Existence of a Heteroclinic Solution for a~Double Well Potential Equation in an Infinite Cylinder of ℝN

Claudianor O. Alves

doi:10.1515/ans-2018-2024

Article Open Access

Existence of a Heteroclinic Solution for a~Double Well Potential Equation in an Infinite Cylinder of ℝ^N

Claudianor O. Alves

Published/Copyright: September 11, 2018

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Advanced Nonlinear Studies Volume 19 Issue 1

Abstract

This paper is concerned with the existence of a heteroclinic solution for the following class of elliptic equations:

- Δ ⁢ u + A ⁢ ( ϵ ⁢ x , y ) ⁢ V ′ ⁢ ( u ) = 0 in ⁢ Ω ,

where ϵ>0, Ω=ℝ×𝒟 is an infinite cylinder of ℝN with N≥2. Here, we consider a large class of potentials V that includes the Ginzburg–Landau potential V⁢(t)=(t2-1)2 and two geometric conditions on the function A. In the first condition we assume that A is asymptotic at infinity to a periodic function, while in the second one A satisfies

0 < A 0 = A ⁢ ( 0 , y ) = inf ( x , y ) ∈ Ω ⁡ A ⁢ ( x , y ) < lim inf | ( x , y ) | → + ∞ ⁡ A ⁢ ( x , y ) = A ∞ < ∞ for all ⁢ y ∈ 𝒟 .

Keywords: Heteroclinic Solutions; Variational Methods; Double Potential; Critical Points

MSC 2010: 26A33; 34C37; 35A15; 35B38

1 Introduction

This paper is concerned with the existence of a heteroclinic solution for the following class of elliptic equations:

(PDE) - Δ ⁢ u + A ⁢ ( ϵ ⁢ x , y ) ⁢ V ′ ⁢ ( u ) = 0 in ⁢ Ω ,

together with the Neumann boundary condition

(NC) ∂ ⁡ u ∂ ⁡ ν ⁢ ( x , y ) = 0 , x ∈ ℝ , y ∈ ∂ ⁡ 𝒟 ,

where N≥2, ϵ>0, Ω is an infinite cylinder of the type Ω=ℝ×𝒟 with 𝒟⊂ℝN-1 being a smooth bounded domain and ν=ν⁢(y) is the normal vector outward pointing to ∂⁡𝒟. Related to the functions A:Ω¯→ℝ and V:ℝ→ℝ, we are assuming the following conditions.

Conditions on V:

V ∈ C 1 ⁢ ( ℝ , ℝ ) ,
V ⁢ ( - 1 ) = V ⁢ ( 1 ) = 0 and V⁢(t)≥0 for all t∈ℝ,
V ⁢ ( t ) > 0 for all t≠-1,1.

An example of V satisfying (V1)–(V3) is the Ginzburg–Landau potential V⁢(t)=(t2-1)2.

Conditions on A:

Throughout this paper, A is a C1-function that belongs to one of the following classes:

Class 1.

The function A is asymptotic at infinity to a periodic function.

In this class, we assume that there exists a C1-function Ap:Ω¯→ℝ, which is 1-periodic in x, such that

| A ⁢ ( x , y ) - A p ⁢ ( x , y ) | → 0 as |(x,y)|→+∞,
0 < A 0 = inf ( x , y ) ∈ Ω ⁡ A ⁢ ( x , y ) ≤ A ⁢ ( x , y ) < A p ⁢ ( x , y ) for all (x,y)∈Ω.

This type of condition is well known when we are working with periodic asymptotically problem of the type

- Δ ⁢ u + A ⁢ ( x ) ⁢ u = f ⁢ ( u ) in ⁢ ℝ N ,

see for example Alves, Carrião and Miyagaki [7], Jianfu and Xiping [16] and their references.

Class 2.

The function A satisfies the Rabinowitz condition.

In this class of functions, we suppose that

0 < inf ( x , y ) ∈ Ω ¯ ⁡ A ⁢ ( x , y ) ≤ max y ∈ 𝒟 ¯ ⁡ A ⁢ ( 0 , y ) < lim inf | ( x , y ) | → + ∞ ⁡ A ⁢ ( x , y ) = A ∞ < ∞ .

A condition like the above has been introduced by Rabinowitz [10, Theorem 4.33] to study the existence of solution for a PDE of the type

- ϵ 2 ⁢ Δ ⁢ u + A ⁢ ( x ) ⁢ u = f ⁢ ( u ) in ⁢ ℝ N ,

where ϵ>0, f:ℝ→ℝ is a continuous function with subcritical growth and A:ℝN→ℝ is a continuous function satisfying

0 < inf x ∈ ℝ N ⁡ A ⁢ ( x ) < lim inf | x | → ∞ ⁡ A ⁢ ( x ) .

By using variational methods, more precisely the mountain pass theorem, Rabinowitz has established the existence of solution for ϵ small enough. For this reason, throughout this article, we will call (A3) of Rabinowitz’s condition.

By (V1)–(V3), V is a double well potential and we are interested in the existence of solutions for (PDE) and (NC) that are heteroclinic in x from 1 to -1. A heteroclinic solution from 1 to -1 is a function u∈C2⁢(Ω¯,ℝ) verifying (PDE)–(NC) with

u ⁢ ( x , y ) → 1 as ⁢ x → - ∞ and u ⁢ ( x , y ) → - 1 as ⁢ x → + ∞ , uniformly in ⁢ y ∈ 𝒟 .

In [11], Rabinowitz has proved the existence of a heteroclinic solution for elliptic equations of the type

- Δ ⁢ u = g ⁢ ( x , y , u ) in ⁢ Ω ,

together with the boundary condition (NC) and also with the Dirichlet boundary condition, that is,

u ⁢ ( x , y ) = 0 , x ∈ ℝ , y ∈ ∂ ⁡ 𝒟 .

In order to prove the existence of heteroclinic solution, in [11, Section 2], Rabinowitz has used variational methods by supposing on g the conditions below:

g ∈ C 1 ⁢ ( Ω ¯ × ℝ , ℝ ) .
g ⁢ ( x , y , t ) is even and 1-periodic in x.

In [11, Section 3], Rabinowitz has considered some conditions on g that permit to study other classes of nonlinearity. From these comments, we see that if

g ⁢ ( x , y , t ) = A ⁢ ( x , y ) ⁢ V ′ ⁢ ( t ) ,

Rabinowitz has studied the case when A⁢(x,y) is 1-periodic in x, see [11, Section 2]. Here, we continue this study, because we will work with two new classes of function A that were not considered in that paper, more precisely Classes 1 and 2.

After Byeon, Montecchiari and Rabinowitz [8] have established the existence of heteroclinic solution u:Ω→ℝm for a large class of elliptic system like

- Δ ⁢ u + V u ⁢ ( x , u ) = 0 in ⁢ Ω ,

together with the boundary condition (NC) by supposing the following conditions on potential V:

V ∈ C 1 ⁢ ( Ω ¯ × ℝ m , ℝ ) and V⁢(x1+1,x2,…,xN,y)=V⁢(x,y), i.e., V is 1-periodic in x1.
There are points a-≠a+ such that V⁢(x,a±)=0 for all x∈Ω and V⁢(x,y)>0 otherwise.
There is a constant V¯>0 such that lim inf|t|→∞⁡V⁢(x,t)≥V¯ uniformly in x∈Ω.
For N≥2, there exist constants c1,C1>0 such that

| V u ⁢ ( x , t ) | ≤ c 1 + C 1 ⁢ | t | p ,

where 1<p<N+2N-2 for N≥3 and there is no upper growth restriction on p if N=2.

In the present paper, we are working with the potential V⁢(x,y,u)=A⁢(x,y)⁢V⁢(u), with A belonging to Classes 1 or 2 and V satisfying (H1)–(H4). Our paper also continues the study made in [8] for m=1, because we are working with other classes of function A. Here, it is very important to mention that the study of elliptic system as above is very subtle because some arguments used for the scalar case m=1 cannot be used for general case m>1 as for example maximum principle.

In the literature we also find interesting papers that study the existence of heteroclinic solution for elliptic equations in whole ℝN like

- Δ ⁢ u ⁢ ( x , y ) + A ⁢ ( x , y ) ⁢ V ′ ⁢ ( u ⁢ ( x , y ) ) = 0 , ( x , y ) ∈ ℝ N ,

by supposing different conditions on A and V, see for example Alessio and Montecchiari [5, 6], Alessio, Jeanjean and Montecchiari [4, 3], Alessio, Gui and Montecchiari [2], Rabinowitz [12], Rabinowitz and Stredulinsky [15, 13, 14] and their references. The reader can find versions for elliptic systems of the above equation in Alama, Bronsard and Gui [1], Alessio, Jeanjean and Montecchiari [4], Montecchiari and Rabinowitz [9] and the references therein.

Motivated by papers [8] and [11], we intend to establish the existence of a heteroclinic solution for equation (PDE) under the Neumann boundary conditions by working with Classes 1 and 2. As in the above papers, we have used variational method, more precisely minimization technical on a special set, however new ideas have been introduced in the study of the problem, see for example Proposition 3.1. The regularity and behavior of the heteroclinic are obtained by using the same arguments found in [8].

Our main results are the following:

Theorem 1.1.

Assume (V1)–(V3), ϵ=1 and that A belongs to Class 1. Then problem (PDE)–(NC) has a heteroclinic solution from 1 to -1.

Theorem 1.2.

Assume (V1)–(V3) and that A belongs to Class 2. Then there is a constant ϵ0>0 such that problem (PDE)–(NC) possesses a heteroclinic solution from 1 to -1 for all ϵ∈(0,ϵ0).

The plan of the paper is as follows: In Section 2, we prove some technical results, which will be useful to prove the above theorems. In Section 3 we prove Theorem 1.1, while in Section 4 we prove Theorem 1.2.

2 Preliminary Results

Consider problem (PDE)–(NC) with ϵ=1, more precisely,

{ - Δ ⁢ u + A ⁢ ( x , y ) ⁢ V ′ ⁢ ( u ) = 0 , ( x , y ) ∈ Ω = ℝ × 𝒟 , ∂ ⁡ u ∂ ⁡ ν ⁢ ( x , y ) = 0 , x ∈ ℝ , y ∈ ∂ ⁡ 𝒟 .

In the sequel, we define the set

Γ = { U ∈ W loc 1 , 2 ⁢ ( Ω ) : | ∇ ⁡ U | ∈ L 2 ⁢ ( Ω ) , ∥ P k ⁢ U - 1 ∥ L 2 ⁢ ( Ω 1 ) → 0 ⁢ as ⁢ k → - ∞ , ∥ P k ⁢ U + 1 ∥ L 2 ⁢ ( Ω 1 ) → 0 ⁢ as ⁢ k → + ∞ } ,

where Ω1=(0,1)×𝒟 and

P k ⁢ U ⁢ ( x , y ) = U ⁢ ( x + k , y ) for ⁢ ( x , y ) ∈ Ω ⁢ and ⁢ k ∈ ℤ .

It is very important to observe that Γ≠∅, because the function Φ given by

(2.1) Φ ⁢ ( x , y ) = { 1 , if ⁢ x ≤ j , y ∈ 𝒟 , 2 ⁢ j + 1 - 2 ⁢ x , if ⁢ j < x ≤ j + 1 , y ∈ 𝒟 , - 1 , if ⁢ j + 1 < x , y ∈ 𝒟 ,

belongs to Γ. Furthermore, we also fix

ℒ ⁢ ( u ) = 1 2 ⁢ | ∇ ⁡ u | 2 + A ⁢ ( x , y ) ⁢ V ⁢ ( u ) ,

and the functionals J:Γ→ℝ∪{+∞} given by

(2.2) J ⁢ ( U ) = ∑ k ∈ ℤ I k ⁢ ( U )

and Ik:W1,2⁢((k,k+1)×𝒟)→ℝ defined by

I k ⁢ ( U ) = ∫ k k + 1 ∫ 𝒟 ℒ ⁢ ( U ) ⁢ 𝑑 x ⁢ 𝑑 y .

Associated with functional J we have the number

(2.3) Θ * = inf ⁡ { J ⁢ ( U ) : U ∈ Γ } .

By (2.1), Φ∈Γ, then Θ*<+∞. By the definition of Θ*, there exists a minimizing sequence (Un)⊂Γ for J, that is,

(2.4) J ⁢ ( U n ) → Θ * as ⁢ n → ∞ .

Without loss of generality, we can assume that (Un) verifies

(2.5) - 1 ≤ U n ⁢ ( x , y ) ≤ 1 for all ⁢ ( x , y ) ∈ Ω .

Indeed, for each n∈ℕ let us consider

U ~ n ⁢ ( x , y ) = { - 1 , if ⁢ U n ⁢ ( x , y ) ≤ - 1 , U n ⁢ ( x , y ) , if - 1 ≤ U n ⁢ ( x , y ) ≤ 1 , 1 , if ⁢ U n ⁢ ( x , y ) ≥ 1 .

It is easy to check that U~n∈Wloc1,2⁢(Ω) with

| U ~ n ⁢ ( x , y ) - 1 | ≤ | U n ⁢ ( x , y ) - 1 | for all ⁢ ( x , y ) ∈ Ω ,

and

| U ~ n ⁢ ( x , y ) + 1 | ≤ | U n ⁢ ( x , y ) + 1 | for all ⁢ ( x , y ) ∈ Ω .

Hence (U~n)⊂Γ, and so

Θ * ≤ J ⁢ ( U ~ n ) for all ⁢ n ∈ ℕ .

Since

J ⁢ ( U ~ n ) ≤ J ⁢ ( U n ) for all ⁢ n ∈ ℕ ,

it follows that

Θ * ≤ J ⁢ ( U ~ n ) ≤ J ⁢ ( U n ) = Θ * + o n ⁢ ( 1 ) ,

thereby showing that (U~n) is also a minimizing sequence for J on Γ with

- 1 ≤ U ~ n ⁢ ( x , y ) ≤ 1 for all ⁢ ( x , y ) ∈ Ω .

From (2.4)–(2.5), there is M>0 independent of k and m such that

∥ | ∇ ⁡ U m | ∥ L 2 ⁢ ( ( k , k + 1 ) × 𝒟 ) + ∥ U m ∥ L 2 ⁢ ( ( k , k + 1 ) × 𝒟 ) ≤ M for all ⁢ m ∈ ℕ ⁢ and ⁢ k ∈ ℤ .

Consequently, (Un) is bounded in Ek=W1,2⁢((k,k+1)×𝒟), endowed with the usual norm, for all k∈ℤ. Then for some subsequence, there is U∈Wloc1,2⁢(Ω) such that

(2.6) U n ⇀ U ⁢ in ⁢ E k ⁢ for all ⁢ k ∈ ℤ ,

(2.7) U n → U ⁢ in ⁢ L 2 ⁢ ( ( k , k + 1 ) × 𝒟 ) ⁢ for all ⁢ k ∈ ℤ ,

and

(2.8) U n ⁢ ( x , y ) → U ⁢ ( x , y ) a.e. in ⁢ Ω .

Therefore, from (2.4)–(2.8),

(2.9) J ⁢ ( U ) ≤ Θ * and - 1 ≤ U ⁢ ( x , y ) ≤ 1 a.e. in ⁢ Ω .

In the next section, our main goal is to prove that U is the desired heteroclinic solution, and in this point, the conditions on the function A play their role. However, before doing that, we need to say that if A is 1-periodic in x, the same arguments explored in [8] guarantee the existence of a heteroclinic solution W* from 1 to -1.

3 Proof of Theorem 1.1: A is Asymptotic at Infinity to a Periodic Function

By hypothesis,

A ⁢ ( x , y ) < A p ⁢ ( x , y ) for all ⁢ ( x , y ) ∈ Ω .

If W*∈Γ is a heteroclinic solution for the periodic case, we must have

Θ * ≤ J ⁢ ( W * ) < J p ⁢ ( W * ) = Θ p * ,

that is,

(3.1) Θ * < Θ p * .

Here Jp:Γ→ℝ∪{+∞} is given by

J p ⁢ ( U ) = ∑ k ∈ ℤ I k , p ⁢ ( U )

and Ik,p:W1,2⁢((k,k+1)×𝒟)→ℝ is defined by

I k , p ⁢ ( U ) = ∫ k k + 1 ∫ 𝒟 ℒ p ⁢ ( U ) ⁢ 𝑑 x ⁢ 𝑑 y ,

where

ℒ p ⁢ ( u ) = 1 2 ⁢ | ∇ ⁡ u | 2 + A p ⁢ ( x , y ) ⁢ V ⁢ ( u ) .

Moreover, Θp* is the real number given by

Θ p * = inf ⁡ { J p ⁢ ( U ) : U ∈ Γ } .

In what follows, (Um)⊂Γ is a minimizing sequence for J with

- 1 ≤ U m ⁢ ( x , y ) ≤ 1 for all ⁢ ( x , y ) ∈ Ω ,

and U∈Wloc1,2⁢(Ω) satisfies (2.6)–(2.9).

The next proposition is a key point in our approach.

Proposition 3.1 (Main Proposition).

For fixed τ∈(0,|Ω1|), there is j0∈N such that

(3.2) ∥ U - 1 ∥ L 2 ⁢ ( ( - j , - j + 1 ) × 𝒟 ) ≤ τ 𝑎𝑛𝑑 ∥ U + 1 ∥ L 2 ⁢ ( ( j , j + 1 ) × 𝒟 ) ≤ τ for all ⁢ j ≥ j 0 .

We will assume for a moment that Proposition 3.1 is proved and show Theorem 1.1.

Proof of Theorem 1.1.

From the limit J⁢(Un)→Θ*, we get

(3.3) ∑ j ∈ ℤ I j ⁢ ( U ) = J ⁢ ( U ) ≤ Θ * ,

from where it follows that

I j ⁢ ( U ) → 0 as ⁢ j → ± ∞ ,

or equivalently,

∫ 0 1 ∫ 𝒟 | ∇ ⁡ P j ⁢ U | 2 ⁢ 𝑑 x ⁢ 𝑑 y + ∫ 0 1 ∫ 𝒟 A ⁢ ( x , y ) ⁢ V ⁢ ( P j ⁢ U ) ⁢ 𝑑 x ⁢ 𝑑 y → 0 as ⁢ j → ± ∞ .

As U∈L∞⁢(Ω), we have that (P-j⁢U) is a bounded sequence in W1,2⁢(Ω1). Thus, there are a subsequence (P-jk⁢U) of (P-j⁢U) and U^∈W1,2⁢(Ω1) such that

P - j k ⁢ U ⇀ U ^ ⁢ in ⁢ W 1 , 2 ⁢ ( Ω 1 ) ⁢ ⁢ as ⁢ j k → + ∞

P - j k ⁢ U → U ^ ⁢ in ⁢ L 2 ⁢ ( Ω 1 ) ⁢ ⁢ as ⁢ j k → + ∞

and

P - j k ⁢ U ⁢ ( x , y ) → U ^ ⁢ ( x , y ) a.e. in ⁢ Ω 1 as ⁢ j k → + ∞ .

From this,

∫ 0 1 ∫ 𝒟 V ⁢ ( U ^ ) ⁢ 𝑑 x ⁢ 𝑑 y = 0 ,

then

U ^ = 1 or U ^ = - 1 ,

and so

P - j k ⁢ U → 1 or P - j k ⁢ U → - 1 in ⁢ L 2 ⁢ ( Ω 1 ) as ⁢ j k → + ∞ .

Since τ∈(0,|Ω1|), these limits combine with (3.2) to give

P - j k ⁢ U → 1 in ⁢ L 2 ⁢ ( Ω 1 ) as ⁢ j k → + ∞ .

The above argument also yields

P - j ⁢ U → 1 in ⁢ L 2 ⁢ ( Ω 1 ) as ⁢ j → + ∞ .

Similar reasoning proves

P j ⁢ U → - 1 in ⁢ L 2 ⁢ ( Ω 1 ) as ⁢ j → + ∞ .

Consequently, U∈Γ and -1≤U⁢(x,y)≤1 for all (x,y)∈Ω¯. Moreover, by (3.3),

J ⁢ ( U ) = Θ * .

Now, we claim that for each ϕ∈C0∞⁢(Ω¯), we have ∂⁡J∂⁡ϕ⁢(U)=0, where ∂⁡J∂⁡ϕ⁢(U) denotes the directional derivative of J at U in the direction of ϕ∈C0∞⁢(Ω¯), where

C 0 ∞ ⁢ ( Ω ¯ ) = { ϕ : Ω ¯ → ℝ : there exists ⁢ ψ ∈ C 0 ∞ ⁢ ( ℝ N , ℝ ) ⁢ such that ⁢ ψ ⁢ ( x ) = ϕ ⁢ ( x ) ⁢ for all ⁢ x ∈ Ω ¯ } .

Indeed, taking w=U+t⁢ϕ with ϕ∈C0∞⁢(Ω¯) and t∈ℝ, we derive that for k large enough, let us say, |k|>ℓ0, we have

I k ⁢ ( U + t ⁢ ϕ ) = I k ⁢ ( U ) for all ⁢ | k | > ℓ 0 .

Thereby

J ⁢ ( U + t ⁢ ϕ ) - J ⁢ ( U ) t = 1 t ⁢ ∑ k ∈ ℤ ( I k ⁢ ( U + t ⁢ ϕ ) - I k ⁢ ( U ) ) = ∑ k = - ℓ 0 ℓ 0 ( I k ⁢ ( U + t ⁢ ϕ ) - I k ⁢ ( U ) t ) ,

and so

∂ ⁡ J ∂ ⁡ ϕ ⁢ ( U ) = lim t → 0 ⁡ J ⁢ ( U + t ⁢ ϕ ) - J ⁢ ( U ) t = ∑ k = - ℓ 0 ℓ 0 I k ′ ⁢ ( U ) ⁢ ϕ .

As w∈Γ and J⁢(U)≤J⁢(w), a standard argument ensures that ∂⁡J∂⁡ϕ⁢(U)=0 for all ϕ∈C0∞⁢(Ω¯). Therefore,

∫ Ω ∇ ⁡ U ⁢ ∇ ⁡ ϕ ⁢ d ⁢ x + ∫ Ω A ⁢ ( x , y ) ⁢ V ⁢ ( U ) ⁢ ϕ ⁢ 𝑑 x = 0 for all ⁢ ϕ ∈ C 0 ∞ ⁢ ( Ω ¯ ) .

From this, U is a weak solution of (PDE). A regularity argument from [8, Section 6] implies that U∈C2⁢(Ω¯,ℝ), and that U is a classical solution of

- Δ ⁢ U + A ⁢ ( x , y ) ⁢ V ′ ⁢ ( U ) = 0 in ⁢ Ω and ∂ ⁡ U ∂ ⁡ ν = 0 , x ∈ ℝ , y ∈ ∂ ⁡ 𝒟 ,

with

U ⁢ ( x , y ) → 1 as ⁢ x → - ∞ and U ⁢ ( x , y ) → - 1 as ⁢ x → + ∞ , uniformly in ⁢ y ∈ 𝒟 .

From this, U is a heteroclinic solution from 1 to -1, which finishes the proof of Theorem 1.1. ∎

Proof of Proposition 3.1.

Arguing as in the proof of Theorem 1.1, there is a sequence (Pjk⁢U) such that

P j k ⁢ U → 1 in ⁢ L 2 ⁢ ( Ω 1 ) or P j k ⁢ U → - 1 in ⁢ L 2 ⁢ ( Ω 1 ) as ⁢ j k → + ∞ .

Claim 1.

The limit Pjn⁢U→-1 in L2⁢(Ω1) as jn→+∞ holds.

If the claim is not true, we must have

(3.4) P j n ⁢ U → 1 in ⁢ L 2 ⁢ ( Ω 1 ) as ⁢ j n → + ∞ .

Hence, as Um→U in L2⁢([j,k]×𝒟) for all j,k∈ℤ with j<k and Um∈Γ, there are a subsequence of (Um), still denoted by itself, i*,km∈ℕ with i*<km and km→+∞ such that

(3.5) ∥ U m ( ⋅ + j , y ) - 1 ∥ L 2 ⁢ ( Ω 1 ) < τ and ∥ U m ( ⋅ + k m , y ) - 1 ∥ L 2 ⁢ ( Ω 1 ) ≥ τ for all j ∈ [ i * , k m - 1 ] ∩ ℕ .

Indeed, by (3.4), there is i*∈ℕ such that

(3.6) ∥ P j n ⁢ U - 1 ∥ L 2 ⁢ ( Ω 1 ) < τ for all ⁢ j n ≥ i * ,

in particular

∥ U - 1 ∥ L 2 ⁢ ( [ i * , i * + 1 ] × 𝒟 ) < τ .

From this, there exists m1∈ℕ such that

∥ U m 1 - 1 ∥ L 2 ⁢ ( [ i * , i * + 1 ] × 𝒟 ) < τ .

Since Um1∈Γ, let us fix k1∈ℕ and k1≥i*+1 as the first number satisfying

∥ U m 1 - 1 ∥ L 2 ⁢ ( [ j , j + 1 ] × 𝒟 ) < τ and ∥ U m 1 - 1 ∥ L 2 ⁢ ( [ k 1 , k 1 + 1 ] × 𝒟 ) ≥ τ for all ⁢ j ∈ [ i * , k 1 ] ∩ ℕ .

By (3.6), we also have

∥ U - 1 ∥ L 2 ⁢ ( [ i * , i * + 1 ] × 𝒟 ) and ∥ U - 1 ∥ L 2 ⁢ ( [ i * + 1 , i * + 2 ] × 𝒟 ) < τ .

Hence, there exists m2∈ℕ such that

∥ U m 2 - 1 ∥ L 2 ⁢ ( [ i * , i * + 1 ] × 𝒟 ) and ∥ U m 2 - 1 ∥ L 2 ⁢ ( [ i * + 1 , i * + 2 ] × 𝒟 ) < τ .

Using the fact that Um2∈Γ, we fix k2∈ℕ and k2≥i*+2 as the first number such that

∥ U m 1 - 1 ∥ L 2 ⁢ ( [ j , j + 1 ] × 𝒟 ) < τ and ∥ U m 2 - 1 ∥ L 2 ⁢ ( [ k 2 , k 2 + 1 ] × 𝒟 ) ≥ τ for all ⁢ j ∈ [ i * , k 2 - 1 ] .

Now, using the fact the inequalities below hold

∥ U - 1 ∥ L 2 ⁢ ( [ i * , i * + 1 ] × 𝒟 ) , ∥ U - 1 ∥ L 2 ⁢ ( [ i * + 1 , i * + 2 ] × 𝒟 ) < τ and ∥ U - 1 ∥ L 2 ⁢ ( [ i * + 2 , i * + 3 ] × 𝒟 ) < τ ,

there exists m3∈ℕ such that

∥ U m 3 - 1 ∥ L 2 ⁢ ( [ i * , i * + 1 ] × 𝒟 ) , ∥ U m 3 - 1 ∥ L 2 ⁢ ( [ i * + 1 , i * + 2 ] × 𝒟 ) < τ and ∥ U m 3 - 1 ∥ L 2 ⁢ ( [ i * + 2 , i * + 3 ] × 𝒟 ) < τ .

The fact that Um3∈Γ yields that there is k3∈ℕ and k3≥i*+3 such that

∥ U m 1 - 1 ∥ L 2 ⁢ ( [ j , j + 1 ] × 𝒟 ) < τ and ∥ U m 3 - 1 ∥ L 2 ⁢ ( [ k 3 , k 3 + 1 ] × 𝒟 ) ≥ τ for all ⁢ j ∈ [ i * , k 3 - 1 ] .

Repeating the above argument, we will find sequences (Ums) and (ks) verifying (3.5). We would like point out that ks→+∞ as s→+∞, because ks≥i*+s for all s∈ℕ.

Since ∥Um∥∞≤1 and (∥|∇⁡Um|∥L2⁢((k,k+1)×𝒟)) is a bounded sequence, the sequence Qm⁢(x,y)=Um⁢(x+km,y) is bounded in Ek, for all k∈ℤ. Thus, for some subsequence, there is W∈Wloc1,2⁢(Ω) such that

(3.7) Q m ⇀ W in ⁢ E k ⁢ for all ⁢ k ∈ ℤ ,

(3.8) Q m → W in ⁢ L 2 ⁢ ( ( k , k + 1 ) × 𝒟 ) ⁢ for all ⁢ k ∈ ℤ ,

(3.9) Q m ⁢ ( x , y ) → W ⁢ ( x , y ) a.e. in ⁢ Ω ,

and

(3.10) - 1 ≤ W ⁢ ( x , y ) ≤ 1 a.e. in ⁢ Ω .

A simples change of variables gives us

(3.11) ∑ k ∈ ℤ I ~ k ⁢ ( Q m ) = J ⁢ ( U m ) = Θ * + o m ⁢ ( 1 ) ≤ Θ * + 1 ,

where

I ~ k ⁢ ( U ) = ∫ k k + 1 ∫ 𝒟 ( 1 2 ⁢ | ∇ ⁡ u | 2 + A ⁢ ( x + k m , y ) ⁢ V ⁢ ( u ) ) ⁢ 𝑑 x ⁢ 𝑑 y .

Consequently, Fatou’s Lemma together with (A1) and (3.7)–(3.11) provide

(3.12) J p ⁢ ( W ) ≤ Θ * ,

which gives

(3.13) I j , p ⁢ ( W ) → 0 as ⁢ j → ± ∞ .

Setting for each j∈ℕ the function W~j=P-j⁢W, the fact that W∈L∞⁢(Ω) together with the Sobolev embeddings guarantee the existence of W0∈L2⁢(Ω1), and a subsequence of (W~j), still denoted by itself, such that

W ~ j → W 0 in ⁢ L 2 ⁢ ( Ω 1 ) ,

that is,

∥ W ~ j - W 0 ∥ L 2 ⁢ ( Ω 1 ) → 0 .

By (3.5), for each j∈ℕ, there is m*=m*⁢(j)∈ℕ such that

∥ P - j ⁢ Q m - 1 ∥ ≤ τ for all ⁢ m ≥ m * .

Therefore, taking the limit of m→+∞, we get

∥ W ~ j - 1 ∥ ≤ τ for all ⁢ j ∈ ℕ .

Now, taking the limit of j→+∞, it follows that

∥ W 0 - 1 ∥ L 2 ⁢ ( Ω 1 ) ≤ τ .

On the other hand, by (3.13),

I 0 , p ⁢ ( W 0 ) = 0 ,

from where it follows that W0=1 or W0=-1. As τ∈(0,|Ω1|), we must have W0=1. Thereby,

∥ W ~ j - 1 ∥ L 2 ⁢ ( Ω 1 ) → 0 as ⁢ j → + ∞ .

Now, fixing Wj=Pj⁢W for j∈ℕ, the same reasoning works to show that there exist W^0∈L2⁢(Ω1) and a subsequence of (Wj), still denoted by itself, such that

W j → W ^ 0 in ⁢ L 2 ⁢ ( Ω 1 ) as ⁢ j → ∞ ,

or equivalently,

∥ W j - W ^ 0 ∥ L 2 ⁢ ( Ω 1 ) → 0 .

This information gathering with (3.13) leads to W^0=1 or W^0=-1. Next we are going to show that W^0=-1. To see why, assume by contradiction that W^0=1. From (3.5), there is j1∈ℕ such that

∥ W - 1 ∥ L 2 ⁢ ( ( j 1 - 1 , j 1 ) × 𝒟 ) ≥ τ and ∥ W - 1 ∥ L 2 ⁢ ( ( j 1 , j 1 + 1 ) × 𝒟 ) ≤ τ .

As Qm→W in L2⁢((j1-1,j1+1)×𝒟), there is m0∈ℕ satisfying

∥ Q m - 1 ∥ L 2 ⁢ ( ( j 1 - 1 , j 1 ) × 𝒟 ) ≥ τ 2 and ∥ Q m - 1 ∥ L 2 ⁢ ( ( j 1 , j 1 + 1 ) × 𝒟 ) ≤ 2 ⁢ τ for all ⁢ m ≥ m 0 .

In what follows, we denote by β=β⁢(τ) the real number given by

β A ~ 0 = inf u ∈ 𝒩 τ ⁡ I * , τ ⁢ ( u ) ,

where A~0=min⁡{1,A0},

𝒩 τ = { u ∈ W 1 , 2 ⁢ ( ( - 1 , 1 ) × 𝒟 ) : ∥ u ∥ L ∞ ⁢ ( ( - 1 , 1 ) × 𝒟 ) ≤ 1 , ∥ u - 1 ∥ L 2 ⁢ ( ( - 1 , 0 ) × 𝒟 ) ≥ τ 2 ⁢ and ⁢ ∥ u - 1 ∥ L 2 ⁢ ( ( 0 , 1 ) × 𝒟 ) ≤ 2 ⁢ τ }

and I*,τ:W1,2⁢((-1,1)×𝒟)→ℝ is defined by

I * , τ ⁢ ( u ) = ∫ - 1 1 ∫ 𝒟 ( | ∇ ⁡ u | 2 + V ⁢ ( u ) ) ⁢ 𝑑 x ⁢ 𝑑 y .

Hence, by a simple change of variable

(3.14) ∫ j 1 - 1 j 1 + 1 ∫ 𝒟 ( | ∇ ⁡ Q m | 2 + V ⁢ ( Q m ) ) ⁢ 𝑑 x ⁢ 𝑑 y ≥ β A ~ 0 for all ⁢ m ≥ m 0 .

Here we would like point out that the same arguments found in [8, Proposition 2.14] work to show that β>0. Having this in mind, we can assume without loss of generality that

(3.15) J ⁢ ( U m ) ≤ Θ * + β 4 for all ⁢ m ≥ m 0 .

In the sequel, for each j≥j1+2 and m≥m0, let us consider the function

Z j , m ⁢ ( x , y ) = { 1 , if ⁢ x ≤ j , y ∈ 𝒟 , ( ( j + 1 ) - x ) + ( x - j ) ⁢ Q m ⁢ ( x , y ) , if ⁢ j < x ≤ j + 1 , y ∈ 𝒟 , Q m ⁢ ( x , y ) , if ⁢ j + 1 < x , y ∈ 𝒟 .

By a direct computation, we see that Zj,m∈Γ and

J p ⁢ ( Z j , m ) = I p , j ⁢ ( Z j , m ) + ∑ k = j + 1 ∞ I p , k ⁢ ( Q m ) = I p , j ⁢ ( Z j , m ) + ∑ k = j + 1 + k m ∞ I p , k ⁢ ( U m ) ,

and so

Θ p * ≤ J p ⁢ ( Z j , m ) = I p , j ⁢ ( Z j , m ) + ∑ k = j + 1 + k m ∞ I p , k ⁢ ( U m ) .

As A verifies (A1)–(A2) and (J⁢(Un)) is bounded, increasing m0 if necessary, we have

(3.16) ∑ k = j + 1 + k m ∞ I p , k ⁢ ( U m ) ≤ ∑ k = j + 1 + k m ∞ I k ⁢ ( U m ) + β 4 for all ⁢ m ≥ m 0 .

Here, we have used the fact km→+∞. Now, as j≥j1+2, (3.16) implies in the inequality

Θ p * ≤ I p , j ⁢ ( Z j , m ) + J ⁢ ( U m ) - A ~ 0 ⁢ ∫ j 1 - 1 j 1 + 1 ∫ 𝒟 ( | ∇ ⁡ Q m | 2 + V ⁢ ( Q m ) ) ⁢ 𝑑 x ⁢ 𝑑 y + β 4 ,

which combined with (3.14)–(3.15) gives

(3.17) Θ p * ≤ I p , j ⁢ ( Z j , m ) + Θ * + β 4 - A ~ 0 ⁢ β A ~ 0 + β 4 = I p , j ⁢ ( Z j , m ) + Θ * - β 2 .

Since

- 1 ≤ W j ⁢ ( x , y ) ≤ 1 and W j → 1 in ⁢ W 1 , 2 ⁢ ( Ω 1 ) as ⁢ j → + ∞ ,

it is easy to check that

lim j → + ∞ ∫ 0 1 ∫ 𝒟 A ( x + j , y ) V ( ( - x + 1 + x W j ) d x d y = 0

and

lim j → + ∞ ⁡ ∫ 0 1 ∫ 𝒟 | 1 - W j | 2 ⁢ 𝑑 x ⁢ 𝑑 y = 0 .

Thus, given δ>0, there is j0=j0⁢(δ)>j1+2, which is independent of m, such that

∫ 0 1 ∫ 𝒟 A ⁢ ( x + j , y ) ⁢ V ⁢ ( - x + 1 + x ⁢ W j ) ⁢ 𝑑 x ⁢ 𝑑 y < δ for all ⁢ j ≥ j 0

and

(3.18) ∫ 0 1 ∫ 𝒟 | 1 - W j | 2 ⁢ 𝑑 x ⁢ 𝑑 y < δ for all ⁢ j ≥ j 0 .

To continue, we further claim that there are j=j⁢(m)≥j0 and m≥m0 such that

(3.19) I p , j ⁢ ( Z j , m ) = ∫ j j + 1 ∫ 𝒟 ℒ p ⁢ ( Z j , m ) ⁢ 𝑑 x ⁢ 𝑑 y < β 2 .

If the claim does not hold, for each j≥j0, there exists m1=m1⁢(j)≥m0 verifying

∫ j j + 1 ∫ 𝒟 ℒ p ⁢ ( Z j , m ) ⁢ 𝑑 x ⁢ 𝑑 y ≥ β 2 for all ⁢ m ≥ m 1 .

From the definition of Zj,m and condition (A2),

∫ j j + 1 ∫ 𝒟 | ∇ ⁡ Z j , m | 2 ⁢ 𝑑 x ⁢ 𝑑 y ≥ β 2 - ∫ j j + 1 ∫ 𝒟 A ⁢ ( x , y ) ⁢ V ⁢ ( ( j + 1 ) - x + ( x - j ) ⁢ Q m ) ⁢ 𝑑 x ⁢ 𝑑 y .

Recalling that

lim m → + ∞ ⁡ ∫ j j + 1 ∫ 𝒟 A ⁢ ( x , y ) ⁢ V ⁢ ( ( j + 1 ) - x + ( x - j ) ⁢ Q m ) ⁢ 𝑑 x ⁢ 𝑑 y = ∫ 0 1 ∫ 𝒟 A ⁢ ( x + j , y ) ⁢ V ⁢ ( - x + 1 + x ⁢ W j ) ⁢ 𝑑 x ⁢ 𝑑 y < δ ,

for j≥j0 and δ<β4, there exists m2=m2⁢(j)≥m1⁢(j) such that

∫ j j + 1 ∫ 𝒟 | ∇ ⁡ Z j , m | 2 ⁢ 𝑑 x ⁢ 𝑑 y ≥ β 4 for all ⁢ m ≥ m 2 .

By using again the definition of Zj,m, there is a constant C>0 such that

∫ j j + 1 ∫ 𝒟 | ∇ ⁡ Z j , m | 2 ⁢ 𝑑 x ⁢ 𝑑 y ≤ C ⁢ ( ∫ 0 1 ∫ 𝒟 | 1 - P j ⁢ ( Q m ) | 2 ⁢ 𝑑 x ⁢ 𝑑 y + ∫ j j + 1 ∫ 𝒟 | ∇ ⁡ Q m | 2 ⁢ 𝑑 x ⁢ 𝑑 y ) .

Now, fixing δ<β8⁢C in (3.18), we obtain

∫ j j + 1 ∫ 𝒟 | ∇ ⁡ Q m | 2 ⁢ 𝑑 x ⁢ 𝑑 y ≥ β 8 for all ⁢ m ≥ m 2 ⁢ ( j ) .

Let l∈ℕ such that

( l + 1 ) ⁢ β 8 > Θ * + 1

and fix m>max⁡{m2⁢(j):j0≤j≤j0+l}. Then

∑ k ∈ ℤ I ~ k ⁢ ( Q m ) ≥ Θ * + 1 ,

which contradicts (3.11), thereby showing (3.19). Thus, by (3.17) and (3.19),

Θ p * < Θ * ,

contrary to (3.1). This ensures that W^0=-1. From the above study, we deduce that W∈Γ, then by (3.12),

Θ p * ≤ J ⁢ ( W ) ≤ Θ * ,

which is absurd. This proves Claim 1. As a byproduct of Claim 1, there is j*∈ℕ such that

∥ P j ⁢ U + 1 ∥ L 2 ⁢ ( Ω 1 ) < τ for all ⁢ j ≥ j * .

A similar argument works to prove that

∥ P j ⁢ U - 1 ∥ L 2 ⁢ ( Ω 1 ) ≤ τ for all ⁢ j ≤ - k * .

Therefore, Proposition 3.1 follows with j0=max⁡{j*,k*}. ∎

4 Proof of Theorem 1.2: A Verifies Rabinowitz’s Condition

In this section we establish the existence of a heteroclinic solution for Class 2. In what follows, we are considering the equation

- Δ ⁢ u + A ⁢ ( ϵ ⁢ x , y ) ⁢ V ′ ⁢ ( u ) = 0 in ⁢ Ω ,

together with the Neumann boundary condition

∂ ⁡ u ∂ ⁡ ν ⁢ ( x , y ) = 0 , x ∈ ℝ , y ∈ ∂ ⁡ 𝒟 ,

where ϵ is a positive parameter and A satisfies

0 < inf ( x , y ) ∈ Ω ¯ ⁡ A ⁢ ( x , y ) ≤ max y ∈ 𝒟 ¯ ⁡ A ⁢ ( 0 , y ) < lim inf | ( x , y ) | → + ∞ ⁡ A ⁢ ( x , y ) = A ∞ < ∞ .

From now on, we are denoting by Jϵ,J∞:Γ→ℝ∪{+∞} the functionals

J ϵ ⁢ ( U ) = ∑ k ∈ ℤ I ϵ , k ⁢ ( U ) and J ∞ ⁢ ( U ) = ∑ k ∈ ℤ I ∞ , k ⁢ ( U ) ,

where Iϵ,k,I∞,k:Ek→ℝ are given by

I ϵ , k ⁢ ( U ) = ∫ k k + 1 ∫ 𝒟 ( | ∇ ⁡ U | 2 + A ⁢ ( ϵ ⁢ x , y ) ⁢ V ⁢ ( U ) ) ⁢ 𝑑 x ⁢ 𝑑 y and I ∞ , k ⁢ ( U ) = ∫ k k + 1 ∫ 𝒟 ( | ∇ ⁡ U | 2 + A ∞ ⁢ V ⁢ ( U ) ) ⁢ 𝑑 x ⁢ 𝑑 y .

Moreover, we denote by Θϵ and Θ∞ the following numbers:

Θ ϵ = inf ⁡ { J ϵ ⁢ ( U ) : U ∈ Γ } and Θ ∞ = inf ⁡ { J ∞ ⁢ ( U ) : U ∈ Γ } .

From Section 2, we know that there are Wmax,W∞∈Γ verifying Jmax⁢(Wmax)=Θmax and J∞⁢(W∞)=Θ∞. Here

J max ⁢ ( U ) = ∑ k ∈ ℤ I max , k ⁢ ( U ) and I max , k ⁢ ( U ) = ∫ k k + 1 ∫ 𝒟 ( | ∇ ⁡ U | 2 + A 0 ⁢ V ⁢ ( U ) ) ⁢ 𝑑 x ⁢ 𝑑 y

with A0=maxy∈𝒟¯⁡A⁢(0,y). Moreover,

Θ max = inf ⁡ { J max ⁢ ( U ) : U ∈ Γ } .

This fact permits us to prove the following lemma:

Lemma 4.1.

We have lim supϵ→0⁡Θϵ≤Θmax and Θmax<Θ∞.

Proof.

For each ϵ>0,

Θ ϵ ≤ J ϵ ⁢ ( W max ) .

Since

lim ϵ → 0 ⁡ J ϵ ⁢ ( W max ) ≤ J max ⁢ ( W max ) = Θ max ,

it follows that

lim sup ϵ → 0 ⁡ Θ ϵ ≤ Θ max .

On the other hand, by (A3),

Θ max ≤ J max ⁢ ( W ∞ ) < J ∞ ⁢ ( W ∞ ) = Θ ∞ ,

which shows the lemma. ∎

In the sequel, we fix ϵ0>0 small enough such that

(4.1) Θ ϵ < Θ ∞ for all ⁢ ϵ ∈ ( 0 , ϵ 0 ) .

4.1 Proof of Theorem 1.2

Arguing as in Section 2, for each ϵ>0 there is a minimizing sequence (Un)⊂Γ with -1≤Un⁢(x,y)≤1 for all (x,y)∈Ω and U∈Wloc1,2⁢(Ω) such that

J ϵ ⁢ ( U n ) → Θ ϵ ,

U n ⇀ U in ⁢ E k ⁢ for all ⁢ k ∈ ℤ ,

U n → U in ⁢ L 2 ⁢ ( ( k , k + 1 ) × 𝒟 ) ⁢ for all ⁢ k ∈ ℤ ,

U n ⁢ ( x , y ) → U ⁢ ( x , y ) a.e. in ⁢ Ω ,

- 1 ≤ U ⁢ ( x , y ) ≤ 1 for all ⁢ ( x , y ) ∈ Ω ,

and

J ϵ ⁢ ( U ) ≤ Θ ϵ .

Proposition 4.1.

For fixed τ∈(0,|Ω1|) and ϵ∈(0,ϵ0), there is j0∈N such that

∥ U - 1 ∥ L 2 ⁢ ( ( - j , - j + 1 ) × 𝒟 ) ≤ τ 𝑎𝑛𝑑 ∥ U + 1 ∥ L 2 ⁢ ( ( j , j + 1 ) × 𝒟 ) ≤ τ for all ⁢ j ≥ j 0 .

Proof.

For fixed ϵ∈(0,ϵ0), arguing as in the proof of Proposition 3.1, there is a sequence (Pjk⁢U) such that

P j k ⁢ U → 1 in ⁢ L 2 ⁢ ( Ω 1 ) or P j k ⁢ U → - 1 in ⁢ L 2 ⁢ ( Ω 1 ) as ⁢ j k → + ∞ .

Claim 2.

The limit Pjk⁢U→-1 in L2⁢(Ω1) as jk→+∞ holds.

If the claim is not true, we must have

P j k ⁢ U → 1 in ⁢ L 2 ⁢ ( Ω 1 ) as ⁢ j k → + ∞ .

Hence, as Um→U in L2⁢([j,k]×𝒟) for all j,k∈ℤ with j<k, there are a subsequence of (Um), still denoted by itself, i*,km∈ℕ with i*<km and km→+∞ such that

(4.2) ∥ U m ( ⋅ + j , y ) - 1 ∥ L 2 ⁢ ( Ω 1 ) < τ and ∥ U m ( ⋅ + k m , y ) - 1 ∥ L 2 ⁢ ( Ω 1 ) ≥ τ for all j ∈ [ i * , k m - 1 ] ∩ ℕ .

From the definition of (km), we have that km→+∞ as m→+∞. Since ∥Um∥∞≤1 and (∥|∇⁡Um|∥L2⁢((k,k+1)×𝒟)) is a bounded sequence, the sequence Qm⁢(x,y)=Um⁢(x+km,y) is bounded in Ek for all k∈ℤ. Thus, for some subsequence, there is W∈Wloc1,2⁢(Ω) such that

Q m ⇀ W ⁢ in ⁢ E k ⁢ for all ⁢ k ∈ ℕ ,

Q m ⁢ ( x , y ) → W ⁢ ( x , y ) ⁢ a.e. in ⁢ Ω ,

and

- 1 ≤ W ⁢ ( x , y ) ≤ 1 a.e. in ⁢ Ω .

By a simple change variable,

(4.3) ∑ k ∈ ℤ I ~ k ⁢ ( W m ) ≤ J ϵ ⁢ ( U m ) = Θ ϵ + o m ⁢ ( 1 ) ,

where

I ~ ϵ , k ⁢ ( U ) = ∫ k k + 1 ∫ 𝒟 ℒ ~ ϵ , m ⁢ ( U ) ⁢ 𝑑 x ⁢ 𝑑 y

with

ℒ ~ ϵ , m ⁢ ( u ) = 1 2 ⁢ | ∇ ⁡ u | 2 + A ⁢ ( ϵ ⁢ x + ϵ ⁢ k m , y ) ⁢ V ⁢ ( u ) .

Now, Fatou’s Lemma combined with (4.3) lead to

J ∞ ⁢ ( W ) ≤ Θ ϵ .

Then

(4.4) I ∞ , j ⁢ ( W ) → 0 as ⁢ j → ± ∞ .

Setting for each j∈ℕ the function W~j=P-j⁢W, the fact that W∈L∞⁢(Ω) implies that there are W0∈Wloc1,2⁢(Ω,ℝ) and a subsequence of (W~j), still denoted by itself, such that

W ~ j → W 0 in ⁢ W 1 , 2 ⁢ ( Ω 1 ) as ⁢ j → + ∞ ,

and so

∥ W ~ j - W 0 ∥ L 2 ⁢ ( Ω 1 ) → 0 .

Arguing as in the proof of Proposition 3.1, the first inequality in (4.2) leads to

∥ W 0 - 1 ∥ L 2 ⁢ ( Ω 1 ) ≤ τ .

On the other hand, by (4.4),

I ∞ , 0 ⁢ ( W 0 ) = 0 ,

which gives W0=1 or W0=-1. As τ∈(0,|Ω1|), we must have W0=1. Then

∥ W j - 1 ∥ L 2 ⁢ ( Ω 1 ) → 0 as ⁢ j → + ∞ .

By using the same type of argument, fixing Wj=Pj⁢W for j∈ℕ, it is possible to prove that there exist W^0∈Wloc1,2⁢(Ω¯) and a subsequence of (Wj), still denoted by itself, such that

W j → W ^ 0 in ⁢ W 1 , 2 ⁢ ( Ω 1 ) as ⁢ j → - ∞ ,

and so

∥ W j - W ^ 0 ∥ L 2 ⁢ ( Ω 1 ) → 0 .

Thereby, W^0=1 or W^0=-1. Here, as in the previous section, we have that W^0=-1. Indeed, assuming by contradiction that W^0=1, we set the function

H j ⁢ ( x , y ) = { 1 , x ≤ j , y ∈ 𝒟 , ( ( j + 1 ) - x ) + ( x - j ) ⁢ Q m ⁢ ( x , y ) , j < x ≤ j + 1 , y ∈ 𝒟 , Q m ⁢ ( x , y ) , j + 1 < x , y ∈ 𝒟 .

Arguing again as in the proof of Proposition 3.1, we will find

Θ ∞ ≤ Θ ϵ ,

which contradicts (4.1), and then W^0=-1. Now we follow the same idea explored in Proposition 3.1 to conclude the proof. ∎

Now, we are ready to prove Theorem 1.2.

Proof of Theorem 1.2.

As an immediate consequence of the last proposition, for each ϵ∈(0,ϵ0), there is j0∈ℕ such that

∥ U - 1 ∥ L 2 ⁢ ( ( - j , - j + 1 ) × 𝒟 ) < τ and ∥ U + 1 ∥ L 2 ⁢ ( ( j , j + 1 ) × 𝒟 ) < τ for all ⁢ j ≥ j 0 .

Now, arguing as in the proof of Theorem 1.1, it follows that U∈C2⁢(Ω¯,ℝ). Moreover, U is a classical solution of

- Δ ⁢ U + A ⁢ ( ϵ ⁢ x , y ) ⁢ V ′ ⁢ ( U ) = 0 in ⁢ Ω and ∂ ⁡ U ∂ ⁡ ν = 0 , x ∈ ℝ , y ∈ ∂ ⁡ 𝒟

with

U ⁢ ( x , y ) → 1 as ⁢ x → - ∞ and U ⁢ ( x , y ) → - 1 as ⁢ x → + ∞ , uniformly in ⁢ y ∈ 𝒟 .

From this, U is a heteroclinic solution from 1 to -1, which finishes the proof of Theorem 1.2. ∎

Communicated by Paul Rabinowitz

Funding statement: The research of the author was partially supported by CNPq/Brazil 304804/2017-7.

Acknowledgements

The author would like to warmly thank Professor Olimpio Hiroshi Miyagaki for several discussions about this subject, and also to Professor Rabinowitz by his comments that were very important to improve this manuscript. Moreover, the author would like to thank to the referee for his/her very nice remarks and suggestions, which were very important to improve this manuscript.

References

[1] S. Alama, L. Bronsard and C. Gui, Stationary layered solutions in ℝ2 for an Allen–Cahn system with multiple well potential, Calc. Var. Partial Differential Equations 5 (1997), no. 4, 359–390. 10.1007/s005260050071Search in Google Scholar

[2] F. Alessio, C. Gui and P. Montecchiari, Saddle solutions to Allen–Cahn equations in doubly periodic media, Indiana Univ. Math. J. 65 (2016), no. 1, 199–221. 10.1512/iumj.2016.65.5772Search in Google Scholar

[3] F. Alessio, L. Jeanjean and P. Montecchiari, Stationary layered solutions in ℝ2 for a class of non autonomous Allen–Cahn equations, Calc. Var. Partial Differential Equations 11 (2000), no. 2, 177–202. 10.1007/s005260000036Search in Google Scholar

[4] F. Alessio, L. Jeanjean and P. Montecchiari, Existence of infinitely many stationary layered solutions in ℝ2 for a class of periodic Allen–Cahn equations, Comm. Partial Differential Equations 27 (2002), no. 7–8, 1537–1574. 10.1081/PDE-120005848Search in Google Scholar

[5] F. Alessio and P. Montecchiari, Entire solutions in ℝ2 for a class of Allen–Cahn equations, ESAIM Control Optim. Calc. Var. 11 (2005), no. 4, 633–672. 10.1051/cocv:2005023Search in Google Scholar

[6] F. Alessio and P. Montecchiari, Layered solutions with multiple asymptotes for non autonomous Allen–Cahn equations in ℝ3, Calc. Var. Partial Differential Equations 46 (2013), no. 3–4, 591–622. 10.1007/s00526-012-0495-2Search in Google Scholar

[7] C. O. Alves, P. C. Carrião and O. H. Miyagaki, Nonlinear perturbations of a periodic elliptic problem with critical growth, J. Math. Anal. Appl. 260 (2001), no. 1, 133–146. 10.1006/jmaa.2001.7442Search in Google Scholar

[8] J. Byeon, P. Montecchiari and P. H. Rabinowitz, A double well potential system, Anal. PDE 9 (2016), no. 7, 1737–1772. 10.2140/apde.2016.9.1737Search in Google Scholar

[9] P. Montecchiari and P. H. Rabinowitz, On the existence of multi-transition solutions for a class of elliptic systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 1, 199–219. 10.1016/j.anihpc.2014.10.001Search in Google Scholar

[10] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), no. 2, 270–291. 10.1007/BF00946631Search in Google Scholar

[11] P. H. Rabinowitz, Solutions of heteroclinic type for some classes of semilinear elliptic partial differential equations, J. Math. Sci. Univ. Tokyo 1 (1994), no. 3, 525–550. Search in Google Scholar

[12] P. H. Rabinowitz, A new variational characterization of spatially heteroclinic solutions of a semilinear elliptic PDE, Discrete Contin. Dyn. Syst. 10 (2004), no. 1–2, 507–515. 10.3934/dcds.2004.10.507Search in Google Scholar

[13] P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen–Cahn type equation, Comm. Pure Appl. Math. 56 (2003), no. 8, 1078–1134. 10.1002/cpa.10087Search in Google Scholar

[14] P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen–Cahn type equation. II, Calc. Var. Partial Differential Equations 21 (2004), no. 2, 157–207. 10.1007/s00526-003-0251-8Search in Google Scholar

[15] P. H. Rabinowitz and E. Stredulinsky, On a class of infinite transition solutions for an Allen–Cahn model equation, Discrete Contin. Dyn. Syst. 21 (2008), no. 1, 319–332. 10.1007/s11784-008-0091-4Search in Google Scholar

[16] J. F. Yang and X. P. Zhu, On the existence of nontrivial solution of a quasilinear elliptic boundary value problem for unbounded domains. I. Positive mass case, Acta Math. Sci. (English Ed.) 7 (1987), no. 3, 341–359. 10.1016/S0252-9602(18)30457-0Search in Google Scholar

Received: 2018-03-13

Revised: 2018-05-21

Accepted: 2018-05-25

Published Online: 2018-09-11

Published in Print: 2019-02-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2018-2024

Keywords for this article

Heteroclinic Solutions; Variational Methods; Double Potential; Critical Points

Creative Commons

BY 4.0