A Singularly Perturbed Elliptic Partial Differential Equation with an Almost Periodic Term

A Singularly Perturbed EllipticPartial Differential Equation with an

Almost Periodic Term

Gregory S. Spradlin

United States Military Academy

West Point, New York, USA

1. Introduction

In [STT], a Hamiltonian system of the form

(1.0) −u′′ + u = h(t)∇F (u)

was studied, where h is an almost periodic (defined in a moment) function, and F : Rn → R a “su-

perquadratic” potential. That is, F (q) behaves like q to a power greater than 2, with F (q)/|q|2 → 0 as

|q| → 0 and F (q)/|q|2 → ∞ as |q| → ∞. For example, F (q) = |q|p−1q with p > 1 would qualify. The

authors found that (1.0) must have a nonzero solution homoclinic to zero. Since this result, many papers

(see [CMN], [R1], and [ACM], for example) have been written concerning Hamiltonian systems with almost

periodic terms.

As we will see, it is natural to extend the definition of almost periodic to functions on Rn, n > 1, or

even to more general topological groups. Thus one can write a PDE version of (1.0),

(1.1) −∆u + u = h(x)f(u),

wherein h is almost periodic and the primitive F of f satisfies appropriate superquadraticity and growth

conditions. Then one may ask, does (1.1) have a “homoclinic-type” solution? That is, is there a nonzero

solution u with |u(x)|+ |∇u(x)| → 0 as |x| → ∞? Here we take a step towards answering in the affirmative.

Let us define an almost periodic function on Rn (R is a special case, and defining an a.p. function on

other topological groups is an obvious generalization). First, a set A ⊂ Rn is relatively dense if there exists

L > 0 such that for every x ∈ Rn, there exists y ∈ A with |x − y| < L. Next, for ε > 0, ~v ∈ Rn, and

h : Rn → R, we say ~v is an ε-almost period of h if for all x ∈ Rn, |h(x + ~v) − h(x)| < ε. Finally, h is

defined to be almost periodic if for every ε > 0, there exists a relatively dense set A ≡ A(ε) ⊂ Rn such that

for all a ∈ A, a is an ε-almost period of h. For properties of almost periodic functions (many properties of

a.p. functions on R extend to a.p. functions on Rn), see [Be], [Bo], [C], [Z].

We will look at an equation similar to (1.1), of the form

(1.2) −ε2∆u + V (x)u = f(u)

on Rn. Equations like (1.2) arise in the study of the nonlinear Schrodinger equation and have been the

subject of much study recently (see [R2], [FdP1-3], [Li] and the references therein). We will assume that V

and f satisfy the following conditions:

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(V1) V ∈ C1(Rn,R),

(V2) 0 < V− ≤ infRn V ≤ supRn V ≤ V + < ∞(V3) V is almost periodic.

(f1) f ∈ C1(R+,R)

(f2) f ′(0) = 0 = f(0).

(f3) There exist A, s > 0 such that |f ′(q)| ≤ A(1 + |q|s−1) for all q ≥ 0. If n ≥ 3, then s < 4/(n− 2).

(f4) For some µ > 2, 0 < µF (q) ≤ f(q)q for all q > 0, where F (ξ) ≡∫ ξ0 f(t) dt.

(f5) The function q 7→ f(q)/q is increasing on (0,∞).

(f6) For every a > 0, the equation −∆u + au = f(u) has a unique (modulo translation) positive solution.

(f1)− (f6) are satisfied if, for example, f(q) = qs with s as in (f3) (for verification of (f6), see [GNN1-2]

and [Y]). (f1) − (f4) give the “superquadratic” character of f . (f5) is a useful convexity assumption found

in many papers such as [R2], [WZ], and [FdP1-3]. In [FdP2] it was shown that, under conditions weaker

than (V1) − (V2) and (f1) − (f6), if V has a “topologically nontrivial” set of critical points, then for small

enough ε, (1.2) has a positive solution u with |u(x)| + |∇u(x)| → 0 as |x| → ∞. A topologically nontrivial

set of critical points is a compact set of critical points of V obtained by a topological “linking.” Such a set

of critical points has the property that if V is perturbed a little in C1, there are still critical points nearby.

Let us define such a linking: suppose there exists a bounded open set Λ and closed sets B0 and B with B

nonempty and connected, and B0 ⊂ B ⊂ Λ. Let Γ be the class of all continuous functions φ : B → Λ with

the property that φ(y) = y for all y ∈ B0. Define the minimax value c as

(1.3) c = infφ∈Γ

supy∈B

V (φ(y)),

and assume

(1.4) B 6= ∅ ⇒ supB0

V < c

and

(1.5) For all φ ∈ Γ, φ(B) ∩ y ∈ Λ | V (y) ≥ c 6= ∅.

In the language of the calculus of variations, the sets B0, B, and V ≥ c link in Γ. Also assume

(1.6) For all y ∈ ∂Λ, ∂τV (y) 6= 0,

where ∂τ denotes tangential derivative. For example, such B0, B, Λ exist if V has a strict local maximum

or minimum; that is, if there exists a bounded open set O with max∂O V < supO V or inf∂O V < infO V .

Another example is a “saddle point”. For example, (0, 0) is a topologically stable critical point of V (x1, x2) =

x21− x2

2. In [FdP2] it was shown, under weaker conditions than (V1)− (V2), (f1)− (f6), and (1.3)-(1.6), that

for small enough ε, (1.2) has a positive homoclinic-type solution.

We would like to show that (1.2) has a nontrivial homoclinic-type solution for small enough ε if we

do not assume (1.3)-(1.6) but assume instead that V is almost periodic. It is easy to see that an almost

periodic function on R is either constant or has an infinite number of topologically nontrivial local maxima

and minima. However, an a.p. function on Rn need not have a topologically nontrivial critical point. For

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example, consider V (x1, x2) = 2 + sin x1. If this V occured in (1.2), then the variational arguments can be

made one-dimensional, so the equation with this V would not be too challenging. For a more interesting

example, let g : R → R be almost periodic, and define V (x) = 2+sin(x1−g(x2)). Then V is almost periodic,

nonconstant and has no topologically stable critical points. Indeed, an a.p. function of several variables need

not have any local minimum at all, let alone a topologically nontrivial one ([S]). So the work of [FdP] will

not give the desired result.

We will prove the following:

Theorem 1.7 Let V and f satisfy (V1)− (V3) and (f1)− (f6), and assume as well that V satisfies one of

the following three cases:

(I) V ≡ constant,

(II) there exists a bounded open set Λ ⊂ Rn and closed sets B0 ⊂ B ⊂ Λ satisfying (1.3)-(1.6).

(III) there exists an open set O ⊂ Rn with inf∂O V > infO V and ~v ∈ Rn \ 0 with

sup~u · x | x ∈ O, ~u ⊥ ~v, ‖~u‖ = 1 < ∞.

Then there exists ε0 > 0 such that if ε ≤ ε0, then (1.2) has a positive homoclinic-type solution uε. uε has

exactly one local maximum (hence, global maximum) point zε ∈ Rn. Further, there exist α, β > 0 with

uε(z) ≤ α exp(−βε |z − zε|) for ε ≤ ε0. In Case III, V (zε) → infO V as ε → 0.

More detailed conclusions for Case II are given in [FdP2]. O above can be thought of as a “tube” that is

bounded in all but possibly one direction in Rn. While the above result is strong, it is especially interesting

because one of Cases I-III is automatically satisfied when n = 2:

Theorem 1.8 If V : R2 → R is continuous and almost periodic, then V satisfies one of Cases (1.7)I-III.

Note: Theorem 1.8 does not hold for n ≥ 3; consider V (x1, x2, x3) = 2 + sin x1. Putting Theorems 1.7

and 1.8 together yields:

Corollary 1.9 Let n = 2 and let V and f satisfy (V1) − (V3) and (f1) − (f6). Then there exists ε0 > 0

such that if ε ≤ ε0, then (1.2) has a nontrivial homoclinic-type solution u.

Let us compare equations (1.1) and (1.2). In (1.2), the coefficient function V is placed differently, in

order to take advantage of recent results for equations of the same general form as (1.2). If V were moved in

front of f(u), the solution techniques would be essentially the same, so this difference is not very important.

Knowing that one of Cases I-III hold (or simply that n = 2) is essential for this proof. The most troubling

restriction, however, is the presence of ε2 in front of ∆u. This is equivalent to dilating V in the plane.

Despite these limitations, Theorem 1.7 is not easy to prove and seems markedly different from anything in

the literature.

Proof of n = 1 result

Before outlining the proof of Theorem 1.3, let us outline the proof of [STT]’s result to show why it

cannot be applied here. That proof fails because of differences in the topology of R and Rn for n > 1. It is

because of those differences that we impose the limitations described above.

For equation (1.0), let E = W 1,2(R) and define I ∈ C1(E,R) by I(u) = 12‖u‖

2 −∫

h(t)F (u) dt. Then

critical points of I are homoclinic solutions of (1.0). Using variational mountain-pass techniques, [STT]

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construct a sequence (um) ⊂ E with I ′(um) → 0, I(um) → b > 0, and ‖um − um−1‖ → 0 as m →∞. Then

they show that there exist δ > 0 and a sequence (tm) ⊂ R with |um(tm)| > δ for all m and |tm− tm−1| → 0.

They use (um) to construct a Palais-Smale sequence which “concentrates” near some value of t and therefore

has a nonzero weak limit. If (tm) has a bounded subsequence then this occurs already. So assume |tm| → ∞.

For s ∈ R, let τs denote the translation operator given by τs(u)(t) = u(t − s). Since |tm − tm−1| → 0, for

each l = 1, 2, 3, . . . there exists a 1/l-period sl of h and ml ∈ N with |ml − sl| < 1/l. Define vl = τ−sluml .

It is straightforward to show that I ′(vl) → 0. The um’s are uniformly continuous independently of m, so

it follows that lim inf l→∞ |vl(0)| > 0. Now it is standard to check that along a subsequence, (vl) converges

weakly to a nonzero critical point of I.

For a multidimensional version of (1.0), such as (1.1), it is still possible to define a functional I corre-

sponding to (1.1), and find a Palais-Smale sequence (um) with the above properties, with (xm) ⊂ Rn having

similar properties to (tm). But even though |xm − xm−1| → 0, it is no longer the case that xm must pass

arbitrarily close to any ε-periods of h for ε small. Here the attempt to copy [STT]’s proof breaks down.

Variational Framework and Plan of Proof

By a change of coordinates, we can recast (1.2) as

(1.10) −∆u + V (εx)u = f(u).

We will deal with this version of (1.2) exclusively. Extend f and F to the negative reals by defining

f(−q) = f(q). Let E = W 1,2(Rn) and define Iε ∈ C1(E,R) by Iε(u) =∫

Rn12 (‖∇u‖2 +V (εx)(u)2)−F (u) dt.

We will search for positive critical points of Iε. By elliptic regularity theory, such points are classical (C2)

solutions of (1.10). If V is a constant (Case I), it is well known that such a solution exists. If Case II holds,

then work of [FdP2] gives the result. For Case III, we employ an original argument based on the idea of

the n = 1 proof described above. We construct a sequence (um) with similar properties, and show it is

“confined” to the “tube” O/ε. Then we find a nozero critical point v of Iε with the property that, along a

subsequence, um is close to a translate of v.

Organization of Paper

In Section 2, we prove Theorem 1.8. Only basic real analysis techniques are used. Section 3 contains

some technical results, primarily about sequences of the form (um), where Iεm ′(um) → 0 and εm → 0.

Section 4 contains the proof of Theorem 1.7.

2. Almost Periodic Functions on R2

In this section we prove Theorem 1.8. Note that Theorem 1.8 is false for n ≥ 3; consider the counterex-

ample V (x1, x2, x3) = 2+sin x1. Though this theorem is a simple result, the proof is difficult. We will prove

the following stronger result:

Theorem 2.0 Let V : R2 → R be almost periodic. Then one of the following four alternatives holds:

(i) V ≡ constant

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(ii) There exists a bounded open set O ⊂ R2 with inf∂O V > infO V

(iii) There exists a bounded open set O ⊂ R2 with sup∂O V < supO V

(iv) There exist b > infR2 V , a nonzero vector ~u ∈ R2, and a connected set C ⊂ R2 and such that V (x) > b

for all x ∈ C, C · ~u ≡ x · ~u | x ∈ C is bounded, and if ~v 6= 0 with ~v · ~u = 0, then C · ~v = R.

This gives Theorem 1.8 for the following reason: in Case(2.0)ii, (1.7)(II) is satisfied; pick z ∈ O with

V (z) = minO V and take B0 = ∅, B = z, and Λ = O, choosing O to be smooth enough. In Case(2.0)(iii),

(1.7)(II) and (III) both hold. For Case (1.7)(iv), let b2 ∈ (infR2 V, b). Let R be large enough so that every

ball of radius R in R2 contains a point z with V (z) < b2. This is possible because V is a.p. Let a ∈ R2 be

a b2 − b-almost period of V with a · ~u large enough so that R2 \ (C ∪ (C + a)) contains a component which

is bounded in the ~u direction, and which contains a ball of radius R. Let z be a point in that ball with

V (z) < b2. Let O be the component of V b2 on C ∪ (C + a), O

is bounded in the ~u direction.

We will need to use another definition of almost periodic which is equivalent ([Be]) to that given in

the Introduction. “Almost periodic” is easily generalized to functions from Rn into an arbitrary Banach

space (the domain may also be generalized, but we need not consider this). For V : Rn → X and x ∈ Rn,

define τxV : Rn → X by τxV (y) = V (y − x). That is, τxV is V translated by x. Let C(Rn, X) denote

the Banach space of bounded, continuous functions from Rn to X, with the uniform norm. A continuous

function V : Rn → X is almost periodic if and only if the set of translates τxV | x ∈ Rn is precompact in

C(Rn, X); that is, if any sequence (xm) ⊂ Rn has a subsequence (xm) with τxmV convergent in C(Rn, X).

This alternate definition will help us prove the following lemma, which proves the less-than-obvious fact

that if V : Rn → R is almost periodic, then it is almost periodic “in any direction:”

Lemma 2.1 Let V : Rn → R be almost periodic, ~u ∈ Rn, and ε > 0. Then there exists a relatively dense

set A ⊂ R such that for all a ∈ A, a~u is an ε-period of V .

Proof: without loss of generality let ~u = (1, 0, 0, . . . , 0). Define V : R → C(Rn−1,R) by V(x)(y) =

V (x, y). We will show V is almost periodic. Let (xm) ⊂ R. Since V is a.p., τ(xm,0)V | m ≥ 1 is

precompact in C(Rn,R). Take a subsequence of (xm) and V ∈ C(Rn,R) with τ(xm,0)V → V in C(Rn,R).

Define V ∈ C(R, C(Rn−1,R)) by V(x)(y) = V (x, y). It is easy to see that τxmV → V in C(R, C(Rn−1,R)).

Therefore, V is almost periodic, and there exists a relatively dense set A ⊂ R such that for all a ∈ A,

‖τaV − V‖C(R,C(Rn−1,R)) < ε. From this it is clear that for all (x, y) ∈ Rn, |V (x − a, y) − V (x, y)| < ε.

Therefore (a, 0) is an ε-period of V .

We need several lemmas to complete the proof of Theorem 2.0. First,

Lemma 2.2 Let V : R2 → R be almost periodic and not satisfy any of Cases (2.0)(i)-(iii). Suppose that

there exist ~u ∈ R2 \ (0, 0), and an unbounded, connected set C ⊂ R2 such that C · ~u is bounded, and

infC V > infR2 V or supC V < supR2 V . Then (2.0)(iv) holds.

Proof: Let ~v ∈ R2 \ 0 with ~v ⊥ ~u. First let us take the case infR2 C > b > b1 > b2 > infR2 V . Since

C is unbounded but C · ~u is bounded, C · ~v is unbounded (and connected). C · ~v contains an interval of

the form (−∞, λ) or (λ,∞). Assume without loss of generality that the former occurs. Let (am) ⊂ R be a

sequence with am →∞ and am~v a (b− b1)-almost period of V for each m. This is possible by Lemma 2.1.

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Let R > 0 be large enough so that C · ~u ⊂ (−R,R). For large m, C + am~v ≡ x + am~v | x ∈ C intersects

the segment −R~u R~u. Also, V > b1 on C +am~v. Let ρm ∈ (−R, R) with ρm~v ∈ −R~u R~u∩ (C +am~v). Take

a subsequence along which ρm converges to ρ ∈ [−R, R]. V (ρ~u) ≥ b1.

Let r > 0 be small enough so that V > b2 on Br(ρ~u). Take a “tail” of the sequence (am), so that

ρm~v ∈ Br(ρ~u) for all m. Define

(2.3) A = Br(ρ~u) ∪∞⋃

m=1

(C + am~v).

Now A · ~u is bounded, because C · ~u is bounded and ~v ⊥ ~u. A · ~v = R, because C · ~v is unbounded in the

negative direction and am → ∞. V > b2 on A, since V > b2 on Br(ρ~v) and V > b1 > b2 on C + am~v for

each m. A is connected because C + am~v intersects Br(ρ~v) for all m. (2.0)(iv) follows.

Now for the case supC V < supR2 V . By the above argument, we can construct C2 ⊂ R2 with C2

connected, supC2V < supR2 V , C2 · ~u bounded, and C2 · ~v = R. Also, there exists a translate of C2, called

C3, with the property that R2 \ (C2 ∪ C3) has a component which is bounded in the ~u direction and which

contains a point z with V (z) > b > supC2∪C3V . Let U denote the component of V > b containing z.

U is sandwiched between C2 and C3, so U · ~u is bounded. Since (2.0)(iii) does not hold, U is unbounded.

Applying the first part of Lemma 2.2, just proven, (2.0)(iv) follows.

Next, define V b = x ∈ R2 | V (x) > b. We say that x, y ∈ V <b are

“b-connected” if they are in the same component of V <b. Then,

Lemma 2.4 Let V : R2 → R be almost periodic, and not satisfy any of 2.0(i)-(iv).

Let b ∈ (infR2 V, supR2 V ). Then there exists R = R(b, V ) with the following property: if x, y ∈ V 0 big enough so that for all a ∈ R2,

BR/2(a) contains a (b2 − b)-almost period of V .

Let x, y ∈ V <b be b-connected, and z ∈ xy. Suppose that BR(z) contains no points in V <b that are

b-connected with x and y. Let γ : [0, 1] → V <b \BR(z) be a continuous path with γ(0) = x and γ(1) = y.

Let ~v = (y − x)/|y − x| and |~u| = 1 with ~u ⊥ ~v. By taking a subset of γ([0, 1]), we can find a path

γ : [0, 1] → R2 \BR(z) and points x′, y′ with

γ(0) = x′, γ(1) = y′(2.5)(i)

x′ and y′ are on the line↔xy(ii)

(γ(t)− z) · ~u 6= 0 for all t ∈ (0, 1),(iii)

(x′ − z) · ~v ≤ −R, (y′ − z) · ~v ≥ R.(iv)

By (2.5)(iii) we may assume without loss of generality that (γ(t)− z) · ~u > 0 for all t ∈ (0, 1).

Let U be any component of V>b2 . By Lemma 2.2 and the assumption that (2.0)(iv) is false, U is

unbounded in every direction, in particular the direction ~u. Let D = max|γ(t1)−γ(t2)| | t1, t2 ∈ [0, 1], the

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diameter of γ([0, 1]). Let g : [0, 1] → U with (g(1)− g(0)) · ~u > D. By taking a subset of g([0, 1]), we obtain

a path g : [0, 1] → U ⊂ V>b2 and w0, w1 ∈ U with

w0 = g(0), w1 = g(1)(2.6)(i)

(w1 − w0) · ~u > D(ii)

(g(t)− w0) · ~u > 0 for all t ∈ (0, 1].(iii)

Let P denote the closed half-plane ξ | (ξ− z) ·~u ≥ 0. P ∩BR(z) contains a ball of radius R/2. By the

choice of R, there exists y ∈ P ∩ BR(z) with y − w0 a (b2 − b)-almost period of V . Define g2 : [0, 1] → R2

by g2(t) = g(t) + (y − w0). Then V > b2 − (b2 − b) = b on g2([0, 1]). By (2.6)(iii), g2([0, 1]) lies in the half-

plane P . γ([0, 1]) separates P into two components. One is a bounded component that contains BR(z)∩ P .

g2(0) = y ∈ BR(z) ∩ P . By (2.6)(ii), g2(1) belongs to the unbounded component of P \ γ([0, 1]). Therefore

the path g2 must cross the path γ. This is impossible, since V b on g2([0, 1]).

Lemma 2.4 is proven.

Lemma 2.4 helps prove the following lemma:

Lemma 2.7 Let V : R2 → R be almost periodic, and not satisfy any of (2.0)(i)-(iv). Then there exist

b ∈ (infR2 V, supR2 V ) and an unbounded sequence of colinear points in V <b that are b-connected.

Proof: let b ∈ (infR2 V, supR2 V ). Assume, by translating V if necessary, that V (0, 0) < b. Since (2.0)(ii)

is false, there exists an unbounded sequence of points (xm) ⊂ V 2lR and there exists zm ∈BR(2lR xm

|xm| ) that is b-connected with (0, 0). For large enough m, zm ∈ BR(2lR xm|xm| ) ⊂ B2R(2lR, 0). In this

manner, construct a sequence of distinct points (zm) that are b-connected to (0, 0), with zm ⊂ B2R(2lmR, 0)

and lm →∞.

Let zm = (z(1)m , z(2)

m ). z(2)m ∈ (−2R, 2R), so we can extract a subsequence (wm) = (w(1)

m , w(2)m ) ⊂ (zm) with

(wm) unbounded, each wm b-connected to (0, 0), and w(2)m → ρ ∈ [−2R, 2R] as m →∞. Let b2 ∈ (b, supR2 V ).

V is uniformly continuous (see [Be]), so let r > 0 be small enough so that V < b2 on Br(wm) for all m. For

large m, we may select ym ≡ (y(1)m , y(2)

m ) ∈ Br(wm) with y(2)m = ρ, V (ym) < b2, and ym b2-connected to wm,

hence to (0, 0). The lemma is proven (with b2 in place of b).

We state the following result and postpone the proof until later:

Lemma 2.8 Let 0 < r < l and M ≥ 4l2/r, and let γ ∈ C([0, 1],R2) be a one-to-one path with γ(0) = (0, 0)

and γ(1) = (M, 0). Then there exist t1, t2 ∈ [0, 1] with

|γ(t1)− (γ(t2) + (l, 0))| ≤ r.

Now we complete the proof of Theorem 2.0. Suppose none of Cases (2.0)(i)-(iv) hold. By Lemma 2.7,

there exist b ∈ (infR2 V, supR2 V ) and an unbounded sequence (xm) of colinear points in R2 with V (xm) < b

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for all m and all the xm’s b-connected to each other. Without loss of generality, by rotating and translating

V , taking a subsequence, and abusing notation, we may assume that xm = (xm, 0), x1 = (0, 0), and xm →∞.

Let b2 ∈ (b, supR2 V ) and let r > 0 be small enough so that for all x, y ∈ R2,

(2.9) |x− y| ≤ r ⇒ |V (x)− V (y)| < (b2 − b)/2.

Let L > 0 be large enough so that for all t ∈ R, there exists t0 ∈ (t, t + L/2) with (t0, 0) a (b2 − b)/2-almost

period of V . This is possible by Lemma 2.1.

Fix an m with xm > 4L2/r. Let C be the image of a one-to-one path connecting (0, 0) and (xm, 0) in

V <b. By the choice of L above, there exists a sequence (am) ⊂ R with am → ∞, |ai+1 − ai| < L for all

i ≥ 1, and (ai, 0) a (b2 − b)/2-almost period of V for all i.

Define

(2.10) A =∞⋃

i=1

Nr(C + (ai, 0)).

A is unbounded, since (ai) is unbounded. A is bounded in the x2-direction, since C is bounded. If z ∈ A,

then z has the form z = x + (ai, 0) + y for some x ∈ C and y ∈ Br((0, 0)), with (ai, 0) a (b2 − b)/2-almost

period of V . By (2.9), V (z) ≤ V (x) + (b2 − b)/2 + (b2 − b)/2 4L2/r. By Lemma 2.8, C and C + (ai+1 − ai, 0) are within distance r of each other. Thus

C + (ai, 0) and C + (ai+1, 0) are within distance r of each other for each i ≥ 1. Thus Nr(C + (ai, 0))

and Nr(C + (ai+1, 0)) intersect, and A is connected. By Lemma 2.2, (2.0)(iv) holds. This contradicts our

assumption that (2.0)(i)-(iv) do not hold, proving Theorem 2.0.

Proof of Lemma 2.8:

If suffices to prove the result for l = 1, because if l′ 6= 1 and 0 < r′ < l′, we may apply the lemma with

r = r′/l′, l = 1, and M = 4/r′, then rescale by l′ to obtain the desired result.

From now on we assume that the conclusion of Lemma 2.8 does not hold, that is,

(2.11) C and C + (1, 0) are separated by distance at least r.

We will obtain a contradiction. Assume without loss of generality that

(2.12) γ([0, 1]) ∩ (((−∞, 0) ∪ (M,∞))× 0) = ∅.

If this were not true, then we could take x− = minx | (x, 0) ∈ γ([0, 1]) ≤ 0 and x+ = maxx | (x, 0) ∈γ([0, 1]) ≥ 1 and apply the conclusion of Lemma 2.8 to the “sub-arc” of γ connecting (x−, 0) and (x+, 0)

(shifted |x−| units to the right if necessary).

Define projection operators π1, π2 : R2 → R by π1(x, y) = x, π2(x, y) = y. Define

ymax = maxπ2(γ(s)) | x ∈ [0, 1] and ymin = minπ2(γ(s)) | x ∈ [0, 1]. Now ymax > 0 > ymin; other-

wise, γ([0, 1]) and γ([0, 1]) + (1, 0) would intersect, contradicting (2.11).

Let t0 and t1 satisfy π2(γ(t0)) = ymax, π2(γ(t1)) = ymin, and π2(γ(t)) ∈ (ymin, ymax) for all t strictly

between t0 and t1. Assume without loss of generality that t0 < t1.

8

γ([t0, t1]) separates the strip R × [ymin, ymax] into two components. Let UL denote the component of

(R× [ymin, ymax]) \ γ([t0, t1]) that is unbounded in the negative x1-direction, and UR, the other component

of (R× [ymin, ymax]) \ γ([t0, t1]) (which is unbounded in the positive x1-direction).

We claim that for some s ∈ (t0, t1), γ(s) ∈ (0, 1) × 0. Proof: otherwise, (2.12) implies that (1, 0)

lies in UL. (1, 0) and γ(t0) + (1, 0) both are on the path γ + (1, 0). γ([0, 1]) + (1, 0) is disjoint with

γ([t0, t1]). Thus (1, 0) and γ(t0) are in the same component of (R× [ymin, ymax]) \ γ([t0, t1]). Clearly, by the

definition of UR, γ(t1) + (1, 0) ∈ UR. This is a contradiction, and the claim is proven. By similar reasoning,

γ([t0, t1]) ∩ ((M − 1,M)× 0) 6= ∅.Next, we claim that for all x ∈ (1,M − 2),

(2.13) ((x, x + 1)× 0) ∩ γ([t0, t1]) 6= ∅.

To prove, suppose that for some x ∈ (1,M − 2), (2.13) does not hold. Let a ≤ x and b ≥ x + 1 with

(a, 0), (b, 0) ∈ γ([t0, t1]) but (y, 0) 6∈ γ([t0, t1]) for all y ∈ (a, b). This is possible because γ([t0, t1]) intersects

both (0, 1)× 0 and (M − 1, M)× 0. Let ta and tb satisfy γ(ta) = (a, 0), γ(tb) = (b, 0), and π2(γ(t)) 6= 0

for all t strictly between ta and tb. π2(γ(t)) has a single, nonzero sign for all t strictly between ta and tb.

Since b− a ≥ 1, γ([ta, tb]) must intersect γ([ta, tb]) + (1, 0), violating (2.11). Claim (2.13) is proven.

For m = 0, 1, 2, . . ., define Am ⊂ [0,M ] by

(2.14) Am = x ∈ [0,M ] | (x, 0) ∈ UL + (m, 0).

By assumption (2.11), and the definition of t0 and t1, it is apparent that

(2.15) Nr(Am) ∩ [0, M ] ⊂ Am+1

for m = 0, 1, 2, . . .. Let m∗ = b2/rc + 1, where ‘bxc’ denotes the greatest integer less than or equal to x.

Since m∗ ≥ 2/r, (2.13) and (2.15) imply that Am∗ = [0,M ]. Now m∗ + 1 = b2/rc+ 2 ≤ 2/r + 2 < 4/r ≤ M ,

so m∗ + 1 ∈ Am∗ . By definition of Am∗+1, (m∗ + 1, 0) ∈ UL + (m∗, 0), so (1, 0) ∈ UL. We have seen that

this is false. Therefore, assumption (2.11) is false, proving Lemma 2.8.

3. Technical Results

To prove Theorem 1.7 we seek critical points of Iε, as defined in the Introduction. This will require a

detailed study of Palais-Smale sequences of Iε, that is, sequences (um) with Iε(um) convergent and Iε′(um) →0 as m → ∞. It is well-known that Iε badly fails the “Palais-Smale condition,” that is, a Palais-Smale

condition need not be precompact. We will be able to examine the structure of such sequences, however,

employing concentration-compactness ideas like those developed in [Lio]. We need to obtain estimates on

such sequences that are independent of ε. Therefore we will examine sequences of the form (um; Iεm), where

Iεm(um) converges, Iεm ′(um) → 0, and εm → 0. We will obtain results similar to those in [CR2], which

involved Palais-Smale sequences of a functional containing a periodic coefficient function. Our proofs will

be very similar. Because the functional Iεm varies, however, extra care is required, so we give the proofs in

some detail.

9

If V : Rn → R+ is measurable, bounded, and bounded away from zero, define (·, ·)V : E × E → R by

(u,w)V =∫

Rn∇u · ∇w + V (x)uw dx. (·, ·)V is equivalent to the usual inner product on W 1,2(Rn). Also

define ‖u‖V =√

(u, u)2V , and I[V ] : E → R by I[V ](u) = 12‖u‖

2V −

∫

RnF (u). Then,

Proposition 3.0 Let f satisfy (f1) − (f4). Let (Vm) be positive, uniformly continuous functions on Rn

that are uniformly bounded and uniformly bounded away from zero, and define Im = I[Vm]. Let (um) be a

sequence in E with (Im(um)) bounded and I ′m(um) → 0 as m → ∞. Then there exists a subsequence (also

denoted (um)) such that either (a) um → 0, or (b), there exists u∞ ∈ E \ 0, a sequence (xm) ⊂ Rn, and

V∞ : Rn → R satisfying

(i) τ−xmVm → V∞ locally uniformly as m →∞

and defining I∞ = I[V∞],

(ii) I ′∞(u∞) = 0

(iii) I ′m(um − τxmu∞) → 0

(iv) Im(um − τxmu∞) → limm→∞

Im(um)− I∞(u∞)

Proof: Suppose (a) does not hold, and (‖um‖) is bounded away from zero. By arguments from [CR1-2],

(um) is bounded. Also by [CR2], there exists ρ > 0, a sequence (xm) ⊂ R2, and a subsequence of (um)

(also denoted (um)), such that ‖um‖L2(B1(xm)) ≥ ρ for all m. By translating the um’s and the Vm’s, we may

assume that xm ≡ 0. Assume that n ≥ 3 (the n = 1, 2 arguments are very similar). Since (um) is bounded,

there exists u∞ ∈ E such that, along a subsequence, umu∞ weakly in L2(R2) and in W 1,2(Rn), and

um → u∞ in Lrloc for every r ∈ [1, 2n/(n − 2)). Therefore ‖u∞‖L2(B1(0)) ≥ ρ, and u∞ 6= 0. Since the Vm’s

are uniformly continuous and uniformly bounded, there exist V∞ and a subsequence of (Vm) with Vm → V∞locally uniformly in R2.

To prove (3.0)(iv), take a subsequence of (um) along which Im(um) converges and let

b = limm→∞ Im(um). Modifying Lemma 1.21 of [CR1] slightly shows b > 0. By the dominated conver-

gence theorem, Im(u∞)−Im(um) → 0 and Im(u∞) → b as m →∞. So to prove (3.0)(iv), it suffices to show

b− (Im(u∞) + Im(um − u∞)) → 0. Defining (u,w)m ≡ (u,w)Vm and ‖u‖m ≡√

(u, u)m,

b−(Im(u∞) + Im(um − u∞)) = Im(um)− (Im(u∞) + Im(um − u∞)) =(3.1)

=12(‖um‖2m − (‖u∞‖2m + ‖um − u∞‖2m)) +

∫

RnF (u∞) + F (um − u∞)− F (um) =

= (u∞, um − u∞)m +∫

RnF (u∞) + F (um − u∞)− F (um).

The inner product goes to zero as m →∞ because um−u∞0 in E. Let R > 0. By (f3), Sobolev estimates,

and estimates in [R3],∣

∣

∫

RnF (u∞) + F (um − u∞)− F (um)

∣

∣ =∣

∣

∫

BR(0)F (u∞)− F (um) +

∫

BR(0)F (um − u∞)+(3.2)

+∫

BR(0)CF (um − u∞)− F (um) +

∫

BCR (0)

F (u∞)∣

∣ ≤

≤ o(m) +

∣

∣

∣

∣

∣

∫

BCR (0)

F (um − u∞)− F (um) +∫

BR(0)CF (u∞)

∣

∣

∣

∣

∣

≤

≤ o(m) + C‖u∞‖W 1,2(BR(0)C) +∫

BR(0)CF (u∞)

10

with o(m) → 0 as m → ∞, for some C independent of R and m, using the fact that (um) is bounded.

Letting R be large, we can make lim supm→∞ |∫

RnF (u∞) + F (um − u∞) − F (um)| as small as we like,

proving (3.0)(iv).

To help prove both (3.0)(ii) and (iii), we will show

(3.3) I ′m(u∞)− I ′∞(u∞) → 0

as m →∞. Let ε1 > 0. Let R > 0 be large enough so that ‖u∞‖W 1,2(BR(0)C) < ε1. Let z ∈ E. Then

|(I ′m(u∞)− I ′∞(u∞))z| = |∫

Rn(Vm − V∞)u∞z| ≤(3.4)

≤ |∫

BR(0)(Vm − V∞)u∞z|+ |

∫

BR(0)C(Vm − V∞)u∞z| ≤

≤∫

BR(0)|Vm − V∞||u∞||z|+ V +

∫

BR(0)C|u∞||z| ≤

≤ ( supx∈BR(0)

|Vm(x)− V∞(x)|)‖u∞‖L2(BR(0))‖z‖L2(BR(0))+

+ V +‖u∞‖L2(BR(0)C)‖z‖L2(BR(0)C) ≤

≤ ( supx∈BR(0)

|Vm(x)− V∞(x)| · ‖u∞‖+ V +ε1)‖z‖.

Since Vm → V∞ locally uniformly, lim supm→∞ ‖I ′m(u∞)−I ′∞(u∞)‖ ≤ V +ε1. ε1 is arbitrary, so (3.3) follows.

Now to prove (3.0)(ii), note that

‖I ′∞(u∞)‖ = ‖I ′m(um) + (I ′m(u∞)− I ′m(um)) + (I ′∞(u∞)− I ′m(u∞))‖ ≤(3.5)

≤ ‖I ′m(u∞)− I ′m(um)‖+ o(m)

by (3.4) and the fact that I ′m(um) → 0. Let z ∈ E. We must show (I ′m(u∞)− I ′m(um))z → 0:

|(I ′m(u∞)− I ′m(um))z| = |(u∞ − um, z)m +∫

Rn(f(um)− f(u∞))z| ≤

≤ |(u∞ − um, z)m|+∫

Rn|f(um)− f(u∞)||z|.

Since u∞um weakly in E, (u∞ − um, z)m → 0. We must show∫

Rn |f(um) − f(u∞)||z| → 0. Let R > 0.

Then∫

Rn|f(um)− f(u∞)||z| =

∫

BR(0)|f(um)− f(u∞)||z|+

∫

BR(0)C|f(um)− f(u∞)||z|.

Since um → u∞ in Lrloc(R

2) for all r ∈ [1, 2n/(n − 2)), (f3) implies that∫

BR(0) |f(um) − f(u∞)||z| → 0 as

m →∞. The last integral also approaches 0 as R →∞, independently of m, by arguments of [R3]. (3.0)(ii)

follows.

To prove (3.0)(iii), it suffices, by (3.0)(ii), (3.3) and the fact that I ′m(um) → 0, to prove

(3.6) I ′m(um)− (I ′m(u∞) + I ′m(um − u∞)) → 0

11

as m →∞. Let ε1 > 0 and let R be large enough so that ‖u∞‖L2(BR(0)C) < ε1. Let z ∈ E. Then

[I ′m(um)−(I ′m(u∞) + I ′m(um − u∞))]z =∫

Rn(f(u∞) + f(um − u∞)− f(um))z =

=∫

BR(0)(f(u∞)− f(um))z +

∫

BR(0)f(um − u∞)z+

+∫

BR(0)C(f(um − u∞)− f(um))z +

∫

BR(0)Cf(u∞)z =

= o(m)‖z‖+∫

BR(0)C(f(um − u∞)− f(um))z +

∫

BR(0)Cf(u∞)z.

By (f3) and arguments of [R3] again, we obtain

‖I ′m(um)− (I ′m(u∞) + I ′m(um − u∞))‖ ≤ o(m) + C‖u∞‖W 1,2(BR(0)C)

for some C independent of m and R. Letting R →∞, (3.0)(iii) is proven.

From Proposition 3.0 comes:

Proposition 3.7 Let f , (Vm) and Im be as in Proposition 3.0. Let (um) be a sequence in E with Im(um) →b > 0 and I ′m(um) → 0 as m → ∞. Then there exist a subsequence of (um) (also denoted (um)), k ∈ N,

sequences (xim)i=1,...k

m∈N , u1∞, . . . , uk

∞ ∈ E \ 0, and functions V i∞

i=1,...k satisfying

(i) |xim − xj

m| → ∞ for i 6= j as m →∞

(ii) τ−xim

V i∞ → V i

∞ locally uniformly as m →∞

and defining Ii∞ = I[V i

∞],

(iii) Ii∞′(ui∞) = 0 for all i

(iv) ‖um −k

∑

i=1

τxim

ui∞‖ → 0 as m →∞

(v)k

∑

i=1

Ii∞(ui

∞) = b

Proof: By [CR1-2], (um) is a bounded sequence. Apply Proposition 3.0 to (um), obtaining (x1m) ≡ (xm) and

u1∞ ≡ u∞ as in the conclusion of Proposition 3.0. Let wm = um − τx1

mu1∞. Then by Proposition 3.0(iii),

I ′m(wm) → 0, so we may apply Proposition 3.0 again, obtaining (x2m) and u2

∞ satisfying 3.0(i)-(iv). And so

on. The process ceases after finitely many steps, because there exists b = b(V−, V +, f) with the property

that if (um; Vm) is a sequence as in Proposition 3.0, then either (um) goes to zero along a subsequence or

lim inf Im(um) ≥ b (to prove this, modify [CR1], Lemma 1.21). Therefore we obtain finite k in (3.7)(v) with

k ≤ b/b.

Let us apply Proposition 3.7 to the situation at hand. Define Vε by

(3.8) Vε(x) = V (εx).

Then Iε = I[Vε].

12

Proposition 3.9 Let f satisfy (f1)− (f4) and V satisfy (V1)− (V2). Let εm → 0. Let (um) be a sequence

in E with Iεm(um) → b > 0 and Iεm ′(um) → 0 as m → ∞. Then there exist k ∈ N, ai, . . . , ak ∈ [V−, V +],

a subsequence of (um) (also denoted (um)), v1, . . . , vk ∈ E \ 0, and sequences (xim)i=1,...k

m∈N satisfying

(i) |xim − xj

m| → ∞ as m →∞ for i 6= j

(ii) Vεm(xim) → ai

(iii) I[ai]′(vi) = 0

(iv) ‖um −k

∑

i=1

τxim

vi‖ → 0

(v)k

∑

i=1

I[ai](vi) = b

We will need the following simple lemma in Section 4. It estimates the error incurred when transposing

a cutoff function from one side of an inner product to another.

Lemma 3.10 Let Ω be an open subset of Rn, and ϕ ∈ W 1,∞(Ω) with ‖∇ϕ‖L∞(Ω) ≤ d. Let V be a

measurable function on Ω that is bounded and bounded away from zero. Define (·, ·)V ;Ω : W 1,2(Ω)2 → R by

(u, w)V ;Ω =∫

Ω∇u · ∇v + V (x)uw dx. Then

|(ϕu,w)V ;Ω − (u, ϕw)V ;Ω| ≤ d‖∇u‖W 1,2(Ω)‖∇w‖W 1,2(Ω).

for all u,w ∈ W 1,2(Ω).

Proof:

(ϕu, w)V ;Ω − (u, ϕw)V ;Ω =∫

Ω∇u · [(∇ϕ)w + (∇w)ϕ] + uϕw dx

−∫

Ω[(∇ϕ)u + (∇u)ϕ] · ∇w + ϕuw dx =

=∫

Ω(∇ϕ) · [(∇u)w − (∇w)u] dx,

so

|(ϕu,w)V ;Ω − (u, ϕw)V ;Ω| = |∫

Ω(∇ϕ) · [(∇u)w − (∇w)u] dx|

≤ ‖∇ϕ‖L∞(Ω)

∫

Ω|∇u||w|+ |∇w||u| dx ≤ d(‖∇u‖L2(Ω)‖w‖L2(Ω) + ‖∇w‖L2(Ω)‖u‖L2(Ω)) ≤

≤ ‖∇ϕ‖L∞(Ω)

√

‖∇u‖2L2(Ω) + ‖u‖2L2(Ω)

√

‖∇w‖2L2(Ω) + ‖w‖2L2(Ω) =

(by the Cauchy-Schwartz Inequality)

= d‖u‖W 1,2(Ω)‖w‖W 1,2(Ω).

13

The Gradient Vector Flow

For V : Rn → R, define WV : E → E to be the gradient of I[V ] with respect to the inner product

(·, ·)V . That is, (WV (u), w)V = I[V ]′(u)w for all u,w ∈ E. Then define ηI ≡ ηI[V ] to be the solution of the

initial value problem

(3.11)dηI

dt= −WV (η); ηI(0, u) = u.

It is easy to show that if f satisfies (f1) − (f3), then I ′ is locally Lipschitz continuous, so WV is locally

Lipschitz continous. Thus for every u ∈ E, ηI(t, u) exists at least for small |t|. We will work with a vector

flow of this form in Section 4. We would like to obtain estimates on the norm of ηIε(t, u) that are independent

of ε, and to find conditions under which ηIε(t, u) is well-defined for all positive t. The latter is not an easy

question, since WVε is not bounded on E, or even on sublevel sets of Iε.

Lemma 3.12 Let B0 > 0. There exists B2 = B2(V +, µ,B0) with the property that if V satisfies (V2),

I ≡ I[V ], and u ∈ E with ‖u‖V ≤ B0, then for all t ≥ 0,

(3.13) I(ηI(t, u)) ≥ 0 ⇒ ‖ηI(t, u)‖V ≤ B2.

Thus for all u ∈ E, either I(ηI(t, u)) < 0 for some t > 0, or ηI(t, u) is well-defined and I(ηI(t, u)) ≥ 0 for

all t > 0.

Proof: To prove the latter claim, suppose I(ηI(t, u)) ≥ 0 for all t > 0 in the interval of definition of

ηI(·, u). If ηI(·, u) has an interval of definition of the form (a, b) with b < ∞, then there exist (tm) with

tm ↑ b and ‖WV (ηI(tm, u))‖ → ∞. Since I ′ is bounded on bounded subsets of E, ‖ηI(tm, u)‖ → ∞. Since

I(ηI(tm, u)) ≥ 0 for all m, this contradicts (3.13).

To prove (3.13), let B0 > 0. Let B1 > B0 and be big enough so

(3.14) x ≥ B1 ⇒ (µ2− 1)x2 − x > µV +B2

0 .

We claim that if ‖w‖ ≥ B1 and I(w) ≤ V +B20 , then

(3.15) ‖I ′(w)‖V ≡ supI ′(w)y | ‖y‖V = 1 > 1;

to prove, suppose ‖w‖V = x ≥ B1, I(w) ≤ V +B20 , and ‖I ′(w)‖V ≤ 1. Then

−x = −‖w‖V ≤ I ′(w)w = ‖w‖2V −∫

Rnf(w)w ≤ ‖w‖2V − µF (w) = µI(w)− (

µ2− 1)‖w‖2V ≤

≤ µV +B20 − (

µ2− 1)‖w‖2V = µV +B2

0 − (µ2− 1)x2.

This contradicts (3.14). (3.15) is proven.

Define

(3.16) B2 = B1 + V +B20 + 1.

14

Suppose u ∈ E with ‖u‖V ≤ B0, and for some t∗ > 0, ‖ηI(t∗, u)‖V > B2 and I(ηI(t∗, u)) ≥ 0. Denote

η ≡ η(t) ≡ ηI(t) ≡ ηI(t, u). Let t1, t2 ∈ (0, t∗) with ‖η(t1)‖V = B1, ‖η(t2)‖V = B2, and ‖η(t)‖V ∈ (B1, B2)

for all t ∈ (B1, B2). Then I(η(t)) ∈ [0, I(u)] ⊂ [0, V+B20 ] for all t ∈ (t1, t2), so

V +B20 ≥ I(η(t1))− I(η(t2)) = −

∫ t2

t1

ddt

I(η) =∫ t2

t1‖I ′(η)‖2V ≥

≥∫ t2

t1‖I ′(η)‖V =

∫ t2

t1

∥

∥

∥

∥

dηdt

∥

∥

∥

∥

V≥ ‖η(t2)− η(t1)‖V ≥ B2 −B1 = V +B2

0 + 1.

This is impossible. (3.13) is proven.

4. Proof of Theorem 1.7

Here we complete the proof of Theorem 1.7, Case III. We follow the plan outlined in the Introduction.

Define the ground energy function C as follows: for b > 0, C(b) is the smallest positive critical value of

the functional I[b]. By arguments of [R2], C(b) is well defined (even without (f5)). Because of (f5), C is

continuous and strictly increasing on (0,∞) (see [WZ]).

Choose b0, b1, b2, b3, b4 satisfying

(4.0) infO

V = b0 < b1 < b2 < b3 < b4 = inf∂O

V

and

(4.1) C(b2) <32C(b0).

Let Oin ⊂ O with

(4.2) infO\Oin

V > b3

and

(4.3) d(Oin, OC) > 0,

where d(A,B) ≡ sup|x− y| | x ∈ A, y ∈ B. (4.2)-(4.3) are possible because V is uniformly continuous.

The most difficult task of this section is to find a Palais-Smale sequence (um) for Iε such that ‖um −um+1‖ → 0 and (um) is “confined to O/ε”. That is, ‖um‖W 1,2((O/ε)C) stays below a certain threshold. Define

Θ to be the family of open subsets U of Rn with the property that U = int(C), the interior of C, where

C is a union of closed n-cubes with side a and vertices in the lattice aZn for some a ≥ 1. Sets in Θ satisfy

a uniform cone condition. For U ⊂ Rn, define the inner product (u,w)Vε;U =∫

U ∇u · ∇w + Vε(x)uw dx on

W 1,2(U), and the norm ‖u‖2Vε;U = (u, u)Vε;U . Then by (f1) − (f3), and arguments from [R3], there exists

r0 > 0 such that if U ∈ Θ, ε > 0, and u ≥ 0 with ‖u‖Vε;U ≤ 4r0, then

(4.4)∫

U|f(u)||w| dx ≤ 1

4‖w‖Vε;Ur0.

15

Assume also that r0 is small enough so that if (um) is as in Proposition 3.7, then for large enough m there

exists xm with

(4.5) ‖um‖W 1,2(B1(xm)) > 2r0.

This is possible by Proposition 3.7.

Next, let ε1 > 0 and δ0 > 0 satisfy the following: if ε ≤ ε1, u ∈ E, I(u) < C(b2), and ‖u‖Vε;(O\Oin)/ε ≥ r0,

then

(4.6) ‖I ′(u)‖Vε > δ0.

To prove this is possible, assume the contrary. Then there exist εm → 0 and (um) ⊂ E with I(um) <

C(b2), ‖um‖W 1,2((O\Oin)/εm) ≥ ‖um‖Vε;(O\Oin)/εm/V + ≥ r0/V +, and ‖Iεm ′(um)‖Vεm(defined as in (3.15))

→ 0. Since min(1, V−)‖z‖2W 1,2(Rn) ≤ ‖z‖2Vε≤ V +‖z‖2W 1,2(Rn) for all ε > 0, ‖Iεm ′(um)‖ → 0. Apply

Proposition 3.9 to obtain k ≥ 1, sequences (xim), (ai)i=1,...,k, and (vi)i=1,...,k satisfying 3.9(i)-(v). Since

‖um‖W 1,2((O\Oin)/εm) ≥ r0/V +, it follows for at least one i, the distance d(xim, (O \ Oin)/εm) is bounded

along a subsequence of m. Since V > b3 on O \ Oin, and Vεm(xim) → ai, ai ≥ b3. Since the ground energy

function C is monotone, I[ai](vi) ≥ C(b3). By Proposition 3.9(v), lim infm→∞ Iεm(um) ≥ C(b3) > C(b2).

This contradicts the assumption I(um) < C(b2), proving (4.6).

Let

(4.7) M ∈ N ∩ (C(b2)δ0r0

,∞).

Let ρ = ρ(f, V, M) > 0 and open sets O0, O1, · · · , OM satisfy

(4.8)(i) Oin ⊂ O0 ⊂ O0 ⊂ O1 ⊂ O1 ⊂ . . . ⊂ OM ⊂ O.

(ii) Oi is a union of closed n-cubes with side ρ and vertices in the lattice ρZn.

(iii) Oi = intOi (the interior of Oi) for all i.

Assume without loss of generality that

(4.9) 0 ∈ Oin, V (0) = b1.

By (f5), I[b1] has a critical point ω ∈ E with I[B1](ω) = C(b1) (see [CR1] for proof) and ω(0) = max ω, and

there exists T > 0 such that I[b1](Tω) = −1 and maxθ∈[0,T ] I[b1](θω) = C(b1) = I[b1](ω). Since Vε → b1

locally uniformly as ε →∞, it is easy to check that for small enough ε, Iε(Tω) < −1/2 and

(4.10) maxθ∈[0,T ]

Iε(θω) < C(b2).

Because of the mountain-pass structure of Iε, it is easy to check that there exists θε ∈ (0, T ) with

(4.11) limt→∞

ηIε(t, θεω) > 0,

where ηIε is as in (3.11). Since ηIε(t, θεω) | t > 0 is bounded (Lemma 3.12), and Iε′′ is bounded on

bounded subsets of E, it is easy to check that

(4.12) ‖Iε′(ηIε(t, θεω))‖Vε → 0

16

as t →∞. See [STT] for a similar argument.

Recall T from (4.10) and let C1 = C1(T, ω, V−, V +, µ) be large enough so that if ε > 0 and ‖u‖ ≤ T‖ω‖,then for all t > 0 and ε > 0,

(4.13) Iε(ηIε(t, u)) ≤ 0 or ‖ηIε(t, u)‖Vε < C1.

This is possible because of Lemma 3.12 and the equivalence of ‖ · ‖Vε and ‖ · ‖W 1,2 independently of ε. Let

C2 = C2(f, V−, V +) be large enough that if ε > 0 and ‖u‖Vε ≤ C1 then

(4.14)∫

Rn|f(u)||w| ≤ C2‖w‖ε

for all w ∈ E.

Assume that ε is small enough so

(4.15) ‖θεω‖W 1,2(Oin/ε) < ‖Tω‖W 1,2(Oin/ε) < r0/V +.

Also let

(4.16) ε < min(r0ρ8C2

,ρ2),

where ρ is as in (4.8).

For convenience let η ≡ η(t) ≡ ηIε(t, θεω). We will show that “η stays inside O/ε,” more precisely, that

(4.17) ‖η(t)‖W 1,2((O/ε)C) < r0 for all t > 0.

To prove this, we assume the contrary. Assume for convenience that

(4.18) V− ≥ 1;

We can do this without loss of generality: if V− < 1, then (1.10) is equivalent to −(ε2/V−)∆u+(V (x)/V−)u =

f(u)/V−. Defining ε = ε/√

V −, V = V/V− and f = f/V−, we obtain −ε2∆u + V u = f(u), where V and f

satisfy (V1)− (V3) and (f1)− (f6).

Since V− ≥ 1, ‖u‖W 1,2(U) ≤ ‖u‖Vε;U for all u ∈ E and U ⊂ Rn. Define Oi,ε = Oi/ε for i = 0, . . . , M ,

where Oi are from (4.8). Assuming (4.17) is false, by (4.8) and (4.15) we may define 0 < t1 ≤ t2 ≤ . . . ≤ tMby

(4.19) ti = mint | ‖η(t)‖Vε;OCi,ε

= r0.

We claim that for all i = 1, . . . ,M ,

(4.20) ‖η(ti)‖Vε;Oi,ε\Oi−1,ε > 2r0.

To prove, first note that by definition of ti,

0 ≤(

ddt‖η‖2Vε;OC

i,ε

)

t=ti

= 2(

(η,dηdt

)Vε;OCi,ε

)

t=ti

= −2(η(ti), WVε(η(ti)))Vε;OCi,ε

,(4.21)

17

where WVε , the gradient of Iε, is defined in (3.11). For u ∈ E, define F(u) ∈ E by (F(u), w)Vε =∫

Rnf(u)w dx

for all w ∈ E. Then WVε(u) = u − F(u) for all u ∈ E. Let z = η(ti), and suppose ‖z‖Vε;Oi,ε\Oi−1,ε ≤ 2r0.

Then ‖z‖Vε;OCi−1,ε

≤ 3r0 by definition of ti. Let ϕ ∈ C∞(Rn,R) with ϕ ≡ 0 on Oi−1,ε, ϕ ≡ 1 on OCi,ε, and

‖∇ϕ‖L∞(Rn) < 2ε/ρ < r0/4C2. This is possible by (4.8) and (4.16). Then,

‖F(z)‖Vε;OCi,ε

= ‖ϕF(z)‖Vε;OCi,ε≤ ‖ϕF(z)‖Vε = sup

‖w‖Vε≤1(ϕF(z), w)Vε ≤(4.22)

≤ sup‖w‖Vε≤1

[

(F(z), ϕw)Vε +r0

4C2‖F(z)‖W 1,2(Rn)‖w‖W 1,2(Rn)

]

≤

(by Lemma 3.10)

≤ r0

4C2‖F(z)‖W 1,2(Rn) + sup

‖w‖Vε≤1(F(z), ϕw)Vε ≤

≤ r0

4C2‖F(z)‖Vε + sup

‖w‖Vε≤1

∫

Rnf(z)ϕw ≤

(by V− ≥ 1)

≤ r0/4 + sup‖w‖Vε≤1

∫

OCi−1,ε

|f(z)||w| ≤ r0/4 + r0/4 = r0/2.

In the last line we used (4.13)-(4.14) on the first part. On the second part we used (4.16) and (4.8) to show

that OCi−1,ε belongs to the family of sets Θ described before (4.4). Then we invoked (4.4). (4.22) implies

(z, WVε(z))Vε;OCi,ε

= (z, z −F(z))Vε;OCi,ε

= ‖z‖2Vε;OCi,ε− (z,F(z))Vε;OC

i,ε≥(4.23)

≥ ‖z‖2Vε;OCi,ε− ‖z‖Vε;OC

i,ε‖F(z)‖Vε;OC

i,ε= r0(r0 − ‖F(z)‖Vε;OC

i,ε) ≥ r0(r0 − r0/2) > 0.

This contradicts (4.21). (4.20) is proven.

For i = 1, . . . , M , define

(4.24) si = maxs < ti | ‖η(s)‖Vε;Oi,ε\Oi−1,ε = r0.

By the definition of ti, and (4.20), si is well-defined, with ti−1 < si < ti. For all i = 1, . . . ,M and t ∈ [si, ti],

(4.25) ‖η(t)‖Vε;(O\Oin)/ε ≥ ‖η(t)‖Vε;OM,ε\O0,ε ≥ ‖η(t)‖Vε;Oi,ε\Oi−1,ε ≥ r0,

so ‖I ′(η(t))‖Vε > δ0 by (4.6). Also, for all i = 1, . . . , M , (4.20) and (4.24) give

‖η(ti)− η(si)‖Vε ≥ ‖η(ti)− η(si)‖Vε;Oi,ε\Oi−1,ε ≥(4.26)

≥ ‖η(ti)‖Vε;Oi,ε\Oi−1,ε − ‖η(si)‖Vε;Oi,ε\Oi−1,ε ≥ 2r0 − r0 = r0.

Therefore,

C(b2) > Iε(η(0)) ≥ Iε(η(t0)) ≥ Iε(η(t0))− Iε(η(tM )) =(4.27)

= −∫ tM

t0

ddt

Iε(η) =∫ tM

t0‖Iε′(η)‖2Vε

≥M∑

i=1

∫ ti

si

‖Iε′(η)‖2Vε≥

≥ δ0

M∑

i=1

∫ ti

si

‖Iε′(η)‖Vε = δ0

M∑

i=1

∫ ti

si

‖dηdt‖Vε ≥ δ0

M∑

i=1

‖η(ti)− η(si)‖Vε ≥ r0δ0M.

18

This is impossible by definition of M ((4.7)). (4.17) is proven.

To complete the existence portion of the proof of Theorem 1.7, Case III, let tm →∞ with tm+1−tm → 0

as m →∞. Let um = η(tm). Then Iε′(um) → 0 and ‖um+1 − um‖ → 0 as m →∞.

Define Hull(Vε) to be the set of all limit points, under the uniform norm, of sequences of the form

(τxmVε). Let ε be small enough so that if V ∈ Hull(Vε), V(0) ≥ b0, and v is a nonzero critical point of I[V]

with ‖v‖W 1,2(B1(0)) ≥ r0, then

(4.28) I[V](v) ≥ 45C(b0).

It is easy to show this is possible, using Proposition 3.9.

If we apply Proposition 3.7 to any subsequence of (um), we obtain k, (xim), and (ui

∞) as in Proposi-

tion 3.7(i)-(v). Assume that all the ui∞’s are “normalized” so ‖ui

∞‖W 1,2(B1(0)) > r0 (see (4.5)) for all i. Then

by (4.17), xim ∈ N1(O/ε) for all i, for large enough m, so Ii

∞(vi) ≥ 45C(b0) for all i by (4.28) and (4.0), and

limm→∞ Iε(um) ≥ 45kC(b0). Since Iε(um) < C(b2) < 3

2C(b0) for all m (see (4.10), (4.1)), k = 1. That is,

(um) does not “split into more than one piece”.

Choose (xm) ⊂ N1(O/ε) with ‖um‖W 1,2(B1(xm)) > r0 for large m. Since k = 1 in the conclusion of

Proposition 3.7, it is easy to verify that |xm − xm+1| is bounded in m. Let R > 0 with |xm − xm+1| < R

for large m. Recall ~v from Theorem 1.7(III), and let T = sup~u · x | x ∈ O/ε, |~u| = 1, ~u ⊥ ~v < ∞.

Then there exists a sequence (δm) with δm → 0, a sequence (tm) ⊂ R, and a subsequence of (um) (also

denoted (um)), with tm~v a δm-almost period of V and |xm − tm~v| < R + T + 1. Define zm = τ−tm~vum and

Vm = τ−tm~vVε. Then Vm → Vε locally uniformly and I[Vm]′(zm) → 0. Also ‖zm‖BR+T+1(y) ≥ r0. In fact,

we may select a subsequence of (zm) and a point x ∈ BR+T+1(0) with lim infm→∞ ‖zm‖W 1,2(B1(x)) ≥ r0.

Applying Proposition 3.7 (and noting that k = 1 in Proposition 3.7(v) by the above argument), zm converges

strongly to a nonzero critical point z of Iε, with ‖z‖W 1,2(B1(x)) ≥ r0 and C(b0) ≤ Iε(z) < C(b2). It is not

apparent whether or not x must belong to O/ε.

Concentration of Solutions

We have proven the existence portion of Theorem 1.7, Case III, but not the positivity and exponential

decay properties. In the proofs in this section, we may replace O by O ∩ y | V (y) < b0 + ε for any

ε > 0, and the proof works the same. Thus for any sequence (εm) with εm → 0, we can obtain a sequence

(um) of critical points of Iεm , with Iεm(um) → C([b0]), and points xm ∈ Rn with V (xm) → b0 and

‖um‖W 1,2(B1(xm/εm)) ≥ r0. By Proposition 3.9, (τ−xm/εmum) converges along a subsequence in W 1,2(Rn) to

a solution ω of −∆ω + V (b0)ω = f(ω) with I[b0](ω) = C[b0]. By arguments of [CR1], ω does not change

sign. Without loss of generality, ω is positive. By elliptic theory, um converges pointwise to ω. A maximum

principle argument shows that for large m, um is nonnegative. Namely, if um is negative somewhere for

large enough m, it has a negative global minimum at y ∈ Rn, with |um(y)| small. However, the equation

−∆u + V (εmx)u = f(u) implies that ∆um(y) > 0 (if m is chosen to make |um(y)| small enough). Thus

um ≥ 0. A slight refinement of the above argument shows that in fact um > 0.

Arguments of [GNN1-2] show that ω above has a unique local maximum, which is nondegenerate.

Since (τ−xm/εmum) converges to ω locally in C2, a similar maximum principle argument, found in [FdP1],

19

shows that um has a unique local maximum xm for large m. We have shown that there exists a sequence

(xm) ⊂ Rn with lim infm→∞ ‖um‖W 1,2(B1(xm)) > 0. It is clear that |xm − xm| is bounded in m. Therefore,

since Vεm(xm) → b0, Vεm(xm) → b0 as well. The exponential decay of um is given by a standard maximum

principle argument found in [FdP1]. The proof of Theorem 1.7 is complete. Note that is not apparent

whether xm belongs to O/εm; therefore in the statement of Theorem 1.7, it is not apparent whether zε

belongs to O.

Open Questions

There are many open questions associated with (1.2) and related equations. It is unknown whether

the ε is really necessary in (1.2) or in similar elliptic PDE containing almost periodic terms, even in special

cases such as n = 2, or V quasiperiodic rather than merely almost periodic. It is also unclear whether the

convexity condition (f5) is necessary in equations like (1.2), even when V satisfies stronger global conditions

than (V1)− (V3).

20

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A Singularly Perturbed Elliptic Partial Differential Equation with an Almost Periodic Term

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