A BSTRACT . We consider a class of periodic Allen-Cahn equations (1) − ∆u(x, y) + a(x, y)W

Doubly Periodic Media

F RANCESCA A LESSIO , C HANGFENG G UI & P IERO

M ONTECCHIARI

A ^BSTRACT . We consider a class of periodic Allen-Cahn equations (1) − ∆u(x, y) + a(x, y)W

(u(x, y)) = 0 , (x, y) ∈ R ² , where _{a ∈ C(R} ² ) is an even, periodic, positive function repre- senting a doubly periodic media, and W : R → R is a classi- cal double well potential such as the Ginzburg-Landau potential W (s) = (s ² − 1 ² ) ² . We show the existence and asymptotic behav- ior of a saddle solution on the entire plane, which has odd symme- try with respect to both axes, and even symmetry with respect to the line _{x = y} . This result generalizes the classic result on saddle solutions of Allen-Cahn equation in a homogeneous medium.

1. I NTRODUCTION

In this paper, we deal with a class of semilinear elliptic equations of the form (E) − ∆u(x, y) + a(x, y)W

(u(x, y)) = 0 , (x, y) ∈ R ² , where we assume the following:

(H 1 ) _{a ∈ C(R} ² ) is strictly positive, and the following hold:

(i) a(x + 1 , y) = a(x, y) = a(x, y + 1 ⁾ for all (x, y) ∈ R ² ;

(ii) a(x, y) = a(x, −y) = a(−x, y) , a(x, y) = a(y, x) for each (x, y) ∈ R ² .

(H 2 ) _{W ∈ C} ² (R) satisfies _{W (±} 1 _{) =} 0, W

(± 1 ) > 0 and W (s) > 0 for any s ∈ (− 1 , 1 ) , and W (s) = W (−s) for _{s ∈ R} .

1 (actual pages TBD)

Indiana University Mathematics Journal c , Vol. 65, No. 1 (2016)

The equations may be regarded as generalizations of the classical Allen-Cahn model of phase transition to an inhomogeneous medium, periodic in both coor- dinates on a plane. The main result of the present paper concerns the existence of saddle solutions to (E) together with the characterization of their asymptotic behaviour.

The problem of existence and stability of saddle solutions and various gener- alizations has been widely studied in the homogeneous case. It was introduced by H. Dang, P. C. Fife, and L. A. Peletier in [10], where the authors consider the case in which ^a is a positive constant and the potential ^W satisfies (H

) and suitable monotonicity properties. By using a sub-supersolution method, they show that, in these cases, (E) has a unique saddle solution _{v ∈ C} ² (R ² ) (i.e., a bounded solu- tion which has the same sign of the product function xy ), is odd in each variable, and is symmetric with respect to the diagonals _{y = ±x} . The saddle solution is asymptotic at infinity to _± 1 along any directions not parallel to the coordinate axes, and it may represent a phase transition with cross interfaces.

Stability properties of the saddle solution were studied by M. Schatzman in [25], where it is shown that its Morse index is at least equal to one, and, more recently, by M. Kowalczyk and Y. Liu in [18], where non-degeneracy properties of the saddle solution are studied.

The saddle solutions are examples of 2k-end solutions, that is, planar solutions whose nodal set is asymptotic at infinity to 2k half lines. These solutions of (E) have been studied in the homogeneous case in [12], [19], [16], and [20] (see also [11]). In particular, in [16] (see also [15]), it is proved that any four-end solution is, modulo roto-translations, even in its variables, and monotonic in x and y on the first quadrant. In [20], it is shown that the set of even four-end solutions constitutes a continuum containing the saddle solution. General 2k-end solutions with nearly parallel ends have been obtained in [12] for every k ∈ N . Other 2k-end solutions with diedral symmetry have been built in [2]. Along directions parallel to the end lines, the 2k-end solutions are asymptotic to the unique odd one-dimensional connection ^θ 1 between ± 1, suitably reflected and rotated; and along different directions, they tend to ± 1 in an alternate way with respect to the end lines.

Further generalizations of the study of saddle-type solutions have been made in higher dimensions. X. Cabr´e and J. Terra in [8] and [9] found, in all even dimensions, saddle-type solutions (vanishing on the Simons cones) characterizing their instability for dimensions _{n =} 4 , 6 (see also [7] and [22]); and we refer to [4] and [5] for the study of saddle-type solutions on R ³ , as well as their asymp- totic behaviour. A vectorial version of the saddle solution has been obtained by S. Alama, L. Bronsard, and C. Gui in [1], where systems of autonomous Allen- Cahn equations have been considered on the plane (see [17] and [6] for related studies on R ³ ).

In the present work, we introduce and study the problem of existence of

saddle-type solutions for the non-autonomous Allen-Cahn equation. We use a

global variational approach suitable to recovering the existence of a saddle-type solution of (E), providing a characterization of its asymptotic behaviour.

As we have recalled above, in the autonomous case, the saddle solution is asymptotic as _{y → +∞} to the unique odd one-dimensional solution θ ₁ (x) of (E), such that lim

θ ₁ (x) = ± 1. When a is periodically dependent on the space variables, the analogues of the one-dimensional connection θ ₁ are solutions to (E) on R ² that are one-periodic in the variable ^y , and asymptotic as x → ±∞ to the minima of the potential ± 1.

As a preliminary step in our work, we first consider in Section 2 the problem of finding this kind of solution to (E) on R ² , that is, solutions to the problem

(1.1)

 

 

 



−∆u(x, y) + a(x, y)W

(u(x, y)) = 0 , (x, y) ∈ R ² , u(x, y) = u(x, y + 1 ), (x, y) ∈ R ² ,

x→±∞

lim u(x, y) = ± 1 , uniformly in _{y ∈ R.}

Following [23] and [3], we use variational methods, and, letting S ₀ = R × [ 0 , 1 ] , we look for minima of the action functional

ϕ(u) = Z

S0

1

2 ^{|∇u(x, y)|}

2 + a(x, y)W (u(x, y)) dx dy

on the class E ₀ = n

u ∈ H loc ¹ (R × [ 0 , 1 _{]) |}

u(x, y) = −u(−x, y) for _{x ∈ R,} and 1 ≥ u(x, y) ≥ 0 for x > 0 ^o . Denoting by K the set of minima of ^ϕ on ^E 0 , we show in Section 2 that K is not empty, is compact, and is constituted by classical solutions to (1.1), which can be extended to solutions to (1.1). In Section 3, we study other compactness properties of _K useful to prove, in Section 4, our main result.

Theorem 1.1. There exists _{v ∈ C} ² (R ² ) , a solution of (E) on R ² that verifies the following:

(a) v(x, y) > 0 on the first quadrant in R ² ;

(b) v(x, y) = −v(−x, y) = −v(x, −y) and v(x, y) = v(y, x) on R ² ; and such that

dist

1

(v, K) → 0 as j → +∞.

The function ^v is a saddle-type solution to the situation where (E) is odd

with respect to each of its variables, symmetric with respect to the diagonal, and

strictly positive on its first quadrant. Note also that v has the same sign of the

product function xy , and its nodal set coincides with the union of the coordinate

axes. Analogously to the autonomous case, along directions parallel to the axes,

v is uniformly asymptotic to the set _K that is constituted by two-dimensional solutions, periodic in the variable y , which connects as _{x → ±∞} the pure phases

± 1.

Adapting to the present context some of the ideas developed in [1] and [2], the proof of Theorem 1.1 uses variational methods to study an auxiliary problem.

Indeed, using the symmetry properties of the functions a and W , we can restrict ourselves to consider the region T = {(x, y) ∈ R ² | 0 ≤ |x| ≤ y} , where we look for solutions _{u ∈ C} ² _{(T )} of (E) that satisfy the Neumann boundary condition on

∂T , are odd in the variable x , are positive for x > 0, and are asymptotic to _K as y → +∞ . Then, the saddle solution ^v on R ² is obtained by recursive reflections of ^u with respect to the faces of P . To solve the auxiliary problem, we build up in Section 4 a renormalized minimization procedure inspired by the one used in [3] (related variational renormalization procedures are used by P. H. Rabinowitz and E. Stredulinsky in different settings; see [24]) that takes into account refined properties of the functional ϕ studied in Sections 2 and 3.

Remark 1.2. Note that, by (H

), we have that _{a =} min

a(x, y) > 0.

Moreover, by (H

), there exist w ₀ > 0 and ρ ₀ ∈ ( 0 , ¹ ₆ ) such that (1.2) If _{s ∈ (} 1 ₋ 2 ρ ₀ , 1 ₊ 2 ρ ₀ ),

then W

(s) ≥ 2 w ₀ , |W

(s)| ≥ 2 w ₀ | 1 _{− s|} and _{W (s) ≥ w} 0 | 1 _{− s|} ² . We derive also that there exists w > 0 such that, if 0 _{≥ s ≥} 1, we have that W (s) ≥ w| 1 _{− s|} ² .

2. T HE P ERIODIC H ETEROCLINIC P ROBLEM

In this section, we study the problem

(2.1)

 

 

 



−∆u(x, y) + a(x, y)W

(u(x, y)) = 0 , (x, y) ∈ R ² , u(x, y) = u(x, y + 1 ), (x, y) ∈ R ² ,

x→±∞

lim u(x, y) = ± 1 , uniformly in _{y ∈ R.}

We are particularly interested in some variational properties related to (2.1). More precisely, letting ^S 0 = R × [ 0 ^, 1 ^] , in this section we will study the functional

ϕ(u) = ZZ

S0

1

2 ^{|∇u(x, y)|}

2 + a(x, y)W (u(x, y)) dx dy

on the class E ₀ = n

u ∈ H loc ¹ (S ₀ ) |

u(x, y) = −u(−x, y) for _{x ∈ R} and 1 ≥ u(x, y) ≥ 0 for x > 0 ^o .

2.1. Minimizing ϕ on E ₀ : existence. As a first step in our study, we show that ϕ admits a minimum on E ₀ that will be a solution of (2.1).

We note that ϕ : E ₀ → [ 0 _{, +∞]} , and it is sequentially lower semicontinuous with respect to the weak H _loc ¹ (S ₀ ) topology. Moreover, note that if _{u ∈ E} 0 , then u(·, y) ∈ H loc ¹ (R) for almost every _{y ∈ (} 0 , 1 ) , and so, given x ₁ < x ₂ ∈ R , we obtain

Z 1

0 |u(x 2 , y) − u(x 1 , y)| ² d y = Z 1

0

Z

x₁

∂

u(x, y) dx    

2

d y

≤ |x 2 − x 1 | Z 1

0

 Z

|∂

u(x, y)| ² d x

 d y

≤ 2 ϕ(u)|x 2 − x 1 |.

By the previous estimate, we can deduce that, if ϕ(u) < +∞ , then the function x ∈ R ֏ u(x, ·) ∈ L ² ( 0 ^, 1 ⁾ is H¨older continuous, verifying

(2.2) u(x ₂ , ·) − u(x 1 , ·) ²

₀

₁

≤ 2 _ϕ(u)|x 2 − x 1 |, ∀ x 1 , x ₂ ∈ R.

Hence, we obtain the following result.

Lemma 2.1. For all ^{r >} 0, there exists ^µ

> 0 such that, if u ∈ E 0 satisfies ku(x, ·) − 1 _k

0

1

≥ r for almost every x ∈ (x 1 , x ₂ ) ⊂ ( 0 _{, +∞)} , then

Z

Z ₁

0

1 2 ^|∇u|

2 + a(x, y)W (u) dy dx (2.3)

≥ 1

2 (x ₂ − x 1 )

u(x 2 , ·) − u(x 1 , ·) ²

₀

₁

+ 1 2 ^µ

r

2 (x ₂ − x 1 )

≥ µ

ku(x 1 , ·) − u(x 2 , ·)k

0

1

. Proof. Let us consider on ^{H 1 (} 0 ^, 1 ⁾ the functional

F(q) = Z 1

0

1

2 ^| _q(y)| ˙ ² + aW (q(y)) dy.

Note that if (q

) ⊂ H ¹ ( 0 , 1 ) is such that 1 _{≥ q}

≥ 0 on ( 0 , 1 ) and F(q

) → 0, then

(2.4) _kq

− 1 _k

0

1

→ 0 as _{n → ∞.}

Indeed, if F(q

) → 0, we have Z 1

0 | q ˙

| ² d y → 0 and Z 1

0

W (q

) dy → 0. Since 1 ≥ q

≥ 0, by Remark 1.2 we deduce

Z 1

0 |q

(x) − 1 | ² d y ≤ 1 w

Z 1 0

W (q

) dy → 0 ^,

and (2.4) follows.

By (2.4), we deduce that, for any r > 0, there exists µ

> 0 such that, if q ∈ H ¹ ( 0 , 1 ) , 1 _{≥ q ≥} 0 on ( 0 , 1 ) , and _{kq −} 1 _k

0

1

≥ r , then _{F(q) ≥} ¹ ₂ µ ²

.

If _{u ∈ E} 0 , we have 1 ≥ u(x, ·) ≥ 0 on ( 0 , 1 ) for almost every x > 0, and so, if ku(x, ·)− 1 _k

0

1

≥ r for almost every _{x ∈ (x} 1 , x ₂ ) , we have F(u(x, ·)) ≥ ¹ ₂ µ

² for almost every x ∈ (x 1 , x ₂ ) . Then, we derive

Z

Z 1 0

1 2 ^|∇u|

2 + a(x, y)W (u) dy dx

≥ Z

x1

1 2

Z 1

0 |∂

u(x, y)| ² d y + F(u(x, ·)) dx

≥ 1

2 (x ₂ − x 1 )

u(x ₂ , ·) − u(x 1 , ·) ²

₀

₁

+ 1 2 ^µ

r

2 (x ₂ − x 1 )

≥ µ

ku(x 2 , ·) − u(x 1 , ·)k

0

1

,

and the lemma follows. ^❐

As a consequence of Lemma 2.1, we can easily characterize the asymptotic behaviour of the function _{u ∈ E} 0 such that ϕ(u) < +∞ .

Lemma 2.2. Let _{u ∈ E} 0 with ϕ(u) < +∞ . Then, _{ku(x, ·) −} 1 _k

0

1

→ 0 as _{x → +∞} .

Proof. Note that, by Lemma 2.1, we have lim inf

_{ku(x, ·)−} 1 _k

0

1

= 0, and we have to show that also lim sup

_{ku(x, ·) −} 1 _k

0

1

= 0.

If not, there exists _{r ∈ (} 0 , ¹ ₄ ) such that lim sup

_{ku(x, ·)−} 1 _k

0

1

> 2 r . Then, by (2.2), there exists a sequence of disjoint intervals ^(σ

, τ

) with 0 ^{< σ}

<

τ

< σ

₁ → +∞ , as i → +∞ , such that

ku(τ

, ·) − u(σ

, ·)k

0

1

≥ t and r ≤ ku(x, ·) − 1 k

0

1

≤ 2 ^{r ,} for x ∈ S

i

(σ

, τ

) . By (2.3), we obtain the contradiction ϕ(u) ≥

X

1

Z

σi

Z 1 0

1 2 ^|∇u|

2 + a(x, y)W (u) dy dx

≥ µ

X

1

ku(τ

, ·) − u(σ

, ·)k

0

1

= +∞. ❐

We define _{c =} inf

ϕ and _{K = {u ∈ E} 0 | ϕ(u) = c} . We then apply the direct method of the calculus of variations to show that _K is not empty.

Lemma 2.3. _{K ≠ ∅} and any _{u ∈ K} is a classical solution of _−∆u+aW

_{(u) =}

0 on S ₀ with ∂

u(x, 0 _{) = ∂}

u(x, 1 _{) =} 0 for all _{x ∈ R} . Moreover, u(x, y) → ± 1

as _{x → ±∞} , uniformly for _{y ∈ [} 0 , 1 ] .

Proof. We let {v

} ⊂ E 0 be such that ϕ(v

) → c . Since we have that k∇u

k ²

≤ 2 ^ϕ(u

) and ku

k

≤ 1, there then exists {u

} ⊂ {u

} and u ∈ H loc ¹ (S 0 ) such that u

→ u weakly in H _loc ¹ (S 0 ) and u

(x, y) → u(x, y) for almost every (x, y) ∈ S 0 . By pointwise convergence, we then obtain that kuk

≤ 1, u(x, y) = −u(−x, y) on S ₀ and _{u(x, y) ≥} 0 for x > 0.

Hence, _{u ∈ E} 0 , and since by semicontinuity we have _{ϕ(u) ≤ c} , we conclude u ∈ K 6= ∅ .

To prove that _{u ∈ K} is a classical solution of _{−∆u + aW}

_{(u) =} 0 on S ₀ that verifies ∂

u(x, 0 _{) = ∂}

u(x, 1 _{) =} 0 for all _{x ∈ R} , it is sufficient to show that (2.5)

Z

S₀

∇u · ∇ψ + aW

(u)ψ dx dy ≥ 0 for any _{ψ ∈ C 0}

(R ² ).

Let us consider first the case _{ψ ∈ C 0}

(R ² ) such that ψ(x, y) = −ψ(−x, y) . If t ∈ ( 0 , ρ ₀ /kψk

) , we have _{ku + tψk}

≤ 1 _{+ ρ} 0 , and defining

φ(x, y) =

 

 

 



u(x, y) + tψ(x, y) if x > 0 and u(x, y) + ψ(x, y) ≥ 0 ,

−u(x, y) − tψ(x, y) if ^{x >} 0 and u(x, y) + ψ(x, y) ≤ 0 ^,

−φ(−x, y) if x < 0 , we derive

kφk

≤ 1 _{+ ρ} 0 , φ(x, y) = −φ(−x, y)

and _{φ(x, y) ≥} 0 if x > 0 and, by (H

), ϕ(φ) = ϕ(u + tψ) . Considering the function

φ(x, y) = ˜ max _{− 1 , min _{ 1 , φ(x, y)}}, we recognize that

φ ∈ E ˜ 0 , |∇ φ(x, y)| ≤ |∇(u + tψ)(x, y)| ˜ and (by Remark 1.2)

W ( φ(x, y)) ≤ W ((u + tψ)(x, y)) ˜ for almost every _{(x, y) ∈ S} 0 . Then, we derive

(2.6) ϕ(u + tψ) = ϕ(φ) ≥ ϕ( _{φ) ≥ ϕ(u).} ˜

Given now any _{ψ ∈ C 0}

(R ² ) , we set ψ _o (x, y) = ¹ 2 (ψ(x, y) − ψ(−x, y)) , ψ _e (x, y) = ¹ ₂ (ψ(x, y)+ψ(−x, y)) , and, observing that the functions _∇u·∇ψ e

and _∇ψ o · ∇ψ e are odd in the variable x , we recover (2.7)

ϕ(u+tψ)−ϕ(u+tψ o ) = Z

S0

t ² 2 ^{|∇ψ e |}

2 +a(W (u+tψ)−W (u+tψ o )) dx dy.

By (2.6), we have _{ϕ(u + tψ} o ) − ϕ(u) ≥ 0 for t small, and so, using (2.7), we derive

Z

S0

∇u · ∇ψ + aW

(u)ψ dx dy

= lim

t→

0

1

t (ϕ(u + tψ) − ϕ(u)) ≥ lim

t→

0

1

t (ϕ(u + tψ) − ϕ(u + tψ o ))

≥ lim

t→

0

Z

S₀

a W (u + tψ) − W (u)

t + a W (u) − W (u + tψ o )

t d x dy

= Z

S0

aW

(u)ψ _e d x dy,

which implies (2.5), since the function aW

(u)ψ _e is odd in the variable x . It remains to show that if _{u ∈ K} , then u(x, y) → ± 1 as _{x → ±∞} , uniformly in _{y ∈ [} 0 , 1 ] .

To this end, we observe that, since _kuk

≤ 1, _{−∆u + aW}

_{(u) =} 0 on S ₀ and ∂

u(x, 0 _{) = ∂}

u(x, 1 _{) =} 0 for all _{x ∈ R} , there exists ^C > 0 such that kuk

≤ C . Assume by contradiction that u does not verify u(x, y) → + 1 as _{x → +∞} uniformly with respect to _{y ∈ [} 0 , 1 ] . Then, there exist δ > 0 and a sequence (x

, y

) ∈ S 0 with x

→ +∞ and _|u(x

, y

) − 1 _{| ≥} 2 δ for all _{n ∈ N} . The ^{C 2} estimate above implies that there exists ρ > 0 such that

|u(x, y) − 1 | ≥ δ ∀ (x, y) ∈ B

(x

, y

) ∩ S 0 , n ∈ N.

Along a subsequence, y

→ y 0 , and thus we have _{|u(x, y) +} 1 _{| ≥ δ} for all (x, y) ∈ B

2 (x

, y ₀ ) ∩ S 0 , whenever n is large enough. But this is not possible since, because of Lemma 2.2, we already know that ku(x, ·) ∓ 1 k

0

1

→ 0 as

x ± ∞ . ^❐

We shall now use the reversibility condition a(x, y) = a(x, −y) , to show, following [23], that if u ∈ K , then it is in fact a solution of (2.1). Let

c

= inf

Ep

ϕ, E

= {u ∈ E 0 | u(x, 0 _{) = u(x,} 1 ) for almost every _{x ∈ R}.}

Then, we have the following result.

Lemma 2.4. It holds that c

= c . Moreover, any _{u ∈ K} must satisfy u(x, 0 _{) =} u(x, 1 ) for all _{x ∈ R} .

Proof. Clearly, c

≥ c . To prove the opposite inequality, we show that, for any _{u ∈ E} 0 , there exists _{v ∈ E}

with ϕ(v) ≤ ϕ(u) . For that, we write

ϕ(u) = Z

R

 Z 1

2 0

1 2 ^|∇u|

2 + aW (u) dy

 d x

+ Z

R

 Z 1 1

2

1 2 ^|∇u|

2 + aW (u) dy



d x = I 1 + I 2 ,

and we define

(2.8)

v(x, y) =

( u(x, y) 0 _{≤ y ≤} ¹ ₂ ,

u(x, 1 − y) ¹ ₂ ≤ y ≤ 1 ^, if I ₁ ≤ I 2 , v(x, y) =

( u(x, 1 _{− y)} 0 _{≤ y ≤} ¹ ₂ ,

u(x, y) ¹ ₂ ≤ y ≤ 1 ^, if I ₁ ≥ I 2 ,

observing that _{v ∈ E}

. Moreover, since a is 1-periodic and a(x, y) = a(x, −y) , we have a(x, y) = a(x, 1 _{− y)} for any _{y ∈ [} 0 , ¹ ₂ ] , so that ϕ(v) ≤ ϕ(u) as we claimed.

To conclude the proof, we have now to show that any _{u ∈ K} belongs to E

. This follows as in [23] (see also [3]), and we give here the proof for the sake of completeness.

Suppose that u ∈ K , and define ^v as in (2.8). Since u ∈ K , v ∈ E

, and ϕ(v) ≤ ϕ(u) , we derive that _{v ∈ K} too. Then, both u and v are classical solutions of

( −∆u + aW

(u) = 0 , on S ₀ ,

∂

u(x, 0 _{) = ∂}

u(x, 1 _{) =} 0 , for _{x ∈ R.}

Letting _{w = u − v} , we obtain

( −∆w(x, y) = b(x, y)w(x, y), on ^S 0 ,

∂

w(x, 0 _{) = ∂}

w(x, 1 _{) =} 0 , for _{x ∈ R,} where

b(x, y) =

 



 

a(x, y)[W

(v(x, y)) − W

(u(x, y))]

u(x, y) − v(x, y) if u(x, y) ≠ v(x, y),

−a(x, y)W

(u(x, y)) if u(x, y) = v(x, y).

Note that ^b is continuous and w(x, y) = 0 for (x, y) ∈ R × [ 0 ^{, 1} ₂ ^] . But then a local unique continuation theorem for elliptic equations (see, e.g., [21]) and a continuation argument imply _{w(x, y) =} 0 for _{(x, y) ∈ S} 0 , and _{u = v ∈ E}

follows. ^❐

Remark 2.5. From the above results, we know that the set _K is non-empty and that any _{u ∈ K} satisfies u(x, 0 _{) = u(x,} 1 ) and ∂

u(x, 0 _{) = ∂}

u(x, 1 _{) =} 0 for all _{x ∈ R} . Therefore, we can extend u by periodicity on R ² , obtaining a classical solution of (2.1). In the sequel, the elements of _K will be considered extended on R ² .

Remark 2.6. Any function _{u ∈ K} is such that u(x, y) > 0 for x > 0.

Indeed, we already know that _{u(x, y) ≥} 0 for x > 0 since _{u ∈ E} 0 ∩ C ² (S ₀ ) .

Moreover, setting

b(x, y) =

 



 

a(x, y)W

(u(x, y))

u(x, y) if u(x, y) > 0 ,

−a(x, y)W

( 0 ) if _{u(x, y) =} 0 , we know that u satisfies the differential inequality

−∆u + max _{{b(x, y),} 0 _{}u ≥} 0 on S ₀ ,

when ^{x >} 0. Since ^b is continuous, the maximum principle tells us that either u ≡ 0 or ^{u >} 0 on ^S 0 when ^{x >} 0, and so, since u(x, y) → 1 as x → +∞ , we derive u > 0 on S ₀ when x > 0.

2.2. Further compactness properties of ϕ . For _{L ∈ (} 0 _{, +∞]} , we shall let S ₀

= {(x, y) ∈ S 0 : |x| < L} and

ϕ

(u) = ZZ

S0,L

1 2 ^|∇u|

2 + a(x, y)W (u) dx dy.

Note that ϕ

is well defined on E ₀ and weakly lower semicontinuous with respect to the H _loc ¹ (S ₀ ) topology. Moreover, if 0 < L ₁ < L ₂ ∈ R , we have ϕ

(u) ≤ ϕ

(u) ≤ ϕ

(u) ≡ ϕ(u) for any _{u ∈ E} 0 . Given δ > 0, we set, moreover,

(2.9) λ

= 1 2 ^δ

2 + max

(x,y)∈R²

a(x, y) · max

|s−

1

2

W (s) and ℓ

= c + 1 µ

,

where µ

is defined by Lemma 2.1. We have the following result.

Lemma 2.7. There exists ^δ 0 ∈ ( 0 , ρ ₀ / 2 ) such that, for any δ ∈ ( 0 , δ ₀ ) , if u ∈ E 0 , L ∈ (ℓ

+ 1 , +∞] , and ^ϕ

(u) ≤ c + λ

, then the following hold:

(i) There exists ^x

∈ ( 0 , ℓ

) such that ku(x

, ·) − 1 _k

0

1

< δ . (ii) _{ku(x, ·) −} 1 _k

0

1

≤ ρ 0 for any x ∈ (x

, L) , and

Z

 Z 1 0

1 2 ^|∇u|

2 + aW (u) dy

 d x ≤ 3

2 ^λ

^. Proof. Since λ

→ 0 as _{δ →} 0, we can fix δ ₀ ∈ ( 0 , ρ ₀ / 2 ) such that (2.10) ^λ

< min { 1 ^{, µ}

2 ρ ₀ / 3 } for any δ ∈ ( 0 ^{, δ} 0 ),

where ρ ₀ is defined in (1.2) and µ

₂ by (2.3) in correspondence to _{r = ρ} 0 / 2.

Let _{δ ∈ (} 0 , δ ₀ ) , _{L ∈ (ℓ}

+ 1 _{, +∞]} , and _{u ∈ E} 0 such that ϕ

(u) ≤ c + λ

.

The point (i) readily follows by the definition of ℓ

in (2.9). Indeed, if ku(x, ·) − 1 _k

0

1

≥ δ for almost every _{x ∈ (} 0 , ℓ

), by Lemma 2.1 we have

c + λ

< c + 1 = µ

ℓ

≤ Z

0

Z 1 0

1 2 ^|∇u|

2 + aW (u) dx dy ≤ ϕ

(u).

To prove (ii), note that by (i) there exists ^x

∈ ( 0 ^{, ℓ}

) such that ku(x

, ·) − 1 _k

0

1

< δ.

We define

u(x, y) = ˜

 

 

 



 

 

 

1 if x > x

+ 1 ,

(x − x

) + (x

+ 1 _{− x)u(x}

, y) if x

≤ x ≤ x

+ 1 ,

u(x, y) if 0 _{≤ x ≤ x}

,

− _{u(−x, y)} ˜ if x < 0 ,

and note that, since ˜ _{u ∈ E} 0 , ϕ( _{u) = ϕ} ˜

( _{u) ≥ c} ˜ . Hence, recalling that kqk

0

1

≤ √

2 kqk

0

1

for any q ∈ H ¹ ( 0 ^, 1 ⁾ , a direct estimate gives Z

1

x₊

 Z 1 0

1

2 ^|∇ _u| ˜ ² _{+ aW (} u) dy ˜

 d x

≤ 1 2 ^δ

2 + max

(x,y)∈R²

a(x, y) · max

|s−

1

2

W (s) = λ

. Since

c ≤ ϕ

( _{u) = ϕ} ˜

(u) − 2 Z

x₊

Z 1 0

1 2 ^|∇u|

2 + aW (u) dy dx + 2

Z

1

Z 1 0

1

2 ^|∇ _u| ˜ ² _{+ aW (} u) dy dx ˜

≤ c + 3 λ

− 2 Z

x₊

Z ₁

0

1 2 ^|∇u|

2 + aW (u) dy dx,

we obtain _Z

L x₊

Z 1 0

1 2 ^|∇u|

2 + aW (u) dy dx ≤ 3

2 ^λ

^,

which proves the second part of (ii).

To conclude the proof of (ii), assume that there exists _{τ ∈ (x}

, L) such that ku(τ, ·) − 1 _k

0

1

≥ ρ 0 . Since _ku(x

_{, ·) −} 1 _k

0

1

< δ < ρ ₀ / 2, there exists σ ∈ (x

, τ) such that

.ku(x, ·) − 1 _k

0

1

≥ ρ ₀

2 for x ∈ (σ , τ) and ku(τ, ·) − u(σ , ·)k

0

1

≥ ρ ₀

2 ^. By (2.3), we get Z

x₊

 Z 1 0

1 2 ^|∇u|

2 + aW (u) dy



d x ≥ µ

2

ρ ₀ 2 ^,

and so µ

2 ρ ₀ / 2 _≤ ³ ₂ λ

, in contradiction with (2.10). ^❐ In particular, we derive the following result.

Lemma 2.8. For all ε > 0, there exist ¯ λ

> 0 and ¯ ℓ

> 0 such that if _{u ∈ E} 0

and ϕ(u) ≤ c + _λ ¯

, then u − 1 ²

¯

0

1

≤ ε . Proof. Let ^{ε >} 0, and choose _{δ ∈ (} 0 , δ ₀ ) such that

3 2 ^λ

^≤

ε

max _{ 2 , 1 _{/(a w)}} ^.

Denoting ¯ ^λ

= λ

and ¯ ^ℓ

= ℓ

, by Lemma 2.7, we have that if u ∈ E 0 and ϕ(u) ≤ c + _λ ¯

, then

Z

ℓ

¯

Z 1 0

1 2 ^|∇u|

2 + a(x, y)W (u) dy dx ≤ 3 2 ^λ

^≤

ε

max { 2 ^, 1 /(a w)} . Using Remark 1.2, we derive

u − 1 ²

¯

0

1

≤ 2 Z

ℓ

¯

Z 1 0

1 2 ^|∇u|

2 d y dx + Z

ℓ

¯

Z 1 0

1

a w a(x, y)W (u) dy dx

≤ max (

2 ^, 1 a w

) Z

ℓ

¯

Z 1 0

1 2 ^|∇u|

2 + a(x, y)W (u) dy dx ≤ ε,

and the lemma follows. ^❐

As a consequence, we obtain the following result.

Lemma 2.9. If _{u

} ⊂ E ⁰ is such that ϕ(u

) → c , then there exists u 0 ∈ K

such that, along a subsequence, ku

− u 0 k

→ 0.

Proof. Let {u

} ⊂ E 0 be such that ϕ(u

) → c . As in the proof of Lemma 2.3, we obtain that there exists u ₀ ∈ K and _{u

} ⊂ {u

} such that u

→ u 0 weakly in H _loc ¹ (S ₀ ) .

Since ϕ(u ₀ ) = c and ϕ(u

) → c , by Lemma 2.8, we derive that, for all ε > 0, there exists ¯ ℓ

2 > 0 and ¯ k such that

u ₀ − 1 ²

¯

ε/2,+∞)×(

0

1

≤ ε 2

and

u

− 1 ²

¯

ε/2,+∞)×(

0

1

≤ ε

2 for any k ≥ _k, ¯ and so

u

− u 0

²

_¯

ε/2,+∞)×(

0

1

≤ 2 ε for any _{k ≥ k.} ¯ (2.11)

Since we already know that u

→ u 0 weakly in H _loc ¹ (S ₀ ) , by (2.11) and since ε is arbitrary, we obtain that u

− u 0 → 0 strongly in L ² (S ₀ ) .

Since _k∇u

k ²

≤ 2 ϕ(u

) → 2 c , and since u

→ u 0 weakly in H _loc ¹ (S ₀ ) , we have also _∇u

→ ∇u 0 weakly in L ² (S ₀ ) . Then, to prove that u

− u 0 → 0 strongly in H ¹ (S ₀ ) , it is sufficient to show that _k∇u

k

→ k∇u 0 k

. For that, we observe that by semicontinuity we have

Z

S₀

|∇u 0 | ² d y dx ≤ lim inf

k→+∞

Z

S₀

|∇u

| ² d y dx

and _Z

S₀

a(x, y)W (u ₀ ) dy dx ≤ lim inf

k→+∞

Z

S₀

a(x, y)W (u

) dy dx.

Then, since lim

ϕ(u

) = ϕ(u 0 ) , we derive Z

S₀

|∇u 0 | ² d x dy ≤ lim inf

k→+∞

Z

S₀

|∇u

| ² d x dy ≤ lim sup

k→+∞

Z

S₀

|∇u

| ² d x dy

= lim sup

k→+∞



2 ϕ(u

) − 2 Z

S0

a(x, y)W (u

) dx dy



= 2 ϕ(u ₀ ) − 2 lim inf

k→+∞

Z

S0

a(x, y)W (u

) dx dy

≤ 2 ϕ(u ₀ ) − 2 Z

S₀

a(x, y)W (u ₀ ) dx dy = Z

S₀

|∇u 0 | ² d x dy,

and the lemma follows. ^❐

As a consequence of Lemma 2.9, using the fact that any u ∈ K solves (E), we finally have the following result.

Lemma 2.10. _K is compact with respect to the ^{H 1 (S} 0 ) and ^{C 2 (S} 0 ) topology.

Remark 2.11. Note that, using Lemma 2.3 and the periodicity in the variable y , we recognize that any _{u ∈ K} is such that lim

u(x, y) = ± 1 uniformly with respect to _{y ∈ R} . By Lemma 2.9, we furthermore have uniformity in the convergence, that is, for any ε > 0, there exists L > 0 such that

sup

u∈K

| 1 − u(x, y)| ≤ ε for x > L, sup

u∈K

| 1 + u(x, y)| ≤ ε for x < −L.

3. T ^HE A PPROXIMATING F ^UNCTIONALS

Given _{j ∈ N ∪ {} 0 _} , we define S

= R × (j, j + 1 ) , T

= {(x, y) ∈ S

: _{|x| ≤ y}} , and

E

= {u ∈ H ¹ (T

) | u(−x, y) = −u(x, y), 1 ≥ u(x, y) ≥ 0 for x > 0 _}.

On E

, we consider the functional ϕ

(u) =

ZZ

Tj

1 2 ^|∇u|

2 + a(x, y)W (u) dx dy,

which is lower semicontinuous with respect to the weak H ¹ (T

) topology. Setting c

= inf

Ej

ϕ

(u) and _K

= {u ∈ E

| ϕ

(u) = c

},

we have that c

≤ c

1 < c for any _{j ∈ N ∪ {} 0 _} , and, moreover, we obtain the following result.

Lemma 3.1. _K

6= ∅ and any _{u ∈ K}

satisfies _{u ∈ C} ² (T

) , _{−∆u +} aW

(u) = 0 on ^T

, and ^∂

u(x, j) = 0 for |x| < j , ^∂

u(x, j + 1 ) = 0 for all |x| < j + 1 and ^∂

u(x, x) = ∂

u(x, x) for x ∈ (j, j + 1 ⁾ .

Proof. Let ^u

∈ E

be such that ^ϕ

(u

) → c

. Since ku

k

≤ 1 and since _k∇u

k

≤ 2 ϕ

(u

) → c

, we derive that _{u

} is a bounded sequence in H ¹ (T

) . Then, there exists a subsequence _{u

} ⊂ {u

} and u ₀ ∈ H ¹ (T

) such that ^u

→ u 0 weakly in ^{H 1 (T}

) and for almost every (x, y) ∈ T

. By pointwise convergence, we get ^u 0 ∈ E

, and by semicontinuity, ^ϕ

(u ₀ ) ≤ c

. We conclude that ^u 0 ∈ K

. To show that

Z

Tj

∇u∇ψ + a(x, y)W

(u)ψ dx dy = 0 for any ψ ∈ C 0

(R ² ) , we proceed as in the proof of Lemma 2.3, and the lemma

follows by classical arguments. ^❐

Remark 3.2. As in Remark 2.6, we can prove that any u ∈ K

satisfies u(x, y) > 0 on ^T

for ^{x >} 0. Since kuk

≤ 1, elliptic regularization argu- ments show, moreover, that there exists a constant ¯ ^C 0 > 0 such that

(3.1) kuk

≤ C ¯ ₀ for any u ∈ K

and j ∈ N.

Lemma 3.3. There exists ^{L >} 0 such that, for any j ∈ N , ^{j > L} , if u ∈ K

, then _{|u(x, y) −} 1 _{| ≤ ρ} 0 for any _{(x, y) ∈ T}

with j > x > L .

Proof. Arguing by contradiction, assume that there exists a sequence of indices j

→ +∞ , a sequence ^u

∈ K

, and a sequence of points

(x

, y

) ∈ (−j

, j

) × (j

, j

+ 1 ⁾

with ^x

→ +∞ such that 1 − ρ 0 > u

(x

, y

) > 0. By (3.1), we deduce that there exists ^{η >} 0 such that 1 − ρ 0 / 2 ^{> u}

(x, y) > 0 for (x, y) ∈ B

(x

, y

) . Therefore, there exists r > 0 such that _ku

(x, ·) − 1 _k

1

≥ r for any x ∈ (x

− η/ 2 , x

) and _{n ∈ N} . We consider the function

v

(x, y) = u

(x, y + j

).

In particular, we have _kv

(x, ·) − 1 _k

0

1

≥ r for any _{x ∈ (x}

− η/ 2 , x

) and _{n ∈ N} , and, by Lemma 2.1, we derive that

(3.2) Z

x_jn−η/

2

Z 1 0

1 2 ^|∇u

^|

2 + a(x, y)W (u

) dy dx ≥ 1 4 ^µ

r

2 η for any _{n ∈ N.}

On the other hand, since for any _{n ∈ N} we have ϕ

(v

) =

Z ₁

0

Z

−jⁿ

1 2 ^|∇v

^|

2 +a(x, y)W (v

) dx dy ≤ ϕ

(u

) = c

≤ c,

we can use Lemma 2.7 to derive that there exist ^{ℓ >} 0 and ¯ n ∈ N such that, for any _{n ≥} n ¯ ,

Z

ℓ

Z 1 0

1 2 ^|∇v

^|

2 + a(x, y)W (v

) dx dy < 1 4 ^µ

r

2 η,

which contradicts (3.2), proving the lemma. ^❐

Lemma 3.4. There exists ¯ ^C 1 > 0 and ¯ ^C 2 > 0 such that 0 ≤ c − c

≤ C ¯ ₁ e

^¯

for all j ∈ N .

Proof. Let j > 1 and _{u ∈ K}

. By Lemma 3.1, u satisfies the boundary

conditions ∂

u(x, j) = 0 and ∂

u(x, j + 1 _{) =} 0 for all _{|x| < j} . This allows us

to extend ^u to a classical solution of −∆u+aW

(u) = 0 on the the set (−j, j)×R .

Indeed, we first reflect with respect to the axis y = j , by setting, with abuse of

notation, u(x, y) = u(x, 2 j −y) for x ∈ (−j, j) and y ∈ (j − 1 , j + 1 ⁾ . Noting

now that _{u(x, j −} 1 ) = u(x, j + 1 ) and ∂

u(x, j − 1 _{) = ∂}

u(x, j + 1 _{) =} 0, we

extend u by periodicity on the entire set (−j, j) × R by setting _{u(x, y +} 2 _{k) =}

u(x, y) for any _{k ∈ Z} and (x, y) ∈ (−j, j) × (j − 1 _{, j +} 1 ) . Then, u is a classical

solution of _{−∆u + aW}

_{(u) =} 0 on (−j, j) × R , which is periodic of period 2 in the variable y , and is even with respect to any axis _{y = k} for _{k ∈ Z} . In particular,

Z

1

Z

−j

1 2 ^|∇u|

2 + aW (u) dx dy (3.3)

= Z ₁

0

Z

−j

1 2 ^|∇u|

2 + aW (u) dx dy, ∀ j > 1 . By Lemma 3.3, we obtain that, if j > L , we have

|u(x, y) − 1 _{| ≤ ρ} 0 for any _{y ∈ R} and x ∈ (L, j).

Defining ψ(x, y) = (u(x, y) − 1 ) ² , by (1.2) we have ∆ψ ≥ aW

(u)(u − 1 _{) ≥} 2 aw ₀ ψ on _{(L, j) × R} (where _{a =} min

a(x, y) ). Considering the function

ξ(x, y) = ρ ² 0

cosh ^{( p} 2 ^aw 0 (x − (j + L)/ 2 ⁾⁾ cosh ( p

2 aw ₀ (j − L)/ 2 ) ,

we have that ξ(L, y) = ξ(j, y) = ρ 0 ² for any y ∈ R and ∆(ψ−ξ) ≥ 2 ^aw 0 (ψ−ξ) on _{(L, j) × R} . We then deduce that _{ψ − ξ ≤} 0 on _{(L, j) × R} . Indeed, the function ψ − ξ is periodic in the variable y , (ψ − ξ)(L, y) ≤ 0 and (ψ − ξ)(j, y) ≤ 0 for _{y ∈ R} . If _{ψ − ξ} assumes a positive value on _{(L, j) × R} , we recover that it has a positive maximum on _{(L, j) × R} , which is not possible since _{∆(ψ − ξ) ≥} 2 _{w(ψ − ξ)} .

In particular, setting x

= (j + L)/ 2, we have

|u(x

, y) − 1 _| ² _{≤ ξ(x}

_{, y) = ρ 0} ² 1 cosh ( p

2 aw ₀ (j − L)/ 2 ) (3.4)

< ρ ₀ ² e

√

2

2 ≡ ρ

² , ∀ y ∈ R.

Note that, if j is sufficiently large, we have x

+ ρ

≤ j , and for such values of j , we set

u(x, y) = ˜

 

 

 

 

 



 

 

 

 

 

1 if x > x

+ ρ

, y ∈ R, u(x

, y) + 1

ρ

(x − x

)( 1 _{− u(x}

, y)) if x

≤ x ≤ x

+ ρ

, y ∈ R, u(x, y) if 0 _{≤ x ≤ x}

_{, y ∈ R,}

−u(−x, y) if x < 0 _{, y ∈ R.}

We explicitely note here that, by using (3.1) and (3.4), there exist ^C 1 > 0 and

C ₂ > 0, independent of j , such that, if _{x ∈ (x}

, x

+ ρ

) and _{y ∈ R} , we have

|∂

u(x, y)| = ( ˜ 1 /ρ

)| 1 _{− u(x}

_{, y)| ≤} 1, _|∂

_{u(x, y)| ≤} ˜ 2 _|∂

u(x

, y)| ≤ C 1 , a(x, y)W ( u(x, y)) ≤ C ˜ 2 | 1 _{− u(x}

_{, y)|} ² . We derive that

Z

x₊

 Z 1 0

1

2 ^|∇ _u| ˜ ² _{+ aW (} u) dy ˜



d x ≤ ρ

2 ⁽ 1 + 4 ^C ₁ ² ) + C 2 ρ ³

≤ C 3 ρ

for a constant ^C 3 > 0 independent of j and u .

By construction, we have ϕ

( _{u) ≤ ϕ} ˜

(u) + 2

Z

x₊

 Z 1 0

1

2 ^|∇ _u| ˜ ² _{+ aW (} u) dy ˜



d x ≤ c

+ C 3 ρ

,

and, since we recognize that ˜ _{u ∈ E} 0 , by (3.3), we have also _{c ≤ ϕ( u) = ϕ} ˜

( u) ˜ . We then conclude that c

≤ c ≤ c

+ C 3 ρ

= c

+ ρ 0 C ₃ e

√ ₂

0(j−L)/

4 , and the

lemma follows. ^❐

Lemma 3.5. Let j

→ +∞ and u

∈ E

be such that ϕ

(u

) − c

→ 0 as n → ∞ . Then, dist

(u

, K) → 0 as _{n → +∞} .

Proof. For _{n ∈ N} , let v

(x, y) = u

(x, y +j

) for _{y ∈ [} 0 , 1 ] and _{|x| < j}

. Since j

→ +∞ , we have c

→ c , and so ϕ

(v

) ≤ ϕ

(u

) = c + o( 1 ) . By Lemma 2.10, we can then derive that there exists ^x

∈ (j

/ 2 ^{, j}

) such that δ

= kv

(x

, ·) − 1 _k

0

1

→ 0. Analogous to the definition of ˜ u in the proof of the preceding lemma, we now define

˜

v

(x, y) =

 

 

 

 

 



 

 

 

 

 

1 if ^{x > x}

+ δ

, y ∈ ( 0 ^, 1 ^), v

(x

, y) + 1

δ

(x − x

)( 1 − u

(x

, y))

if x

≤ x ≤ x

+ δ

, y ∈ ( 0 , 1 ), v

(x, y) if 0 ≤ x ≤ x

, y ∈ ( 0 ^, 1 ^),

− v ˜

(−x, y) if x < 0 _{, y ∈ (} 0 , 1 ).

recognizing again that ˜ v

∈ E 0 and so _{c ≤ ϕ(} v ˜

) . Moreover, since we have δ

= kv

(x

, ·) − 1 _k

0

1

→ 0, we derive

Z

x_+,n

Z 1 0

1

2 ^|∇ v ˜

| ² + aW ( v ˜

) dy dx → 0 ^, and so

c ≤ ϕ( v ˜

) ≤ ϕ

(v

) + 2

Z

Z 1 0

1

2 ^|∇ v ˜

| ² + aW ( v ˜

) dy dx

≤ ϕ

(u

) + o( 1 _{) = c + o(} 1 ).