Spatially heteroclinic solutions for a semilinear elliptic P.D.E.

URL:http://www.emath.fr/cocv/ DOI: 10.1051/cocv:2002027

SPATIALLY HETEROCLINIC SOLUTIONS FOR A SEMILINEAR ELLIPTIC P.D.E.

Paul H. Rabinowitz

Abstract. This paper uses minimization methods and renormalized functionals to find spatially heteroclinic solutions for some classes of semilinear elliptic partial differential equations

Mathematics Subject Classification. 35J20, 35J60, 58E15.

Received January 16, 2002.

1. Introduction

In an earlier paper [6], the equation (PDE) −∆u=g(x, u), x∈Ω⊂Rⁿ

was studied where Ω is a cylindrical domain inRⁿ,i.e. Ω =R× DwithDa bounded open set inRⁿ⁻¹having a smooth boundary. Forx∈Ω, setx= (x₁, y) withx₁∈Randy∈ D. The functiong satisfies

• (g₁)g∈C¹(Ω×R,R) and is 1-periodic inx₁;

• (g₂)G(x, z) =R_z

0 g(x, s)dsis 1-periodic inz;

• (g₃)g is even inx₁.

Letting ν(x) denote the outward pointing normal to∂Ω, the boundary condition taken for (PDE) in [6] was (BC) ∂u

∂ν = 0, x∈∂Ω.

Let

L(u) = 1

2|∇u|²−G(x, u) and Ω_j = [j, j+ 1]× D forj∈Z. By minimizing the functional

I₀(u) = Z

Ω0

L(u)dx over

E₀={u∈W^1,2(Ω₀)|u is 1-periodic in x₁},

Keywords and phrases:Spatially heteroclinic solutions, minimization methods, renormalized functional.

∗This research was sponsored in part by the National Science Foundation under grant number #MCS-8110556. Any repro- duction for the purposes of the U.S. Government is permitted.

1Mathematics Department, University of Wisconsin–Madison, U.S.A.; e-mail: [email protected]

c EDP Sciences, SMAI 2002

it was shown in [6] that hypotheses (g₁,g₂) imply there is a classical solutionv of (PDE) minimizingI₀ onE₀. If in addition (g₃) holds,vis even inx₁ and

c₀≡inf

E0

I₀=I₀(v) = inf

W^1,2(Ω0)I₀≡c. (1.1)

Set

M={v∈E₀|I₀(v) =c₀}·

Note that by (PDE) and (BC), v∈ M impliesv+j ∈ Mfor all j ∈Z. The main result in [6] was if (g1–g₃) hold andMconsists of isolated points, then for anyv∈ M, there is aw∈ M\{v}and a solution,U, of (PDE) and (BC) such thatU is heteroclinic inx₁fromvtow,i.e. U(x)−v(x)→0 asx₁→ −∞andU(x)−w(x)→0 as x₁→ ∞. It was also shown that the same result obtains for (PDE) under boundary conditions for different hypotheses ongincluding a class of water wave problems studied by Kirchgassner [3], Turner [8], and Mielke [4].

The goal of this paper is to improve on [6] in 3 ways. First the structure of Mwill be clarified. It will be shown thatMis an ordered set,i.e. v, w∈ Mandv6≡wimpliesv < w orv > w. Secondly the condition that Mconsists of isolated points will be weakened. Since ifv∈ M, so isv+ 1, either these two functions are part of a continuum inM, or not. It will be assumed that the latter alternative holds and in particularvandware adjacent members ofMwithv < w.

Thirdly, and this is the main contribution of the current paper, condition (g₃) will be dropped. This hy- pothesis was crucial in [6]. Indeed the approach taken in [6] was to find U as the minimizer of an appropriate functional. The natural functional corresponding to (PDE) is

I(u) = Z

Ω

L(u)dx

but whenc₀6= 0,I(u) will not be finite on the class of admissible functions. Therefore a renormalized functional was introduced in [6]. More precisely, set

Γ =

nu∈W_loc^1,2(Ω)|u(x)−v(x)→0 as x₁→ −∞ and u(x)−w(x)→0 as x₁→ ∞for some w∈ M\{v}o ,

and for u∈Γ, define the renormalized functional for (PDE)via J(u) =X

j∈Z

a_j(u) (1.2)

where

a_j(u) = Z

Ωj

L(u)dx−c₀. Setting τ_ju(x₁, y) =u(x₁−j, y),it follows that

a_j(u) =I₀(τ_−ju)−c₀. Since τ_−ju∈W^1,2(Ω₀), by (1.1),

a_j(u)≥0 (1.3)

and J is a nonnegative functional. In [6],U was obtained as the minimizer ofJ on Γ. Dropping (g₃), it is no longer the case thatc₀=c; in generalc₀> c. Hence (1.3) fails and it is not clear thatJ is bounded from below on Γ. Replacingc₀ bycwill not help since then J will be infinite on the natural class of admissible functions.

Thus another approach is needed here. For such an approach, we were motivated by some recent work [7] on

an Allen–Cahn model equation which in turn has antecedents in work of Moser [5] and Bangert [1] on minimal laminations of a torus and of Bosetto and Serra [2] on an ODE problem related to [1]. Γ will be replaced by a class of functions asymptotic from v tow and having an additional monotonicity property andJ by a related function J^∗. Minimizing J^∗ on Γ^∗ will produce the desired heteroclinic joining v and w. The proof will be carried out in Section 2 with some details postponed until Section 3. Then in Section 4, it will be indicated how the current approach also applies to the water wave problem of Kirchg¨assner [3].

2. The main results

In this section, our main results will be formulated and proved. As was noted in Section 1, it was shown in [6] thatM 6=φand the elements ofMare classical solutions of (PDE) and (BC) which are 1-periodic inx₁. The first result uses the variational structure of (PDE) and the Maximum Principle to say more aboutM.

Proposition 2.1. Ifg satisfies (g₁,g₂),M is an ordered set.

Proof. An argument essentially due to Moser [5] will be employed. Suppose v6=w∈ Mand v andw are not ordered. Setϕ= max(v, w) andψ= min(v, w). Thenϕ, ψ∈E₀. For functionsa, bon Ω₀,

{a≤b} ≡ {x∈Ω₀|a(x)≤b(x)} and {a < b}, the same with strict inequality. Hence

2c₀≤I₀(ϕ) +I₀(ψ) = Z

{w≥v}

L(w)dx+ Z

{v>w}

L(v)dx+ Z

{w≥v}

L(v)dx+ Z

{v>w}

L(w)dx= 2c₀. (2.1)

Thereforeϕ, ψ∈ Mand by [6] are classical solutions of (PDE) and (BC). Sincevandware not ordered, without loss of generality there are pointsx, x∈Ω₀such thatv(x)> w(x) andv(x) =w(x). Considerχ≡ϕ−w. Then χ≥0 in Ω₀andχ(x)>0. Observe thatχsatisfies a linear elliptic partial differential equation in Ω₀:





−∆χ=a(x)χ, x∈Ω₀

∂χ

∂ν = 0 x∈[0,1]×∂D (2.2)

where





a(x) = g(x, ϕ(x))−g(x, w(x))

ϕ(x)−w(x) ifϕ(x)> w(x),

=g_u(x, ϕ(x)) ifϕ(x) =w(x).

Writinga(x) =a⁺(x)−a⁻(x) where

a⁺= max(a,0); a⁻= max(a,0), equation (2.3) leads to





−∆χ+a⁻χ=a⁺χ≥0, ifx∈Ω₀

∂χ

∂ν = 0, ifx∈[0,1]×∂D. (2.3)

By the Maximum Principle,χ cannot have an interior minimum in Ω₀ unless χ ≡ constant which is not the case becausev andware not ordered. Sinceχ(x) = 0, x∈[0,1]×∂D. But then by the Maximum Principle,

∂χ

∂ν(x)<0, (2.4)

contrary to (2.3). Thusv andware ordered.

To continue, as was mentioned in the introduction, M may be a connected set. E.g. ifg ≡ 0, M =R. For the existence argument that will be given shortly, it is necessary thatMis not connected. Thus for what follows, it will be assumed that:

(∗) there are adjacent membersv < wof M.

Next it will be shown that (∗) implies there is a solution of (PDE) and (BC) heteroclinic in x₁ from v to w and similarly fromwto v. This will be done by minimizing a variant of the renormalized functionalJ of (1.2) over an appropriate class of sets, Γ^∗. To motivate the choice of Γ^∗, proceeding formally, suppose thatJ has a minimizer in the class ofW_loc^1,2(R×D) functions heteroclinic inx₁fromvtow. Suppose further the minimizer is a classical solution of (PDE) and (BC). Then repeating the proof of Proposition 2.1 for this new setting shows M^∗, the set of minimizers of J, is an ordered set. Moreover if U ∈ M^∗, (g₁) implies J(τ_jU) = J(U) for all j ∈ Z. Hence τ_jU ∈ M^∗. The ordering property of M^∗ then impliesτ₋₁U > U, i.e. U has a monotonicity property. This formal argument will be exploited by building the monotonicity into the class of functions Γ^∗ and it will be crucial for what follows. Define

Γ^∗=

u∈W_loc^1,2(R× D)|v≤u≤τ₋₁u≤w, and τ_−ju

Ω0 →v (resp.w) inL²(Ω₀) as j→ −∞(resp.∞)

·

To introduce the analogue ofJ of (1.2) that will be employed here, letm, n∈Zwithm≤n. Foru∈Γ^∗, set J_m,n(u) =Xⁿ

m

a_j(u)

and define

J^∗(u) = lim_m→−∞J_m,0(u) + lim_n→∞J_1,n(u). (2.5) Set

c^∗= inf

u∈Γ^∗J^∗(u). (2.6)

The main result of this section is:

Theorem 2.2. Letg satisfy (g₁,g₂) and let (∗) hold. Then there is aU ∈Γ^∗such thatJ^∗(U) =c^∗. Moreover U is a classical solution of (PDE) and (BC).

The proof of Theorem 2.2 is divided into several steps. So as not to delay the exposition, the details of some of the steps will be postponed until Section 3. Although the setting is different, the structure of the argument is very close to that of [7] which we will strongly follow.

To begin, note thatc^∗<∞. Indeed if

ϕ(x) =v, x₁≤0, (2.7)

=x₁w+ (1−x₁)v, 0≤x₁≤1

=w, 1≤x₁, then ϕ∈Γ^∗ andc^∗≤J^∗(ϕ)<∞.

Next it will be shown thatJ_m,nis bounded from below on Γ^∗. This requires a preliminary result.

Lemma 2.3. Letn∈NandΓ_n ={u∈W^1,2

loc(R× D)|uisnperiodic inx₁}. If c_n= inf

u∈Γn

Z _n

0

Z

D

L(u)dx. (2.8)

Then c_n=nc₀and is achieved by v∈ M.

Proof. The existence of a minimizer,v of (2.8) follows as in [6] for the case ofn= 1. It remains to prove that Z _n

0

Z

D

L(v)dx=nc₀. (2.9)

By (g₁), τ₋₁v is also a minimizer of (2.8) and the proof of Proposition 2.1 shows that the set of minimizers of (2.8) is ordered. Therefore either (i)τ₋₁v≡v, (ii)τ₋₁v < v, or (iii) τ₋₁v > v. Ife.g. (ii) is valid,

v(x₁+ 1, y)< v(x₁, y)< v(x₁−(n−1), y) =v(x₁+ 1, y), (2.10) a contradiction. Likewise (iii) cannot hold. Thus (i) is valid,i.e. v is 1-periodic inx₁,c_n=nc₀, and the result is proved.

Now it follows thatJ_m,nis bounded from below:

Proposition 2.4. There is a constantK >0 such that for all u∈Γ^∗ andm≤n,

J_m,n(u)≥ −K. (2.11)

The proof of Proposition 2.4 will be given in Section 3. The proposition implies an upper bound forJ_m,n(u):

Lemma 2.5. For allm≤nandu∈Γ^∗,

J_m,n(u)≤J(u) + 2K. (2.12)

Proof. By (2.11),

J(u) = lim_i→−∞J_i,0(u) + lim_i→∞J_1,i(u) =J_m,n(u) + lim_i→−∞J_i,m−1(u) + lim_i→∞J_n+1,i(u)≥J_m,n(u)−2K

from which (2.12) follows.

The next step in the proof is to show that the finiteness ofJ^∗(u) implies some strong asymptotic properties for u:

Proposition 2.6. Letu∈Γ^∗ andJ^∗(u)<∞. Then J^∗(u) = lim

m→−∞n→∞

J_m,n(u), (2.13)

i.e. the lim’s in (2.5) are limits, and

m→−∞lim ku−vk_W^1,2_(Ωm)= 0, (2.14)

n→∞lim ku−wk_W^1,2_(Ωn)= 0. (2.15)

The proof of this proposition will be given in Section 3.

With the aid of the above preliminaries, the function U of Theorem 2.2 can be obtained. Let (u_k) be a minimizing sequence for (2.6). Then there is anM >0 such that

J(u_k)≤M (2.16)

for all k ∈ N. Since u ∈Γ^∗ impliesτ_ju ∈ Γ^∗ for all j ∈ Z, and J^∗(τ_ju) = J^∗(u), the sequence (u_k) can be normalized by requiring that







 Z

Ω−1

(u_k−v)dx≤1 2

Z

Ω0

(w−v)dx Z

Ω0

(u_k−v)dx≥1 2

Z

Ω0

(w−v)dx. (2.17)

Note that since u_k ≤τ₋₁u_k,







 Z

Ωj

(u_k−v)dx≤ 1 2

Z

Ω0

(w−v)dx, j≤ −1 Z

Ωj

(u_k−v)dx≥ 1 2

Z

Ω0

(w−v)dx, j≥0. (2.18)

By (2.12) and (2.16), 1 2

Z _n+1

m

Z

D|∇u_k|²dx≤ Z _n+1

m

Z

D

G(x, u_k)dx+ (n+ 1−m)c₀+M+ 2K. (2.19)

Thus (2.19) coupled with theL^∞bounds foru_kimply (u_k) is bounded inW_loc^1,2(Ω). Hence there is aU ∈W_loc^1,2(Ω) such that along a subsequence,u_k →U weakly inW_loc^1,2, strongly inL²_loc, and pointwise a.e. Thus

v ≤U ≤τ₋₁U ≤w (2.20)

and U satisfies (2.18). By weak lower semicontinuity and (2.12) again,

J_m,n(U)≤lim_k→∞J_m,n(u_k)≤M + 2K (2.21)

for allm≤nin Z. Hence

J^∗(U)≤M+ 2K <∞. (2.22)

To show thatU ∈Γ^∗, it remains to prove thatU is heteroclinic fromv tow. Towards that end, forx∈Ω₀, set V_i =τ_iU. Then (V_i) is a monotone nonincreasing sequence asi→ ∞andV_i is bounded inW^1,2(Ω₀)via(2.21) with m=n=−i. Therefore there is aV ∈W^1,2(Ω₀) such that V_i →V weakly in W^1,2(Ω₀) and strongly in L²(Ω₀) asi→ ∞. Since

i→∞lim V_i=V = lim

i→∞τ₁V_i=τ₁V, (2.23)

V ∈E₀. We claimV ∈ M. If not, there is aρ >0 such that

I₀(V)> c₀+ρ. (2.24)

Since

I₀(V)≤ lim

i→∞I₀(V_i), (2.25)

for all largei,

I₀(V_i) =I₀(τ_iU)≥c₀+ρ/2. (2.26) But then

J_i,0(U)→ ∞ (2.27)

as i→ −∞, contrary to (2.21). ThusV ∈ M and by (2.20) lies betweenv andw. Moreover by (2.18) with u_k replaced byU, letting j→ −∞shows

Z

Ω0

(V −v)dx≤1 2

Z

Ω0

(w−v)dx. (2.28)

Consequently by condition (∗),V =v. Similarly V_i→wasi→ −∞andU ∈Γ^∗. The next step in our argument is:

Proposition 2.7. J^∗(U) =c^∗.

The proof of this proposition will be given in Section 3.

The final step in the proof of Theorem 2.2 is to verify thatU is a classical solution of (PDE) and (BC). This is a consequence of local minimization properties that the global minimizer,U of (2.6) possesses. To be more precise, letB_r(z) denote an open ball of radiusraboutz∈Rⁿ. Suppose B_2r(z)⊂Ω. Set

A_r(z) ={u∈W^1,2(B_2r(z))|u=U in B_2r(z)\B_r(z)} · (2.29) Foru∈A_r(z), define

F_r(u) = Z

Br(z)L(u)dx. (2.30)

The local minimization property for interior points is:

Proposition 2.8. For eachz∈Ωand0< r < ¹₂ such thatB_2r(z)⊂Ω,U minimizesF_r overA_r(z).

This result will also be proved in Section 3. Since F_r(U) = inf

u∈Ar(z)F_r(u)

and, by standard elliptic regularity arguments, any minimizer ofF_ronA_r(z) is a classical solution of (PDE) in B_r(z),U is a classical solution of (PDE) in Ω, and is even inC_loc^2,α(Ω) for anyα∈(0,1).

It remains only to show thatU is smooth up to∂Ω and (BC) holds. This follows as above by a slight variant of Proposition 2.8. Let z∈∂Ω andr >0. Consider

Fˆ_r(u) = Z

Br(z)∩Ω

L(u)dx (2.31)

on

Aˆ_r(z) ={u∈W^1,1(B_2r(z)∩Ω)|u=U in (B_2r(z)\B_r(z))∩Ω} · (2.32) The analogue of Proposition 2.8 here is:

Proposition 2.9. For eachz∈∂Ωand0< r <¹₂,U minimizesFˆ_r onAˆ_r(z).

The proof of this result will be sketched after that of Proposition 2.8.

By the regularity of U given by Proposition 2.9, U is a C^2,α function up to ∂Ω and ^∂U_∂ν = 0 on ∂Ω. The proof of Theorem 2.2 is complete.

3. Proofs of the technical results

This section is devoted to the proof of Propositions 2.4, and 2.6–2.9.

Proof of Proposition 2.4. Define

ϕ(x) =







(x₁−m)u(x) + (m+ 1−x₁)v(x), m≤x₁≤m+ 1,

u(x), m+ 1≤x₁≤n

(x₁−n)v(x) + (n+ 1−x₁)u(x), n≤x₁≤n+ 1

(3.1)

and continueϕtoR× Das ann−m+ 1 periodic function. Then by Lemma 2.3,

J_m,n(ϕ)≥0. (3.2)

By (3.1), Z

Ωm

L(ϕ)dx= Z

Ωm

1

2|(x1−m)∇u+ (m+ 1−x₁)∇v|²+ (x₁−m)u_x1+ (m+ 1−x₁)v_x1(u−v) +1

2(u−v)²+G(x, ϕ)

dx. (3.3)

The terms in (3.3) will be estimated separately. First,

|(x1−m)∇u+ (m+ 1−x₁)∇v|²≤[(x₁−m)²+ (m+ 1−x₁)²](|∇u|²+|∇v|²)≤ |∇u|²+|∇v|². (3.4) Next

T = (x₁−m)u_x1+ (m+ 1−x₁)v_x1(u−v) = (x₁−m) ∂

∂x₁

(u−v)²

2 + (u−v)v_x1. (3.5)

Therefore, sinceu−v≤w−v≤1, Z

Ωm

Tdx≤ Z

D

(x₁−m)(u−v)² 2

^m+1

m dy+1 2

Z

Ωm

(u−v)²dx+1 2

Z

Ωm

v_x²

1dx (3.6)

≤ 1 2|D|+1

2|D|+1 2

Z

Ωm

v²_x1 ≡K₁

where|D|denotes the measure ofD. The last integral in (3.3) can be estimatedvia Z

Ωm

G(x, ϕ)dx= Z

Ωm

(G(x, ϕ)−G(x, v) +G(x, v))dx≤M₁ Z

Ωm

(ϕ−v)dx+ Z

Ωm

G(x, v)dx (3.7)

≤M₁|D|+ Z

Ωm

G(x, v)dx≡K₂

whereM₁ is a Lipschitz constant forG(x, z) on Ω×[minv,maxw]. Combining (3.3–3.7) gives Z

Ωm

L(ϕ)dx≤ Z

Ωm

1

2(|∇u|²+|∇v|²)dx+K₁+K₂≤ Z

Ωm

L(u)dx+| Z

Ωm

G(x, u)dx|+K₃. (3.8) As in (3.7),

Z

Ωm

G(x, u)dx ≤M₁

Z

Ωm

(u−v)dx+ Z

Ωm

G(x, v)dx

. (3.9)

Hence

a_m(ϕ)≤a_m(u) +K₄. (3.10)

Combining (3.2, 3.10), and a similar estimate fora_n(ϕ) yields

J_m,n(u) = J_m,n(ϕ) +a_m(u)−a_m(ϕ) +a_n(u)−a_n(ϕ)≥ −K (3.11)

and Proposition 2.4 is proved.

Proof of Proposition 2.6. Setu_k =τ_ku. As following (2.22),u_k→v (resp.w) as k→ ∞(resp.− ∞) weakly in W^1,2(Ω₀) and strongly inL²(Ω₀). Moreover as in (2.24–2.27),

lim

|k|→∞

I₀(u_k) =c₀. (3.12)

Now (3.12) will be used to prove (2.13–2.15). Choose a sequencek_i→ ∞such that

i→∞lim I₀(u_ki) =c₀. (3.13)

We claim

i→∞lim ku_ki−vk_W^1,2_(Ω0)= 0. (3.14) By the convergence already established for u_ki, it suffices to show

i→∞lim k∇u_ki− ∇vk_L²_(Ω0)= 0. (3.15)

To verify (3.15), note first that

i→∞lim k∇u_kik_L²_(Ω0)=k∇vk_L²_(Ω0) (3.16) for if not, by the weak lower semicontinuity of k · k_L²_(Ω0), there is aδ >0 such that

1

2k∇vk²_L2(Ω0)+δ≤1 2 lim

i→∞k∇u<sub>`i</sub>k<sup>2</sup><sub>L</sub>2(Ω0) (3.17) where `_i is a subsequence ofk_i. But then

I₀(v) +δ≤ lim

i→∞

1

2k∇u_`ik²_L2(Ω0)+ lim_i→∞

Z

Ω0

G(x, u_{`i</sub>)dx= lim<sub>i→∞</sub>I<sub>0</sub>(u<sub>`i}) =c₀ (3.18)

contrary to (3.13). Now asi→ ∞,

k∇u_ki− ∇vk²_L2(Ω0)=k∇u_kik²_L2(Ω0)+k∇vk²_L2(Ω0)−2 Z

Ω0

∇u_ki· ∇vdx→0

via (3.16) and the weak convergence of inL²(Ω) of∇u_ki to ∇v. Thus (3.15) holds.

Next (3.12) will be strengthened to show that

k→∞lim I₀(u_k) =c₀. (3.19)

Then the arguments in (3.13–3.18) show (2.14) holds and the analogue of (3.19) fork→ −∞yields (2.15). To get (3.19), it will be shown that each of the following limits exist:

(i) lim

k→−∞J_k,0(u); (ii) lim

k→∞J_1,k(u). (3.20)

Then (3.20) (i) impliesa_k(u)→0 ask→ −∞and therefore (3.19) is valid.

To prove (3.20), (i), set

N⁻ ={n∈ −N |a_n(u)≤0} · (3.21)

Suppose first thatN⁻is finite. ThenJ_n,0(u) is an increasing sequence which is bounded from above asn→ −∞

via(2.12). Thus (3.20) (i) is verified. On the other hand ifN⁻ is an infinite set, the sequence (n_i) it represents must satisfy (3.13) so by (3.14) asi→ ∞,

ku_ni−vk_W^1,2_(Ω0)→0. (3.22)

If the limit (3.20) (i) does not exist, then `<sup>−</sup>< `⁺ where

`<sup>−</sup>≡lim<sub>n→−∞</sub>J<sub>n,0</sub>(u); `⁺= lim_n→−∞J_n,0(u). (3.23) It will be shown that`<sup>−</sup> < `⁺is not possible.

Choose satisfying

0< < `<sup>+</sup>−`⁻

3 · (3.24)

By (3.23), there are sequences (p_k), (q_k)⊂ −Nwith p_k, q_k → −∞as k→ ∞ and such that q_k > p_k > q_k+1, and

J_pk,0(u)→`<sup>−</sup>; J<sub>qk,0</sub>(u)→`⁺ (3.25) as k→ ∞. Hence fork large,

J_pk,0(u)≤`<sup>−</sup>+ < `⁺−≤J_qk,0(u). (3.26) Sets_k the largestq∈ N⁻ such thatq < q_k and setr_k the smallestp∈ N⁻ such that p≥p_k. As a function of t,J_sk+t,0(u) is increasing in (0, q_k−1−s_k] whileJ_pk+t,0(u) decreases in [0, r_k−p_k]. Therefore

J_rk,0(u)≤`<sup>−</sup>+ < `⁺−≤J_sk,0(u) (3.27) and

J_rk,sk−1(u) =J_rk,0(u)−J_sk,0(u)≤ −(`<sup>+</sup>−`⁻) + 2. (3.28) Now by (3.22), uis close to vin W^1.2(Ω_{`</sub>) for`=s_k and`=r<sub>k</sub> and klarge. This allows us to modifyuin Ω<sub>`} so as to produce a function,ψ, which equalsv in [r_k, r_k+¹₂] and in [s_k+¹₂, s_k+ 1]. Henceψextends to Ω as a s_k−r_k+ 1 periodic function inx₁ so

J_rk,sk(ψ)≥0 (3.29)

via Lemma 2.3. More precisely,ψis given by

























ψ(x) =v(x), r_k ≤x₁≤r_k+1 2

= 2(x₁−

r_k+1 2

u(x) + 2((r_k+ 1)−x₁)v(x), r_k+1

2 ≤x₁≤r_k+ 1

=u(x), r_k+ 1≤x₁≤s_k

= 2(x₁−s_k)v(x) + 2

s_k+1 2 −x₁

u(x), s_k≤x₁≤s_k+1 2

=v(x), s_k+1

2 ≤x₁≤s_k+ 1.

(3.30)

Then fork large enough,

|a_rk(u)|+|a_rk(ψ)|+|a_sk(u)|+|a_sk(ψ)|< (3.31) via (3.22) and straightforward estimates. Consequently by (3.29) and (3.31),

J_rk,sk−1(u) =J_rk,sk(ψ)−a_rk(ψ) +a_rk(u)−a_sk(ψ)≥ −. (3.32) Combining (3.32) and (3.28) shows

`<sup>+</sup>−`⁻≤3 (3.33)

contrary to (3.24). Thus (3.20) (i) has been verified and (3.20) (ii) follows similarly. The proof of Proposition 2.6

is complete.

Proof of Proposition 2.34. This proof has some elements in common with that of Proposition 2.6. Suppose J^∗(U)6=c^∗. Then sinceU ∈Γ^∗, by (2.6), there is aσ >0 such that

J^∗(U) =c^∗+ 9σ. (3.34)

By Proposition 2.6,

J^∗(U) = lim

n→∞J_−n,n(U). (3.35)

Hence there is ann₀∈Nsuch that ifn≥n₀,

J_−n,n(U)≥c^∗+ 8σ. (3.36)

By the weak lower semicontinuity ofJ_−n,n and (3.36), there is ak₀(n)∈Nsuch that ifk≥k₀,

J_−n,n(u_k)≥J_−n,n(U)−σ≥c^∗+ 7σ. (3.37) Moreover since (u_k) is a minimizing sequence for (2.6), forklarge,

J^∗(u_k)≤c^∗+σ. (3.38)

LetT_n(u) denote the tail ofJ^∗(u),i.e.

T_n(u) =J_{−∞,−n−1}(u) +J_n+1,∞(u)≡T_n⁻(u) +T_n⁺(u).

Therefore by (3.37, 3.38) for n≥n₀andk large,

c^∗+σ≥J_−n,n(u_k) +T_n(u_k)≥c^∗+ 7σ+T_n(u_k) or

T_n(u_k)≤ −6σ. (3.39)

Next it will be shown that (3.39) is not possible,i.e. the tail ofu_k is uniformly small provided thatnandkare sufficiently large. It suffices to show that

T_nⁱ(u_k)>−3σ, i= +,−. (3.40)

The−case will be verified; the + case is handled in the same way.

Letδ >0, free for the moment. Since U ∈Γ^∗ andJ^∗(U)<∞, by (2.14) forn=n(δ) large enough,

kU−vk_W^1,2_{(Ω−n−1)}≤δ. (3.41)

We claim

ku_k−Uk_W^1,2_{(Ω−n−1)}≤δ (3.42)

for n(δ) large and appropriatek. Assuming (3.42) for now, by (3.41, 3.42),

ku_k−vkW^1,2(Ω−n−1)≤2δ. (3.43)

LetN ∈NwithN > n+ 1 and letψbe as in (3.30) withureplaced byu_k,rbyN, andsby−n−1. Then (3.29) becomes

J_{−N,−n−1}(ψ)≥0. (3.44)

Therefore forδ=δ(σ) sufficiently small, as in (3.31),

|a_−n−1(u_k)|+|a_−n−1(ψ)|< σ. (3.45) ForN =N(k) near∞, by (2.13) again,

ku_k−vk_W^1,2_(Ω−N)≤δ. (3.46)

Hence

|a_−N(u_k)|+|a_−N(ψ)|< σ. (3.47) Finally as in (3.32, 3.44, 3.45) imply

T_n⁻(u_k) =J_{−N,−n−1}(u_k) +T_N⁻(u_k)≥ −2σ+T_N⁻(u_k). (3.48) Since J^∗(u_k)<∞, forN possibly still larger,

|T_N⁻(u_k)|< σ. (3.49)

Thus (3.40) is a consequence of (3.48, 3.49) andJ^∗(U) =c^∗.

It remains to verify (3.42). Sinceu_k→U in L²(Ω_−n−1) ask→ ∞, it suffices to show

k∇u_k− ∇Uk_L²_{(Ω−n−1)}≤δ/2 (3.50)

for n(δ) large and appropriatek. Since by weak lower semicontinuity, Z

Ω−n−1

L(U)dx≤ lim

k→∞

Z

Ω−n−1

L(u_k)dx, (3.51)

0≤ρ_n= lim_k→∞

Z

Ω−n−1

L(u_k)dx− Z

Ω−n−1

L(U)dx.

Now (3.50) is a consequence of the following result:

Lemma 3.1. ρ_n →0 asn→ ∞and

lim_k→∞k∇(u_k−U)k²_L2(Ω−n−1)≤2ρ_n. (3.52) Proof.

lim

k→∞

Z

Ω−n−1

|∇(u_k−U)|²dx= lim

k→∞

Z

Ω−n−1

(|∇u_k|²−2∇u_k· ∇U+|∇U|²)dx

= lim

k→∞

Z

Ω−n−1

|∇u_k|²dx− Z

Ω−n−1

|∇U|²dx.

Thus to get (3.52), it suffices to prove

k→∞lim Z

Ω−n−1

|∇u_k|²dx≤ Z

Ω−n−1

|∇U|²dx+ 2ρ_n. (3.53)

But

k→∞lim Z

Ω−n−1

L(u_k)dx= Z

Ω−n−1

L(U)dx+ρ_n= lim

k→∞

Z

Ω−n−1

1

2|∇u_k|²dx+ Z

Ω−n−1

G(x, U)dx (3.54)

so (3.53) holds. Finally to show thatρ_n →0 asn→ ∞, by (2.11, 2.12) and (2.16)

−K≤J<sub>−j,−`</sub>(U) = X−`

i=−j

lim

k→∞a_i(u_k)−ρ_−i

≤ lim

k→∞J<sub>−j,−`</sub>(u<sub>k</sub>)− X−`

−j

ρ_−i≤M+ 2K− X−`

−j

ρ_−i. (3.55)

Thus letting `= 0 andj → −∞shows

X∞ 0

ρ_n≤M + 3K so ρ_n→0 asn→ ∞.

Proof of Proposition 2.8. Set

c(B_r(z)) = inf

u∈Ar(z)F_r(u). (3.56)

SinceA_r(z) is closed and convex, it is weakly closed. The functionalF_r is weakly lower semicontinuous. Hence there is aV ∈A_rsuch thatF_r(V) =c(B_r(z)). Standard regularity arguments showV is a classical solution of (PDE) and even in C^2,α(B_r(z)) for anyα∈(0,1). Let

M(B_r(z)) ={W ∈A_r(z)|F_r(W) =c(B_r(z))}·

Then M(B_r(z)) is an ordered setvia the proof of Proposition 2.1.

We claim each V ∈ M(B_r(z)) satisfiesv≤V ≤w. If not, supposeV(x)> w(x) for somex∈B_r(z). Set B ={x∈B_r(z)|V(x)> w(x)}·

Then ϕ≡min(w, V)∈A_r(z) so

F_r(V)≤F_r(ϕ) (3.57)

and therefore

Z

B

L(V)dx≤ Z

B

L(w)dx. (3.58)

Forx∈Ω andj∈Z, setx_j =x+ (j,0). Define





ψ(x) = max(w(x), V(x)), x∈B_2r(z) ψ(x_j) =ψ(x), x∈B_2r(z_j), j∈Z

ψ(x) =w(x), x∈Ω\S

j∈ZB_r(z_j). (3.59)

Then ψ∈E₀, so

I₀(ψ)≥I₀(w). (3.60)

Hence

Z

BL(V)dx≥ Z

BL(w)dx. (3.61)

Thus by (3.58) and (3.61),

Z

B

L(V)dx= Z

B

L(w)dx (3.62)

and

I₀(V) =I₀(w) =c₀. (3.63)

Consequently by (3.59) and (3.62), ψ∈ M. Butψ andw are not ordered, contrary to Proposition 2.1. Thus V ≤wand similarly,V ≥v.

It remains to show thatF_r(U) =c(B_r(z)). For j∈Z, let V_j(x) = inf

V∈M(Br(zj))V(x).

Then V_j(x)∈ M(Br(z_j)). Define

U^∗(x) =





V_j(x), x∈B_r(z_j) U(x), x∈Ω\ [

j∈Z

B_r(z_j). (3.64)

We claim

U^∗≤τ₋₁U^∗. (3.65)

If so,U^∗∈Γ^∗ and

J^∗(U)≤J^∗(U^∗). (3.66)

Now (3.64) and (3.66) imply

Z

Br(z)

L(U)dx≤ Z

Br(z)

L(V₀)dx. (3.67)

Hence by the minimization property ofV₀, Z

Br(z)

L(U)dx= Z

Br(z)

L(V₀) =c(B_r(z)). (3.68)

Lastly to verify (3.65), suppose it is false. Then for somej∈Z, there is an (x, y)∈B_r(z_j) such that

V_j(x₀, y₀)> V_j+1(x₀+ 1, y₀). (3.69)

Define functions onB_2r(z_j) as follows: ψ(x, y) =V_j+1(x₁, y),ϕ=V_j, χ= max(ϕ, ψ),ζ= min(ϕ, ψ). Then on B_2r(z_j)\B_r(z_j),

ζ=ϕ=U ≤τ₋₁U =ψ=χ and

ζ∈A_r(z_j), τ₁χ∈A_r(z_j+1).

As in (2.1),

F_r(δ) +F_r(χ) =F_r(ϕ) +F_r(ψ) =c(B_r(z_j)) +c(B_r(z_j+1)) (3.70) which implies thatζ∈ M(B_r(z_j)) andτ₁χ∈ M(B_r(z_j+1)). Thusζ≥ϕ=V_j and in particular at (x₀, y₀),

V_j(x₀, y₀)≤ζ(x₀, y₀)≤ψ(x₀, y₀) =V_j+1(x₀+ 1, y₀). (3.71) But (3.71) contradicts (3.69). Hence (3.65) holds and Proposition 2.8 is proved.

Proof of Proposition 2.9. The proof here follows the same lines as that of Proposition 2.8 and will be omitted.

4. Final remarks

The results of Section 2 extend to other situations such as those treated in Section 3 of [6] provided that some basic properties of M carry over to these new settings. In particular it is required that: (i) M has at least two members so that heteroclinics are possible, (ii)Mis an ordered set, and (iii)Mis not too big,i.e.(*) holds. It will briefly be indicated why this is the case for a class of equations which arise in a water wave model and treated by Kirchg¨assner [3] near a bifurcation point using center manifold methods.

Consider

−∆u=λa(y)u−f(x, y, u, λ) (4.1)

for (x, y)∈R² with|y|<1 with the Dirichlet boundary conditions:

u(x,±1) = 0. (4.2)

It is assumed that a(y) isC¹and positive,f isC¹in its arguments, 1-periodic inx, andf(x, y, z, λ) =o(|z|) as z→0. Then the linearization of (4.1) about u= 0 gives a linear eigenvalue problem:

−∆ϕ=λaϕ (4.3)

with boundary conditions (4.2) and ϕ1-periodic in x. The smallest eigenvalue λ₁ is positive and simple and there is a corresponding positive eigenfunctionϕ₁. Chooseλ > λ₁. Then – seee.g.[6] –

I₀(u) = Z

Ω0

1

2|∇u|²−λ 2u²

dx+o(kuk²_W1,2(Ω0)) (4.4)

as u→0. Since Z

Ω0

|∇ϕ₁|²dx=λ₁ Z

Ω0

ϕ²₁dx, equation (4.4) shows that ifuis a small multiple ofϕ₁,

I₀(u) = 1 2

1− λ

λ₁ Z

Ω0

|∇u|²dx+o k∇uk²_L2Ω2

<0. (4.5)

Consequentlyc₀=c₀(λ)<0 forλ > λ₁. Of course it is possible thatc₀(λ) =−∞. Some conditions were given in [6] where this is not the case. E.g. c₀(λ) > −∞if the primitive, F(x, y, z, λ) = R_z

0 f(x, y, t, λ)dt of f is a bounded function ofz or ifF(x, y, z, λ)→ ∞as|z| → ∞.

Assuming thatc₀(λ) is finite, suppose further thatf is odd inz. ThenF is even inz. Henceu∈ Mimplies

−u ∈ M and by (4.5), 0 6∈ M. Furthermore the argument of Proposition 2.1 shows M is an ordered set.

Therefore if v, −v ∈ M, without loss of generality,−v <0 < v in Ω₀. Consequently our requirements (i–iii) hold here. In particular if v is the smallest positive member ofM, there is a pair of solutions of (4.1, 4.2) heteroclinic inx, one from−v tov and the other fromv to−v.

References

[1] V. Bangert, On minimal laminations of the torus.Ann. Inst. H. Poincar´e Anal. Non Lin´eaire6(1989) 95-138.

[2] E. Bosetto and E. Serra, A variational approach to chaotic dynamics in periodically forced nonlinear oscillators.Ann. Inst. H.

Poincar´e Anal. Non Lin´eaire17(2000) 673-709.

[3] K. Kirchg¨assner, Wave-solutions of reversible systems and applications.J. Differential Equations45(1982) 113-127.

[4] A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications. Math. Mech. Appl. Sci. 10 (1988) 51-66.

[5] J.K. Moser, Minimal solutions of variational problems on a torus.Ann. Inst. H. Poincar´e Anal. Non Lin´eaire3(1986) 229-272.

[6] P.H. Rabinowitz, Solutions of heteroclinic type for some classes of semilinear elliptic partial differential equations.J. Fac. Sci.

Univ. Tokyo1(1994) 525-550.

[7] P.H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen–Cahn type equation.Comm. Pure Appl. Math.(to appear).

[8] R.E.L. Turner, Internal waves in fluids with rapidly varying density.Ann. Scuola Norm. Sup. Pisa Cl. Sci. Ser. 4 8(1981) 513-573.