x. We mention also the paper by Alama, Bronsard and Gui, [1], where the authors exhibit systems of autonomous Allen Cahn equations which admit solutions without one dimensional symmetry.

The major difference between the case of autonomous equations and the cases consid-

ered in [3] and [1] lies in the different structure of the set of solutions of the corresponding

one dimensional problems. While in the case of a single autonomous equation the set of one dimensional solutions is a continuum homeomorphic to R, being constituted by the translations of a single heteroclinic solution (see Lemma 2.1 below), in both the works [3]

and [1] the construction of two dimensional solutions is strongly related with the discrete structure of the set of one dimensional solutions.

To be more precise, let us briefly describe the case in which the function a is a reg- ular, T -periodic, positive and not constant function, as in [3]. Let us consider the action functional

F (q) =

�

R 1

2 | ˙q(x)| ² + a(εx)W (q(x)) dx on the class

Γ = {q ∈ H loc ¹ (R) / �q� L

( R) = 1 and q − z ⁰ ∈ H ¹ (R)}.

where z 0 ∈ C ^∞ (R) is any fixed increasing function such that z 0 (x) → ±1 as x → ±∞

and |z ⁰ (x)| = 1 for any |x| ≥ 1. Set c = inf Γ F and K = {q ∈ Γ / F (q) = c}, noting that K is the minimal set of F on Γ consisting of one dimensional solutions of (1.3). It can be proved that, when a is not constant and ε is sufficiently small, the following non-degeneracy condition holds:

(∗) there exists ∅ �= K 0 ⊂ K such that, setting K ^j = {q(· − j ^T _ε ) | q ∈ K 0 } for j ∈ Z, there results

(i) K 0 is compact with respect to the H ¹ (R) topology,

(ii) K = ∪ j ∈Z K ^j and there exists d 0 > 0 such that if j �= j ^� then d(K ^j , K ^j

) ≥ d 0

where d(A, B) = inf{�q ¹ (x) − q ² (x)� L

( R) / q 1 ∈ A, q ² ∈ B}, A, B ⊂ Γ.

The existence of solutions depending on both the planar variables can be achieved pre- scribing different asymptotic behaviors of the solutions as y → ±∞. Indeed, in [3], so- lutions to problem (1.3) which are asymptotic as y → ±∞ to different minimal sets K ^j

, K ^j

are found. More precisely, letting

H = {u ∈ H loc ¹ (R ² ) / �u� L

( R

) ≤ 1 and u − z ⁰ ∈ ∩ ^(ζ

,ζ

) ⊂R H ¹ (R × (ζ 1 , ζ 2 ))}, in [3], it is shown that fixed j ₋ = 0 there exists at least two different values of j + ∈ Z\{0}

for which there exists a solution u ∈ H of (1.3) such that d(u(·, y), K ^j

) → 0, as y → ±∞

(in fact, as shown in [17], there results j + = ±1).

The variational argument used in [3] is related to a renormalization procedure intro- duced by P.H. Rabinowitz in [19], [20]. The solutions are found as minima of the renor- malized action functional

ϕ(u) =

�

R 1

2 �∂ ^y u( ·, y)� ² L

( R) + (F (u(·, y)) − c) dy on the set

M ^j

,j

= {u ∈ H | lim

y→±∞ d(u( ·, y), K ^j

) = 0}.

Note that, since c is the minimal level of F on Γ, the function y → F (u(·, y)) − c is non negative and the functional ϕ is well defined with values in [0, +∞] for any u ∈ M ^j

,j

. Moreover, the property (∗) of K implies that if u ∈ M j

,j

and ϕ(u) < +∞

then sup _{y ∈R} d(u( ·, y), K ^j

) < +∞. This is basic to avoid ‘sliding’ phenomena for the minimizing sequences (see [1]) possibly due to the unboundedness of the domain in the x-direction and so to recover for suitable values of the index j + the sufficient compactness to obtain the existence of a minimizer of ϕ in M ^j

,j

.

In the present paper we generalize the result in [3]; studying the existence of two di- mensional solutions to (1.3) in the case in which a is almost periodic, as stated in (H 1 ).

The structure of the set of one dimensional solutions of (1.3) when a is almost periodic is different from the one of the periodic case and the arguments used in [3] are no more useful in the almost periodic context.

To describe these differences, we first note that if a is almost periodic, setting c = inf Γ F (q) , we cannot say in general that c is attained, or, in other words, that the set K is not empty. Anyhow, thanks to the almost periodicity of the function a, if we look at the set of local minima of F on Γ, we see that it has discreteness and local compactness properties similar to the ones described by (∗) for the periodic case. In section 2 we prove in fact that there exist c ^∗ > c, δ 0 ∈ (0, 1/4) and ε ⁰ > 0 such that for any ε ≤ ε 0 there is a family of open intervals {A j = (t ⁻ _j , t ⁺ _j ) / j ∈ Z} verifying that

there exists T ₋ , T + > 0 such that ^T _ε

≤ t ⁺ j − t ⁻ j ≤ ^T ε

and t ⁺ _j = t ⁻ _j+1 for any j ∈ Z, and for which, setting for j ∈ Z

Γ j = {q ∈ Γ / 1 − |q(t)| ≤ δ ⁰ for any t ∈ R \ A ^j }, c ^j = inf

Γ

F (q) and K ^j = {q ∈ Γ j / F (q) = c j }

we have

i) {q ∈ Γ / F (q) ≤ c ^∗ } = ∪ ^j ∈Z Γ j and Γ j �= ∅ for any j ∈ Z,

ii) ∃d ⁰ , D > 0 such that if j �= j ^� then d(Γ j , Γ j

) ≥ d 0 and sup _q

_,q

_∈Γ

�q ¹ −q ² � L

( R) ≤ D for any j ∈ Z,

iii) K ^j is not empty, is compact, with respect to the H ¹ (R) topology, and constituted by one dimensional solution of (1.3).

In other words, if ε is sufficiently small, the sublevel set {q ∈ Γ / F (q) ≤ c ^∗ } splits in the disjoint union of the uniformly separated and uniformly bounded subsets Γ j . Each set Γ j contains a nonempty set K ^j , compact with respect to the H ¹ (R) topology, consisting of local minimizers of F at the level c j = inf Γ

F (q) ∈ [c, c ^∗ ). For our purposes this is a good generalization of the property (∗) and, following [3], we can try to look for two dimensional solutions of (1.3) prescribing to the solution to be asymptotic as y → ±∞ to different minimal sets K j

or, more simply, to different sets Γ j

.

Following this line, fixed j ∈ Z, we may look for this kind of solutions in the set M ^∗ j = {u ∈ H / lim

y→−∞ d(u( ·, y), K ^j ) = 0, lim inf

y→+∞ d(u( ·, y), K ^j ) > 0}.

Differently from the periodic case, since the set K ^j is not minimal for F on Γ, if u ∈ M ^∗ j , the function y → F (u(·, y)) − c ^j is indefinite in sign and the natural renormalized functional

ϕ j (u) = �

R 1

2 �∂ ^y u( ·, y)� ² L

( R) + (F (u(·, y)) − c j ) dy

is not a priori well defined. To overcome this difficulty we take advantage of a tech- nique already introduced in [5] to find, in the periodic case, two dimensional solutions emanating as y → −∞ from k-bump one dimensional solutions. As in [5] we observe that if u ∈ H is a solution of (1.2) on the half plane R × (−∞, y ⁰ ), which satisfies

�

R×(−∞,y

) |∂ ^y u(x, y) | ² dx dy < + ∞ and d(u(·, y), K ^j ) → 0 as y → −∞, then the energy E u (y) = ¹ ₂ �∂ ^y u( ·, y)� ² L

( R) − F (u(·, y)) is conserved along the trajectory y ∈ (−∞, y 0 ) → u(·, y) ∈ Γ and precisely the following identity holds true

F (u( ·, y)) = c ^j + 1

2� ∂ y u( ·, y)� ² L

( R) , ∀y ∈ (−∞, y ⁰ ). (1.4) In particular (1.4) implies that F (u(·, y)) ≥ c ^j for any y ∈ (−∞, y ⁰ ) and suggests that we consider as variational space the set

M ^j = {u ∈ M ^∗ j / inf

y ∈R F (u( ·, y)) ≥ c ^j }

on which we can look for minima of the natural renormalized functional ϕ j which is well defined there.

We note that if u ∈ M ^j and ϕ j (u) < +∞ we have lim inf

y →+∞ F (u( ·, y)) = c ^j and lim inf

y →+∞ d(u( ·, y), K ^j ) > 0.

As showed in Section 3, thanks to the properties (i) − (ii) of the sublevel set {q ∈ Γ / F (q) ≤ c ^∗ }, this implies that the trajectory y ∈ R → u(·, y) ∈ Γ definitively en- ters in a set Γ j

as y → +∞ for an index j ^∗ �= j such that c ^j

≤ c ^j . That condition on the index j is therefore necessary for the minimum problem inf _M

ϕ j (u) to be meaningful and we denote the set of admissible indices

J = {j ∈ Z / there exists i ∈ Z \ {j} such that c ⁱ ≤ c ^j }.

Note that J is an infinite set being either J = Z or there exists an index j ⁰ ∈ Z such that c j

< c j for any j ∈ Z \ {j ⁰ }, i.e., J = Z \ {j ⁰ }.

In section 3, for fixed ε ∈ (0, ε ⁰ ) and j ∈ J , setting m j = inf _M

ϕ j (u), we show

that M ^j �= ∅ and m ^j < + ∞, and so that the minimum problem is well posed. As

in the periodic case, thanks to the discreteness property (i) − (ii) of the sublevel set

{q ∈ Γ / F (q) ≤ c ^∗ } we can prove moreover that if u ∈ M ^j and ϕ j (u) < +∞ then,

sup _{y ∈R} d(u( ·, y), K ^j ) < +∞. That allows us to avoid sliding phenomena for the minimiz-

ing sequences of ϕ j on M ^j obtaining suitable convergence properties, up to subsequences

and y-translations, to two dimensional functions. In fact, as limit of minimizing sequences,

we prove the existence of functions u j ∈ H verifying

y→−∞ lim d(u j (·, y), K j ) = 0 and there exists j ^∗ ∈ Z \ {j} such that

y→+∞ lim d(u j (·, y), Γ j

) = 0.

A problem arises since we can not say that the limit functions u j are minima of ϕ j

on M ^j , indeed the constraint F (u j (·, y)) ≥ c ^j is not necessarily satisfied for any y ∈ R.

However, as limit of a minimizing sequence, u j inherits sufficient minimality properties to construct from it a two dimensional solution of (1.3). More precisely, we consider the restriction of u j on the domain R × (−∞, y ⁰ ) where y 0 is defined by

y 0 = sup{¯y ∈ R / u j (·, y) ∈ K j or F (u j (·, y)) > c j for any y < ¯y}.

If y 0 = +∞, then inf ^y ∈R F (u j (·, y)) ≥ c ^j and we conclude that u j is actually a minimum of ϕ j in M ^j . It is not difficult to show that in this case, u j ∈ C ² (R ² ) is a classical entire solution to (1.2). Concerning the asymptotic behavior of u j as y → +∞

we can say moreover that there exists

K ˜ ^j

⊂ Γ ^j

∩ {q ∈ Γ / F (q) = c ^j },

compact with respect to the H ¹ (R) topology, consisting of one dimensional solutions of (1.3) and such that lim y →+∞ d(u( ·, y), ˜ K ^j

) = 0. We refer to this kind of solutions as heteroclinic type solutions.

If y 0 ∈ R, we prove that

� y

−∞

1

2 �∂ ^y u j � ² L

( R) + (F (u j (·, y)) − c j ) dy = m j , lim _{y →y}

0

F (u j (·, y)) = c j , and u j (·, y 0 ) ∈ Γ j

.

Then, we derive that u j is minimal for ϕ j with respect to small variations with compact support in R × (−∞, y ⁰ ). That says that u j solves (1.2) in the half plane R × (−∞, y ⁰ ).

Using the conservation of Energy (1.4), since lim _y→y

0

F (u j (·, y)) = c j , we derive that u j satisfies lim _{y →y}

�∂ ^y u( ·, y)� ^L

⁽ R) = 0 and so that u j verifies the Neumann boundary condition ∂ y u j (·, y 0 ) ≡ 0. This allows us to recover, by reflection, an entire solution to (1.2) even in this case. In fact, setting

v j (x, y) =

� u j (x, y), if y ≤ y ⁰ ,

u j (x, 2y 0 − y), if y > y ⁰ , if y 0 ∈ R,

we obtain that v j is a classical entire solution to (1.2), which is truly two dimensional since it verifies

y →±∞ lim d(v j (·, y), K j ) = 0 and v j (·, y 0 ) ∈ Γ j

. We refer to these kinds of solutions as homoclinic type solutions.

In both the cases, the solutions satisfy the boundary conditions lim x→±∞ u(x, y) = ±1 uniformly with respect to y ∈ R and they actually solve (1.3). That proves our main result.

Theorem 1.1 There exists ε 0 > 0 such that for any ε ∈ (0, ε ⁰ ) the problem (1.3)

admits infinitely many geometrically distinct two dimensional solutions.

We finally refer the interested reader to the related “fully” non autonomous case, where a is periodic in both the variables x and y. For that kind of problem, even in more general settings, we quote the papers [15], [7], [4], [17], [18], [10] (and the references therein) where the existence of a wide variety of solutions has been shown.

Some constants and notations. We now make precise some basic consequences of the assumptions (H 1 ) and (H 2 ), fixing some constants which will remain unchanged in the rest of the paper.

First of all we note that , since W ∈ C ² (R) and W ^�� (±1) > 0, there exists δ ∈ (0, ¹ ₄ ) and w > w > 0 such that

w ≥ W ^�� (s) ≥ w for any |s| ∈ [1 − 2δ, 1 + 2δ]. (1.5) In particular, since W (±1) = W ^� (±1) = 0, setting χ(s) = min{|1 − s|, |1 + s|}, we have that

if |s| ∈ [1 − 2δ, 1 + 2δ], then w

2 χ(s) ² ≤ W (s) ≤ w

2 χ(s) ² and |W ^� (s)| ≤ wχ(s). (1.6) Note that by (1.6), since W (s) > 0 for any s ∈ (−1, 1), a continuity argument gives rise to the existence of two constants b, b > 0 such that

b χ(s) ² ≤ W (s) and |W ^� (s)| ≤ b χ(s), ∀|s| ≤ 1. (1.7) Moreover, denoting

ω δ = min

|s|≤1−δ W (s), δ ∈ (0, 1), (1.8)

we have that ω δ > 0 for any δ ∈ (0, 1). In relation with δ, defined by (1.5), ω δ and the constants a, a given by (H 1 ), we define the constants

λ = � 2aω _δ δ and λ 0 = min{ 1 2 ,

√ a

√ a − 1}λ, (1.9)

noting that, since by (H 1 ), a > a, we have λ 0 > 0.

Finally, for a given q ∈ L ² (R) we denote �q� ≡ �q� L

( R) .

To conclude, we recall that a function a is almost periodic if for any η > 0 there exists L η > 0 such that any real interval of length L η contains an η-period of a, that is, a τ ∈ R for which sup _{t ∈R} |a(t − τ) − a(t)| ≤ η (see e.g. [13]).

2 The one dimensional problem

In this section we display some variational properties of the one dimensional solutions to (1.3), i.e. solutions to the heteroclinic problem

� −¨q(t) + a ^ε (t)W ^� (q(t)) = 0, t ∈ R,

lim t →±∞ q(t) = ±1, (2.10)

where a ε (t) = a(εt) for t ∈ R.

Let z 0 ∈ C ^∞ (R) be an increasing function such that z 0 (t) → ±1 as t → ±∞ and

|z ⁰ (t)| = 1 for any |t| ≥ 1. We define the class

Γ = {q ∈ H loc ¹ (R) / �q� L

( R) = 1 and q − z 0 ∈ H ¹ (R)}, on which we consider the action functional

F a

(q) = �

R 1

2 | ˙q(t)| ² + a ε (t)W (q(t)) dt.

Moreover, if I is an interval in R, we set F ^a

,I (q) = �

I 1

2 | ˙q| ² + a ε (t)W (q) dt.

Note that the functionals F a

and F a

,I are well defined on H _loc ¹ (R) with value in [0, +∞] and are Lipschitz continuous on Γ endowed with the H ¹ (R) metric.

For future references we introduce also the set

Γ = {q ∈ L ^∞ (R) / �q� L

( R) = 1 and q − z 0 ∈ L ² (R)}

which is the completion of Γ with respect to the metric d(q 1 , q 2 ) = �q ¹ − q ² � ^L

⁽ R) .

Denoting, for c ∈ R, {F ^a

≤ c} = {q ∈ Γ / F ^a

(q) ≤ c}, we remark that if (q n ) ⊂ {F ^a

≤ c} then �q ⁿ � L

( R) ≤ 1 and � ˙q ⁿ � ≤ 2c for any n ∈ N. Then, by the semicontinuity of the L ² -norm with respect to the weak convergenze and the Fatou Lemma, we recover the following property.

Lemma 2.1 Let (q n ) ⊂ {F ^a

≤ c} for some c > 0. Then, there exists q ∈ H loc ¹ (R) with �q� L

( R) ≤ 1 such that, along a subsequence, q ⁿ → q in L ^∞ loc (R), ˙q n → ˙q weakly in L ² (R) and moreover F a

(q) ≤ lim inf n →∞ F a

(q n ).

Given a positive constant b, we denote F b (q) = �

R 1

2 | ˙q| ² + bW (q) dt. Moreover, if I is an interval in R, we set F ^b,I (q) = �

I 1

2 | ˙q| ² + bW (q) dt. We recall here below some known properties of the functional F b with b ∈ [a, a], the range of the function a ^ε . Note first of all that the following property holds true (see [5], Lemma 2.2).

Proposition 2.1 For every positive constant b the problem

� −¨q(t) + bW ^� (q(t)) = 0, t ∈ R

lim t →±∞ q(t) = ±1. (P b )

admits a unique solution in Γ, modulo time translation. That solution is increasing and denoting with q b the unique solution to (P b ) in Γ which verifies q b (0) = 0, then q b is a minimum of F b on Γ. Moreover, setting c b = inf Γ F b , there results c b = √

bc 1 .

Since the functional F a,I bounds from below the functional F a

,I for any ε > 0 and any real interval I, it is useful to observe that if δ ∈ (0, 1) and q ∈ Γ is such that |q(t)| ≤ 1 − δ for any t ∈ (σ, τ) ⊂ R, then by (1.8)

F a,(σ,τ ) (q) ≥ _2(τ−σ) ¹ |q(τ) − q(σ)| ² + aω δ (τ − σ) ≥ �

2aω δ |q(τ) − q(σ)|. (2.11) As direct consequence of (2.11) and Proposition 2.1 we have the following result.

Lemma 2.2 There exists η 0 ∈ (0, ^{a −a} 8 ) such that c a+η

− c ^a ≤ ^λ 6

and c _a−η

− c ^a ≥ 3λ 0 .

Proof. Given any q ∈ Γ, since δ ≤ ¹ ₄ (see (1.5)) and q is continuous, there exists (σ, τ) ⊂ R such that |q(t)| ≤ 1 − δ for any t ∈ (σ, τ) and |q(τ) − q(σ)| ≥ 6δ. Then, by (2.11) and recalling the definition of λ given in (1.9), we derive that F a (q) ≥ 6λ for any q ∈ Γ and so that c a ≥ 6λ. From this last estimate, since by Proposition 2.1 we have c ^a = ^√ √ ^a a c a , by (1.9) we recover that c a − c ^a ≥ 6( ^{√ √ a} a − 1)λ ≥ 6λ ⁰ . Then, since by Proposition 2.1 the function b → c b is continuous on [a, a], the Lemma follows.

Given δ ∈ [0, 1), assume that a function q ∈ H loc ¹ (R) verifies q(σ) = −1 + δ and q(τ ) = 1 − δ for certain σ < τ ∈ R. If δ > 0, one easily obtains that F b,(σ,τ ) (q) ≥ c ^b − o ^δ with o δ → 0 as δ → 0. The following Lemma (see [5], Lemma 2.3) fix a δ ⁰ > 0 such that o δ

is comparable with the constant λ 0 fixed in (1.9) for any b ∈ [a, a].

Lemma 2.3 There exists δ 0 ∈ (0, δ) such that if q ∈ Γ verifies q(σ) = −1 + δ ⁰ and q(τ ) = 1 − δ ⁰ for some σ < τ ∈ R then, F b,(σ,τ ) (q) ≥ c b − ^λ 8

for every constant b ∈ [a, a].

In relation with ρ 0 and δ 0 we set ε 0 = 1

2 ρ 0 min{1, a

2c a ω δ

}. (2.12)

Remark 2.1 Note that, by (2.11) and (2.12), if q ∈ Γ is such that |q(t)| < 1 − δ ⁰ for every t ∈ I, where I is an interval with length |I| ≥ ^ρ ε

, then

F a,I (q) = �

I 1

2 | ˙q| ² + aW (q) dt ≥ aω δ

|I| ≥ 2c ^a .

We fix here any ε ∈ (0, ε 0 ) and it will remain unchanged in the rest of the section.

In the sequel we will omit the dependence on ε and, in particular, we will denote

F (q) =

� F a

(q), if q ∈ Γ, +∞, if q ∈ Γ \ Γ.

The remaining part of this section studies the behavior of the functional F on its sub- level set {F ≤ c ^∗ }, where

c ^∗ = c a + λ 0 , (2.13)

characterizing concentration properties of the functions in {F ≤ c ^∗ } and related compact- ness properties of the functional F . To this aim, note that since a is continuous, almost periodic and not constant, it is elementary to show the following.

Lemma 2.4 There exists ρ 0 , L 0 > 0 and two bilateral sequences {x j } ^j ∈Z , {x j } ^j ∈Z

such that

i) x _j + 4ρ 0 < x j < x _j+1 − 4ρ ⁰ and x j − x ^j −1 ≤ L ⁰ for any j ∈ Z, ii) max {a(x) / x ∈ [x j − 2ρ ⁰ , x _j + 2ρ 0 ]} ≤ a + η 0 for any j ∈ Z, iii) min {a(x) / x ∈ [x ^j − 2ρ ⁰ , x j + 2ρ 0 ]} ≥ a − η 0 for any j ∈ Z.

Given any j ∈ Z, we define the intervals A j = ( x j −1

ε , x j

ε ) (2.14)

and note that, if i �= j then A ⁱ ∩ A ^j = ∅ and R = ∪ ⁱ ∈Z A ¯ j , where ¯ A j denotes the closure of A j . Moreover, by Lemma 2.4, we have |A ^j | ≤ ^L ε

and

( x _j − 2ρ ⁰

ε , x _j + 2ρ 0

ε ) ⊂ A j for any j ∈ Z, (2.15)

max{a(t) / t ∈ [ x _j − 2ρ ⁰

ε , x _j + 2ρ 0

ε ]} ≤ a + η 0 for any j ∈ Z, (2.16) min{a(t) / t ∈ [ x j − 2ρ ⁰

ε , x j + 2ρ 0

ε ]} ≥ a − η 0 for any j ∈ Z. (2.17) In other words, by (2.15) and (2.16), each interval A j contains a subinterval of length ^4ρ _ε

where the function a ε takes values smaller than or equal to a + η 0 . Moreover, by (2.17), on the intervals of length ^4ρ _ε

and centered on the extremes of A j the function a ε takes values greater than or equal to a − η ⁰ .

The main concentration properties of the functions in {F ≤ c ^∗ } are displayed in the following Lemma which tells us in particular that if F (q) ≤ c ^∗ then, there exists an index j ∈ Z such that 1 − |q(t)| ≤ 2δ for every t ∈ R \ A ^j .

Lemma 2.5 For every q ∈ Γ such that F (q) ≤ c ^∗ there exists a unique interval (σ q , τ q ) ⊂ R such that

i) q(σ q ) = −1 + δ 0 , q(τ q ) = 1 − δ 0 and |q(t)| < 1 − δ 0 for any t ∈ (σ q , τ q ), ii) τ q − σ ^q ≤ ^ρ ε

and (σ q , τ q ) ⊂ ( ^x

ε ^+ρ

, ^x

^−ρ _ε

) ⊂ A ^j for some j ∈ Z iii) if t < σ q then, q(t) ≤ −1 + 2δ and if t > τ ^q then, q(t) ≥ 1 − 2δ.

Proof. Let q ∈ Γ and F (q) ≤ c ^∗ . We set

σ q = sup{t ∈ R / q(t) ≤ −1 + δ 0 } and τ ^q = inf{t > σ q / q(t) ≥ 1 − δ ⁰ }.

Since q(t) → ±1 as t → ±∞ and q is continuous, we have σ ^q < τ q ∈ R, q(σ ^q ) = −1+δ 0 , q(τ q ) = 1 − δ 0 and |q(t)| < 1 − δ ⁰ for any t ∈ (σ ^q , τ q ). Moreover by Remark 2.1, since F (q) ≤ c ^∗ , we have that τ q − σ ^q ≤ ^ρ ε

.

Let j ∈ Z be such that (σ ^q , τ q ) ∩ A j �= ∅. Since τ ^q − σ ^q ≤ ^ρ ε

, if (σ q , τ q ) �⊂

( ^x

_ε ^+ρ

, ^x

^−ρ _ε

) then by (2.17), a(t) ≥ a − η 0 for any t ∈ (σ ^q , τ q ) and so, by Lem- mas 2.3 and 2.2,

F (q) ≥ F ^a −η

,(σ

,τ

) (q) ≥ c a−η

− ^λ 8

≥ c ^a + 3λ 0 − ^λ 8

> c ^∗ , a contradiction. This proves (i) and (ii).

To show that (iii) holds true, assume by contradiction that there exists σ < σ q such that q(σ) > −1+2δ or τ > τ ^q such that q(τ) < 1−2δ. Since |q(σ q )| = |q(τ q )| = 1−δ 0 < 1 −δ we have in both the cases that there exists an interval (t ₋ , t + ) ⊂ R \ (σ q , τ q ) such that

|q(t)| ≤ 1 − δ for any t ∈ (t − , t + ) and |q(t + ) − q(t − )| = δ. Then by (2.11) we have F a,(t

,t

) (q) ≥ 2λ ⁰ and so using Lemma 2.3 and (2.13) we obtain

F (q) ≥ F ^a,(σ

,τ

) (q) + F a,(t

,t

) (q) ≥ c a − ^λ 8

+ 2λ 0 > c ^∗ , a contradiction which proves (iii).

For j ∈ Z we define

Γ j = {q ∈ {F ≤ c ^∗ } | (σ ^q , τ q ) ⊂ A j },

noting that, by Lemma 2.5, we have {F ≤ c ^∗ } = ∪ ^j ∈Z Γ j . Setting c j = inf{F (q) | q ∈ Γ j }, we denote

K ^j = {q ∈ Γ j | F (q) = c ^j }.

Lemma 2.6 For all j ∈ Z there results Γ j �= ∅ and c ^j ≤ c ^∗ − ^λ 8

.

Proof. By Proposition 2.1, the problem (P a+η

) admits a unique solution q 0 ≡ q ^a+η

∈ Γ with q 0 (0) = 0. Since q 0 is increasing and q 0 (0) = 0, we have that there exist σ 0 < 0 < τ 0

such that q 0 (σ 0 ) = −1 + δ 0 , q 0 (τ q

) = 1 − δ 0 . Moreover, since by Lemma 2.2 we have F a (q 0 ) ≤ F a+η

(q 0 ) = c a+η

< 2c a , by Remark 2.1 we obtain τ 0 − σ ⁰ < ^ρ _ε

.

We define the function

Q(t) =

 

 

 

 

 

 

 



−1 if t ≤ σ ⁰ − 1,

q 0 (σ 0 )(t − σ 0 + 1) + (t − σ 0 ) if σ 0 − 1 < t ≤ σ ⁰ , q 0 (t) if σ 0 < t < τ 0 , q 0 (τ 0 )(τ 0 + 1 − t) + (t − τ 0 ) if τ 0 ≤ t < τ ⁰ + 1,

1 if t ≥ τ ⁰ + 1,

noting that, since F a+η

,(σ

,τ

) (Q) ≤ F a+η

(q 0 ) = c a+η

, we have

F a+η

(Q) ≤ F a+η

,(σ

−1,σ

) (Q) + c a+η

+ F a+η

,(τ

,τ

+1) (Q).

Now, let us note that for any t ∈ (τ ⁰ , τ 0 + 1), since q 0 (τ 0 ) = 1 − δ 0 , we have Q(t) = 1 − δ 0 (1 − (t − τ 0 )) and so, by (1.5), W (Q(t)) ≤ ^wδ 2

(1 − (t − τ 0 )) ² . Then, taking δ 0

eventually smaller in Lemma 2.3, we deduce

F a+η

,(τ

,τ

+1) (Q) ≤ ^δ

2

+ ^(a+η

₂ ^{) wδ}

� τ

+1 τ

(1−(t−τ ⁰ )) ² dt = ^δ

₂

+ ^(a+η

₆ ^{) wδ}

≤ λ 0

16 . Similar estimates allow us to conclude that also F a+η

,(σ

−1,σ

) (Q) ≤ ^λ 16

.Given j ∈ Z we now define Q j (t) = Q(t − ^x _ε

) and we note that, since τ 0 − σ ⁰ < ^ρ _ε

, by the definition of Q , we have 1 − |Q(t)| = 0 for any t ∈ R \ ( ^x

^−2ρ ε

, ^x

^+2ρ _ε

). Then, by (2.16) and Lemma 2.2, F (Q j ) ≤ F a+η

(Q j ) = F a+η

(Q) ≤ c a+η

+ ^λ ₈

< c ^∗ − ^λ 8

. Then, by construction and Lemma 2.5, Q j ∈ Γ ^j and the proof is complete.

The next lemma describes some simple metric properties of the sets Γ j as subsets of the metric space Γ. We show that they are uniformly bounded and well separated in Γ and moreover that any bounded set in Γ can intersect at most a finite number of Γ j . Given two subsets U 1 and U 2 of Γ, we denote

d( U ¹ , U ² ) = inf

q

∈U

, q

∈U

�q ¹ − q ² � and diam(U ¹ ) = sup

q

,q

∈U

�q ¹ − q ² �.

Lemma 2.7 There results

i) ∃D ∈ R such that diam(Γ ^j ) ≤ D for all j ∈ Z;

ii) if j, ¯j ∈ Z and j �= ¯j then d(Γ ^j , Γ ¯ j ) ≥ ( ^2ρ ε

) ^1/2 ; iii) d(Γ j , Γ ¯ j ) → +∞ as j → ±∞ for any ¯j ∈ Z.

Proof. To prove (i), let us consider j ∈ Z and q 1 , q 2 ∈ Γ ^j . We clearly have

�

A

|q ¹ − q ² | ² dx ≤ 4|A ^j | ≤ ^4L ε

.

Moreover, since q 1 and q 2 belong to Γ j , by Lemma 2.5 there results |q ¹ (t)| ∈ [1 − 2δ, 1],

|q ² (t)| ∈ [1 − 2δ, 1] for any t ∈ R \ A j . Then by (1.6), we have that |q ¹ − q ² | ² ≤ χ(q 1 ) ² + χ(q 2 ) ² ≤ ^2a w a

(W (q 1 ) + W (q 2 )) for any t ∈ R \ A j . Therefore

�

R\A

|q ¹ − q ² | ² dt ≤ 2 w a

�

R\A

a ε (W (q 1 ) + W (q 2 )) dt ≤ 4c ^∗ w a and (i) follows.

Let now ¯j < j ∈ Z, q ∈ Γ ^j and ¯q ∈ Γ ^¯ j . Then, by Lemma 2.5, we have ¯q(t) > 1 − 2δ

and q(t) ≤ −1 + 2δ for any t ∈ ( ^x

ε ^−ρ

, ^x

^+ρ _ε

). Since δ < ¹ ₄ , we have q(t) − ¯q(t) ≥ 1

for all t ∈ ( ^x

_ε ^−ρ

, ^x

^+ρ _ε

) and so �q − ¯q� ² L

( R) ≥ ^x

^−x

ε ^+2ρ

. Since q and ¯q are

arbitrary this proves that both d(Γ j , Γ ¯ j ) ≥ ( ^2ρ _ε

) ^1/2 for ¯j < j and d(Γ j , Γ ¯ j ) → +∞ for

j → +∞. Analogously, we recover the same result in the cases j < ¯j and j → −∞,

therefore (ii) and (iii) follow.

Remark 2.2 Setting

3d 0 = ( 2ρ 0

ε 0

) ^1/2 ,

then, by Lemma 2.7, d(Γ j , Γ ¯ j ) ≥ 3d 0 for any j, ¯j ∈ Z, j �= ¯j. In particular, if q ∈ Γ is such that 0 < d(q, Γ j ) < 3d 0 for a j ∈ Z, then, by Lemma 2.6, F (q) ≥ c ^∗ > c j .

We can now describe some compactness properties of functional F on the set Γ j . First, by Lemma 2.1 and Lemma 2.5, it is immediate to recognize that the sets Γ j are sequentially compact with respect to the weak topology in H _loc ¹ (R).

Lemma 2.8 If j ∈ Z and (q n ) ⊂ Γ j then, there exists q ∈ Γ j such that, along a subsequence, q n → q in L ^∞ loc (R) and ˙q n → ˙q weakly in L ² (R).

The main compactness property of the functional F is described in the following Lemma.

Lemma 2.9 Let j ∈ Z, (q ⁿ ) ⊂ Γ j , q ∈ Γ j , be such that q n → q in L ^∞ loc (R), ˙q n → ˙q weakly in L ² (R) and F (q n ) → c ∈ [c j , c ^∗ ). If (s n ) ⊂ [0, 1] is such that F (q +s n (q n − q)) = c for any n ∈ N, then we have

(1 − s ⁿ )�q ⁿ − q� H

( R) → 0 as n → ∞.

Proof. We set v n = q n − q. By assumption we have F (q + v ⁿ ) − F (q + s n v n ) → 0 and v n → 0 in L ^∞ loc (R). So we first derive that for any T > 0

(1 − s ² n )

2 � ˙v ⁿ � ² + �

|t|>T

a ε (W (q + v n ) − W (q + s n v n )) dt → 0 as n → ∞. (2.18) Since q and q+v n belong to Γ j , by Lemma 2.5 there exists T 0 > 0 such that if |t| > T ⁰ then

|q(t)| ∈ [1 − 2δ, 1], |q(t) + v ⁿ (t)| ∈ [1 − 2δ, 1] and so also |q(t) + s n v n (t)| ∈ [1 − 2δ, 1].

By convexity of W around the points −1 and 1, we recover that for any |t| ≥ T ⁰ we have W (q(t) + s n v n (t))) ≤ (1 − s n )W (q(t)) + s n W (q(t) + v n (t)) and then, by (2.18), for any T > T 0 we obtain that as n → ∞,

1 − s ² n

2 � ˙v ⁿ � ² + (1 − s n ) �

|t|>T

a ε W (q + v n ) dt ≤ (1 − s n ) �

|t|≥T

a ε W (q) dt + o(1).

Moreover, by (1.6), we obtain that for any |t| ≥ T ⁰ there results a ε (t)W (q(t) + v n (t)) ≥ a wχ(q(t) + v n (t)) ² = a w(χ(q(t)) − sgn(t)v n (t)) ² and so we recover that, as n → ∞, for any T > T 0

1 − s ² n

2 � ˙v ⁿ � ² +(1−s ⁿ )a w �

|t|>T (χ(q)−sgn(t)v ⁿ ) ² ≤ (1−s ⁿ ) �

|t|≥T

a ε W (q)dt+o(1).

Then, since �

R a ε W (q) dt < + ∞, �

R χ(q) ² dt < + ∞ and since �v ⁿ � ≤ diam(Γ ^j ) ≤ D for any n ∈ N, it is immediate to verify that for any η > 0 there exists T ^η > 0 such that

(1 − s ² n )

2 � ˙v ⁿ � ² + (1 − s n )a w �

|t|>T

v _n ² dt ≤ η + o(1) as n → ∞.

This last inequality, since v n → 0 in L ^∞ loc (R), implies

(1 − s ⁿ )�v ⁿ � ² H

( R) → 0 as n → ∞, (2.19) and the Lemma follows.

As a consequence of Lemma 2.9, the next lemma shows in particular that the sets K ^j , j ∈ Z, are compact with respect to the H ¹ (R) metric.

Lemma 2.10 If (q n ) ⊂ Γ j is such that F (q n ) → c j then, there exists q ∈ K j such that, along a subsequence, �q n − q� H

( R) → 0 as n → ∞.

Proof. Let j ∈ Z and (q n ) ⊂ Γ j be such that F (q n ) → c j as n → +∞. By Lemmas 2.1 and 2.8 we obtain that there exist q ∈ Γ j and a subsequence of (q n ), still denoted (q n ), such that q n → q in L ^∞ loc (R), ˙q ⁿ → ˙q weakly in L ² (R) and F (q) ≤ c ^j . Then, since q ∈ Γ ^j , F (q) = c j , i.e., q ∈ K ^j . Since F (q) = c j , defining s n = 0 for any n ∈ N, by Lemma 2.9 we obtain �q ⁿ − q� ² H

( R) → 0 as n → ∞, and the Lemma follows.

Remark 2.3 By Lemma 2.10 in particular we obtain that for every ε ∈ (0, ε 0 ) and j ∈ Z, we have K ^j �= ∅. By standard variational argument it is possible to prove that if q ∈ K j then q ∈ C ² (R) and it is a classical solution to (2.10).

Note that by Lemma 2.10, using a direct contradiction argument, we obtain that for any r > 0 there exists ν r > 0 such that

if q ∈ Γ ^j and inf

¯

q ∈K

�q − ¯q� H

( R) ≥ r then F (q) ≥ c ^j + ν r . (2.20) We end this section with a technical result which will be used in the next section in some comparison argument. In the proof we make use of the following result (see [5], Lemma 2.13).

Lemma 2.11 For any q ∈ Γ there results �

R a ε W (q) dt < + ∞. Moreover for any q 1 , q 2 ∈ Γ we have

�

R a ε |W (q ¹ ) − W (q ² )| dt ≤ ba�q ¹ − q ² �(( b a ¹

�

R a ε W (q 1 ) dt) ^{1 2} + �q ¹ − q ² �).

We then have

Lemma 2.12 Let µ > 0, q n , q ∈ Γ be such that q ⁿ − q → 0 in L ² (R) as n → +∞

and F (q n ) ≥ F (q) + µ for any n ∈ N. Then, setting v n = q n − q, we have that lim inf n →∞ � ˙v ⁿ � ² ≥ 2µ and there exists ¯n ∈ N such that if n ≥ ¯n

F (q n ) − F (q + sv n ) ≥ � ˙v n � ²

4 (1 − s), ∀s ∈ [0, 1].

Proof. Since sup n ∈N �v ⁿ � < +∞, by Lemma 2.11 we obtain that there exists C > 0 depending on sup n ∈N �v ⁿ �, q and W such that ∀s ∈ [0, 1] and ∀n ∈ N there results

|

�

R a ε (W (q + v n ) − W (q + sv ⁿ )) dt| ≤ (1 − s)C�v ⁿ �. (2.21) By (3.32) and (2.21), since by assumption µ ≤ F (q + v ⁿ ) − F (q), we obtain

µ ≤ 1

2� ˙v n � ² − ( ˙q, ˙v ⁿ ) − �

R a ε (W (q + v n ) − W (q)) dt (2.22) for any n ∈ N and, by (2.22), we plainly obtain that

lim inf

n →+∞ � ˙v ⁿ � ² ≥ 2µ. (2.23)

Indeed, if we assume by contradiction that there exists a subsequence (v n

) of (v n ) and µ 0 < µ such that lim j →+∞ � ˙v ⁿ

� ² = 2µ 0 , then since v n

→ 0 in L ² (R) we have also that

˙v n

→ 0 weakly in L ² (R). Then, in this case, ( ˙q, ˙v n

) → 0 and since by (2.21) we have also �

R a ε (W (q + v n

) − W (q)) dt → 0, by (2.22) we reach the contradiction µ ≤ µ 0 . By (2.23) we recover that _{� ˙v} ^v

_� → 0 in L ² (R) as n → +∞ and since the sequence ( _{� ˙v} ^˙v

_� ) is bounded in L ² (R), we deduce that _{� ˙v} ^˙v

_� → 0 weakly in L ² (R) and in particular ^{|( ˙q, ˙v} _{� ˙v}

_�

^{) |} ≤

√ 1 µ |( ˙q, _{� ˙v} ^˙v

_� )| → 0 as j → ∞. That shows that

n→∞ lim ( ˙q, ˙v n )

� ˙v ⁿ � ² = 0 (2.24)

Note now that, thanks to (2.21), for any s ∈ [0, 1] we have F (q + v n ) − F (q + sv n ) = � ˙v n � ²

2 (1 − s ² ) + (1 − s)( ˙q, ˙v n ) + + �

R a ε (W (q + v n ) − W (q + sv n )) dt

≥ � ˙v ⁿ � ² (1 − s)( (1 + s)

2 − |( ˙q, ˙v ⁿ )|

� ˙v ⁿ � ² − C �v ⁿ �

� ˙v ⁿ � ² ).

Then, by (2.23) and (2.24) there exists ¯n ∈ N such that for any n ≥ ¯n we have F (q + v n ) − F (q + sv n ) ≥ � ˙v n � ²

4 (1 − s), ∀s ∈ [0, 1], and the Lemma is proved.

3 Two dimensional solutions

In this section we will show that problem (1.3) admits infinitely many two dimensional solutions for any ε ∈ (0, ε ⁰ ). In fact, for any fixed ε ∈ (0, ε 0 ) and using the notation introduced in the previous section, we set

J = {j ∈ Z / there exists i ∈ Z \ {j} such that c ⁱ ≤ c ^j }

and we will prove

Theorem 3.1 For every j ∈ J there exists v j ∈ C ² (R ² ), solution of (1.3) such that

∂ y v j �≡ 0 and

d(v j (·, y), K j ) → 0 as y → −∞.

Note that, as already noted in the introduction, J is an infinite set being either J = Z or J = Z \ {j ⁰ } for some index j ⁰ ∈ Z.

To prove Theorem 3.1 we make use of a variational framework analogous to the one already introduced in [5]. For (y 1 , y 2 ) ⊂ R, we set S (y

,y

) = R×(y ¹ , y 2 ) and we consider the space

H = {u ∈ H loc ¹ (R ² ) / �u� L

( R

) ≤ 1 and u − z ⁰ ∈ ∩ (y

,y

) ⊂R H ¹ (S (y

,y

) )}.

Note that, using the Fubini Theorem and a Poincar´e type inequality, one plainly obtains that if u ∈ H then, u(·, y) ∈ Γ for a.e. y ∈ R and the function y ∈ R → u(·, y) ∈ Γ defines a continuous trajectory verifying

�u(·, ζ ² ) − u(·, ζ 1 )� ² ≤ �∂ ^y u � ² L

(S(ζ

,ζ

)) |ζ ² − ζ ¹ |, ∀ (ζ ¹ , ζ 2 ) ⊂ R. (3.25) Given j ∈ J , we look for minimal properties of the renormalized action functional

ϕ j (u) = �

R 1

2 �∂ ^y u( ·, y)� ² + (F (u(·, y)) − c j ) dy.

on the set

M ^j = {u ∈ H / lim _y

→−∞ d(u( ·, y), K ^j ) = 0, lim inf

y →+∞ d(u( ·, y), Γ ^j ) ≥ d 0

and inf

y∈R F (u( ·, y)) ≥ c ^j }.

The problem of looking for a minimum of ϕ j on M ^j is well posed. In fact, if u ∈ M ^j then F (u(·, y)) ≥ c ^j for every y ∈ R and so the functional ϕ ^j is well defined on M ^j with values in [0, +∞]. Moreover, as we will prove in the next lemma, for any j ∈ J there results M ^j �= ∅ and m ^j ≡ inf M

ϕ j < + ∞.

Given an interval I ⊂ R we also consider on M ^j the functional

ϕ j,I (u) = �

I 1

2 �∂ ^y u( ·, y)� ² + (F (u(·, y)) − c j ) dy.

Remark 3.1 i) Given any I ⊆ R, the functional ϕ ^j,I (u) is well defined for any u ∈ H such that the set {y ∈ I / F (u(·, y)) < c ^j } has bounded measure.

ii) Letting I ⊆ R and u ∈ H, if ϕ ^j,I (u) is well defined and (u n ) ⊂ M j is such that u n → u weakly in H loc ¹ (R ² ), then ϕ j,I (u) ≤ lim inf _n

→∞ ϕ j,I (u n ) (see e.g. [3],

Lemma 3.1, for a similar argument) .

iii) Given u ∈ H, if (y ¹ , y 2 ) ⊂ R and ν > 0 are such that F (u(·, y)) ≥ c j + ν for any y ∈ (y 1 , y 2 ), then

ϕ j,(y

,y

) (u) ≥ ¹ ₂ � y

y

�∂ ^y u( ·, y)� ² dy + ν(y 2 − y ¹ )

≥ 2(y

¹ −y

)

�

R ( � y

y

|∂ ^y u(x, y) | dy) ² dx + ν(y 2 − y ¹ )

≥ 2(y

¹ −y

) �u(·, y ¹ ) − u(·, y ² )� ² + ν(y 2 − y ¹ )

≥ √

2ν �u(·, y 1 ) − u(·, y 2 )�. (3.26) Lemma 3.1 For every j ∈ J , M ^j �= ∅, m ^j < + ∞ and inf ^j ∈J m j ≥ d ^{0 √} 8 ^λ

.

Proof. Since j ∈ J , there exists i ∈ Z \ {j} such that c i ≤ c ^j . Moreover, we can assume that i is such that

if k ∈ Z is such that min{i, j} < k < max{i, j} then c ^k > c j . (3.27) Pick any couple of functions q j ∈ K ^j and q i ∈ K ⁱ and recall that by Lemma 2.5, we have that max{q ^j (x), q i (x)} ≤ −1 + 2δ for any x ≤ min{σ q

, σ q

} and min{q ^j (x), q i (x)} ≥ 1 − 2δ for any x ≥ max{τ q

, τ q

}. Defining ˜u(x, y) = (1 − y)q ^j (x) + yq i (x), for (x, y) ∈ R × [0, 1], it follows that for any y ∈ [0, 1] there results ˜u(x, y) ≤ −1 + 2δ for any x ≤ min{σ ^q

, σ q

} and ˜u(x, y) ≥ 1 − 2δ for any x ≥ max{τ ^q

, τ q

}. Then,

|˜u(x, y)| − 1 ≤ 2δ for every x ∈ R \ ∪ min {j,i}≤k≤max{j,i} A k and y ∈ [0, 1]. Hence, by Lemma 2.5, we obtain that if y ∈ [0, 1] and F (˜u(·, y)) ≤ c ^∗ then, ˜u(·, y) ∈ Γ ^k for an index k such that min{i, j} ≤ k ≤ max{i, j}. In other words, we have

˜u(·, y) ∈ {F > c ^∗ } ∪ (∪ min{i,j}≤k≤max{i,j} Γ k ) ∀y ∈ [0, 1]. (3.28) Define y ^∗ = sup{y ∈ [0, 1] / F (˜u(·, s)) ≥ c j , ∀s ∈ [0, y]} and note that since the function y ∈ [0, 1] → F (˜u(·, y)) ∈ R is continuous and since F (˜u(·, 1)) = F (q ⁱ ) = c i ≤ c ^j , we have F (˜u(·, y ^∗ )) = c j . Moreover, note that

˜u(·, y ^∗ ) ∈ Γ i . (3.29)

Indeed if y ^∗ = 1 (possible only if c i = c j ) then ˜u(·, y ^∗ ) = q i . If otherwise y ^∗ < 1, then there exists a sequence {y ⁿ } ⊂ (y ^∗ , 1) such that y n → y ^∗ and F (˜u(·, y ⁿ )) < c j . By (3.27) and (3.28) we obtain that ˜u(·, y ⁿ ) ∈ Γ i for any n ∈ N and so, since ˜u(·, y ^∗ ) − ˜u(·, y n ) → 0 in H ¹ (R), by Lemma 2.8, we have ˜u(·, y ^∗ ) ∈ Γ i and (3.29) follows.

Define now

u(x, y) =

 



 

q j (x) for y < 0,

˜u(x, y) for 0 ≤ y ≤ y ^∗ ,

˜u(x, y ^∗ ) for y > y ^∗ .

We plainly have u ∈ H and d(u(·, y), K ^j ) → 0 as y → −∞. Moreover, since by def-

inition of y ^∗ , F (˜u(x, y)) ≥ c ^j for any y ∈ [0, y ^∗ ], we obtain also F (u(x, y)) ≥ c j for

any y ∈ R. Finally, by (3.29) and Remark 2.2, we derive lim inf y→+∞ d(u( ·, y), Γ ^j ) = d(˜ u(x, y ^∗ ), Γ j ) ≥ d(Γ i , Γ j ) ≥ 3d 0 . This implies that u ∈ M ^j , which proves that M ^j �= ∅.

Moreover, one can plainly verifies that m j ≤ ϕ ^j (u) < +∞.

We finally note that, by (3.25), for every u ∈ M j there exists (a, b) ⊂ R such that d(u( ·, y), Γ ^j ) ∈ ( ^d 4

, ^d ₂

) for any y ∈ (a, b) and �u(·, a) − u(·, b)� = ^d 4

. Then, by Remark 2.2 and Lemma 2.6, we recover that F (u(·, y)) ≥ c ^∗ ≥ c ^j + ^λ ₈

for any y ∈ (a, b) and, by (3.26), we obtain

ϕ j (u) ≥

� 2 λ 0

8 d 0

4 = d 0

√ λ 0

8 , ∀u ∈ M ^j concluding the proof of the Lemma.

We characterize here below some properties of the minimizing sequences in M ^j . As first remark, we note that by (3.26), given u ∈ M ^j , any time that the trajectory y ∈ R → u( ·, y) ∈ Γ passes trough an annulus in ¯Γ of thickness d ⁰ , which does not intersect {F ≤ c ^∗ }, then, the value of the functional ϕ ^j increases by the positive uniform amount √

λ 0 d

2 . In fact, by (3.26), if ξ 1 < ξ 2 ∈ R and u ∈ M ^j are such that �u(·, ξ 1 ) − u(·, ξ 2 )� ≥ d 0

and F (u(·, y)) > c ^∗ for any y ∈ (ξ ¹ , ξ 2 ) then ϕ j,(ξ

,ξ

) (u) ≥ � 2(c ^∗ − c ^j )d 0 ≥ √ λ 0 d

2 . Thanks to the discreteness of the set {F ≤ c ^∗ } described in Lemma 2.7, we obtain that if ϕ j (u) < +∞ then, the trajectory y ∈ R → u(·, y) ∈ Γ is bounded. Precisely (for a detailed proof of a similar result see Lemma 3.1 in [5]) we have the following:

Lemma 3.2 There exists C > 0 such that for any j ∈ J , if u ∈ M j and (y 1 , y 2 ) ⊂ R then

�u(·, y ¹ ) − u(·, y 2 )� ≤ C ϕ j (u).

Remark 3.2 We set C j = C(m j +1) and we note that if u ∈ M j and ϕ j (u) ≤ m j +1 then, by Lemma 3.2, we have d(u(·, y), K j ) ≤ C j for any y ∈ R. Then, defining

Ω(j) = {¯j ∈ Z / d(Γ j , Γ ¯ j ) ≤ C j },

by Lemma 2.7, the set Ω(j) is finite and moreover if u ∈ M j and y ∈ R are such that ϕ j (u) ≤ m ^j + 1 and F (u(·, y)) ≤ c ^∗ then, u(·, y) ∈ ∪ ^¯ j∈Ω(j) Γ ¯ j .

Now note that if u ∈ M ^j and ϕ j (u) ≤ m j + 1 we have lim inf y →+∞ F (u( ·, y)) = c ^j and so, by Remark 3.2 and Lemma 2.6, one obtains that there exists ¯j ∈ Ω(j) such that lim inf y →+∞ d(u( ·, y), Γ ^¯ j ) = 0. Using again an annulus type argument, one can easily prove that lim sup y →+∞ d(u( ·, y), Γ ^¯ j ) = 0, obtaining the following characterization of the asymptotic behavior of the trajectory.

Lemma 3.3 If j ∈ J , u ∈ M j and ϕ j (u) ≤ m j + 1 then, there exists ¯j ∈ Ω(j) \ {j}

such that

y →+∞ lim d(u( ·, y), Γ ^¯ j ) = 0.

Remark 3.3 By Lemma 3.3 and Remark 3.2, we obtain that for any j ∈ J there exists j ^∗ ∈ Ω(j) \ {j} such that, setting

M ^j,j

= {u ∈ M j / lim

y →+∞ d(u( ·, y), Γ ^j

) = 0}, we have

m j = inf

u ∈M

ϕ j (u).

To recover some compactness, modulo y-translations, of the minimizing sequences of ϕ j on M ^j,j

, it is useful to study the concentration properties of the trajectories y ∈ R → u( ·, y) ∈ ¯Γ when ϕ ^j (u) is close to m j . To this aim we have the following result:

Lemma 3.4 If j ∈ J , (u n ) ⊂ M j and y 0 ∈ R satisfy F (u ⁿ (·, y 0 )) → c j as n → +∞

and for all n ∈ N, u ⁿ (·, y ⁰ ) ∈ Γ ^¯ j , for some ¯j �= j, then lim inf ϕ j,( −∞,y

) (u n ) ≥ m ^j . Proof. Note that, by the invariance with respect to the y-translation of F , we can assume y 0 = 0. By Lemmas 2.1 and 2.8, there exists q ∈ Γ ^¯ j such that F (q) ≤ c ^j and a subsequence of (u n ), still denoted (u n ), such that, setting v n (·) = u n (·, 0) − q(·), there results v n → 0 in L ^∞ _loc (R) and ˙v n → 0 weakly in L ² (R). We set s n = sup{t ∈ [0, 1] / F (q + tv n ) ≤ c j }, noting that by continuity F (q + s n v n ) = c j for any n ∈ N, and so, by Lemma 2.9, we have

that (1 − s n )�v n � ² H

( R) → 0 as n → ∞. (3.30)

Note that, since F (q + s n v n ) = c j and since q + s n v n is a convex combination of q ∈ Γ ^¯ j

and u n (·, 0) ∈ Γ ^¯ j , by Lemma 2.5 we have q + s n v n ∈ Γ ^¯ j . Hence, considering the new sequence

˜u n (x, y) =

 



 

u n (x, y) if y ≤ 0,

q(x) + (1 − y)v ⁿ (x) if 0 ≤ y ≤ 1 − s n , q(x) + s n v n (x) i if y ≥ 1 − s ⁿ , we have ˜u n ∈ M ^j and so ϕ j (˜u n ) ≥ m j . Then, since

ϕ _j,(−∞,0) (u n ) = ϕ j (˜u n ) − � 1 −s

0 1

2 �v ⁿ � ² + F (q + (1 − y)v n ) − c j dy, the Lemma follows once we prove that, as n → ∞, we have

� 1−s

0 1

2 �v ⁿ � ² + F (q + (1 − y)v ⁿ ) − c ^j dy → 0. (3.31) Indeed, � 1 −s

0 1

2 �v ⁿ � ² dy = ¹ ₂ (1 − s ⁿ )�v ⁿ � → 0 follows by (3.30). Moreover, by Lemma 2.11, since F (q + s n v n ) = c j , one obtains that there exists C > 0 such that

F (q + (1 − y)v ⁿ ) − c j = F (q + (1 − y)v n ) − F (q + s n v n ) ≤ C(1 − s n − y)�v ⁿ � H

( R)

for any n ∈ N and y ∈ [0, 1 − s ⁿ ]. Then, by (3.30), � 1 −s

0 F (q + (1 − y)v ⁿ ) − c j dy → 0

and the Lemma follows.

Lemma 3.4 allows us to prove next result that, together with (3.26), is used in Lemma 3.6 to characterize suitable concentration properties of the minimizing sequences of ϕ j in M ^j,j

.

In the sequel we will denote

� 0 = min{1, ^√ ₄ ^λ

d 0 }.

Lemma 3.5 For all j ∈ J , there exists ν ^j ∈ (0, ^λ 8

] such that if u ∈ M ^j,j

, ϕ j (u) ≤ m j + � 0 and F (u(·, ¯y)) < c ^j + ν j for some ¯y ∈ R, then, either

(i) u(·, ¯y) ∈ Γ j and d(u(·, y), Γ j ) ≤ d 0 for all y ≤ ¯y; or (ii) u(·, ¯y) ∈ Γ j

and d(u(·, y), Γ j

) ≤ d 0 for all y ≥ ¯y.

Proof. We prove (ii), noting that the proof of (i) can be done by a simpler and similar contradiction comparison argument.

We prove that there exists ν j ∈ (0, ^λ 8

] such that if u ∈ M j,j

, ϕ j (u) ≤ m j + � 0 and F (u( ·, ¯y)) < c ^j + ν j for some ¯y ∈ R and u(·, ¯y) /∈ Γ ^j then, there exists ˆj ∈ Ω(j) \ {j}

for which u(·, ¯y) ∈ Γ ^ˆ j and d(u(·, y), Γ ^ˆ j ) ≤ d 0 for all y ≥ ¯y (then ˆj = j ^∗ follows since u ∈ M ^j,j

).

Assume by contradiction that there exist ˆj ∈ Ω(j) \ {j}, a sequence (u n ) ⊂ M j,j

such that ϕ j (u n ) ≤ m j +� 0 , and two sequences (y n,1 ), (y n,2 ) ⊂ R such that for any n ∈ N there results

y n,1 ≤ y ^n,2 , lim

n→∞ F (u n (·, y n,1 )) = c j , u n (·, y n,1 ) ∈ Γ ˆ j and d(u n (·, y n,2 ), Γ ˆ j ) > d 0 . Since u n (·, y n,1 ) ∈ Γ ^ˆ j and d(u n (·, y n,2 ), Γ ˆ j ) > d 0 , by (3.25) there exists ¯y n,1 , ¯ y n,2 ∈ [y n,1 , y n,2 ] such that F (u(·, y)) ≥ c ^∗ for any y ∈ (¯y n , ¯ y n,2 ) and d(u(·, ¯y n,1 ), u(·, ¯y n,2 )) = d 0 . By (3.26), we obtain

ϕ j,(y

,+∞) (u n ) ≥ ϕ j,(¯ y

,¯ y

) (u n ) ≥ ^√ ₂ ^λ

d 0 ≥ 2� ⁰ for any n ∈ N.

Hence, for all n ∈ N we get

ϕ _j,(−∞,y

) (u n ) = ϕ j (u n ) − ϕ j,(y

,+∞) (u n ) ≤ m j − � ⁰

which is a contradiction since, by Lemma 3.4, we have lim inf _n→∞ ϕ j,( −∞,y

) (u n ) ≥ m j .

Now, we consider the minimizing sequences of ϕ j on M j,j

and we can prove that the ones which verifies the condition

d(u n (·, 0), Γ j ) = 3

2 d 0 for any n ∈ N,

which can be always assumed modulo y translations, weakly converge to two dimensional

functions in H.

Lemma 3.6 Let j ∈ J and (u n ) ⊂ M j,j

be such that ϕ j (u n ) → m j as n → ∞ and d(u n (·, 0), Γ j ) = ³ ₂ d 0 for all n ∈ N. Then, there exists u j ∈ H and a subsequence of (u n ), still denoted (u n ), such that

(i) u n → u ^j as n → ∞ weakly in H loc ¹ (R ² ), (ii) d(u j (·, y), Γ j ) ≤ C j for any y ∈ R, (iii) lim

y→−∞ d(u j (·, y), K j ) = 0 and lim sup

y→+∞ d(u j (·, y), Γ j

) ≤ d 0 .

Proof. Since ϕ j (u n ) → m j as n → ∞, we can assume that ϕ ^j (u n ) ≤ m j + � 0 for any n ∈ N. Then for any fixed function q ∈ Γ ^j , by Lemmas 2.7 and Remark 3.2 we have that for any y ∈ R

�u ⁿ (·, y) − q� ≤ d(u n (·, y), Γ j ) + D ≤ C j + D.

Then, �u ⁿ − q� ² L

(S

) ≤ 2k( ¯ C + D) ² for any n ∈ N and k ∈ N. Since moreover

�∇u ⁿ � ² L

(S

) ≤ 2(ϕ ^j (u n ) + 2kc j ) we conclude that the sequence (u n − q), and so the sequence (u n − z ⁰ ), is bounded in H ¹ (S ( −k,k) ) for any k ∈ N. Then, with a diagonal argument, we derive that there exists u j ∈ H loc ¹ (R ² ) such that along a subsequence u n − z 0 → u ^j − z ⁰ weakly in H ¹ (S ( −k,k) ) for any k ∈ N. Then u ^j ∈ H and u ⁿ − u ^j → 0 weakly in H ¹ (S (y

,y

) ) for any (y 1 , y 2 ) ⊂ R and (i) follows.

Since by Remark 3.2 we have d(u n (·, y), Γ j ) ≤ C j for any n ∈ N, y ∈ R there exists q n ∈ Γ ^j such that lim sup _{n →∞} �u ⁿ (·, y) − q n (·)� ≤ C j . By Lemma 2.8 we have that along a subsequence, still denoted (q n ), q n → q ∈ Γ ^j as n → ∞ in L ² loc (R) and so, by the Fatou Lemma, we obtain that for a.e. y ∈ R there results

d(u j (·, y), Γ j ) ≤ �u j (·, y) − q� ≤ lim inf

n→∞ �u ⁿ (·, y) − q n (·)� ≤ C j .

Then d(u j (·, y), Γ j ) ≤ C j for a.e. y ∈ R and since u ∈ H, by (3.25) we obtain that d(u j (·, y), Γ j ) ≤ C j for any y ∈ R and (ii) follows.

Let us finally prove (iii). By (3.26) there exists L > 0 such that, for any n ∈ N there exist y n,1 ∈ (−L, 0) and y ^n,2 ∈ (0, L) for which F (u ⁿ (·, y n,1 )), F (u n (·, y n,2 )) ≤ c j + ν j . By Lemma 3.5 we obtain that for any n ∈ N there results u ⁿ (·, y n,1 ) ∈ Γ j and u n (·, y n,2 ) ∈ Γ j

and

d(u n (·, y), Γ j ) ≤ d 0 for any y ≤ −L and d(u ⁿ (·, y), Γ j

) ≤ d 0 for any y ≥ L.

Hence, as in the proof of (ii), in the limit we obtain

d(u j (·, y), Γ j ) ≤ d 0 for any y ≤ −L and d(u ^j (·, y), Γ j

) ≤ d 0 for any y ≥ L.

This proves in particular that, as stated in (iii), lim sup y →+∞ d(u j (·, y), Γ j

) ≤ d 0 . To complete the proof let us show now that lim y→−∞ d(u j (·, y), K j ) = 0.

We first note that, by Remark 2.2, F (u j (·, y)) ≥ c j for any y ≤ −L. Then, by Re-

mark 3.1 (i), (ii), we have that ϕ j,( −∞,−L) (u j ) is well defined and ϕ j,( −∞,−L) (u j ) ≤

lim inf n →∞ ϕ j (u n ) ≤ m j .

By (2.20) we have that for any r > 0, if y ≤ −L and d(u ^j (·, y), K j ) ≥ r then F (u j (·, y)) ≥ min{c j + ν r , c ^∗ }. Then, since ϕ ^j,( −∞,−L) (u j ) ≤ m j , we deduce that lim inf _y→−∞ d(u j (·, y), K j ) = 0. Finally, if we assume by contradiction that

lim sup

y →−∞ d(u j (·, y), K j ) = r > 0

we obtain the existence of a sequence of intervals (y 1,k , y 2,k ) with y 2,k+1 < y 1,k <

y 2,k < −L for any k ∈ N, y ^2,k → −∞ as k → ∞, �u ^j (·, y 1,k ) − u j (·, y 2,k )� = ^r ₂ and F (u j (·, y)) ≥ min{c j + ν r/4 , c ^∗ } for any y ∈ (y ^1,j , y 2,j ). Then, by (3.26) we obtain

ϕ j,(y

,y

) (u j ) ≥ �

2 min{ν r/4 , c ^∗ − c ^j } r

2 , ∀k ∈ N, a contradiction.

The function u j of Lemma 3.6 does not necessarily satisfy the condition F (u j (·, y)) ≥ c j for any y ∈ R. Therefore, we cannot say that u ^j belongs to M ^j and so that it is a minimum for ϕ j on M ^j . In the following we prove that from these limit points u j it is possible to construct two dimensional solutions of (1.3).

Let us introduce, for j ∈ J , the set of limit points of the minimizing sequences of ϕ ^j in M ^j :

L ^j = {u ∈ H / ∃(u n ) ∈ M j,j

such that d(u n (·, 0), Γ j ) = ³ ₂ d 0 for any n ∈ N, ϕ j (u n ) → m j and u n → u weakly in H loc ¹ (R ² ) as n → ∞}.

Remark 3.4 Note that, by the invariance with respect to the y-translation of ϕ j

and by Lemma 3.6, L j is not empty for any j ∈ J . Moreover, if j ∈ J and u ∈ L j

then, by Lemma 3.6, we have sup

y ∈R

d(u( ·, y), Γ ^j ) ≤ C j , lim

y→−∞ d(u( ·, y), K ^j ) = 0 and lim sup

y→+∞ d(u( ·, y), Γ ^j

) ≤ d 0 . For any j ∈ J and u ∈ L ^j we set

y 0,u = inf{y ∈ R / d(u(·, y), K j ) ≥ d 0 and F (u(·, y)) ≤ c ^j },

where we agree that y 0,u = +∞ whenever F (u(·, y)) > c j for any y ∈ R such that d(u( ·, y), K ^j ) ≥ d 0 . Note that, since by Remark 3.4 lim

y→−∞ d(u( ·, y), K ^j ) = 0, we plainly have y 0,u > −∞.

The following Lemma gives us a lower bound of the value ϕ j,( −∞,y

) (u j ) for the function u j ∈ L.

Lemma 3.7 If u ∈ H, j ∈ J and y 0 ∈ R are such that i) d(u( ·, y), K ^j ) → 0 as y → −∞,

ii) u( ·, y ⁰ ) ∈ Γ ^¯ j for some ¯j �= j,