Entire solutions in <span class="mathjax-formula">$\mathbb {R}^{2}$</span> for a class of Allen-Cahn equations

October 2005, Vol. 11, 633–672 DOI: 10.1051/cocv:2005023

ENTIRE SOLUTIONS IN R

FOR A CLASS OF ALLEN-CAHN EQUATIONS

Francesca Alessio

and Piero Montecchiari

Abstract. We consider a class of semilinear elliptic equations of the form

−ε²∆u(x, y) +a(x)W(u(x, y)) = 0, (x, y)∈R² (0.1) where ε >0,a :R→Ris a periodic, positive function andW :R→Ris modeled on the classical

two well Ginzburg-Landau potential W(s) = (s²−1)². We look for solutions to (0.1) which verify the asymptotic conditionsu(x, y)→ ±1 asx→ ±∞ uniformly with respect toy∈R. We show via variational methods that if ε is sufficiently small anda is not constant, then (0.1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.

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

Received September 10, 2004.

1. Introduction

In paper we deal with a class of semilinear elliptic equations of the form

−ε²∆v(x, y) +a(x)W(v(x, y)) = 0, (1.1)

((x, y)∈R²) or equivalently (settingu(x, y) =v(εx, εy))

−∆u(x, y) +a(εx)W(u(x, y)) = 0, (1.2) where we assumeε >0 and

(H₁) a:R→Ris not constant, 1-periodic, positive and H¨older continuous,

(H₂) W ∈ C²(R) satisfiesW(s)≥0 for anys∈R,W(s)>0 for anys∈(−1,1),W(±1) = 0 andW(±1)>0.

This kind of equation arises in various fields of Mathematical Physics. As an example, whenW is the classical two well Ginzburg-Landau potential,W(s) = (s²−1)², (1.2) can be viewed as a generalization of the stationary Allen-Cahn equation introduced as a model for phase transitions in binary metallic alloys. Another kind of equation of the Mathematical Physics that fits in our assumption is the stationary version of the so called Sine-Gordon equation, corresponding to takingW(s) = 1 + cos(πs), potential which has been applied to several problems in condensed state Physics like for instance the propagation of dislocations in crystals. The functionv,

Keywords and phrases. Heteroclinic solutions, elliptic equations, variational methods.

∗Supported by MURST Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”.

1 Dipartimento di Scienze Matematiche, Universit`a Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy;

[email protected];[email protected]

c EDP Sciences, SMAI 2005

Article published by EDP Sciences and available at http://www.edpsciences.org/cocvor http://dx.doi.org/10.1051/cocv:2005023

in these models, is considered as an order parameter describing pointwise the state of the material. The global minima ofW represent energetically favoritepure phasesand different values ofvdepict mixed configurations.

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

−∆u(x, y) +a(εx)W(u(x, y)) = 0, (x, y)∈R²

x→±∞lim u(x, y) =±1, uniformly w.r.t. y∈R. (1.3)

That kind of problem has been extensively studied under various points of view. In [11], N. Ghoussoub and C. Gui partially proved a De Giorgi’s conjecture (see [9]) regarding (1.3). The following result is a particular consequence of their study.

Theorem 1.1. Ifa(x) =a₀>0for anyx∈Rand ifu∈ C²(R²)is a solution of (1.3), thenu(x, y) =q(x) for all (x, y)∈R², whereq∈ C²(R)is a solution of the problem

−¨q(x) +a₀W(q(x)) = 0, x∈R

x→±∞lim q(x) =±1.

By Theorem 1.1, when equation (1.2) is autonomous, any solution of (1.3) depends only on thexvariable being in fact a solution of the corresponding ordinary differential equation. We have to remark that the result in [11], as the De Giorgi conjecture, deals with an asymptotic condition weaker than the one in (1.3), asking that the limits ±1 are realized only pointwise with respect to y∈Rand thatuis increasing in the variablex. In such a case in [11] it is proved that the conclusion in Theorem 1.1 is still true modulo a space roto-translation. We mention that in this form the De Giorgi conjecture is still open inRⁿ forn≥4 while it was recently proved in [5] for n= 3 (see also [2]). The assumption, as in the problem (1.3), that the limits are uniform with respect to y ∈ Rⁿ⁻¹, simplifies in fact the matter and the question of De Giorgi, known in this setting as Gibbons conjecture, is nowadays completely solved for anyn≥2, see [7, 8, 10].

All these results show that, in the autonomous case, the problem (1.3) is in fact one dimensional and the set of its solutions can be considered in this sense trivial. This is not the case, in general, for systems of autonomous Allen Cahn equations as shown in [1] and this is not the case for non autonomous x-dependent Allen Cahn type equation as shown in [3]. In fact, in [3] it is proved that introducing in the potential a non trivial periodic dependence on the single variablex, as in (1.2), the one dimensional symmetry of the problem disappears. The existence of at least two solutions of problem (1.3), distinct up to translations, depending on both the planar variables xand y is displayed whenε is sufficiently small. This reveals that for thex-dependent Allen Cahn type equations (1.2) even the weaker Gibbons conjecture decades.

Pursuing the study started in [3], aim of the present paper is to show that the introduction of a space x-dependence leads to a complicated structure of the set of solutions of problem (1.3). In particular we show that, ifεis sufficiently small, it always admits infinitely many solutions depending on both the planar variables and distinct up to space translations.

To state precisely our result it is better to recall some of the properties of the one dimensional problem associated to (1.3),

−q(x) +¨ a(εx)W(q(x)) = 0, x∈R,

x→±∞lim q(x) =±1. (1.4)

As it is nowadays well known, whenais not constant andεis sufficiently small, the problem (1.4) admits the so called multibump dynamics. To be precise, letz₀∈C^∞(R) be an increasing function such thatz₀(x)→ ±1 as x→ ±∞and|z0(x)|= 1 for any|x| ≥1, and define the action functional

F(q) =

R

1

2|q(x)|˙ ²+a(εx)W(q(x)) dx

on the class

Γ ={q∈H_loc¹ (R)/qL^∞(R)= 1 and q−z₀∈H¹(R)}.

Then (see Sect. 2) there exists δ₀ ∈ (0,1/4) and ε₀ > 0 such that for any ε ≤ ε₀ there is a family of open intervals{(t⁻_j, t⁺_j)/ j∈Z} verifying

t⁺_j =t⁻_j+1and t⁺_j −t⁻_j =1

ε, for anyj∈Z,

for which, for any odd integer numberk, for anyp= (p₁, . . . , p_k) withp₁< p₂<· · ·< p_k∈Z, setting c_k,p= inf{F(q)/ q∈Γ, |q(t⁻_pi)−(−1)ⁱ| ≤δ₀and|q(t⁺_pi)−(−1)ⁱ⁺¹| ≤δ₀ fori= 1, . . . , k} and

Kk,p={q∈Γ/ F(q) =c_k,p, |q(t⁻_pi)−(−1)ⁱ| ≤δ₀and |q(t⁺_pi)−(−1)ⁱ⁺¹| ≤δ₀ fori= 1, . . . , k} we have thatKk,p is not empty, and constituted by k-bump solutions of (1.4).

In particular the 1-bump solutions are global minima ofF on Γ at the levelc=c_1,p= min_ΓF(q). Moreover it can be proved that, sinceais not constant, whenεis sufficiently small the following non-degeneracy condition holds:

K={q∈Γ/ F(q) =c}=∪p∈ZK1,p,

where the setsK1,p⊂Γ are compact and uniformly separated in Γ (with respect to theH¹(R) metric).

In [3] the existence of solutions depending on both the planar variables is proved looking for solutions to problem (1.3) which are asymptotic asy→ ±∞to different minimal setsK1,p−,K1,p+. More precisely, letting

H={u∈H_loc¹ (R²)/uL^∞(R²)≤1 andu−z₀∈ ∩(ζ1,ζ2)⊂RH¹(R×(ζ₁, ζ₂))},

in [3] it is shown that fixed p₋ = 0 there exists at least two different values of p₊ ∈ Z\ {0} for which there exists a solutionu∈ Hof (1.3) such that

d(u(·, y),K_1,p±)→0, as y→ ±∞,

where, if q ∈Γ and A ⊂Γ, we denote d(q(·), A) = inf{q−q¯ _L²_(R)/q¯∈ A} (in fact, as shown in [13], there resultsp₊=±1),

In the present paper we strengthen that result showing that (1.3) admits infinitely many solutions u ∈ H ∩C²(R²) with∂_yu= 0, which emanate asy→ −∞from different 3-bump solutions of (1.4). In fact we prove Theorem 1.2. There existsε₀>0for which for anyε∈(0, ε₀)there existsp0∈Nsuch that ifp= (p₁, p₂, p₃)∈ Z³ verifiesmin{p2−p₁, p₃−p₂} ≥p0 then there exists a solution u∈ H ∩C²(R²) of (1.3) such that ∂_yu= 0 andlim_y→−∞d(u(·, y),K3,p) = 0.

We find these solutions using a variational argument which generalizes the one introduced in [3].

In [3], following a renormalization procedure inspired by the one introduced by P.H. Rabinowitz in [15, 16], the solutions are found as minima of the renormalized action functional

ϕ(u) =

R

1

2∂_yu(·, y)²_L2(R)+ (F(u(·, y))−c) dy on the set

Mp−,p+={u∈ H | lim

y→±∞d(u(·, y),K_1,p±) = 0}.

The fact thatcis the minimal level ofF on Γ implies that for anyu∈ Mp−,p+, the functiony→F(u(·, y))−c is non negative and so the functional ϕ is well defined with values in [0,+∞]. Moreover, the discreteness of the minimal setK allows us to show that ifu∈ Mp−,p+ and ϕ(u)<+∞then sup_y∈Rd(u(·, y),K1,p−)<+∞.

This excludes “sliding” phenomena for the minimizing sequences (see [1]) possibly due to the non compactness of the domain in thex-direction. The lack of compactness in they-direction is then overcameviaconcentration compactness techniques.

Following that argument, to find solutions to (1.3) asymptotic to 3-bump solutions, fixed p ∈ P = {(p₁, p₂, p₃)∈Z³/ p₁< p₂< p₃}, we may consider the set

M^∗_p={u∈ H/ lim

y→−∞d(u(·, y),K_3,p) = 0, lim inf

y→+∞d(u(·, y),K_3,p)>0}.

A difficulty arises when we try to define on M^∗_p a suitable renormalized functional. Indeed, differently from the 1-bump case, the set K_3,p is not minimal for F on Γ and if u∈ M^∗_p, the functiony →F(u(·, y))−c_3,p is indefinite in sign. In other word the natural renormalized functional

ϕ_p(u) =

R

1

2∂yu(·, y)²_L2(R)+ (F(u(·, y))−c_3,p) dy

is not well defined on M^∗_p. To overcome that difficulty we make use of a natural constraint of the prob- lem. Indeed, we observe that any solution u ∈ H of (1.2) on the half plane R×(−∞, y₀), which satisfies d(u(·, y),K_3,p)→0 asy→ −∞, and

R×(−∞,y0)|∂yu(x, y)|²dxdy <+∞, verifies the property F(u(·, y)) =c_3,p+1

2∂yu(·, y)²_L2(R), ∀y∈(−∞, y₀), (1.5) a sort of conservation ofEnergy. In particular (1.5) implies thatF(u(·, y))≥c_3,pfor anyy∈(−∞, y₀) and that suggests us to define, givenp∈ P, the set

Mp=

u∈ M^∗_p/ inf

y∈RF(u(·, y))≥c_3,p

on which we look for minima of the natural renormalized functionalϕ_p which is well-defined there.

As in the one bump case, to avoid sliding phenomena, we have to show that ifu∈ Mp and ϕ_p(u)<+∞

then sup_y∈Rd(u(·, y),K3,p)<+∞. This is done studying the discreteness properties of the level set{F =c_3,p} showing that we have sufficient compactness in the problem whenever min{p₂−p₁, p₃−p₂}is sufficiently large, but not in general for anyp∈ P.

That allows us to prove that for such kind of p, the minimizing sequences of ϕ_p on Mp converges up to subsequences, weakly inH_loc¹ (R²). Unfortunately we can not say that the limit functionsu_p are minima ofϕ_p onMpsince the constraint inf_y∈RF(u_p(·, y))≥c_3,pis not necessarily satisfied. However we recover thatu_p∈ H and that lim_y→−∞d(u(·, y),K3,p) = 0. Moreover, setting

y_0,u= inf{y∈R/d(u_p(·, y),K3,p)>0 andF(u_p(·, y))≤c_3,p} we prove that lim inf_y→y−

0,ud(u_p(·, y),K3,p) ≥ d₀ > 0, that lim inf_y→y−

0,uF(u_p(·, y)) = c_3,p and that u_p is a classical solution of (1.2) on the setR×(−∞, y0,u).

If y_0,u = +∞, we conclude that u_p ∈ Mp∩C²(R²) is a solution to (1.2). Differently from the one bump case we are not able to precisely characterize the asymptotic behaviour of this kind of solutions as y →+∞.

Anyway we can say that in this case there exists ap =p∈ P such thatu_p(·, y) remains for large values ofy nearby the setK3,p.

If otherwisey_0,u∈R, we have thatu_psolves (1.2) only on the half planeR×(−∞, y_0,u). Using the conserva- tion ofEnergy(1.5), we show that in such caseu_psatisfies the Neumann boundary condition∂_yu_p(·, y_0,u)≡0.

This will allow us to recover, by reflection, an entire solution to (1.2) even in this case.

More precisely, setting

v_p≡u_p, ify_0,u= +∞, and v_p(x, y) =

u_p(x, y), ify≤y_0,u,

u_p(x,2y_0,u−y), ify > y_0,u, ify_0,u∈R,

we get that v_p ∈ H ∩C²(R) is a classical solution (of homoclinic type if y_0,u ∈R and of heteroclinic type if y_0,u= +∞) of (1.2) verifying∂_yv_p= 0. The fact thatv_pis actually a solution of (1.3) follows now in a standard way since sup_y∈Rd(v_p(·, y),K3,p)<+∞andϕ_p(v_p)<+∞.

We finally want to point out some comments on our result.

First, we note that the proof described above can be adapted to find solutions asymptotic as y → −∞ to k-bump solutions for anyk∈N. Moreover thatEnergyconstraint can be used even in other contests and to find other kind of solutions. We think in particular to the possibility of finding periodic solutions of thebrake orbits type (in the y-variable), and to study the case in which the function ahas more general recurrence properties (e.g. aalmost periodic).

Another remark regards the connection of our result with the ones obtained for the “fully” non autonomous case,i.e.,

−ε²∆u+a(x, y)W(u(x, y)) = 0, (x, y)∈R²

x→±∞lim u(x, y) =±1, uniformly w.r.t. y∈R, (1.6) where a is periodic in both variables. That kind of problem, and even in a more general setting, has been already considered for example in the papers [4, 6, 12–14]. In these papers the existence of a wide variety of solutions has been shown. However, in this setting, the existence of solutions asymptotic to periodic solutions in the variabley and of thek-bump type in the variablexis still an open problem and the present work gives a partial positive answer in that direction.

Some constants and notation. Before starting in our study we fix here some constants and notation which will be used in the rest of the paper.

By (H₁) there existx, x∈[0,1) such that a(x) =a≡min

t∈Ra(t), a(x) =a≡max

t∈R a(t). (1.7)

Considering if necessary a translation of the functiona, we can assume thatx < x.

By (H₂) there exists δ∈(0,¹₄) andw > w >0 such that

w≥W(s)≥wfor any|s| ∈[1−2δ,1 + 2δ]. (1.8) In particular, settingχ(s) = min{|1−s|,|1 +s|}, we have that

if|s| ∈[1−2δ,1], then w

2χ(s)²≤W(s)≤w

2χ(s)² and|W(s)| ≤wχ(s). (1.9) Therefore there existb, b >0 such that

b χ(s)²≤W(s) and|W(s)| ≤b χ(s), ∀|s| ≤1. (1.10) For anyδ∈(0,1) we denote

ω_δ= min

|s|≤1−δW(s) (1.11)

and we note thatω_δ >0 for anyδ∈(0,1). Moreover we define the constants

m=

2aω_δ δ and m₀= min 1

2,

√a

√a−1

m. (1.12)

Note that, sinceais not constant,a > aand som₀>0.

Finally, for a givenq∈L²(R) we denoteq ≡ q_L²_(R).

2. The one dimensional problem

In this section, lettinga_ε(x) =a(εx), we focalize our study to the ODE problem associated to (1.3), namely, −q(t) +¨ a_ε(t)W(q(t)) = 0, t∈R,

lim_t→±∞q(t) =±1. (2.1)

In particular we are interested in some variational aspects of the problem.

Letz₀∈C^∞(R) be an increasing function such thatz₀(t)→ ±1 as t→ ±∞and |z₀(t)|= 1 for any|t| ≥1.

We define the class

Γ =

q∈H_loc¹ (R)/qL^∞(R)= 1 andq−z₀∈H¹(R) , on which we consider the action functional

F_aε(q) =

R

1

2|q(t)|˙ ²+a_ε(t)W(q(t)) dt.

Note that the functionalF_aε is a continuous and in fact a Lipschitz continuous positive functional on Γ endowed with theH¹(R) metric (see Lem. 2.13 in the appendix). Note also that in factF_aε(q) is well defined with value in [0,+∞] wheneverq∈H_loc¹ (R).

For future references we introduce also the set

Γ ={q∈L^∞(R)/q_L^∞_(R)= 1 and q−z₀∈L²(R)}

which is in fact the completion of Γ with respect to the metric d(q₁, q₂) =q₁−q_2L²_(R).

The main objective of this section will be to study some discreteness properties of a particular sublevel ofF_aε in Γ forεsmall enough.

First we remark that, due to the unboundedness ofR, the sublevels ofF_aε in Γ are not precompact in any sense. In fact, it is sufficient to note that given a functionq∈Γ we have that the sequenceq_n(·) =q(· − ⁿ_ε) is such that F_aε(q_n) =F_aε(q) for anyn∈Nandq_n(t)→ −1∈/ Γ for any t∈Ras n→ ∞. Anyway, it is simple to recognize that the sublevels of F_aε are (sequentially) weakly precompact in H_loc¹ (R). In fact, denoting by {Faε≤c}the set{q∈Γ|F_aε(q)≤c} for everyc >0, the following result holds

Lemma 2.1. Let(q_n)⊂ {F_aε ≤c}for somec >0. Then, there existsq∈H_loc¹ (R)withqL^∞(R)≤1such that, along a subsequence,q_n→q inL^∞_loc(R),q˙_n→q˙ weakly inL²(R)and moreover F_aε(q)≤lim inf_n→∞F_aε(q_n) Proof. Let = lim inf_n→∞F_aε(q_n). Up to a subsequence, we can assume thatF_aε(q_n)→ as n→ ∞. Since qnL^∞(R) ≤ 1 and q˙_n ≤ 2c, there existsq ∈ H_loc¹ (R) and a subsequence of (qn), denoted again (q_n), such that q_n → q weakly in H_loc¹ (R) (and so strongly in L^∞_loc(R)) and such that ˙q_n → q˙ weakly in L²(R). Then, qL^∞(R) ≤1. By weak semicontinuity of the norm, we obtain q˙ ≤ lim inf_n→∞q˙_n and by the pointwise convergence and the Fatou Lemma, we get

Ra_εW(q) dt≤lim inf_n→∞

Ra_εW(q) dt.

The autonomous problem

Given a positive continuous function β, we denote F_β(q) =

R1

2|q|˙²+β(t)W(q) dt. Moreover, if I is an interval in Rwe setF_β,I(q) =

I 1

2|q˙|²+β(t)W(q) dt.

It will be useful to recall some properties of the functional F_β when β is a given positive constant. First, setting

c_b= inf

Γ F_b,

using standard argument (see e.g. [3]), it can be proved that ifQ ∈ Γ is such that F_b(Q) = c_b, then it is a classical solution to the autonomous problem

−q(t) +¨ bW(q(t)) = 0, t∈R

lim_t→±∞q(t) =±1. (P_b)

Moreover we have

Lemma 2.2. For every b >0 the problem (P_b) admits a unique solution in Γ, modulo time translation. Such solution is increasing, is a minimum of F_b onΓ andc_b =√

bc₁.

Proof. It is standard to show that (P_b) admits a solutionQ∈Γ which is a minimum on Γ of the functionalF_b. To show thatQis increasing we argue by contradiction assuming that there existσ < τ∈Rsuch thatQ(σ) =Q(τ).

Then the function

ˆ q(t) =

Q(t) ift≤σ,

Q(t+τ−σ) ift > σ, belongs to Γ and moreover c_b ≤F_b(ˆq) =F_b(Q)−_τ

σ 1

2|Q|˙ ²+bW(Q) dt. Then _τ

σ 1

2|Q|˙ ²+bW(Q) dt = 0 and we deduce that Q is constantly equal to 1 or−1 on the interval (σ, τ), a contradiction since Q solves (P_b).

Note now that since the problem (P_b) is invariant by time translations, all the functions Q(· −τ), τ ∈R, are classical solutions to (P_b) and in fact, it is a simple consequence of the maximum principle that all the solutions to (P_b) which are in Γ belong to this family. Indeed, let ¯q ∈Γ be a solution of (P_b) and t₀ ∈Rbe such that

¯

q(t)<−1 + 2δfor anyt < t₀ and ¯q(t₀) =−1 + 2δ. Let moreoverτ₀∈Rbe such thatQ(t₀−τ₀) + 1 = 2δand set h(t) = (¯q(t)−Q(t−τ₀))². We have

¨h(t) = 2(¨q(t)¯ −Q(t¨ −τ₀))(¯q(t)−Q(t−τ₀)) + 2|h(t)˙ |²≥2b(W(¯q(t))−W(Q(t−τ₀)))(¯q(t)−Q(t−τ₀)), and since, by (1.8), there results (W(s₁)−W(s₂))(s₁−s₂)≥w(s₁−s₂)² for anys₁, s₂∈[−1−2δ,−1 + 2δ], we conclude that

¨h(t)≥2bw h(t) ∀t≤τ.

Sinceh(t₀) = 0 and lim_t→−∞h(t) = 0, by the maximum principle we conclude thath(t) = 0 for anyt≤t₀ and so, by uniqueness of the solution of the Cauchy problem, that ¯q(t)≡Q(t−τ₀) for anyt∈R.

Let nowb=d >0. Settingq(t) =Q( ^d_bt) we have

¨ q(t) =d

bQ¨ d

bt

= dW

Q d

bt

= dW(q(t)),

i.e.,q is a solution of problem (P_d). Since, as we have proved, all the solutions of (P_d) in Γ are minima of F_d on Γ, we haveF_d(q) =c_d. Therefore

c_d =

R

1

2|q(t)|˙ ²+ dW(q(t)) dt

= 1 2 d b

R

Q˙

d bt

2

dt+d

RW

Q d

bt

dt=1 2

d b

R|Q(s)|˙ ²ds+√ bd

RW(Q(s)) ds

= d

b 1

2

R|Q(s)˙ |²ds+b

RW(Q(s)) ds

= d

bF_b(Q) = d

b c_b. In particular we obtainc_b=√

b c₁for anyb >0.

In the following, we will denote byq_b the unique solution to (P_b) in Γ which verifiesq_b(0) = 0.

Remark 2.1. Since the equation−¨q(t) +bW(q(t)) = 0 is reversible, the results concerning (P_b) reflect on the

symmetric problem

−¨q(t) +bW(q(t)) = 0, x∈R

lim_t→±∞q(t) =∓1. ( ˜P_b)

In fact, the problem ( ˜P_b) has only one solution in ˜Γ ={q∈H_loc¹ (R)|q(−t)∈Γ}, modulo time translations, and if we denote by ˜q_b the solution which verifies ˜q_b(0) = 0, we have ˜q_b(t) =q_b(−t) for allt∈R. Moreover we have F_b(˜q_b) = inf_˜ΓF_b =c_b.

The constants ρ₀,δ₀, ε₀ and c^∗

Here below we display some estimates concerning the functionals F_b withb ∈[a, a], the range of the func- tion a_ε, and fix some constants which will remain unchanged along the paper. In particular, note that the functionalsF_a andF_a bound respectively from below and above the functionalF_aε for anyε >0.

The basic remark is that for anyδ∈(0,1), ifq∈Γ is such that|q(t)| ≤1−δfor anyt∈(σ, τ)⊂R, then by (1.11)

F_a,(σ,τ)(q)≥ 1

2(τ−σ)|q(τ)−q(σ)|²+aω_δ(τ−σ)≥

2aω_δ |q(τ)−q(σ)|. (2.2) As direct consequence, recalling the definition of δ in (1.8) and the ones of m and m₀ given in (1.12), we recover that if q ∈ Γ is such that |q(t)| ≤ 1 −δ for any t ∈ (σ, τ) and |q(τ)−q(σ)| = δ then, by (2.2), F_a,(σ,τ)(q)≥m≥2m₀. Hence, sinceδ≤ ¹₄, we recognize thatF_a(q)≥6mfor anyq∈Γ. Thenc_a >6m, and, by Lemma 2.2 and the definition ofm₀in (1.12), we obtain

c_a−c_a= √

√a a−1

c_a≥6

√

√a a−1

m≥6m₀.

By the previous estimate, since by Lemma 2.2 the functionb→c_bis continuous on [a, a], we can fixα < α∈(a, a) andρ₀>0 such that

c_α−c_a≤ m₀

6 and c_α−c_a ≥3m₀, (2.3)

a(x+t)≤α and a(x+t)≥α , ∀ |t| ≤2ρ₀. (2.4) Given δ ∈ [0,1), assume that a function q ∈ H_loc¹ (R) verifies q(σ) = (−1)^l(1−δ) and q(τ) = (−1)^l+1(1−δ) for certain σ < τ ∈R, l ∈N. If δ >0, one easily guess thatF_b,(σ,τ)(q)≥c_b−o_δ witho_δ →0 asδ →0. The following lemma fix aδ₀>0 in such a way the o_δ0 is comparable with the constantm₀ fixed in (1.12) for any b∈[a, a].

Lemma 2.3. There existsδ₀∈(0, δ)such that ifq∈Γ verifiesq(σ) = (−1)^l(1−δ₀)andq(τ) = (−1)^l+1(1−δ₀) for someσ < τ ∈R,l∈N, then

F_b,(σ,τ)(q)≥c_b−m₀

8 , ∀b∈[a, a].

Proof. Letδ₀∈(0, δ) such thatλ_δ0 ≡(1 +^{a w}₃ )^δ0₂² < ^m₁₆⁰.We define

ˆ q(t) =

















(−1)^l ift≤σ−1,

q(σ)(t−σ+ 1) + (−1)^l(t−σ) ifσ−1≤t≤σ,

q(t) ifσ < t < τ,

q(τ)(τ+ 1−t) + (−1)^l+1(t−τ) ifτ ≤t≤τ+ 1,

(−1)^l+1 ift≥τ+ 1,

observing that ˆq∈Γ∪Γ and so that˜

c_b ≤F_b(ˆq) =F_{b,(σ−1,σ)}(ˆq) +F_b,(σ,τ)(q) +F_b,(τ,τ+1)(ˆq). (2.5) Now, let us note that since q(τ) = (−1)^l+1(1−δ₀), then |(−1)^l+1−q(t)|ˆ =δ₀(1−(t−τ))≤δ₀ ≤δ for any t∈(τ, τ+ 1) and by (1.8) we obtain

F_b,(τ,τ+1)(ˆq) = δ₀² 2 +

_τ+1

τ

bW(ˆq) dt≤ δ₀²

2 +b wδ₀² 2

_τ+1

τ

(1−(t−τ))²dt

= δ₀²

2 +b wδ₀²

6 ≤λ_δ0. (2.6)

Similar estimates allow us to conclude that alsoF_{b,(σ−1,σ)}(ˆq)≤λ_δ0 and by (2.5) the lemma follows.

In relation with ρ₀ andδ₀we set

ε₀= 1 2ρ₀min

1, a

2c_aω_δ0

. (2.7)

Remark 2.2. Note that, by (2.2) and (2.7), ifq∈Γ is such that|q(t)|<1−δ₀ for everyt∈I, whereI is an interval with length|I| ≥ ^ρ_ε⁰₀, then

F_a,I(q) =

I

1

2|q|˙²+aW(q) dt≥aω_δ0|I| ≥4c_a.

The properties and the constants fixed above exhaust the preliminaries we need to tackle the principal object of the study of this section. In the sequel we will denote

c^∗= 3c_a+m₀. (2.8)

The set{F_aε ≤c^∗}: concentration and local compactness properties

Our goal now is to characterize some discreteness properties of the sublevel {Faε ≤c^∗} which will be basic in the proof of the existence of two dimensional solutions of (1.3) in the next section.

As useful tool to study this problem we first introduce the functionnt: Γ→N, which counts the number of transitions of a functionq∈Γ between the values−1 +δ₀ and 1−δ₀.

Givenq∈Γ, let us consider the set

Dδ0,q={t∈R| |q(t)|<1−δ₀},

the set of times in which q(t) has distance from the equilibria±1 greater than δ₀. The set Dδ0,q is an open subset ofRand so it is the disjoint union of open intervals which we denote by (s_i,q, t_i,q),i∈ I. We note that, by (2.2),F_a(q)≥

i∈I

_ti,q

si,q

12|q˙|²+aW(q) dx≥

i∈Iaω_δ0(t_i,q−s_i,q) =aω_δ0|Dδ0,q|,and so

|Dδ0,q| ≤ 1

aω_δ0F_a(q) ∀q∈Γ. (2.9)

Now, for anyi∈ I, we have|q(si,q)|=|q(ti,q)|= 1−δ₀ and then we define

nt(q,(s_i,q, t_i,q)) =

1 ifq(s_i,q)=q(t_i,q), 0 otherwise.

Moreover, given any intervalA⊂Rwe set

nt(q, A) =

i∈I |(si,q,ti,q)⊂A

nt(q,(s_i,q, t_i,q)).

and finally nt(q) =nt(q,R).

The functionntcounts the number of transitions of the functionqbetween the values−1 +δ₀ and 1−δ₀. If q∈Γ we always have thatnt(q) is an odd number and the space Γ splits in the countable union of the disjoint classes:

Γ_k={q∈Γ|nt(q) =k}, k= 2n+ 1, n∈N.

Ifq∈Γ_k, by definition, there exist{i₁, . . . , i_k} ⊂ I such that

nt(q,(s_i,q, t_i,q)) =

1 ifi∈ {i₁, . . . , i_k}, 0 otherwise.

We can assume that for anyl∈ {1, . . . , k−1}the interval (s_il,q, t_il,q) is on the left of the interval (s_il+1,q, t_il+1,q) and we set

(σ_l,q, τ_l,q) = (s_il,q, t_il,q), l= 1, . . . , k.

With this position we have that for anyl∈ {1, . . . , k},

q(σ_l,q) = (−1)^l(1−δ₀) and q(τ_l,q) = (−1)^l+1(1−δ₀).

Fixed anyε∈(0, ε₀) and givenj∈Z, we define the intervals A_j=

j+x

ε ,j+ 1 +x ε

, O_j=

j+x−ρ₀

ε ,j+x+ρ₀ ε

(2.10) wherexis a maximum foraas defined in (1.7) andρ₀is defined by (2.4). Note that, ifi=j thenA_i∩A_j =∅ and R=∪i∈ZA¯_j, where ¯A_j denotes the closure ofA_j. Moreover for anyj ∈Z, the intervals O_j and O_j+1 are

centered respectively on the left and on the right extreme ofA_j and, by (2.4), we have if sup

s∈Oj

|t−s| ≤ρ₀

ε then a_ε(t)≥α. (2.11)

Note finally that|Aj|=¹_ε and|Oj|= ^2ρ_ε⁰ for anyj∈Z.

We can now describe simple concentration properties of the functions in{Faε ≤c^∗}. Firstly we show that ifq∈ {Faε ≤c^∗}thennt(q)≤3, in other wordsqmakes at most three transitions between the values−1 +δ₀ and 1−δ₀.

Lemma 2.4. If q∈Γ andF_aε(q)≤c^∗ thennt(q)≤3.

Proof. We simply observe that ifnt(q) =k >3 then by Lemma 2.3 we have

F_aε(q)>

4 l=1

F_{a,(σl,q,τl,q)}(q)≥4c_a−m₀ 2 ,

which contradicts the assumptionF_aε(q)≤c^∗ since by definitionc^∗ = 3c_a+m₀ and, as we know, c_a >6m≥

12m₀.

Now, givenq∈ {F_aε≤c^∗}withnt(q) = 3, we show that the intervals of transition (σ_l,q, τ_l,q) can not intersect the set∪j∈ZO_j and outside of these intervalsqis nearby±1 for less than 2δ.

Lemma 2.5. If q∈Γ is such that F_aε(q)≤c^∗ andnt(q) = 3 then (i) |q(t)|>1−2δfor all t∈R\(∪³_l=1(σ_l,q, τ_l,q)).

(ii) (σ_l,q, τ_l,q)∩(∪j∈ZO_j) =∅, for alll∈ {1,2,3}.

Proof. To prove (i), let q ∈ Γ be such that nt(q) = 3 and assume by contradiction that there exists t₀ ∈ R\(∪³_l=1(σ_l,q, τ_l,q)) for which|q(t₀)| ≤ 1−2δ. Since δ₀ < δ we have |q(σl,q)| > 1−δ and |q(τl,q)| > 1−δ for l= 1,2,3. Then, by the intermediate values Theorem, there exists (σ, τ)⊂R\(∪³_l=1(σ_l,q, τ_l,q)) such that

|q(t)| ≤ 1−δ for anyt ∈(σ, τ) and |q(τ)−q(σ)| =δ. Then by (2.2) we haveF_a,(σ,τ)(q)≥2m₀ and so using Lemma 2.3 and (2.8) we obtain

F_aε(q)≥³

l=1

F_{a,(σl,q,τl,q)}(q) +F_a,(σ,τ)(q)≥3c_a−3m₀

8 + 2m₀> c^∗, a contradiction which proves (i).

To prove (ii), first we note that

τ_l,q−σ_l,q≤ ρ₀

ε₀, ∀l∈ {1,2,3}. (2.12) Otherwise, by Remark 2.2 there exists l ∈ {1,2,3} such that F_aε(q) ≥F_{a,(σl,q,τl,q)}(q)≥4c_a > c^∗. By (2.12) and (2.11), if there existsl∈ {1,2,3}such that (σ_l,q, τ_l,q)∩(∪_j∈ZO_j)=∅, we then have thata_ε(t)≥αfor any t∈(σ_l,q, τ_l,q). Therefore, by Lemma 2.3

F_{aε,(σl,q,τl,q)}(q)≥F_{α,(σl,q,τl,q)}(q)≥c_α−m₀ 8

and so, again by Lemma 2.3,

F_aε(q)≥F_{α,(σl,q,τl,q)}(q) +

l=l

F_{a,(σl,q,τl,q)}(q)≥c_α−m₀ 8 +

2c_a−m₀ 4

·

By (2.8), since by (2.3) we havec_α−c_a≥3m₀, this contradicts the assumptionF_aε(q)≤c^∗. These concentration properties allow us to start in discretizing the set {F_aε ≤ c^∗} ∩ {nt = 3}. We let P ={p= (p₁, p₂, p₃)∈Z³|p₁< p₂< p₃}and forp∈ P we define

Γ_3,p={q∈ {Faε≤c^∗} |nt(q) = 3,(σ_l,q, τ_l,q)⊂A_pl, l= 1,2,3}.

We study here below the existence, for allp∈ P, of solutions to the problem (2.1) belonging to the set Γ_3,p. In fact, settingc_3,p= inf{Faε(q)|q∈Γ_3,p}, we will prove that for anyp∈ P the set

K3,p={q∈Γ_3,p|F_aε(q) =c_3,p}

is not empty, compact, with respect to the H¹(R) metric, and consists of solutions of (2.1). The following preliminary result shows in particular that Γ_3,pis not empty for anyp∈ P.

Lemma 2.6. c_3,p≤c^∗−^m₈⁰ for any p∈ P.

Proof. Let us consider the above defined functionq_α, solution to the problems (P_α) withq_α(0) = 0. Sinceq_αis increasing andq_α(0) = 0 we have that there existσ <0< τ such thatDδ0,qα = (σ, τ). Moreover, since by (2.3) we haveF_a(q_α)≤F_α(q_α) =c_α<2c_a, by Remark 2.2 we obtain

τ−σ < ρ₀

ε₀· (2.13)

We define the function

Q(t) =

















−1 ift≤σ−1,

q_α(σ)(t−σ+ 1) + (t−σ) ifσ−1< t≤σ, q_α(t) ifσ < t < τ, q_α(τ)(τ+ 1−t) + (t−τ) ifτ≤t < τ+ 1,

1 ift≥τ+ 1,

noting that, arguing as in the proof of Lemma 2.3,

F_α(Q) =F_{α,(σ−1,σ)}(Q) +F_α,(σ,τ)(q_α) +F_α,(τ,τ+1)(Q)≤c_α+m₀ 8 · Letting p= (p₁, p₂, p₃)∈ P we set

Q_p1(t) =Q

t−p₁+x ε

, Q_p2(t) =Q

−t+p₂+x ε

and Q_p3(t) =Q

t−p₃+x ε

·

SinceF_αis invariant by time translation and reflection, we have F_α(Q_pj) =F_α(Q)≤c_α+m₀

8 j= 1,2,3. (2.14)

We note also that, by (2.13) and (2.4), if |Qpj(t)| = 1 then t ∈ A_pj and a_ε(t) ≤ α (j = 1,2,3). Therefore, by (2.14), we obtain

F_aε,Apj(Q_pj) =F_aε(Q_pj)≤F_α(Q_pj)≤c_α+m₀

8 , j = 1,2,3. (2.15)