A BSTRACT . We consider a class of a semilinear elliptic system of the form

Allen-Cahn Type Equations

F RANCESCA A LESSIO

A ^BSTRACT . We consider a class of a semilinear elliptic system of the form

(0.1) − ∆u(x, y) + ∇W (u(x, y)) = 0 , (x, y) ∈ R ² , where ^W : R ² → R is a double well nonnegative symmetric potential. We show, via variational methods, that if the set of solutions to the one-dimensional system ₋ q(x)+∇W (q(x)) = ¨ 0, _{x ∈ R} , which connect the two minima of W as _{x → ±∞} , has a discrete structure, then (0.1) has infinitely many layered solutions.

1. I NTRODUCTION

We consider semilinear elliptic systems of the form

(1.1) − ∆u(x, y) + ∇W (u(x, y)) = 0 , (x, y) ∈ R ² , where _{W ∈ C} ² (R ² , R) satisfies the following:

(W 1 ) There exist ^a

_{∈ R} ² such that W (a

) = 0, W (ξ) > 0 for every _{ξ ∈} R ² \ {a

} , and such that ^{D 2} W (a

) are positive definite;

(W 2 ) There exists R > 1 such that ∇W (ξ) · ξ > 0 for all _{|ξ| > R} and inf {W (ξ), |ξ| > R} = µ 0 > 0;

(W 3 ) _{W (−x} 1 , x ₂ ) = W (x 1 , x ₂ ) for all (x ₁ , x ₂ ) ∈ R ² .

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

_{= (±} 1 , 0 ) .

System (1.1) is the rescaled stationary system associated with the reaction- diffusion system

(1.2) ∂

u(x, y) − ε ² ∆u(x, y) + ∇W (u(x, y)) = 0 , (x, y) ∈ Ω ⊂ R ² ,

1535

Indiana University Mathematics Journal c , Vol. 62, No. 5 (2013)

which describes two phase physical systems or grain boundaries in alloys. As _{ε →} 0

, solutions to (1.2) tend almost everywhere to global minima of W , and sharp phase interfaces appear (see e.g., [14], [25], and [30]). Then, the expansion of such solutions in a point on the interface presents, as first term, the system (1.1).

From this point of view, two layered transition solutions correspond to solutions u of (1.1) satisfying the asympotic conditions

(1.3) lim

x→±∞

u(x, y) = a

uniformly with respect to _{y ∈ R.}

S. Alama, L. Bronsard, and C. Gui in [1] studied the existence of a solution to (1.1) which satisfies the asymptotic condition (1.3) for x → ±∞ , and which tends, as _{y → ±∞} , to two different one-dimensional stationary waves, that is, solutions to the one-dimensional associated problem

(1.4)

 



− q(x) + ∇W (q(x)) = ¨ 0 ^, x ∈ R,

t→±∞

lim q(t) = a

, which, furthermore, are minima of the action

V (q) = Z

R

1

2 ^| _q| ˙ ² _{+ W (q) dx}

over the class of trajectories connecting ^a

as x → ±∞ . Under conditions (W

)–

(W

), the authors in [1] found such solutions, assuming, furthermore, that the set of one-dimensional minimizing conditions is constituted by a finite number k ≥ 2 of geometrically distinct trajectories. (For related assumptions and existence results, see also the paper by Alikakos and Fusco [8].)

In [29], M. Schatzman proved the same result, considering a nonsymmetric potential, and assuming there exist two geometrically distinct one-dimensional he- teroclinic connections which are supposed to be nondegenerate; that is, the kernel of the corresponding linearized operators is one dimensional.

We cite also the case of a triple-well potential over R ² which was studied by L.

Bronsard, C. Gui, and M. Schatzman [13], where the existence of entire solutions to (1.1) is proved; these are known as triple-junction solutions, and they connect the three global minima of the potential ^W in certain directions at infinity. See also [24] for quadruple junction solutions, and [9] (and the reference therein) for potential defined over R

with general reflection group of symmetry.

If ^u is scalar valued, much is known about the corresponding heteroclinic problem (1.1)–(1.3). In this scalar setting, E. De Giorgi in[15] has conjectured that any entire bounded solution of _{∆u = u} ³ _{− u} with ∂

u(x) > 0 in R

for n ≤ 8 is, in fact, one dimensional; that is, modulo space rotation-traslation, it coincides with the unique solution of the one-dimensional heteroclinic problem (1.5)

( q(x) = q(x) ¨ ³ − q(x), x ∈ R,

q( 0 _{) =} 0 and _{q(±∞) = ±} 1 .

The conjecture has been proved for _{n =} 2 by N. Ghoussoub and C. Gui in [21], and then by L. Ambrosio and X. Cabr`e in [10] (see also [2]) for _{n =} 3, even for more general double well potentials W . A further step in the proof of the De Giorgi conjecture has been done by O. Savin in [28] where, for _{n ≤} 8, the same one-dimensional symmetry is obtained for solutions u such that ∂

u(x) > 0 on R

and lim

u(x) = ± 1 for all (x ₂ , x ₃ , . . . , x

) ∈ R

¹ (see [11], [12], [18], [19], and [20] for related problems). That result is completed in [16], [17], where the existence of entire solutions without any one-dimensional symmetry and which are increasing and asymptotic to _± 1 with respect to the first variable is proved in dimension n > 8.

In this paper, we consider problem (1.1)–(1.3), and, using a global varia- tional procedure, we prove that, if the minimal set of one-dimensional heteroclinic connections satisfies a suitable discreteness assumption, then there exist infinitely many solutions to the problem with prescribed energy.

To explain precisely our result and to give an idea of our procedure, let us begin considering the problem already considered in [1] and [29].

Assuming that the minimal set of one-dimensional heteroclinic connections M satisfies the discreteness assumption

( _∗ ) _{M = M}

_{∪ M}

with dist

(M

, M

) > 0 ,

we will look for a bidimensional solution with prescribed different asymptotes as y → ±∞ :

(1.6) dist

(u(·, y), M

) → 0 as y → ±∞.

q ₂

q ₂

q ₁

q ₁ (− 1 ^, 0 ⁾

( 1 , 0 )

M

M

x y

F ^IGURE 1.1.

Note that condition ( _∗ ) (as the discreteness assumption made in [1] and [29]) does not hold in the scalar case, where the minimal set of one-dimensional solu- tions _M is a continuum homeomorphic to R , being constituted by the translation of the unique heteroclinic solution of (1.5).

Under our assumption, bidimensional solutions satisfying (1.6) can be found by using a global variational approach (instead of the approximating procedure used in [1] and [29]), and by considering a renormalized action functional over a suitable space. Thus, let

Γ = {q − z 0 ∈ H ¹ (R) ² | q(x) 1 = −q(−x) 1 , q(x) ₂ = q(−x) 2 }

be the space of one-dimenstional heteroclinic connections, where we have _{q(x) =} (q(x) ₁ , q(x) ₂ ) , and where z ₀ is fixed in such a way that z ₀ (x) ₁ = −z 0 (−x) 1 , z ₀ (x) ₂ = z 0 (−x) 2 , and z ₀ (x) = ( 1 , 0 ) for x > 1. Setting _{m =} inf

V (q) , we have that

M = {q ∈ Γ | V(q) = m}.

Although there is a canonical Euler Lagrange functional associated with (1.1), such a functional is always infinite over the solutions we seek; thus, as in [26] and [27]

for Hamiltonian ODE systems and in [3] for scalar Allen-Cahn equations, we have that bidimensional heteroclinic solutions of (1.1) may be obtained as global minimizers of the renormalized action functional

ϕ(u) = Z

R

1 2

∂

u(·, y) ²

(R)²

+ (V(u(·, y)) − m) dy, which is well defined on the space

H = {u ∈ H loc ¹ (R ² , R ² ) | u(·, y) ∈ Γ for almost every _{y ∈ R},}

and such that _{ϕ(q) =} 0 for all _{q ∈ M} . We are interested in solutions which satisfy the right asymptotic conditions as y → ± + ∞ , and that can be found as minima of ^ϕ over the space

H

= {u ∈ H | lim inf

y→±∞

dist

(u(·, y), M

) = 0 }.

In fact, for all _{u ∈ H} , we have u(·, y 1 ) − u(·, y 2 ) ²

(R)²

≤ (y 2 − y 1 ) Z

y₁

∂

u(·, y) ²

(R)²

d y, and, in particular, if ϕ(u) < +∞ , then the map y ∈ R → u(·, y) ∈ Γ is continuous with respect to the L ² metric. Moreover, if y ₁ < y ₂ and _{u ∈ H} , then

ϕ(u) ≥



2 1

y ₂ − y 1

Z

y₁

(V (u(·, y)) − m) dy

 1

2

ku(·, y 1 ) − u(·, y 2 )k

.

u(·, y)

Γ

M

M

F ^IGURE 1.2.

By the previous estimate, if u ∈ H

, we have control of the transition time from M

to M

, and so also concentration in the ^y variable. Together with the symmetry in the ^x variable, this allows us to get compactness of minimizing sequences, and hence to prove the existence of at least one bidimensional solution to (1.1) in H

(as in the theorem by Alama, Bronsard, and Gui, but in a slightly more general setting).

Theorem 1.1. If (W

)–(W

) and ( ∗ ) hold, then there exists a u ∈ C ² (R ² , R ² ) that is a solution to (1.1) such that u(x, y) → a

as x → ±∞ uniformly with respect to y ∈ R , and

y→±∞

lim d

(u(·, y), M

) = 0 . Note that if _{u ∈ H} solves the system (1.1), then

∂

² u(x, y) = −∂

² u(x, y) + ∇W (u(x, y))

| {z }

V

(u(·, y))

.

In other words, ^u defines a trajectory y ∈ R ֏ u(·, y) ∈ Γ that is a solution to the infinite dimensional Lagrangian system

d ²

d y ² u(·, y) = V

(u(·, y)),

which has as equilibria all the one-dimensional solutions _{q ∈ M} . From such a point of view, bidimensional solutions in _H

are heteroclinic-type solutions (see Figure 1.2). Note that the energy is conserved; indeed, if u ∈ H solves (1.1) on R × (y ¹ , y 2 ) , then

E

(y) = 1 2

∂

u(·, y) ²

− V(u(·, y))

is constant on (y ₁ , y ₂ ) (see [22] for more general identities of this kind). In

particular, for the heteroclinic-type solution _{u ∈ H}

given in Theorem 1.1, we

have that E

(y) = −m for every _{y ∈ R} , and, as _{y → ±∞} , it connects the two components _M

of the level set {q ∈ Γ | V(q) ≤ m} .

Now, if we take c ∈ (m, m + λ) with λ > 0 small enough, by ( _∗ ) we get that ( ∗

) {q ∈ Γ | V(q) ≤ c} = V

∪ V

with dist

(V

, V

) > 0 ^.

A natural problem, which generalizes the above one, is to look for a solution u ∈ H to (1.1) with energy ^E

= −c which connects in Γ the sets V

and V

. In such a case, V (u(·, y)) = −E

(y) + ¹ ₂ k∂

u(·, y)k ²

≥ c for every _{y ∈ R} , and so solutions with energy −c can be sought as minima of the new renormalized functional

ϕ

(u) = Z

R

1 2

∂

u(·, y) ²

+ (V(u(·, y)) − c) dy

on the space H

= n

u ∈ H | lim inf

y→±∞

dist

(u(·, y), V

) = 0

and V (u(·, y)) ≥ c almost everywhere in R o . The functional ϕ

enjoys most of the properties of the functional ϕ , and the above concentration arguments work even in this setting. In particular, the suit- able y -translated minimizing sequences of ϕ

on _H

weakly converge to func- tions u

∈ H such that dist

(u(·, y), V

) → 0 as _{y → ±∞} . However, we cannot conclude that u

∈ H

since we do not know whether u

satisfies the constraint ^{V (u}

(·, y)) ≥ c for almost every y ∈ R . Thus, setting

s

= sup _{{y ∈ R |} dist

(u

(·, y), V

) ≤ d 0 and V (u

(·, y)) ≤ c} and t

= inf _{{y > s}

| V(u

(·, y)) ≤ c},

what we prove is that if ^[y 1 , y ₂ ] ⊂ (s

, t

) , then inf

V (u

(·, y)) > c , so that u

is a solution of (1.1) on R × (s

, t

) . By regularity, we recover that V (u(·, y)) → c whenever _{y → s}

or _{y → t}

, and the minimality property of u

guarantees that E

(y) = −c for all _{y ∈ (s}

, t

) .

In particular, if s

, t

∈ R , then _{V (u(·, t}

)) = V(u(·, s

)) = c and hence, by the conservation of energy, we obtain that

∂

u(·, t

) = ∂

u(·, s

) ≡ 0 ^,

which allows us to recover by reflection from u

a brake orbit type entire solution.

On the other hand, if ^s

= −∞ (respectively, ^t

= +∞ ), we can prove that the α -limit (respectively, ^ω -limit) of ^u

is constituted by critical points of ^V at level c . Then, if ^s

= −∞ and ^t

∈ R , or ^s

∈ R and ^t

= +∞ , from ^u

we can construct a homoclinic-type solution, again by reflection. Finally, if ^s

= −∞ and t

= +∞ , we have that u

is an entire solution of heteroclinic type connecting V

as _{y → ±∞} (see Figure 1.3). More precisely, we prove the followng result.

Γ

V

V

(a) Brake-orbit solution

Γ

V

V

(b) Homoclinic-type orbit

Γ

V

V

(c) Heteroclinic-type solution

F ^IGURE 1.3.

Theorem 1.2. For every c ∈ (m, m + λ) with ^{λ >} 0 small enough, there exists v

∈ C ² (R ² , R ² ) that is a solution to (1.1)–(1.3) such that ^E

(y) = −c for all y ∈ R . Moreover, setting K

= {q ∈ V

| V

(q) = 0 and V (q) = c} , we have the following results:

(i) If ^t

= +∞ , then dist

(v

(·, y), K

) → 0 as y → +∞ , (ii) If ^s

= −∞ , then dist

(v

(·, y), K

) → 0 as y → −∞ ,

(iii) If ^t

∈ R or ^s

∈ R then, respectively, ^∂

v

(·, t

) ≡ 0 and ^v

(·, t

) ∈ V

, or ^∂

v

(·, s

) ≡ 0 and ^v

(·, s

) ∈ V

.

In particular, if ^c is a regular value for ^V , then ^t

, s

∈ R , and there exists ^T

> 0 such that ^v

(x, y + 2 ^T

) = v

(x, y) for all (x, y) ∈ R ² , ^∂

v

(·, 0 ) ≡ ∂

v

(·, T

) ≡ 0, ^v

(·, 0 ) ∈ V

, and ^v

(·, T

) ∈ V

.

Note that the theorem guarantees the existence of a brake orbit type solution at level ^c whenever c ∈ (m, m+λ) is a regular value of ^V . As a consequence of the Sard Smale Theorem and the local compactness properties of ^V , it can be proved that the set of regular values of ^V is open and dense in [m, m + λ] (see Lemma 2.9 in [6]). Then, Theorem 1.2 provides the existence of an uncountable set of geometrically distinct two-dimensional solutions of (1.1) of brake-orbit type. If c ∈ (m, m + λ) is a critical value of V , then Theorem 1.2 states the existence of an entire solution with energy _−c which can be in this case of homoclinic, heteroclinic, or brake-orbit type, depending on the geometry of the sublevel _V

.

The variational procedure that we use, and, in particular, the new renormal-

ized functional ϕ

, was already introduced and used in the framework of scalar

nonautonomous Allen-Cahn equations in [4] and [5]. There, we proved the ex-

istence of bidimensional solutions that, as y → ±∞ , connect one-dimensional

solutions which (in the sense of Giaquinta and Giusti) are not minima of the cor-

responding one-dimensional Euler Lagrange functional. Energy-prescribed brake-

orbit solutions were introduced and found in [6] for the same kind of nonau-

tonomous scalar equations.

2. T ^HE O ^NE -D ^IMENSIONAL P ^ROBLEM

In this section, we recall some results concerning the one-dimensional heteroclinic problem associated with (1.1), that is, the heteroclinic problem

(2.1)

 



− q(t) + ∇W (q(t)) = ¨ 0 , t ∈ R,

t→±∞

lim q(t) = a

.

Here, we study the problem under the symmetric assumption (W

); we wish also to note that the existence result can, in fact, be proved without any symmetry of the potential (see, e.g., [29] and [7]).

Fix a function ^z 0 ∈ C

(R, R ² ) such that kzk

≤ R , with ^z 0 odd in the first component and even in the second one, and ^z 0 (t) = a

for all t ≥ 1. On the space Γ = z 0 + H ¹ (R) ² , we consider the functional

V (q) = Z

R

1

2 ^| _q(t)| ˙ ² + W (q(t)) dt.

We are interested in the minimal properties of V on _Γ , and we set m = inf

Γ

V .

Endowing _Γ with the Hilbertian structure induced by the map Q : H ¹ (R) ² → Γ , Q(z) = z 0 + z , it is classical to prove that _{V ∈ C} ² _{(Γ )} and that critical points of V are classical solutions to (2.1).

Moreover, if I is an interval in R , we set V

(q) =

Z

I

1

2 ^| _q(t)| ˙ ² + W (q(t)) dt,

noting that V

(q) is well defined on H _loc ¹ (R) ² with values in [ 0 _{, +∞]} .

Finally, for a given _{q ∈ L} ² (R) ² , we denote _{kqk ≡ kqk}

, and given _{A, B ⊂} L ² (R) ² , we denote

d (A, B) = inf _{kq 1 − q 2 k : q ₁ ∈ A, q 2 ∈ B}.

Remark 2.1. We point out some basic consequences of the assumptions (W

)–(W

), fixing some constants. For all x ∈ R ² , we set

χ(x) = min _{{|x − a}

_{|, |x − a}

_|}.

First, we note that, since _{W ∈ C} ² (R) and ^{D 2} W (a

) are positive definite, then

(2.2) _{∀ r >} 0 _{∃ ω}

> 0 such that if _{χ(x) ≤ r} , then _{W (x) ≥ ω}

χ(x) ² .

Then, since W (a

) = 0, ^D W (a

) = 0, and ^{D 2} W (a

) are positive definite, we have that there exists ¯ _{δ ∈ (} 0 , ¹ ₈ ) and two constants ¯ w > w > 0 such that if χ(x) ≤ 2 ¯ δ , then

4 _w|ξ| ² _{≤ D} ² W (x)ξ · ξ ≤ 4 ¯ _w|ξ| ² for all _{ξ ∈ R} ² , (2.3)

and

wχ(x) ² ≤ W (x) ≤ wχ(x) ¯ ² and _{|∇W (x)| ≤} 2 wχ(x).

(2.4)

Remark 2.2. Given q ∈ Γ , we denote q(t) = (q(t) 1 , q(t) ₂ ) , and we set t

= sup {t ∈ R | q(s) 1 < 0 ∀ s < t}

and

t

= inf {t ∈ R | q(s) 1 > 0 ∀ s > t}.

Now, if V

(q) ≤ V

(q) , we set q(t) = ˆ

( q(t + t

) if _{t ≥} 0 , (−q(t

− t) 1 , q(t

− t) 2 ) if ^{t <} 0 ^, while, if ^V

(q) > V

(q) , we set

q(t) = ˆ

( (−q(t

− t) 1 , q(t

− t) 2 ) if _{t ≥} 0 , q(t + t

) if t < 0 .

Then, we obtain that ˆ _{q ∈ Γ} , ˆ q( 0 ) ₁ = 0 and ˆ q(t) ₁ t > 0 for all _{t 6=} 0. Moreover, by definition and the symmetry assumption (W

), we obtain that V ( _{q) ≤ V(q)} ˆ . Indeed, in the first case, by definition and the invariance under translation, we have ^V

0

( _{q) = V} ˆ

(q) and, by (W

),

V

₀

( _{q) =} ˆ Z

0

1

2 ^| q(t ˙

− t)| ² + W (−q(t

− t) 1 , q(t

− t) 2 ) dt

= Z

t₊

1

2 ^| _q(s)| ˙ ² _{+ W (−q(s) 1} , q(s) ₂ ) ds = V

(q).

Hence, ^{V (} _{q) =} ˆ 2 ^V

(q) ≤ V

(q) + V

(q) ≤ V(q) analogously in the second case.

Moreover, note that ˆ ^q(t) is odd in the first component and even in the second one; that is, ˆ q ≡ q ˆ

, where we have denoted

p

(t) = (−p(−t) 1 , p(−t) 2 ), ∀ t ∈ R, p ∈ Γ .

In the sequel, we will denote

Γ

= {q ∈ Γ | q(t) 1 t > 0 _{∀ t 6=} 0 and _{q ≡ q}

_}.

We have then proved that, for all _{q ∈ Γ} , there exists ˆ _{q ∈ Γ}

such that V ( _{q) ≤} ˆ V (q) , and hence that

m = inf

Γ

V = inf

Γ^∗

V .

Remark 2.3. We have that, for all _{δ ∈ (} 0 , 2 ¯ δ) , if _{q ∈ Γ}

and _|q(t 0 )−a

| = δ for some t ₀ > 0, then

V

(q) ≥ m − δ ²

2 ⁽ 1 ₊ 2 ¯ w).

Indeed, if we let

q(t) = ˜

 

 

 



q(t) if _{t ≤ t} 0 ,

(t ₀ + 1 _{− t)q(t} 0 ) + (t − t 0 )a

if t ₀ ≤ t ≤ t 0 + 1 ,

a

if _{t ≥ t} 0 + 1 ,

we then have that ˜ _{q ∈ Γ} , and therefore m ≤ V( q) ˜

= V

(q)+

Z

1

1

2 ^|a

^{− q(t 0 )|}

2 + W ((t 0 + 1 _{− t)q(t} 0 ) + (t − t 0 )a

) dt.

But since χ((t ₀ + 1 _{− t)q(t} 0 ) + (t − t 0 )a

) ≤ δ < 2 ¯ δ , by (2.4) we have W ((t ₀ + 1 _{− t)q(t} 0 ) + (t − t 0 )a

) ≤ wδ ¯ ² ,

and hence we conclude that

m ≤ V

(q) + δ ²

2 ⁽ 1 + 2 ¯ ^w).

Let us fix ^δ 0 ∈ ( 0 ^, _δ) ¯ such that ^δ 0 /( _{δ − δ} ¯ _{0 ) <} 2 ^p 2 ^w/( 1 + 2 ¯ ^w) , so that λ ₀ : ₌ ^q 2 wδ ₀ ( _{δ − δ} ¯ _{0 ) −} ^{δ 2 0}

2 ⁽ 1 ₊ 2 ¯ w) > 0 .

In the sequel, we will study the compactness properties of the sublevel {V ≤ m + λ 0 } : _{= {q ∈ Γ}

| V(q) ≤ m + λ 0 } . First of all, we note that if _{q ∈ H loc} ¹ (R) ² is such that W (q(t)) ≥ µ for all t ∈ (σ , τ) ⊂ R , µ > 0, then

(2.5) V

(q) ≥ 1

2 _{(τ − σ )} |q(τ) − q(σ )| ² + µ(τ − σ ) ≥ q

2 µ |q(τ) − q(σ )|.

As a first consequence, using (W

), we obtain the following estimate.

Lemma 2.4. For all ^{λ >} 0, there exists ^R

> R and ^C

> 0 such that if q ∈ Γ

and V (q) ≤ m + λ , then kqk

≤ R

and kq − z 0 k

≤ C

.

Proof. To prove the first assertion, assume by contradiction that there exists q ∈ Γ

such that V (q) ≤ m + λ and |q(s)| > 2 r for some r > R and _{s ∈ R} . Then, since _{q(t) → a}

as _{t → +∞} , by continuity we have that there exists s <

σ < τ ∈ R such that _{|q(σ )| =} 2 r , _{|q(τ)| = r} , and r < |q(t)| < 2 r for any t ∈ (σ , τ) ; hence, by (W

), we obtain that W (q(t)) ≥ µ 0 for every t ∈ (σ , τ) . Then, by (2.5) we conclude

m + λ ≥ V(q) ≥ V

(q) ≥ q 2 ^µ 0 r , which is impossible for ^{r > R} big enough.

Now, since kqk

≤ R

, by (2.2), there exists ^ω

> 0 such that W (q) ≥ ω

χ ² (q(t)), ∀ t ∈ R,

and, since ^q(t) 1 t > 0 for all t 6= 0, we obtain that W (q(t)) ≥ C

|q(t) − a

| ² for all ^{t >} 0. Hence,

ω

Z

0 |q(t) − a

| ² d t ≤ Z

0 W (q(t)) dt ≤ V(q).

Therefore, by symmetry, we obtain that

kq − z 0 k

≤ C

for some ^C

> 0 . ❐ Moreover, we have the following result.

Lemma 2.5. For all λ ∈ ( 0 , λ ₀ ) there exists ^T

> 0 such that if q ∈ Γ

and V (q) ≤ m + λ , then χ(q(t)) < _δ ¯ for all |t| > T

.

Proof. By Lemma 2.4, _kqk

≤ R

, and by (2.2), let ω

> 0 be such that W (q(t)) ≥ ω

χ(q(t)) ² for all t ∈ R . Then, noting that ^q( 0 ⁾ 1 = 0, ^q(t) 1 > 0 for all t > 0 and _{q(t) → a}

as _{t → +∞} , let t ₀ = min _{{t >} 0 : _{|q(t) − a}

_{| = δ} 0 } . Then, _{|q(t) − a}

_{| ≥ δ} 0 for all _{t ∈ (} 0 , t ₀ ) , and hence, for all _{t ∈ (} 0 , t ₀ ) , we have W (q(t)) ≥ ω

δ ² ₀ > 0; then, by (2.5),

V (q) ≥ V

0

(q) ≥ ω

δ ² ₀ t ₀ .

Therefore, ^t 0 ≤ T

: = (m + λ)/(ω

δ ² ₀ ) . Now, we claim that |q(t) − a

| ≤ _δ ¯ for all ^{t > T}

. Indeed, arguing by contradiction, we have that there exist τ > σ > T

such that ^δ 0 < |q(t) − a

| < _δ ¯ for all t ∈ (σ , τ) , |q(σ ) − a

| = δ 0 , and

|q(τ)−a

| = _δ ¯ . Hence, by (2.4), we have that W (q(t)) ≥ wδ ² 0 for all t ∈ (σ , τ) , and hence, by (2.5),

V

(q) ≥ q

2 wδ ² ₀ |q(τ) − q(σ )| ≥ q

2 wδ ₀ ( _{δ − δ} ¯ _{0 ).}

But since _|q(t 0 ) − a

| = δ 0 , by Remark 2.3 we have that V

(q) ≥ m − δ ² ₀

2 ⁽ 1 + 2 ¯ ^w), and hence we obtain the contradiction

m + λ ≥ V(q) ≥ V

(q) + V

(q)

≥ m − δ ² ₀

2 ⁽ 1 + 2 ¯ w) + q

2 ^wδ 0 ( _{δ − δ} ¯ _{0 ) = m + λ 0 .}

Then, _{|q(t) − a}

_{| ≤ δ} ¯ for all t > T

and, by symmetry, we conclude that

χ(q(t)) ≤ _δ ¯ for all _{|t| > T}

. ^❐

By Lemma 2.4, we obtain the following result.

Lemma 2.6. Let ^(q

) be a sequence in Γ

such that ^{V (q}

) ≤ m + λ for some λ ∈ ( 0 , λ ₀ ) and all n ∈ N . Then, there exists q ∈ Γ

such that, along a subsequence, q

− q → 0 weakly in ^{H 1} (R) ² and V (q) ≤ lim inf

V (q

) .

Proof. By Lemma 2.4 we have that there exist R ₀ > 0 and ^C 0 > 0 such that kq

k

≤ R 0 and _kq

− z 0 k

≤ C 0 for all _{n ∈ N} . Then, up to a subsequence, (q

) weakly converge in Γ and uniformly on a compact subset to some _{q ∈ Γ} . Since (q

) ⊂ Γ

, by uniform convergence we obtain that _{q ∈ Γ}

. Finally, by Fatou’s Lemma, we also obtain that _{V (q) ≤} lim inf

V (q

) . ^❐ In particular, using Remark 2.2, we obtain that _{M = {q ∈ Γ}

| V(q) = m}

is not empty. Then, by standard argument, it can be proved that every _{q ∈ M} is a classical solution to problem (2.1).

The following lemma establishes, in particular, that _M is compact.

Lemma 2.7. Let ^(q

) ⊂ Γ

and q ∈ Γ

be such that ^q

− q → 0 weakly in H ¹ (R) ² , ^{V (q}

) → ℓ < m + λ 0 as n → +∞ , and ^{V (s}

q

+ ( 1 − s

)q) = ℓ for all n ∈ N and some sequence ^(s

) ⊂ [ 0 ^, 1 ^] . Then,

( 1 _{− s}

) q

− q ²

→ 0 as _{n → ∞.}

Proof. Since ^{V (q}

) → ℓ < m + λ 0 , we can assume that V (q

) < c + λ 0 for all _{n ∈ N} . Moreover, since V is weakly lower semicontinuous, we have also that V (q) < m + λ 0 . Then, by Lemma 2.5, there exists T ₀ > 0 such that

(2.6) if |t| > T 0 , then χ(q(t)) < _δ ¯ and ^χ(q

(t)) < _δ ¯ for all n ∈ N, and the same holds true for their convex combination s

q

+ ( 1 _{− s}

)q . Then, by (W

), we obtain that, for all _{|t| > T} 0 , we have the following results:

( 1 _{− s}

)W (q

(t)) ≤ ( 1 _{− s}

)W (q(t)) + W (q

(t)) (2.7)

− W (s

q

(t) + ( 1 _{− s}

)q(t)).

Moreover, by (2.4), we have that

(2.8) wχ(q

(t)) ² ≤ W (q

(t)) ∀ |t| > T 0 .

Since, by assumption, we have q

→ q in L

_loc (R) ² , ˙ q

− _{q →} ˙ 0 weakly in L ² (R) ² , and V (q

) − V(s

q

+ ( 1 _{− s}

)q) → 0, we derive that, for every _{T ≥ T} 0 , (2.9) 1 − s

²

2 ^k q ˙

− _qk ˙ ² ₊ Z

|t|>T

W (q

) − W (s

q

+ ( 1 − s

)q) dt → 0

as n → ∞.

Then, by (2.7) and (2.9), using (2.8) we recover 1 _{− s}

²

2 ^k q ˙

− _qk ˙ ² _{+ (} 1 _{− s}

)w Z

|t|>T

χ(q

) ² d t

≤ ( 1 _{− s}

) Z

|t|>T

W (q) dt + o( 1 ) as _{n → ∞.}

Moreover, by the choice of T ₀ and the symmetric property of q

and q , for _{|t| >}

T ₀ we have

|q

(t) − q(t)| ² ≤ 2 ^(χ(q

(t)) ² + χ(q(t)) ² ), ∀ n ∈ N.

Hence, since Z

R

W (q) dt < +∞ and Z

R

χ(q) ² d t < +∞ , we obtain that, for any η > 0, there exists ^T

> T ₀ such that

( 1 _{− s}

)  k q ˙

− _qk ˙ ²

2 ^{+ w}

Z

|t|>Tη

|q

− q| ² d t



≤ 1 _{− s} ²

2 ^k q ˙

− _qk ˙ ² _{+ (} 1 _{− s}

)w Z

|t|>T^η

|q

− q| ² d t ≤ η + o( 1 ) as _{n → +∞} . Since q

− q → 0 in L

_loc (R) ² , we obtain the claim. ^❐

By Lemma 2.6, choosing s

= 0 in Lemma 2.7 allows us to conclude.

Theorem 2.8. Let ^(q

) be a minimizing sequence for ^V over Γ

. Then, there exists q ∈ M such that, along a subsequence, kq

− qk

→ 0 as n → +∞ .

In particular, by the previous theorem we obtain that, for every ^{r >} 0, there exists ^ν

> 0 such that

(2.10) if _{q ∈}



V ≤ m + λ ₀ 2



and d

(q, M) ≥ r , then V (q) ≥ m + ν

. Now, by condition ( _∗ ), let _M

_{⊂ M} be such that _{M = M}

_{∪ M}

, and let us denote

(2.11) d (M

, M

) : ₌ 5 d ₀ > 0 .

Moreover, using (2.10), we fix m

∈ (m, m + λ 0 2 ) such that (2.12) if _{q ∈ {V ≤ m}

_}, then d

(q, M) ≤ d 0 . Then, we set

V

= {q ∈ Γ

| V(q) ≤ m

and d

(q, M

) ≤ d 0 }.

Plainly, we have that _M

_{⊂ V}

and, by (2.11) and (2.12), we have (2.13) _{{V ≤ m}

_{} = V}

_{∪ V}

and d (V

, V

) ≥ 3 d 0 .

Finally, having fixed any c ∈ [m, m

] , let us denote V

= {u ∈ V

| V(u) ≤ c} , and note that

{V ≤ c} = V

∪ V

and ^d (V

, V

) ≥ 3 ^d 0 . We have the following result.

Lemma 2.9. The sets V

are weakly compact in Γ

.

Proof. Indeed, if ^(q

) ⊂ V

, then d

(q

, M

) ≤ d 0 and V (q

) ≤ m

. Then, by Lemma 2.6, there exists q ∈ {V ≤ c} such that, up to a subsequence, q

− q → 0 weakly in H ¹ (R) ² . Hence, by the weak semicontinuity of the norm, we obtain inf

¯

kq − _qk ¯

_{≤ d 0} , and so _{q ∈ V}

. ^❐

3. T ^WO -D ^IMENSIONAL S ^OLUTIONS

In this section, we prove Theorems 1.1 and 1.2 using a variational approach. First, for (y ₁ , y ₂ ) ⊂ R , we set S

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

H = n

u ∈ H loc ¹ (R ² ) ² | u − z 0 ∈ \

(y₁,y₂)⊂R

H ¹ (S

) ² o .

Note that if _{u ∈ H} , then u(·, y) ∈ Γ for almost every _{y ∈ R} , and we will consider

H

= {u ∈ H | u(·, y) ∈ Γ

for almost every _{y ∈ R}.}

Moreover, note that, setting ¯ Γ = z 0 + L ² (R) ² (the completion of Γ with respect to the L ² -metric), for all _{u ∈ H} , the function y ∈ R ֏ u(·, y) ∈ _Γ ¯ defines a continuous trajectory verifying

(3.1) ku(·, y 2 ) − u(·, y 1 )k ² ≤ ∂

u ²

(y1^,y2⁾)²

|y 2 − y 1 |, ∀ (y 1 , y ₂ ) ⊂ R.

Remark 3.1. Given any _{u ∈ H}

and (y

) ⊂ R such that _{u(·, y}

) ∈ V

and y

→ y 0 as _{n → +∞} , we have that, by (3.1), _{u(·, y}

) → u(·, y 0 ) in ¯ Γ ; and since, by Lemma 2.9, _V

are weakly precompact in Γ

, we conclude that u(·, y

) → u(·, y 0 ) weakly in Γ

and _{u(·, y} 0 ) ∈ V

.

Finally, setting V (u(·, y)) = +∞ whenever _{u(·, y) ∈ Γ \ Γ} ¯ , we note that the function y ∈ R ֏ V(u(·, y)) ∈ [m, +∞] is lower semicontinuous for every given _{u ∈ H} .

Having fixed a value _{c ∈ [m, m}

) , we look for minimal properties of the functional

ϕ

(u) = Z

R

1

2 ^k∂

^{u(·, y)k}

2 + (V(u(·, y)) − c) dy on the set

H

= n

u ∈ H

| lim inf

y→±∞

d (u(·, y), V

) ≤ d 0

and V (u(·, y)) ≥ c almost everywhere in R o . Note that, if _{u ∈ H}

, then V (u(·, y)) ≥ c for almost every _{y ∈ R} , and so the functional ϕ

is well defined on _H

with values in [ 0 _{, +∞]} .

Finally, given an interval _{I ⊂ R} , we also consider on _H

the functional ϕ

(u) =

Z

I

1

2 ^k∂

u(·, y)k + (V(u(·, y)) − c) dy.

Remark 3.2.

(i) Given any _{I ⊆ R} , the functional ϕ

(u) is well defined for all _{u ∈} H

such that the set {y ∈ I | V(u(·, y)) < c} has bounded measure.

Moreover, if (u

) ⊂ H

is such that u

→ u weakly in H _loc ¹ (R ² ) ² , then ϕ

(u) ≤ lim inf ϕ

(u

) (see [3, Lemma 3.1]).

(ii) Given _{u ∈ H}

, if (y ₁ , y ₂ ) ⊂ R and µ > 0 are such that V (u(·, y)) ≥ c + µ for all _{y ∈ (y} 1 , y ₂ ) , then

ϕ

(u) ≥ 1

2 (y ₂ − y 1 ) ku(·, y 1 ) − u(·, y 2 )k ² + µ(y 2 − y 1 ) (3.2)

≥ q

2 µ ku(·, y 1 ) − u(·, y 2 )k.

(iii) As a result, _H

6= ∅ , and by (3.2), inf

ϕ

: _{= m}

≥ p

2 (m

− c)d 0 . We characterize below some concentration and compactness properties of the minimizing sequences in _H

. First, thanks to the discreteness of the set _{{V ≤} m

} described by (2.13), we obtain that if ϕ

(u) < +∞ , then the trajectory y ∈ R → u(·, y) ∈ _Γ ¯ is bounded. More precisely, we have the following result.

Lemma 3.3. There exists ^{C >} 0 such that if u ∈ H

, then ku(·, y 1 ) −

u(·, y 2 )k

≤ Cϕ

(u) for all ^(y 1 , y ₂ ) ⊂ R .

Proof. First note that, by Lemma 2.9, diam _(V

) and diam _(V

) are bounded;

let us denote D the maximum value. Now, we note that _{{V ≤ m}

_{} = V}

_{∪ V}

, and if V (u(·, y)) > m

> c for all y ∈ (σ , τ) ⊆ (y 1 , y ₂ ) , by (3.2) we have

ku(·, σ ) − u(·, τ)k ≤ ϕ

(u) p 2 ^(m

− c) . Hence, for all y ∈ [y 1 , y ₂ ] we have that

d (u(·, y), V

∪ V

) ≤ ϕ

(u) p 2 (m

− c) and then

ku(·, y 1 ) − u(·, y 2 )k ≤ 2 D + ϕ

(u) p 2 (m

− c) . The lemma then follows by our choosing C = max _{ 1 / p

2 (m

− c), 2 D/m

} . ^❐ By the previous result, we get the following characterization of the asymptotic behaviour of the trajectory.

Lemma 3.4. If u ∈ H

and ^ϕ

(u) ≤ m

+ 1, then

y→±∞

lim d (u(·, y), V

) = 0 ^.

Proof. Note that if we have u ∈ H

and ϕ

(u) ≤ m

+ 1, then we have lim inf

V (u(·, y)) = c , and so lim inf

d (u(·, y), {V ≤ c}) = 0. Since {V ≤ c} = V

∪ V

and d (V

, V

) ≥ 3 d ₀ , by the definition of _H

, we get

lim inf

y→±∞

d (u(·, y), V

) = 0 .

Now, by contradiction, assume that lim sup

^d (u(·, y), V

) > 0. Then, by (3.1), the path y ֏ u(·, y) crosses infinitely many times an annulus of positive thickness d > 0 around _V

in the L ² metric. Then, by (3.2), we get the contradic- tion ϕ

(u) = +∞ . Analogously, we obtain lim sup

d (u(·, y), V

) = 0. ^❐ To recover compactness properties (modulo y -translations) of the minimizing sequences of ϕ

on _H

, it is useful to study the concentration properties of the trajectories y ∈ R → u(·, y) ∈ ¯ _Γ when ϕ

(u) is close to m

. To this end, by Lemmas 2.6 and 2.7, we have first the following result.

Lemma 3.5. Let ^(u

) ⊂ H

and ^y

∈ R be such that ^{V (u}

(·, y

)) → c as n → +∞ . Then,

(i) If ^u

(·, y

) ∈ V

for all n ∈ N , then lim inf

ϕ

(u

) ≥ m

.

(ii) If ^u

(·, y

) ∈ V

for all n ∈ N , then lim inf

ϕ

(u

) ≥

m

.

Proof. We give only a sketch of the proof; the details can be found in the proof of Lemma 3.4 in [5]. Let us prove (i). Note that, by the invariance with respect to the y -translation of V , we can assume y

= 0 for all _{n ∈ N} . Setting q

= u

(·, 0 ) , we have q

∈ V

and V (q

) → c ; hence, by Lemmas 2.6 and 2.9, there exists _{q ∈ V}

such that _{V (q) ≤ c} and (up to a subsequence) q

− q → 0 weakly in H ¹ (R) ² . We set s

= sup _{{s ∈ [} 0 , 1 ] : V (sq

+ ( 1 − s)q) ≤ c} , noting that, by continuity, V (s

q

+ ( 1 _{− s}

)q) = c for all _{n ∈ N} . Hence, by Lemma 2.7, we have

(3.3) ⁽ 1 − s

)

q

− q ²

→ 0 as n → ∞.

Now, considering the new sequence

˜

u

(x, y) =

 

 

 



u

(x, y) if _{y ≤} 0 ,

yq(x) + ( 1 − y)q

(x) if 0 ≤ y ≤ 1 − s

, ( 1 _{− s}

)q(x) + s

q

(x) if _{y ≥} 1 _{− s}

, we have ˜ u

∈ H

and so ϕ

( u ˜

) ≥ m

. Then, since

ϕ

₀

(u

)

= ϕ

( u ˜

) − Z 1

0

1 2

q

− q ²

+ (V(yq(x) + ( 1 − y)q

(x)) − c) dy,

the lemma follows once we prove that Z 1

0

1 2

q

− q ²

+ V(yq(x) + ( 1 _{− y)q}

(x)) − c dy → 0 , as _{n → +∞.}

Indeed, Z 1

0 1

2 kq

− qk ² d y = ¹ ₂ ( 1 _{− s}

)kq

− qk ²

→ 0 follows by (3.3), while, by (2.2) (as in [4, Lemma 2.13]), we have that there exists ^C > 0 such that

V (yq + ( 1 _{− y)q}

) − c = V(yq + ( 1 _{− y)q}

) − V(( 1 _{− s}

)q + s

q

)

≤ C( 1 _{− s}

)kq

− qk

for all n ∈ N and y ∈ [ 0 ^, 1 − s

] . Then, by (3.3),

Z ₁

0 V (yq + ( 1 _{− y)q}

) − c dy → 0 ,

and the lemma follows. ^❐

Lemma 3.5 is used to obtain the following result that, together with (3.2), is used in Lemma 3.7 to characterize the concentration properties of the minimizing sequences of ϕ

on _H

. First, denoting

ℓ ₀ = min

 

 1;

s m

− c 2 ^{d 0}

 

 ,

we obtain the following concentration result.

Lemma 3.6. There exists ν ∈ ( 0 , m

− c) such that if u ∈ H

, ^ϕ

(u) ≤ m

+ ℓ 0 and V (u(·, y)) < c + ν ¯ for some ¯ y ∈ R , then

(i) either u(·, _{y) ∈ V} ¯

and ^d (u(·, y), V

) ≤ d 0 for all y ≤ y ¯ ; (ii) or u(·, _{y) ∈ V} ¯

and ^d (u(·, y), V

) ≤ d 0 for all y ≥ y ¯ .

Proof. Here, we prove only (ii), since the proof of (i) is analogous.

Assume by contradiction that there exists a sequence (u

) ⊂ H

such that ϕ

(u

) ≤ m

+ ℓ ⁰ , and two sequences y

1 < y

2 in R such that, for all _{n ∈ N} , we have

n→∞

lim V (u

(·, y

1 )) = c, u

(·, y

1 ) ∈ V

and ^d(u

(·, y

2 ), V

) > d ₀ . Since u

(·, y

1 ) ∈ V

and d(u

(·, y

2 ), V

) > d ₀ , by (3.1), we obtain that, since d (V

, V

) ≥ 3 d 0 and _V

_{∪ V}

_{= {V ≤ m}

_} , there exists ¯ y

1 , y ¯

2 ∈ [y

1 , y

2 ] such that V (u

(·, y)) ≥ m

for every

y ∈ ( y ¯

₁ , y ¯

₂ ) and ^d(u

(·, y ¯

₁ ), u

(·, y ¯

₂ )) = d 0 . By (3.2), we obtain ϕ

(u

) ≥ ϕ

^¯

¯

(u

) ≥ p

2 (m

− c)d 0 ≥ 2 ℓ ₀ for all _{n ∈ N} . Hence, for all _{n ∈ N} , we get

ϕ

(u

) = ϕ

(u

) − ϕ

(u

) ≤ m

− ℓ 0 ,

which is a contradiction since, by Lemma 3.5, lim inf ϕ

(u

) ≥ m

. ^❐ Hence, using Lemmas 3.3, 3.4, and 3.6, arguing as in Lemma 3.6 in [6], we can prove the following result.

Lemma 3.7. Let ^(u

) ⊂ H

be such that ^ϕ

(u

) → m

as n → ∞ and such that ^d(u

(·, 0 _{), V}

_{) = d} 0 for all n ∈ N . Then, there exists ^u

∈ H

such that, up to a subsequence,

(i) u

− u

→ 0 as n → ∞ weakly in ^H _loc ¹ (R ² ) ² ;

(ii) there exists ^L

> 0 such that ^d(u

(·, y), V

) ≤ d 0 for all y ≤ −L

, and

d(u

(·, y), V

) ≤ d 0 for all y ≥ L

.