Multiplicity of layered solutions for Allen–Cahn systems with symmetric double well potential

ScienceDirect

J. Differential Equations 257 (2014) 4572–4599

www.elsevier.com/locate/jde

Multiplicity of layered solutions for Allen–Cahn systems with symmetric double well potential

Francesca Alessio

, Piero Montecchiari

DipartimentodiIngegneriaIndustrialeeScienzeMatematiche,UniversitàPolitecnicadelleMarche, Via Brecce Bianche,I60131Ancona,Italy

Received 12September2013 Availableonline 27September2014

Abstract

Westudytheexistenceofsolutionsu: R³→ R²forthesemilinearellipticsystems

−u(x, y, z) + ∇W

u(x, y, z)

= 0, (0.1)

whereW: R²→ R isadoublewellsymmetricpotential.Weusevariationalmethodstoshow,undergeneric non-degenerate propertiesofthesetofonedimensionalheteroclinicconnectionsbetweenthetwominima a_± of W, that(0.1)hasinfinitelymanygeometricallydistinctsolutionsu∈ C²(R³,R²)whichsatisfy u(x,y,z)→ a_±asx→ ±∞ uniformlywithrespectto(y,z)∈ R²andwhichexhibitdihedralsymmetries withrespecttothevariablesyandz.Wealsocharacterizetheasymptoticbehavior ofthesesolutionsas

|(y,z)|→ +∞.

©2014ElsevierInc.All rights reserved.

MSC: 35J60;35B05;35B40;35J20;34C37

Keywords: Entiresolutions;Semilinearellipticsystems;Variationalmethods

* Correspondingauthor.

E-mailaddresses:[email protected](F. Alessio),[email protected](P. Montecchiari).

1 Partially supported by the PRIN2009 grant “Critical Point Theory and Perturbative Methods for Nonlinear DifferentialEquations”(2009BS3H7C).

http://dx.doi.org/10.1016/j.jde.2014.09.001 0022-0396/© 2014ElsevierInc.All rights reserved.

1. Introduction

We consider semilinear elliptic system of the form

−u + ∇W(u) = 0, on R

(1.1)

where u : R

→ R

and W ∈ C

( R

, R) satisfies

(W

) there exist a

∈ R

such that W (a

) = 0, W(ξ) > 0 for every ξ ∈ R

\ {a

} and the Hessian matrices ∇

W (a

) are definite positive;

(W

) there exists R > 1 such that ∇W(ξ) · ξ > 0 for every |ξ| > R, from which μ

= inf

W (ξ ) > 0;

(W

) W ( −ξ

, ξ

) = W(ξ

, ξ

) = W(ξ

, −ξ

) for all (ξ

, ξ

) ∈ R

.

In the sequel, without loss of generality, we will assume that a

= (±1, 0).

Models of this kind are used in various fields of physics, chemistry or biology to describe the behavior of two phase systems. The two components of the function u represent different order parameters and the two phases a

are energetically favorite equilibria. In particular (1.1) enters in the study of phase transitions. Considering the reaction–diffusion parabolic system ∂

u −ε

u +

∇W(u) = 0, a formal analysis see [19], [31] and [34] shows that, as ε → 0

, its solutions tend pointwise to the global minima of W and sharp phase interfaces are produced. System (1.1) appears as first term in the expansion at any point on the interface and the corresponding limit solutions u of (1.1), named two layered transition solutions, satisfy the asymptotic condition

x→±∞

lim u(x, y, z) = a

uniformly w.r.t. (y, z) ∈ R

. (1.2) We recall that in the scalar situation, when u : R

→ R and W : R → R is a double well potential, problem (1.1)–(1.2) has been intensively investigated in the last years. In particular the Gibbons conjecture, proved in [16,17] and [24], establishes that in the scalar case the whole set of solutions to (1.1)–(1.2), can be obtained by translations of the unique odd solution of the one dimensional heteroclinic problem: − ¨q(x) + W

(q(x)) = 0 for x ∈ R and q(±∞) = ±1 (for the more general De Giorgi conjecture, [20], we refer to [26,15,32,21,22] and the surveys [23] and [25]). These results completely describe the set of solutions to (1.1)–(1.2) in the scalar context, showing that the problem is in fact one dimensional.

As firstly shown by S. Alama, L. Bronsard and C. Gui in [1], the one dimensional symmetry of the solutions to (1.1)–(1.2) is generically lost when one considers the vectorial setting. What is basic in their analysis is the fact that, differently from the scalar situation, in the vectorial case the one dimensional heteroclinic problem can have more geometrically distinct solutions. Assuming (W

), (W

), (W

) (the symmetry of W is in fact required only in the ξ

variable) in [1] the existence of an entire solution to (1.1)–(1.2) on R

, asymptotic as y → ±∞ to two different one dimensional solutions is obtained. This is done assuming that the set of minimal one dimensional heteroclinic connections is constituted, modulo translation, by k ≥ 2 distinct elements.

The symmetry condition on the potential, which assures more compactness in the problem,

was dropped by M. Schatzman in [33], where the same kind of solutions are obtained, assuming

that the set of minimal one dimensional heteroclinic connections consists of exactly two distinct

elements which are supposed to be non-degenerate, i.e. the kernels of the corresponding lin-

earized operators are one dimensional. In [33] it is furthermore shown that this kind of finiteness

and non-degeneracy conditions on the set of minimal connections is generic with respect to the choice of the potential W satisfying (W

) and (W

).

A refined version of the result in [1] is given for symmetric potentials in [13] by N.D. Alikakos and G. Fusco. In that paper also examples of potentials W satisfying the discreteness assump- tions on the minimal connections are given together with numerical simulations (see also in this direction [11], [12] and [14]).

We finally refer to [2] where, adapting to the vectorial case an energy constrained variational argument used in [5–7,10], it is shown that (1.1)–(1.2) admits infinitely many planar solutions whenever the set of one dimensional minimal heteroclinic solutions is not connected. These planar solutions exhibit different behavior with respect to the variable y, being periodic in y or asymptotic as y → ±∞ to one dimensional heteroclinic (not necessarily minimal) connections.

They are classified by different values of an “energy” parameter.

These results tell us that if the set of minimal one dimensional solutions has some discrete- ness properties, then the problem (1.1)–(1.2) admits a wide variety of planar solutions. A natural question is whether (1.1)–(1.2) admits solutions depending on more than two variables. Follow- ing a strategy already used in [8] for non-autonomous equations, aim of the present paper is to show that under suitable discreteness and non-degeneracy properties of the set of minimal one dimensional solutions (1.1)–(1.2) admits in fact a multiplicity of different three dimensional solutions.

To precisely describe our results we introduce some notation. Letting z

∈ C

( R, R

) be any fixed function (odd in the first component and even in the second one) such that z

(x) = a

for all x ≥ 1, we consider on the space

H ˆ

= z

+ 

q ∈ H

( R)

 q

(x)x > 0 ∀x = 0 and q(−x) = 

−q

(x), q

(x) 

,

the action functional

ϕ

(q) =



R

1

2 ˙ q(x) 

+ W  q(x) 

dx.

Due to the symmetry of W , the critical points of ϕ

on ˆ H

(endowed with the H

topology) are classical one dimensional heteroclinic solution of our problem. Denoting

m

= inf

Hˆ1

ϕ

and M

= 

q ∈ ˆ H

 ϕ

(q) = m



we ask that M

is finite, does not contain scalar connections and consists of not degenerate critical point of ϕ

. More precisely, we require

( ∗) M

is finite and q

(0) = 0 for all q ∈ M

;

( ∗∗) there exists ω

> 0 such that for all q ∈ M

there results

ϕ

(q)h · h =



R

˙ h(x)

+ ∇

W  q(x) 

h(x) · h(x) dx ≥ ω

h

, ∀h ∈ ˆ H

.

Note that, by (W

), if q ∈ M

then q

(x) = (q

(x), |q

(x) |) ∈ M

too. By uniqueness of the solution of the Cauchy problem, this implies that q

(x) has constant sign on R. Then, the condition q

(0) = 0 in (∗) is actually equivalent to ask (as in the assumption (H4) in [13]) that any scalar connection Q(x) = (Q

(x), 0) ∈ ˆ H

, which always exists by (W

), is not a mini- mum for ϕ

. The assumption (∗∗) is exactly equivalent to the non-degeneracy requirement made in [33]. The arguments contained in [33] can be used and adapted to the present context to show that (∗) and (∗∗) hold generically (with respect to the C

topology) for potential W satisfying (W

), (W

) and (W

).

Denoting ¯q(x) = (q

(x), −q

(x)) and observing that ¯q ∈ M

for any q ∈ M

we can finally state our main result.

Theorem 1.1. If (W

), (W

), (W

), ( ∗) and (∗∗) hold true, there exist infinitely many solutions of the problem

 −v(x, y, z) + ∇W 

v(x, y, z) 

= 0, (x, y, z) ∈ R

,

x→±∞

lim v(x, y, z) − a

 = 0 uniformly w.r.t. (y, z) ∈ R

. (1.3) More precisely, for every j ≥ 2 there exists q ∈ M

and a solution v

∈ C

( R

, R

) of (1.3) such that, denoting ˇv

(x, ρ, θ ) = v

(x, ρ cos θ, ρ sin θ ), it satisfies

(i) ˇv

is periodic in θ with period

, (ii) lim

ˇv

(x, ρ,

+

(

+ k)) =

q(x), if k is odd,

¯q(x), if k is even, uniformly w.r.t. x ∈ R.

By (i) the solution v

is invariant under rotation with respect to the x axes of angles which are multiple of 2π/j and so exhibits dihedral symmetry with respect to the variables y and z.

Moreover, by (ii), Theorem 1.1 gives information about the asymptotic behavior of the function v

along direction orthogonal to the x axes. Indeed the function ˇv

(x, ρ, θ ) is asymptotic as ρ → +∞ to q or ¯q for θ equal to

+

(

+ k) with k ∈ {0, . . . , 2j − 1} odd or even.

As in [3,9] and especially in [8] (see also [27]), the proof of Theorem 1.1 uses variational methods to study an auxiliary problem. Indeed, given j ∈ N, j ≥ 2, setting ¯z = tan(

)z and

P

= 

(x, y, z) ∈ R

 (x, y) ∈ R × (−¯z, ¯z), z ≥ 0  ,

we look for functions v ∈ C

( P

)

satisfying the symmetry conditions v( −x, y, z) = ( −v

(x, y, z), v

(x, y, z)) and v(x, −y, z) = (v

(x, y, z), −v

(x, y, z)) for (x, y, z) ∈ P

and which, for a certain q ∈ M

, solve the auxiliary problem

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

−v(x, y, z) + ∇W 

v(x, y, z) 

= 0, (x, y, z) ∈ P

,

z→+∞

lim v(x, ¯z, z) = q(x) uniformly w.r.t. x ∈ R

∂

v(x, y, z) = 0 (x, y, z) ∈ ∂P

,

x→±∞

lim v(x, y, z) − a

 = 0, uniformly w.r.t. (y, z).

(1.4)

If v solves (1.4) then as z → +∞ we have v(x, −¯z, z) = (v

(x, ¯z, z), −v

(x, ¯z, z)) → (q

(x), −q

(x)) = ¯q(x) and the entire solution v

on R

is obtained from v by recursive re- flections of the prism P

with respect to its faces.

To solve (1.4) we build up a renormalized variational procedure (see [28,29] and the mono- graph [30] for the use of renormalized functionals in other contexts) which takes into account the information we have on the lower dimensional problems. Even if the general strategy of the proof is similar to the one used in [8], the maximum principle, which leads in the scalar situation to ordering properties of the solutions, is lost in the present setting and a different deeper analysis is needed. The proof of Theorem 1.1 is contained in Section 5 and refers to a list of properties of one dimensional and two dimensional solutions of (1.1)–(1.2) studied in Sections 2, 3 and 4.

Remark 1.2. We precise some basic consequences of the assumptions (W

)–(W

), fixing some constants. For all ξ ∈ R

, we set

χ (ξ ) = min 

|ξ − a

|, |ξ − a

|  .

First we note that, since W ∈ C

( R

, R) and ∇

W (a

) are definite positive, then

∀r > 0 ∃ω

> 0 such that if χ (ξ ) ≤ r then W(ξ) ≥ ω

χ (ξ )

. (1.5) Then, since W (a

) = 0, ∇W(a

) = 0 and ∇

W (a

) are definite positive, we have that there exists δ ∈ (0,

), two constants w > w > 0 such that if χ (ξ ) ≤ 2δ then

2w|η|

≤ ∇

W (ξ )η · η ≤ 2w|ξ|

for all η ∈ R

, (1.6) and

wχ (ξ )

≤ W(ξ) ≤ wχ(ξ)

and ∇ W (ξ )  ≤ 2wχ (ξ ). (1.7) 2. One dimensional solutions

In this preliminary section we recall some well known properties of the one dimensional minimal solutions to (1.1) verifying (1.2), i.e., minimal solution to the problem

− ¨q(x) + ∇W  q(x) 

= 0, for x ∈ R and lim

x→±∞

 q(x) − a

 = 0, (P

) and we display some consequences of the assumptions ( ∗) and (∗∗).

Fixed a function z

∈ C

( R, R

) such that z

≤ R, z

odd in the first component and even in the second one and such that z

(t) = a

for all t ≥ 1, we consider on the space

H

= z

+ H

( R)

, the functional

ϕ

(q) =



R

1

2 ˙ q(x)

+ W  q(x) 

dx.

We will study some properties of the minima of ϕ

on H

and we set m

= inf

H1

ϕ

.

Endowing H

with the hilbertian structure induced by the map Q : H

( R)

→ H

, Q(z) = z

+ z, it is well known that ϕ

∈ C

( H

) and that critical points of ϕ

are classical solutions to (P

). Moreover, given any interval I ⊂ R we set

ϕ

(q) =



I

1

2 ˙ q(x) 

+ W  q(x) 

dx.

Note that, for every I ⊆ R, the functional ϕ

is weakly lower semicontinuous on H

. Remark 2.1. Given q = (q

, q

) ∈ R

we denote ˆq = (−q

, q

). Then, setting

H ˆ

= 

q ∈ H

 q(x)

x > 0 ∀x = 0 and q(−x) = ˆq(x), t ∈ R  ,

as in Remark 2.2 in [2] (see also [27]), we can prove that for all q ∈ H

there exists Q ∈ ˆ H

such that ϕ

(Q) ≤ ϕ

(q) and so that

m

= inf

H1

ϕ

= inf

Hˆ1

ϕ

.

Remark 2.2. If q ∈ ˆ H

, then q(x)

≥ 0 for x ≥ 0 and q(x)

≤ 0 for x < 0. Hence χ(q(x)) =

|q(x) − a

| for x ≥ 0 and χ(q(x)) = |q(x) − a

| for x < 0, so that |q(x) − z

(x) |

≤ 2χ (q(x))

+ 2χ(z

(x))

for any x ∈ R. Then, by (1.5) we derive that there exists a constant C, depending on q

, such that

q − z

≤ C



R

W (q) dx + 2



R

χ (z

)

dx ≤ Cϕ

(q) + 2



R

χ (z

)

dx.

Let us fix δ

∈ (0, δ) such that

λ

:= 

2wδ

(δ − δ

) − δ

2 (1 + 2w) > 0. (2.1)

Moreover note that if q ∈ H

( R)

is such that W (q(t)) ≥ μ for all t ∈ (σ, τ) ⊂ R, μ > 0, then

ϕ

(q) ≥ 1

2(τ − σ ) q(τ ) − q(σ ) 

+ μ(τ − σ ) ≥ 

2μ  q(τ ) − q(σ )  . (2.2) Remark 2.3. As a consequence of Remark 2.2 and (2.2), using (W

), as in Lemmas 2.1 and 2.2 in [2], we obtain that there exist R

, C

, T

> 0 such that if q ∈ ˆ H

and ϕ

(q) ≤ m

+ λ

then

(i) q

≤ R

and q − z

≤ C

;

(ii) |q(x) − a

| ≤ δ for all x > T

.

Using Remark 2.3 we plainly obtain

Lemma 2.4. Let (q

) be a sequence in ˆ H

such that ϕ

(q

) ≤ m

+ λ

for all n ∈ N. Then, there exists q ∈ ˆ H

such that, along a subsequence, q

− q → 0 weakly in H

( R)

and ϕ(q) ≤ lim inf

ϕ(q

).

By Lemma 2.4, we obtain that M

= {q ∈ ˆ H

| ϕ

(q) = m

} is not empty. Moreover, as we recalled above, every q ∈ M

is a classical C

( R) solution to problem (P

). It is simple to show that the elements of M

are uniformly exponentially asymptotic to the points a

.

Lemma 2.5. For every q ∈ M

, |q(x) − a

| ≤ ¯δe

w

, for all x ≥ T

.

Proof. Let q ∈ M

. By Remark 2.3 we have |q(x) − a

| ≤ δ for x > T

. Setting φ(x) =

|q(x) − a

|

− ¯δ

e

2w(x−T0)

and using (1.6) we recover ¨ φ(x) ≥ 2wφ(x), φ(T

) ≤ 0 and lim

φ(x) = 0 which imply φ(x) ≤ 0 for x ≥ T

. 2

Lemma 2.4 establishes that every minimizing sequence for ϕ

over ˆ H

is precompact with respect to the weak H

( R)

topology. As in Lemma 2.4 in [2] the result can be improved and we have

Lemma 2.6. Let (q

) be a minimizing sequence for ϕ

over ˆ H

. Then, there exists q ∈ M

such that, along a subsequence, q

− q

→ 0 as n → +∞.

In particular, by Lemma 2.6, for every r > 0 there exists ν

> 0 such that

if q ∈ ˆ H

and

(q, M

) ≥ r then ϕ

(q) ≥ m

+ ν

. (2.3) We finally discuss some consequences of the assumptions ( ∗) and (∗∗). Recall first the discrete- ness assumption ( ∗) on M

:

( ∗) M

is finite and q(0)

= 0 for all q ∈ M

.

Since (∗) requires that M

is finite, we have in particular that

min

p,q∈M1,p=q

q − p

= 5d

> 0. (2.4) The assumption (∗∗) asks that

( ∗∗) there exists ω

> 0 such that for all q ∈ M

there results

ϕ

(q)h · h =



R

˙ h(x) 

+ ∇

W  q(x) 

h(x) · h(x) dx ≥ ω

h

∀h ∈ ˆ H

.

As consequence, using Taylor’s formula and Lemma 2.6 we obtain the following coerciveness

property of ϕ

:

Lemma 2.7. There exists ν

∈ (0, λ

) such that if q ∈ ˆ H

is such that ϕ

(q) ≤ m

+ ν

, then ϕ

(q) − m

≥ ω

4

(q, M

)

.

Proof. We set W =

max

|D

W (ξ ) | and let c

be the constant of the immersion of H

( R)

into L

( R)

. By (2.3) there exists ν

∈ (0, λ

) such that if q ∈ ˆ H

and ϕ

(q) ≤ m

+ ν

then

q − q

≤ min

d

, ω

8W c

 ,

for some q

∈ M

. Since, by Remark 2.3 we have q

≤ R

and q

≤ R

, by the Taylor formula and (∗∗), we obtain that

ϕ

(q) − m

= ϕ

(q) − ϕ

(q

) = ϕ

(q

)(q − q

) + 1

2 ϕ

(q

)(q − q

)(q − q

) +



R



W (q) − W(q

) − ∇W(q

) · (q − q

) − 1

2 ∇

W (q

)(q − q

) · (q − q

)

 dx

≥ ω

2 q − q

− Wq − q

≥

 ω

2 − Wc

q − q



q − q

and the lemma follows. 2

3. Two dimensional heteroclinic type solutions

In this section we display some results concerning the solutions of the two dimensional prob- lem

 −v(x, y) + ∇W  v(x, y) 

= 0, (x, y) ∈ R

x→±∞

lim v(x, y) − a

 = 0, uniformly w.r.t. y ∈ R, (P

) which are asymptotic as y → ±∞ to M

.

Let us consider the renormalized functional ϕ

(v) =



R

 

R

1

2 ∇ v(x, y)

+ W  v(x, y) 

dx − m

 dy

=



R

1

2  ∂

v( ·, y) 

L²

+ ϕ

 v( ·, y) 

− m

dy

which is well defined on the space H

= 

v ∈ H



R



 v( ·, y) ∈ ˆ H

for almost every y ∈ R 

.

On H

, given any interval I ⊂ R, we will also consider the functional

ϕ

(v) =



I

1

2  ∂

v( ·, y) 

L²

+ ϕ

 v( ·, y) 

− m

dy, v ∈ H

.

Note that ϕ

(v) ≥ 0 for all v ∈ H

, I ⊆ R and if q ∈ M

, then the function v(x, y) = q(x) belongs to H

and ϕ

(v) = 0, i.e., the minimal solutions of (P

) are global minima of ϕ

on H

. We will look for bidimensional solutions of (P

) as minima of ϕ

on suitable subspaces of H

. We recall (see e.g. [4]) that ϕ

(and ϕ

for I ⊆ R) is weakly lower semicontinuous on H

with respect to the H

( R

)

. Concerning the coerciveness of ϕ

, we firstly list some basic estimates.

First we note that if v ∈ H

then ϕ

(v( ·, y)) ≥ m

for almost every y ∈ R and so

∂

v 

≤ 2ϕ

(v) ∀v ∈ H

. (3.1) Moreover, since W (ξ ) ≥ 0 for any ξ ∈ R

, we have

ϕ

(v) ≥

y₁



y₀

ϕ

 v( ·, y) 

− m

dy + 1

2 ∂

v 

≥ 1

2 ∂

v 

+ 1

2 ∂

v 

− m

(y

− y

) from which we derive

∇v

≤ 2ϕ

(v) + 2m

(y

− y

) ∀v ∈ H

, (y

, y

) ⊂ R. (3.2)

Finally, if v ∈ H

then v(x, ·) ∈ H

( R)

for almost every x ∈ R. Therefore, if y

< y

∈ R then v(x, y

) − v(x, y

) = 

y₀

∂

v(x, y) dy holds for almost every x ∈ R, whenever v ∈ H

. Hence, if v ∈ H

and y

< y

∈ R, by (3.1) we obtain

v( ·, y

) − v(·, y

)

L²

≤ |y

− y

|



R y₁



y₀

|∂

v |

dy dx ≤ 2ϕ

(v) |y

− y

|. (3.3)

By (2.3), if (y

, y

) ⊂ R and v ∈ H

are such that

(v( ·, y), M

) ≥ d > 0 for almost every y ∈ (y

, y

), then there exists ν

> 0 such that ϕ

(v( ·, y)) ≥ m

+ ν

and hence

ϕ

(v) ≥ 1 2(y

− y

)



R

 

y₀

 ∂

v(x, y)  dy



dx + ν

(y

− y

)

≥ 1

2(y

− y

) v( ·, y

) − v(·, y

)

L²

+ ν

(y

− y

)

≥ 

2ν

 v( ·, y

) − v(·, y

) 

L²

. (3.4)

As a consequence of (3.3) and (3.4) we get information on the asymptotic behavior as y → ±∞

of the functions in the sublevels of ϕ

. More precisely, as in Lemma 3.2 in [2] and Lemma 3.1

in [8], we have

Lemma 3.1. If v ∈ H

and ϕ

(v) < +∞, then there exists q ∈ M

such that v(·, y) −q

→ 0 as y → +∞.

Considering the symmetry of W , we look for solutions v of (P

) which satisfy the symmetry condition (v(x, −y)

, v(x, −y)

) = (v(x, y)

, −v(x, y)

) and which connect different elements of M

as y → ±∞. Hence we define

H ˜

= 

v ∈ H

 v(x, −y) = ˜v(x, y), (x, y) ∈ R

 where we denote ˜v = (v

, −v

) for every v = (v

, v

) ∈ R

.

We remark that by definition, if v ∈ ˜ H

, then, a.e. in R

there results v( −x, y)

= −v(x, y)

and v( −x, y)

= v(x, y)

, while v(x, −y)

= v(x, y)

and v(x, −y)

= −v(x, y)

,

from which in particular there results v(−x, −y) = −v(x, y). Hence, if v ∈ ˜ H

is such that v(·, y) − q

→ 0 as y → +∞ for some q ∈ M

, then v(·, y) − ¯q

→ 0 as y → −∞, where ¯q(x) = −q(−x).

Remark 3.2. Using the symmetry properties of the functions in ˜ H

we gain coerciveness prop- erty for ϕ

. Precisely, we have that there exists a constant ˜ C > 0 such that if (y

, y

) ⊂ R and if v ∈ ˜ H

is such that v

≤ R then

v − z

≤ ˜Cϕ

(v) + 2 ˜C(y

− y

)m

+ 4(y

− y

)



R

χ (z

)

dx. (3.5)

Indeed, arguing as in Remark 2.2, we have that |v(x, y) − z

(x) |

≤ 2χ(v(x, y))

+ 2χ(z

(x))

on R

. By (1.5), since v

≤ R, there exists a constant ˜C such that |v(x, y) − z

(x) |

≤

˜CW(v(x, y)) + 2χ(z

(x))

and (3.5) plainly follows.

Then, we have

Lemma 3.3. Let (v

) be a sequence in ˜ H

such that ϕ

(v

) ≤ C and v

≤ R for all n ∈ N and some C > 0. Then, there exists v ∈ ˜ H

such that, up to a subsequence, v

− v → 0 weakly in H

( R × (−L, L))

for every L > 0.

Proof. By (3.2) and (3.5), (v

− z

) is bounded in H

( R × (−L, L))

for all L > 0 and so there exists a subsequence (v

) of (v

) and a function v such that v − z

∈ 

L>0

H

( R × (−L, L))

and v

− z

→ v − q

weakly in H

( R × (−L, L))

for every L > 0 and a.e. in R

. Since v − z

∈ 

L>0

H

( R × (−L, L))

we have that v(·, y) − z

∈ H

( R)

for almost every y ∈ R and by pointwise convergence, v ∈ ˜ H

follows. 2

Remark 3.4. Standard semicontinuity arguments (see e.g. the proof of Lemma 3.3 in [8]) show

that if (v

)

⊂ ˜ H

, v ∈ ˜ H

are such that v

−v → 0 weakly in H

( R ×(−L, L))

for all L > 0

and if q ∈ H

then v(·, y) − q

≤ lim inf

v

( ·, y) − q

for almost every y ∈ R.

Remark 3.5. Note that as a consequence of the symmetry properties of the functions in ˜ H

, we have that if v ∈ ˜ H

then

(v( ·, 0), M

) ≥ d

.

Indeed, given any q ∈ M

, considering ¯q(x) = −q(−x), we have ¯q ∈ M

and then, by (∗), q = ¯q and q − ¯q

≥ 2d

. By symmetry, if v ∈ ˜ H

, we have v(−x, 0)

= −v(x, 0)

and v(x, 0)

= 0 and then

v( ·, 0) − ¯q 

L²

=



R

v(x, 0)

− ¯q(x)



+ ¯ q(x)



dx

=



R

 v(x, 0)

− q(x)



+  q(x)



dx =  v( ·, 0) − q 

L²

,

from which we deduce v(·, 0) − q

≥

q − ¯q

≥ d

. Since q ∈ M

is arbitrary, we con- clude that

(v( ·, 0), M

) ≥ d

.

Having in mind Lemma 3.1, given q ∈ M

we look for minima of ϕ

over the class H ˜

= 

v ∈ ˜ H

  lim

 v( ·, y) − q 

L²

= 0  and we set

m

= inf

v∈ ˜H2,q

ϕ

(v) and M

= 

v ∈ ˜ H

 ϕ

(v) = m

 .

Remark 3.6. One can plainly see that m

< +∞ for all q ∈ M

. Moreover, by (∗) and Lemma 3.1, we have m

= inf

2

ϕ

= min

m

. We denote M

= {q ∈ M

| m

= m

} and M

= 

v ∈ ˜ H

 ϕ

(v) = m



noting that M

= 

q∈M^∗1

M

. Finally, we set

˜λ =

min{m

− m

, q ∈ M

\ M

} if M

\ M

= ∅

1 if M

\ M

= ∅ (3.6)

Setting

¯λ = 1

8 min{λ

; ˜λ; d

√

ν

} (3.7)

(where λ

is given in (2.1), ˜λ in (3.6) and ν

by (3.4) corresponding to d =

) and letting r

∈ (0, min{

, √

2¯λ}) such that

sup 

ϕ

(q)  q ∈ ˆ H

,

(q, M

) ≤ 2r

 ≤ m

+ ¯λ, (3.8)

we have the following concentration result:

Lemma 3.7. If L > 0, v ∈ ˜ H

, ϕ

(v) ≤ m

+ ¯λ and v(·, y

) − q

≤ r

for some y

∈ (0, L) and q ∈ M

, then

q ∈ M

and  v( ·, y) − q 

L²

≤ d

for all y ∈ (y

, L).

Proof. Let v, q, L, y

be as in the statement. We define

w(x, y) =

 v(x, y) if x ∈ R and 0 ≤ y ≤ y

v(x, y

)(y

+ 1 − y) + q(x)(y − y

) if x ∈ R and y

≤ y ≤ y

+ 1,

q(x) if x ∈ R and y ≥ y

+ 1,

and w(x, y) = −w(−x, −y) for x ∈ R and y < 0, noting that w ∈ ˜ H

and so ϕ

(w) = 2ϕ

(w) ≥ m

≥ m

. Since w(·, y) − q

≤ 2r

for y ∈ (y

, y

+ 1), by (3.8) we obtain

ϕ

(w) =

y



y0

1 2



R

 v(x, y

) − q(x) 

dx + ϕ

 w(x, y) 

− m

dy ≤ 2¯λ. (3.9)

In particular, by (3.9) and using (3.7), we derive that

ϕ

(w) = ϕ

(v) + 2ϕ

(w) ≤ m

+ 5¯λ < m

+ ˜λ and so, by the definition of ˜λ in (3.6), that q ∈ M

.

Moreover, by (3.9) again, we obtain that

≤ ϕ

(w) ≤ ϕ

(v) − ϕ

(v) + 2¯λ, from which we derive ϕ

(v) ≤ 3¯λ.

If we assume by contradiction that there exists y

∈ (y

, L) such that v(·, y

) − q

> d

, then, by (3.3) there exists (y

, y

) ⊂ (y

, y

) such that v(·, y

) − v(·, y

) 

≥

and

≤ v(·, y) −q

≤ d

for any y ∈ (y

, y

). In particular, since p−q

> d

for all p = q ∈ M

, we obtain

(v( ·, y), M

) ≥

(v( ·, y), M

) = v(·, y) − q

≥

for almost every y ∈ (y

, y

). Then, by (3.4) and (3.7), we get the contradiction 3¯λ ≥ ϕ

(v) ≥ √

2ν

v(·, y

) − v( ·, y

)  ≥ √

2ν

2

≥ 4¯λ which completes the proof. 2

Using (2.3) and (3.4) we fix ¯  > 0 such that if I is any real interval with length |I| ≥ ¯ then if v ∈ ˜ H

and

 v( ·, y), M

 > r

for almost every y ∈ I then ϕ

(v) ≥ m

+ 2¯λ.

(3.10) As a simple consequence of Lemma 3.7, we then obtain

Lemma 3.8. If L ≥ ¯, v ∈ ˜ H

and ϕ

(v) ≤ m

+ ¯λ then there exists q ∈ M

such that v(·, y) − q

≤ d

for all y ∈ ( ¯, L).

Proof. Since, by Remark 3.5,

(v( ·, 0), M

) ≥ d

and since ϕ

(v) ≤ m

+ ¯λ, by defi-

nition of ¯  there exists y

∈ (0, ¯) and q ∈ M

such that v(·, y

) − q

≤ r

. Then the lemma

follows by Lemma 3.7. 2

In particular

Lemma 3.9. If v ∈ ˜ H

and ϕ

(v) ≤ m

+ ¯λ then there exists q ∈ M

such that v(·, y) − q

≤ d

for all y ≥ ¯.

The results stated in Lemmas 3.1 and 3.9 imply

 v ∈ ˜ H

 ϕ

(v) ≤ m

+ ¯λ 

⊂ 

q∈M^∗1

 v ∈ ˜ H

 ϕ

(v) ≤ m

+ ¯λ 

and Lemma 3.9, Lemma 3.3 allow us to use the direct method of the Calculus of Variation to show that ϕ

admits a minimum in every class ˜ H

with q ∈ M

.

Proposition 3.10. For every q ∈ M

there exists v ∈ M

.

Proof. Fixed q ∈ M

, let (v

)

be a minimizing sequence for ϕ

in ˜ H

. Then, by (2.4), the definition of ˜ H

and Lemma 3.9, for all n ∈ N there results

d_L2

 v

( ·, y), M

 = v

( ·, y) − q 

L²

≤ d

for y ≥ ¯. (3.11) Now, by Lemma 3.3, there exists v ∈ ˜ H

such that, along a subsequence, v

→ v weakly in H

( R × (−L, L))

for every L > 0 and a.e. in R

. By semicontinuity, ϕ

(v) ≤ m

and by (3.11) and Remark 3.4, v(·, y) − q

≤ d

for y ≥ ¯. Then, by Lemma 3.1 since

(q, M

\ {q}) ≥ 2d

, we obtain v(·, y) − q

→ 0 as y → +∞. Hence v ∈ ˜ H

and ϕ

(v) = m

follows. 2 Remark 3.11. By symmetry of W , as in Lemma 5.5 below, it can be proved that every v ∈ M

is a weak solution of −v + ∇W(v) = 0 on R

. By (3.2) and (3.5), using bootstrap arguments, we can conclude that v is a C

solution which satisfies the symmetry conditions

v( −x, y) = ˆv(x, y) and v(x, −y) = ˜v(x, y) for all (x, y) ∈ R

. Moreover, given v ∈ ˜ H

, let P (v) be defined as

P (v)(x, y) =



|v(x,y)|

if |v(x, y)| > R, v(x, y) if |v(x, y)| ≤ R.

The assumption (W

) guarantees that ϕ

(P (v)) < ϕ

(v) whenever v ∈ ˜ H

is such that v

> R, hence, since P (v) ∈ ˜ H

for every v ∈ ˜ H

, we derive that v

≤ R for every v ∈ M

. Hence, by (1.7) and classical Schauder estimates, we can conclude that sup

v

< +∞.

Remark 3.12. For every v ∈ M

we have |v(x, y) − a

| → 0 as x → ±∞ uniformly w.r.t.

y ∈ R. Indeed, assume by contradiction that v ∈ M

and there exist δ > 0 and a sequence (x

, y

) ∈ R

with x

→ +∞ such that |v(x

, y

) − a

| ≥ 2δ for all n ∈ N. By Remark 3.11 there exists ρ > 0 such that |v(x, y) − a

| ≥ δ for all (x, y) ∈ B

(x

, y

), n ∈ N. If along a sub- sequence we have y

→ y

, then |v(x, y) −a

| ≥ δ for all (x, y) ∈ B

2

(x

, y

) and n large, which

is not possible since v(·, y) ∈ H

= z

+ H

( R)

for almost every y ∈ R. If y

→ +∞ (anal- ogous to y

→ −∞) along a subsequence, we plainly obtain lim sup

v(·, y) − q

> 0 which contradicts v ∈ M

.

By definition, we have that if v ∈ M

then |v(·, y) − q| → 0 as y → +∞ with respect to the L

metric. We can in fact say more:

Lemma 3.13. If v ∈ M

for some q ∈ M

, then v(·, y) − q

→ 0 as y → +∞.

Proof. If v ∈ M

, we know that v solves (P

) and by Remark 3.11 that v

< +∞. If by contradiction lim sup

v(·, y) − q

≥ 2ρ

> 0 then, there exists a sequence (x

, y

) ∈ R

with y

→ +∞ such that |v(x

, y

) − q(x

) | ≥ ρ

. Then, since v − q

< +∞, there exists r

> 0 such that |v(x, y

) − q(x)| ≥ ρ

/2 whenever |x − x

| ≤ r

and n ∈ N. Hence we get the contradiction v(·, y

) − q

≥ r

ρ

/2 for all n ∈ N. 2

As last step in this section we use condition ( ∗∗) to obtain L

compactness of the minimizing sequences in ˜ H

.

Lemma 3.14. Let q ∈ M

and (v

) be a minimizing sequence in ˜ H

with v

≤ R. Then

(v

, M

) → 0 as n → +∞.

Proof. Let (v

) ⊂ ˜ H

be such that ϕ

(v

) → m

. By Lemma 3.3 and arguing as in the proof of Proposition 3.10, given any subsequence of (v

), there exists v ∈ M

and a sub-subsequence, denoted again by (v

), such that v

− v → 0 weakly in H

( R × (−L, L))

for any L > 0. To prove the lemma it will be sufficient to show that v

− v

→ 0.

Firstly note that, if F , F

and F

are lower semicontinuous functionals such that F = F

+ F

, if v

→ v and F (v

) → F (v) then also F

(v

) → F

(v) and F

(v

) → F

(v). Iteratively ap- plying this property in our case, since ϕ

(v

) → ϕ

(v), we obtain that for any L, M > 0, as n → +∞, we have



−L



M

W (v

) dx dy →



−L +∞



M

W (v) dx dy (3.12)

+∞



L

ϕ



v

( ·, y) 

− m dy →



L

ϕ

 v( ·, y) 

− m dy (3.13)

We use (3.12) to deduce that

v

− v → 0 strongly in L



R × (−L, L) 

for any L > 0. (3.14) Indeed, given L > 0 and ε > 0 we fix M > 0 such that |v(x, y) − a

| < 1 for x > M and |y| < L and moreover 

−L



M

W (v) dx < ε. By (3.12), there exists ¯n ∈ N such that



−L



M

W (v

) dx dy < 2ε for n ≥ ¯n. Using (1.5), we recover that there exists C > 0 such

that χ (ξ ) ≤ CW(ξ) for any |ξ| ≤ R so that, since v

, v

≤ R, we deduce that for

any x > M and |y| < L there results |v

(x, y) − v(x, y)|

≤ 2(χ(v

(x, y))

+ χ(v(x, y))

) ≤ 2C(W (v

(x, y)) + W(v(x, y))). Then v

− v

≤ 4C 

−L



|x|>M

W (v

) + W (v) dx dy ≤ 12Cε, and since ε is arbitrary and v

→ v in L

( [−M, M] × [−L, L])

, (3.14) follows.

To conclude the proof we show that

∀ε > 0 ∃L

> 0, ¯n ∈ N such that v

− v

≤ ε, ∀n ≥ ¯n. (3.15)

It is not restrictive to assume that ϕ

(v

) ≤ m

+ ¯λ for any n ∈ N, so that, by Lemma 3.9 we have

v

( ·, y) − q 

L²

≤ d

for all n ∈ N and y ≥ ¯ (3.16) Letting ε > 0, we fix L

≥ ¯ such that 

L_ε

ϕ

(v( ·, y)) − m

dy <

and, by (3.13), we fix also n

∈ N such that



Lε

ϕ

 v

( ·, y) 

− m

dy < ε for all n ≥ n

. (3.17)

Then, letting ν

> 0 as in Lemma 2.7 and denoting A

= {y > L

| ϕ

(v

( ·, y)) − m

> ν

}, by (3.17) we have meas(A

) ≤

for any n ≥ n

. Since L

≥ ¯, by (3.16),



An

 v

( ·, y) − q 

L²

dy ≤ εd

ν

for any n ≥ n

, (3.18)

while, since by Lemma 2.7 for y ∈ (L

, +∞) \ A

and n ≥ n

we have

 v

( ·, y) − q 

L²

=

 v

( ·, y), M



≤ 4 ω

 ϕ



v

( ·, y) 

− m  , we recover



(Lε,+∞)\An

v

( ·, y) − q 

L²

dy ≤ 4ε

ω

for any n ≥ n

. (3.19)

By (3.18) and (3.19) for every n ≥ n

we obtain 

(L_ε,+∞)

v

( ·, y) − q

dy ≤ ε(

+

) and, by semicontinuity, the same estimate holds with v

replaced by v. Hence 

(Lε,+∞)

v

( ·, y) − v( ·, y)

dy ≤ 2ε(

+

) for any n ≥ n

and (3.15) follows by the symmetry of v and v

. 2 Remark 3.15. By (2.4) and Lemma 3.9 we have that

( M

, M

) ≥ 3d

for all p = q ∈ M

and L > ¯  + 1.

Indeed, if v ∈ M

and w ∈ M

, by Lemma 3.9 we have p − q

≤ 2d

+ v(·, y) −

w( ·, y)

for all y ≥ ¯, and so, by (2.4), v(·, y) − w(·, y)

≥ 3d

for all y ≥ ¯. Then, for