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SADDLE TYPE SOLUTIONS FOR A CLASS OF REVERSIBLE ELLIPTIC EQUATIONS

Francesca Alessio, Giuseppina Autuori, and Piero Montecchiari Dipartimento di Ingegneria industriale e Scienze Matematiche

Universit` a Politecnica delle Marche Via Brecce Bianche 12, 06131 Ancona, Italy

(Submitted by: Giuseppe Mingione)

Abstract. This paper is concerned with the existence of saddle type solutions for a class of semilinear elliptic equations of the type

−∆u(x) + F

(x, u) = 0, x ∈ R

, n ≥ 2, (P DE) where F is a periodic and symmetric nonlinearity. Under a non degen-

eracy condition on the set of minimal periodic solutions, saddle type solutions of (P DE) are found by a renormalized variational procedure.

1. Introduction

The present paper is devoted to the existence and the asymptotic char- acterization of saddle solutions of a class of reversible symmetric semilinear elliptic equations in R ⁿ . More precisely, we consider equations

(P DE) −∆u(x) + F u (x, u) = 0, x ∈ R ⁿ , n ≥ 2, where F satisfies

(F ₁ ) F ∈ C ² ( R ⁿ × R) is 1-periodic in all its variables;

(F ₂ ) F is even in all its variables;

(F ₃ ) F has flip symmetry with respect to the first two variables, i.e., F (x ₁ , x ₂ , x ₃ , ..., x _n , u) = F (x ₂ , x ₁ , x ₃ , ..., x _n , u) on R ⁿ × R.

Equations like (P DE) are particular cases of a much more general class of equations studied by Moser in [35], where, in the spirit of the Aubry-Mather Theory ([15, 33, 34]), he established various qualitative properties of certain minimal and not self intersecting solutions. He obtained in particular that the set M 0 of minimal periodic solutions is a non empty ordered set. In

AMS Subject Classifications: 35J60, 35B05, 35B40, 35J20, 34C37.

Accepted for publication: July 2015.

1

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the same framework Bangert in [23] studied the problem in which M 0 con- tains a gap pair, i.e., a couple v < w ∈ M 0 such that no other members of M 0 lie between them. He showed that, in this case, there exists another ordered family M 1 of minimal and not self intersecting solutions which are heteroclinic in one space variable to v and w and periodic in the remaining ones. He also iterates the argument to find more complex ordered classes of minimal heteroclinic type solutions. In [41], Rabinowitz and Stredulin- sky studied (PDE) obtaining Bangert’s results by minimization arguments.

Variational gluing arguments were then employed to construct various kinds of homoclinic and heteroclinic type solutions as local minima of renormalized functionals associated to (PDE), see [42]. Other extensions of Moser’s work have been developed in [16, 17, 18, 27, 44] (see also [39, 40, 36] for systems of (PDE)).

A simple example of (PDE), arising in phase transition theory, is the stationary sine-Gordon type equation ∆u = −a(x) sin(2πu), where a is a positive and periodic function. In this case

F (x, u) = _2π ¹ a(x) cos(2πu),

and M 0 = {u k = ¹ ₂ + k | k ∈ Z} consists of the set of pure phases of the system. In correspondence to any gap pair u _k , u _k+1 , the Bangert and Rabinowitz-Stredulinsky results provide the existence of the ordered set M 1

of basic transition solutions between u _k and u _k+1 . If M 1 contains itself a gap pair, new transition and (at least in the reversible case) multitransition solutions, shadowing formal concatenations of the basic ones, appear.

Analogous results have been independently obtained using variational methods for Allen-Cahn equations (or systems) of the type

−∆u + a(x)W ⁰ (u) = 0, x ∈ R ⁿ , (1.1) where W is a double well potential and a a periodic positive function, mainly in the planar case. Again, the set M 1 of solutions which are heteroclinic in the first variable to the two minima of the potential (the analogues of the periodic minimal solutions of (PDE)) is not empty and, if it has gaps (or enjoys suitable discreteness property for systems), different types of transi- tion and multitransition solutions appear (see [1, 2, 5, 6, 7, 8, 9, 13, 14, 19, 20, 21, 22, 37]).

Another kind of transition solutions for (1.1), was first studied by Dang,

Fife and Peletier in [26]. They showed that, in the planar case, when W is

an even double well potential and a is a positive constant, the corresponding

Allen-Cahn equation has a unique bounded solution v ∈ C ² ( R ² ) with the

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same sign as x 1 x 2 , odd in both the variables x 1 and x 2 and symmetric with respect to the diagonals x ₂ = ±x 1 . The saddle solution v is asymptotic to the minima of the potential W along any directions not parallel to the coordinate axes representing a phase transition with cross interface.

We note that the saddle solution is related to the set M 1 of minimal transitions, being asymptotic, as x 2 → +∞, to the unique odd heteroclinic solution of the equation, but it does not enjoy the same minimality properties (see [43, 31]).

Many extensions or generalizations for Allen-Cahn models have been ob- tained in different directions for the autonomous case. We refer to [3, 28, 32, 29] for the study of the class of k end solutions, to [24, 25, 10, 11, 38]

for equations in higher dimensions and to [1, 30, 12] for analogous results in the case of systems of autonomous Allen-Cahn equations.

The recent study [4] proves the existence of saddle type solutions for non autonomous Allen-Cahn equations in the planar case. In [4], it is shown that if W is an even double well potential and if a ∈ C(R ² ) is positive, even, periodic and symmetric with respect to the plane diagonal x ₂ = x ₁ , then (1.1) has a saddle type solution v ∈ C ² ( R ² ). As in the autonomous case, the solution v is odd with respect to both its variables, symmetric with respect to the diagonal, strictly positive on the first quadrant and it is asymptotic to the minima of W along any directions not parallel to the coordinate axes.

Moreover, in [4], it is proved that, as x 2 → +∞ (uniformly with respect to x ₁ ∈ R), v is asymptotic to the set M + of odd minimal heteroclinic solutions of (1.1). More precisely, any u ∈ M + is asymptotic to the minima of the potential W as x ₁ → ±∞, periodic in the variable x 2 and minimal with respect to C ₀ ^∞ perturbations which are odd in the variable x ₁ (note that, differently from the autonomous case, the set M + is in general not contained in the set M 1 of minimal heteroclinics).

In the present paper, we study the existence of saddle type solutions for (PDE) providing a characterization of their asymptotic behavior, through a renormalized variational procedure which refines the approach in [4] to the variational framework used in [42].

To be more precise in stating and describing our results it is useful to introduce some notation. Following [42], we are interested in the structure of the minimal set

M 0 = {u ∈ W ^1,2 ( T ⁿ ) | I 0 (u) = c ₀ }, where denoted with

L(u) = ¹ ₂ |∇u| ² + F (x, u),

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the Lagrangian associated to (PDE), I ₀ (u) =

Z

[0,1]

L(u) dx, c ₀ = inf

u ∈W

(T

) I ₀ (u), and

W ^1,2 ( T ⁿ ) = {u ∈ W ^1,2 ( R ⁿ ) | u is 1–periodic in x 1 , x ₂ , ..., x _n }.

As proved in [42], M 0 is a non empty ordered set, consisting of basic pe- riodic solutions of (PDE). As described above, we think to the elements of M 0 as the analogues of the pure phase solutions in the Allen-Cahn model and, inspired by this analogy, we ask for the existence of a gap pair in M 0

symmetric with respect to the origin. Precisely, we assume that (N ₀ ) v(0) 6= 0 for every v ∈ M 0 .

Since M 0 is ordered and F even, condition (N 0 ) is equivalent to the existence of a symmetric gap pair in M 0 around the origin, in the sense that there exist v ^± ∈ M 0 , such that v ⁻ (x) = −v ⁺ (x) < 0 for any x ∈ R ⁿ , and for every v ∈ M 0 \ {v ^± } there results either v < v ⁻ or v > v ⁺ in R ⁿ , see Lemma 2.8.

To recover compactness in the problem, and again in analogy with the Allen-Cahn case, we strengthen our assumption (N ₀ ) asking that the ex- tremal solutions v ^± are not degenerate:

(N 1 ) there exists ω > 0 such that for all h ∈ W ^1,2 ([0, 1] ⁿ ) I ₀ ⁰⁰ (v ⁺ )h · h =

Z

[0,1]

|∇h(x)| ² + F _u,u (x, v ⁺ (x)) |h(x)| ² dx ≥ 4ωkhk ² _L

_([0,1]

₎ . Note that, by the minimality of v ⁺ , the assumption (N 1 ) is equivalent to ask that the spectrum of the linearized operator about v ⁺ , that is,

L : W ^2,2 ([0, 1] ⁿ ) ⊂ L ² ([0, 1] ⁿ ) → L ² ([0, 1] ⁿ ), given by

Lh = −∆h + F u,u (v ⁺ )h, does not contain 0.

The couple v ^± establishes a global bound for any minimal odd heteroclinic solution of (P DE) (see Lemma 3.1). More precisely, if v ∈ W _loc ^1,2 ( R×[0, 1] ^{n −1} ) is odd in the variable x ₁ and it is not negative for x ₁ ≥ 0, there exists ˆv in the same class such that 0 ≤ ˆv(x) ≤ v ⁺ (x) for x ₁ ≥ 0 and J(ˆv) ≤ J(v), where

J(u) = X

p ∈Z

 Z

T

L(u) dx − c 0



, T _p,0 = [p, p + 1] × [0, 1] ^{n −1} ,

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is the renormalized functional introduced in [41]. Setting E ⁺ = {u ∈ W _loc ^1,2 ( R × [0, 1] ^{n −1} ) | u is odd in x 1 , with 0 ≤ u(x) ≤ v ⁺ (x) for x ₁ ≥ 0} and c = min _u∈E

J(u), we have that M ⁺ = {u ∈ E ⁺ | J(u) = c} 6= ∅, it is compact with respect to the W ^1,2 ( R × [0, 1] ⁿ⁻¹ ) metric (see Lemma 3.10) and it consists of classical solutions to (PDE), which are odd in x 1 , even and 1-periodic in x ₂ , ..., x _n and satisfy the asymptotic condition

ku − v ⁺ k W

(T

) → 0 as p → +∞.

Our main result can now be stated as follows:

Theorem 1.1. Assume (F 1 ) − (F 3 ) and (N 0 ) − (N 1 ). Then, there exists a classical solution w of (P DE) such that

(i) 0 ≤ w(x) ≤ v ⁺ (x) for x ₁ x ₂ > 0;

(ii) w is odd in x ₁ and x ₂ , 1-periodic in x ₃ , ..., x _n ; (iii) w(x 1 , x 2 , x 3 , ..., x n ) = w(x 2 , x 1 , x 3 , ..., x n ) in R ⁿ . Moreover, the solution w satisfies the asymptotic condition (S) dist _W

(S

) (w, M ⁺ ) → 0, as k → +∞, where S _k = R × [k, k + 1] × [0, 1] ^{n −2} .

Thanks to (ii) and (iii) the solution w has the symmetry of a saddle type solution in the variables x ₁ and x ₂ , being periodic in the remaining ones. It has the same sign as the product function x 1 x 2 and it is pinched between v ⁻ and v ⁺ , i.e., v ⁻ (x) ≤ w(x) ≤ v ⁺ (x) on R ⁿ . By (S), as x ₂ → +∞, w is asymptotic to the set M ⁺ of minimal odd heteroclinic type solutions. The symmetry of w and the compactness of M ⁺ imply that w is asymptotic to v ⁺ or v ⁻ along any direction not parallel to the planes x ₁ = 0, x ₂ = 0, representing again a transition with cross interface.

We remark that, even if the assumptions (N ₀ ) and (N ₁ ) are not easy to

check for general functions F satisfying (F 1 )–(F 3 ), it is possible to prove

that they hold generically in the same class of functions. Indeed, as noted in

Remark 2.7, (N 0 ) is equivalent to require that the trivial constant solution

v ₀ ≡ 0 is not minimal (v 0 ∈ M / 0 ). If v ₀ ∈ M 0 , it is possible to locally slightly

modify F around 0 ∈ R ⁿ⁺¹ (and periodically around any k ∈ Z ⁿ⁺¹ ) altering

the minimality property of v ₀ but maintaining the conditions (F ₁ )–(F ₃ ), so

that the new equation satisfies (N ₀ ). Moreover, since v ^± are periodic, even

and symmetric with respect to the diagonal (see Lemma 2.4), it is possible

to gain (N ₁ ) following Proposition 3.56 in [42], obtaining a new function ˜ F

near F , satisfying (F ₁ )–(F ₃ ) for which both (N ₀ ) and (N ₁ ) hold true.

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It is also possible to give simple perturbative examples in which (N 0 ) and (N ₁ ) are satisfied. Consider for example the sine-Gordon type autonomous equation ∆u = − sin(2πu). As explained above M 0 = {u k = ¹ ₂ + k | k ∈ Z}

and in this case (N ₀ ) and (N ₁ ) are satisfied with v ₀ ^± = ± ¹ ₂ . If F (x, u) is a periodic function satisfying (F ₁ )-(F ₃ ) the application of the Implicit Function Theorem shows that for  sufficiently small, the perturbed equation

∆u = − sin(2πu) + F u (x, u) satisfies (N ₀ ) and (N ₁ ) relative to a new gap pair v ^± nearby v ^± ₀ .

The proof of Theorem 1.1 uses variational methods adapting some ideas developed in [4] to the setting in [42]. The symmetry properties of F and the nondegeneracy assumption (N ₁ ) allow us to built up a renormalized variational procedure to find a minimal solution u of (PDE) on the triangular set T = {x ∈ R ⁿ | x 2 ≥ |x 1 |} which is odd in the variable x 1 , positive for x 1 > 0 and asymptotic to M + as x 2 → +∞. The symmetries of the problem are then used to obtain the saddle type solution w on R ⁿ applying consecu- tive reflections of u with respect to the faces of T .

We conclude with a brief outline of the paper. In Section 2, we present a list of preliminary properties of the set of minimal periodic solutions M 0 . Section 3, is devoted to the study of the set M ⁺ and its compactness proper- ties connected with (N ₁ ). The variational study of the existence of minimal odd solutions of (PDE) on T and the construction of a saddle type solution is then developed in Section 4.

2. The periodic solutions

In this section, following [41], we recall some of the results regarding the set M 0 of minimal periodic solutions to (P DE). We moreover, study symmetry properties of the functions u ∈ M 0 , related to the flip symmetry assumption (F ₃ ).

Eventually, adding a constant to F , it is not restrictive to assume, here and in the following, that

(F 4 ) F ≥ 0 on R ⁿ × R.

Consider the functional space

E ₀ = {u ∈ W ^1,2 ( R ⁿ ) | u is 1-periodic in each variable}, embedded with the norm

kuk =  Z

[0,1]

( |∇u| ² + |u| ² )dx 

.

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For any u ∈ E 0 let

L(u) = ¹ ₂ |∇u| ² + F (x, u), and define

I ₀ (u) = Z

[0,1]

L(u) dx and c ₀ = inf

u ∈E

I ₀ (u).

As a particular case of the results established in [35], we have:

Lemma 2.1. If (F ₁ ) holds, then M 0 = {u ∈ E 0 | I 0 (u) = c ₀ } is a not empty ordered set, i.e., u, v ∈ M 0 implies u < v, u > v, or u ≡ v. Moreover, if u ∈ M 0 , then u is a classical periodic solution of (P DE).

We note that the functional I ₀ is well defined in W ^1,2 ([0, 1] ⁿ ). The next Lemma, whose proof is based on the reversibility property of F and can be found in [41] (see also [42]), describes the minimality and compactness properties of M 0 in the whole space W ^1,2 ([0, 1] ⁿ ).

Lemma 2.2. The following results hold.

(i) For any r > 0 there exists λ(r) > 0 such that if u ∈ W ^1,2 ([0, 1] ⁿ ) and verifies dist(u, M 0 ) _W

([0,1]

) > r, then

I 0 (u) ≥ c 0 + λ(r).

(ii) Any bounded minimizing sequence (u _k ) ⊂ W ^1,2 ([0, 1] ⁿ ) has a conver- gent subsequence in W ^1,2 ([0, 1] ⁿ ).

(iii) Any u ∈ M 0 is a minimizer of I(u) =

Z

[0,1/2]

L(u) dx,

in W ^1,2 ([0, 1/2] ⁿ ). Moreover, every u ∈ M 0 is symmetric in x i about x _i = 0 and x _i = 1/2 for any i ∈ {1, . . . , n}.

(iv) It results that c 0 = inf

u∈W

([0,1]

) I 0 (u). Furthermore, if u ∈ W ^1,2 ([0, 1] ⁿ ) verifies I ₀ (u) = c ₀ , then u ∈ M 0 .

The even parity of F with respect to the variable u plainly implies that I ₀ (u) = I ₀ ( −u) = I 0 ( |u|) for any u ∈ W ^1,2 ([0, 1] ⁿ ).

As a consequence, we derive that the elements in M 0 have definite sign.

Lemma 2.3. If u ∈ M 0 , then −u ∈ M 0 and either u ≥ 0 or u ≤ 0 on R ⁿ .

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We now use the assumption (F 3 ) to obtain, by Lemma 2.2, flip symmetry properties for the elements in M 0 . In the following, we denote

T ⁺ = {x ∈ [0, 1] ⁿ | x 1 ≤ x 2 }, and given u ∈ W ^1,2 (T ⁺ ), let

˜ u(x) =

( u(x), x ∈ T ⁺ ,

u(x ₂ , x ₁ , x ₃ , . . . , x _n ), x ∈ [0, 1] ⁿ \ T ⁺ , (2.1) and note that ˜ u ∈ W ^1,2 ([0, 1] ⁿ ).

Lemma 2.4. If u ∈ M 0 , then u ≡ ˜u in [0, 1] ⁿ . Proof. Take u ∈ M 0 and suppose first that

Z

T

L(u)dx ≤ Z

[0,1]

\T

L(u)dx. (2.2)

Since ˜ u ∈ W ^1,2 ([0, 1] ⁿ ), by Lemma 2.2-(iv), we have I 0 (˜ u) ≥ c 0 . On the other hand, (2.2) gives

c ₀ = I ₀ (u) = Z

T

L(u)dx + Z

[0,1]

\T

L(u)dx ≥ 2 Z

T

L(u)dx = I ₀ (˜ u) ≥ c 0 . Hence, I ₀ (˜ u) = c ₀ and so ˜ u ∈ M 0 , again by Lemma 2.2-(iv). This implies that ˜ u = u in [0, 1] ⁿ , being ˜ u = u in T ⁺ and M 0 an ordered set. Then the claim is proved under (2.2). We can argue in a similar way if

Z

T

L(u)dx ≥ Z

[0,1]

\T

L(u)dx,

concluding the proof. 

By Lemma 2.2 and Lemma 2.4, we obtain Lemma 2.5. There results

u∈W min

(T

)

Z

T

L(u) dx = c ₀

2 . (2.3)

Moreover, if u ∈ W ^1,2 (T ⁺ ) verifies Z

T

L(u) dx = c ₀

2 ,

then ˜ u ∈ M 0 .

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Proof. Given u ∈ W ^1,2 (T ⁺ ), since ˜ u ∈ W ^1,2 ([0, 1] ⁿ ), by Lemma 2.2–(iv), we have

I ₀ (˜ u) = 2 Z

T

L(˜ u)dx ≥ c 0 . Hence,

u∈W inf

(T

)

Z

T

L(u)dx ≥ c 0

2 . On the other hand, if u ∈ M 0 by Lemma 2.4, we have

Z

T

L(u)dx = c ₀ /2, and so

min

W

(T

)

Z

T

L(u) dx = min

M

Z

T

L(u) dx = c ₀ 2 . Finally, if u ∈ W ^1,2 (T ⁺ ) verifies

Z

T

L(u) dx = c ₀ 2 ,

then I 0 (˜ u) = c 0 and, again by Lemma 2.2–(iv), ˜ u ∈ M 0 .  Remark 2.6. For future references, it is important to know that, by Lemma 2.5, setting

σ 0 = {x ∈ R × [0, 1] ^{n −1} | x 2 − 1 ≤ x 1 ≤ x 2 }, we have that

min

W

(σ

)

Z

σ

L(u) dx = c ₀ , and that any u ∈ M 0 minimizes

Z

σ

L(u) dx,

on W ^1,2 (σ ₀ ). Moreover, if u ∈ W _loc ^1,2 ( R × [0, 1] ⁿ⁻¹ ) is such that Z

σ

L(u) dx = c ₀ , then ˜ u ∈ M 0 and u ≡ ˜u on σ 0 .

Finally, by symmetry, using Lemma 2.2-(i), one can prove that if u ∈ W ^1,2 (σ ₀ ) verifies Z

σ

L(u) dx − c 0 ≤ λ(r), then

dist(u, M 0 ) _W

(σ

) ≤ r.

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We finally display, in the last part of this preliminary section, some con- sequences of our assumption

(N ₀ ). v(0) 6= 0 for every v ∈ M 0 .

Remark 2.7. Note that, assumption (F 2 ) and the fact that M 0 is ordered, guarantee that condition (N ₀ ) is equivalent to require that v ₀ ≡ 0 is not a minimal solution, i.e., v ₀ 6∈ M 0 . Indeed, clearly, if (N ₀ ) holds, then v ₀ 6∈

M 0 . To show that if v ₀ 6∈ M 0 and v ∈ M 0 , then v(0) 6= 0, assume by contradiction that v(0) = 0. We then have −v(0) = 0 = v(0) and since, by Lemma 2.3, −v ∈ M 0 , by Lemma 2.1, we obtain −v(x) = v(x) for all x ∈ R ⁿ , and so v = v ₀ ∈ M 0 , a contradiction.

In particular, using Lemma 2.3, we get that if v ∈ M 0 , then either v > 0, or v < 0 on R ⁿ .

The compactness of I 0 , Remark 2.7 and the ordering property of M 0 , imply that (N ₀ ) is equivalent to the presence of a gap pair in M 0 which is symmetric with respect to 0.

Lemma 2.8. There exist v ^± ∈ M 0 such that v ⁻ = −v ⁺ , v ⁻ (x) < 0 < v ⁺ (x) on R ⁿ and for every v ∈ M 0 \ {v ^± } there results either v < v ⁻ or v > v ⁺ in [0, 1] ⁿ .

Proof. Set ξ 0 = inf {v(0) | v ∈ M 0 , v ≥ 0 in [0, 1] ⁿ } and observe that ξ 0 ≥ 0.

Fix any sequence (v _k ) ⊂ M 0 such that v _k (0) → ξ 0 and v _k+1 (0) ≤ v k (0) for all k ∈ N. Since M 0 is ordered, using Remark 2.7, we obtain that 0 < v _k < v 1

on [0, 1] ⁿ for any k ∈ N. Then, since k∇v k k ² _L

_([0,1]

₎ ≤ 2c 0 , the sequence (v _k ) is bounded in W ^1,2 ([0, 1] ⁿ ) and so in C ² ([0, 1] ⁿ ), since any v _k solves (P DE).

By Lemma 2.2–(ii) and the Ascoli-Arzel` a Theorem, there exists v ⁺ ∈ M 0

with v ⁺ (0) = ξ ₀ such that, up to a subsequence, v _k → v ⁺ in C ¹ ([0, 1] ⁿ ). By (N ₀ ), we then have ξ ₀ > 0 and by definition of ξ ₀ , if v ∈ M 0 is such that v > 0 in [0, 1] ⁿ , we have 0 < ξ 0 = v ⁺ (0) ≤ v(0). Since M 0 is ordered, we conclude that 0 < v ⁺ < v in [0, 1] ⁿ for all v ∈ M 0 \{v ⁺ } with v > 0 in [0, 1] ⁿ . By applying Lemma 2.3, the proof is completed setting v ⁻ = −v ⁺ .  Lemma 2.8 says that the couple v ^± is a gap pair in M 0 , that is, no other member of M 0 lies between v ⁻ and v ⁺ . For future reference, we set

r ₀ = kv ⁺ − v ⁻ k L

([0,1]

) . (2.4)

As last remark of this section, the following Lemma gives a characteriza-

tion of the minimizing sequences of I ₀ which are pinched between v ₋ and

v ₊ .

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Lemma 2.9. If (u _k ) ∈ W ^1,2 ([0, 1] ⁿ ) is such that 0 ≤ u k ≤ v ⁺ a.e. in [0, 1] ⁿ for all k ∈ N, with I 0 (u _k ) → c 0 , as k → +∞, then

ku k − v ⁺ k W

([0,1]

) → 0, as k → +∞.

Proof. Let (u _k ) ⊂ W ^1,2 ([0, 1] ⁿ ) be as in the statement, and let (u _k

) be a subsequence of (u _k ). First observe that (u _k

) is bounded in W ^1,2 ([0, 1] ⁿ ), since 0 ≤ u k

≤ v ⁺ a.e. in [0, 1] ⁿ and I 0 (u _k

) → c 0 as j → +∞. Then, by Lemma 2.2–(ii), there exists a subsequence of (u _k

), still denoted by (u _k

), such that u _k

converges to some v ∈ M 0 in W ^1,2 ([0, 1] ⁿ ), with 0 ≤ v ≤ v ⁺ a.e. in [0, 1] ⁿ . By Remark 2.7 and Lemma 2.8, it follows that v ≡ v ⁺ a.e. in [0, 1] ⁿ . The generality of the subsequence assures that the entire sequence (u _k ) converges to v ⁺ in W ^1,2 ([0, 1] ⁿ ) as k → +∞ and the Lemma

is proved. 

3. Heteroclinic solutions and compactness properties In this section, following some arguments in [41], we display properties of the set M + of the minimal odd solutions to (PDE) which are asymptotic to v ^± as x ₁ → ±∞. In particular, we study some compactness properties of M + related to the non degeneracy assumption (N ₁ ), which will be used in the next section to construct saddle type solutions.

Setting

E = {u ∈ W _loc ^1,2 ( R × [0, 1] ^{n −1} ) | u is odd in x 1 , with u(x) ≥ 0 for x 1 ≥ 0}, for any u ∈ E, we consider the functional

J(u) = X

p∈Z

J _p,0 (u),

where, denoting T _p,0 = [p, p + 1] × [0, 1] ⁿ⁻¹ , J _p,0 (u) =

Z

T

L(u) dx − c 0 , ∀p ∈ Z.

Note that, by Lemma 2.2-(iv), we have J _p,0 (u) ≥ 0 for any u ∈ E and p ∈ Z.

Then J is not negative on E and clearly, it is weakly lower semicontinuous with respect to the weak W _loc ^1,2 topology.

We are interested in the minimal properties of J on E and the first im-

portant observation is that, thanks to the following Lemma, the minimizing

sequences of J can be always chosen to be made by functions in E which are

pinched between v ⁻ and v ⁺ .

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Lemma 3.1. For every v ∈ E there exists ˆv ∈ E such that 0 ≤ ˆv(x) ≤ v ⁺ (x) for x ₁ ≥ 0 and J(ˆv) ≤ J(v).

Proof. Let v ∈ E. For x 1 ≥ 0, we set ˆv(x) = min{v(x); v ⁺ (x) } and extend ˆ

v as an odd function for x ₁ < 0. Note that ˆ v ∈ E and 0 ≤ ˆv ≤ v ⁺ a.e. in [0, + ∞) × [0, 1] ⁿ⁻¹ . We claim that J _p,0 (ˆ v) ≤ J p,0 (v) for every index p ≥ 0, from which J(ˆ v) ≤ J(v) will follow.

To this aim, fixed p ≥ 0, let A p = {x ∈ T p,0 | v ⁺ (x) ≤ v(x)}. By Lemma 2.2–(iv), setting w(x) = max {v(x); v ⁺ (x) } for x ∈ T p,0 , we have

c ₀ = Z

T

L(v ⁺ ) dx ≤ Z

T

L(w) dx = Z

A

L(v) dx + Z

T

\A

L(v ⁺ ) dx,

from which Z

A

L(v ⁺ ) dx ≤ Z

A

L(v) dx.

Therefore, since ˆ v(x) = v ⁺ (x) for x ∈ A p , we get Z

T

L(ˆ v) dx = Z

A

L(v ⁺ )dx + Z

T

\A

L(v) dx

≤ Z

A

L(v) dx + Z

T

\A

L(v) dx = Z

T

L(v)dx,

and the claim follows. 

By Lemma 3.1, setting

E ⁺ = {u ∈ E | 0 ≤ u(x) ≤ v ⁺ (x) for x 1 ≥ 0}, we have

c = inf

u ∈E J(u) = inf

u ∈E

J(u),

and we can restrict ourself to study the minimal properties of J on E ⁺ . As first remark, the following lemma describes the weak compactness of the sublevels of the functional J in E ⁺ .

Lemma 3.2. Let (u _k ) ⊂ E ⁺ be such that J(u _k ) ≤ λ for all k ∈ N, for some λ > 0. Then, there exists u ∈ E ⁺ such that, up to a subsequence, u _k → u weakly in W _loc ^1,2 ( R × [0, 1] ^{n −1} ), as k → +∞ and J(u) ≤ λ.

Proof. Since u _k ∈ E ⁺ , we have 0 ≤ u k (x) ≤ v ⁺ (x) for x ₁ ≥ 0 and so

ku k k L

(T

) ≤ kv ⁺ k L

([0,1]

)

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for any p ∈ Z and k ∈ N. Moreover, since J p,0 (u _k ) ≤ λ, for all p ∈ Z and k ∈ N, we have

k∇u k k ² _L

_(T

₎ ≤ 2 Z

T

L(u _k )dx ≤ 2(λ + c 0 ).

Thus, setting R _L = [ −L, L] × [0, 1] ^{n −1} for L ∈ N, we obtain

ku k k ² _L

_(R

₎ + k∇u k k ² _L

_(R

₎ ≤ 2L(kv ⁺ k ² _L

_([0,1]

₎ + 2(λ + c ₀ )), ∀k ∈ N.

This proves that (u _k ) is bounded in W ^1,2 (R _L ) for any L ∈ N. A diagonal argument leads to the existence of u ∈ W _loc ^1,2 ( R × [0, 1] ^{n −1} ) such that, up to a subsequence, u _k → u weakly in W _loc ^1,2 ( R × [0, 1] ⁿ⁻¹ ) and a.e. on R × [0, 1] ⁿ⁻¹ . By semicontinuity, we have J(u) ≤ λ and by pointwise convergence u ∈

E ⁺ . 

The weak compactness of the sublevels of J on E + and its weak semicon- tinuity property allow to apply the Weierstrass Theorem to obtain:

Lemma 3.3. M ⁺ = {u ∈ E ⁺ | J(u) = c} = {u ∈ E | J(u) = c} 6= ∅.

It can be proved (see e.g. the argument in Lemma 3.3 of [3] or Lemma 5.2 of [10]) that every u ∈ M ⁺ is a classical C ² solution of (PDE) on S ₀ = R × [0, 1] ⁿ⁻¹ satisfying Neumann conditions on ∂S ₀ . Moreover, the reversibility property of F and its periodicity imply, as in Lemma 2.2-(iii), that any u ∈ M ⁺ is symmetric in x _i about x _i = 0 and x _i = 1/2 for any i ∈ {2, . . . , n}.

Then, any u ∈ M ⁺ can be periodically extended to a classical C ² solution of −∆u + F u (x, u) = 0 in the entire R ⁿ , which is odd in x ₁ , as well as even and 1–periodic in x ₂ , ..., x _n .

Lemma 3.4. Every u ∈ M ⁺ satisfies −∆u + F u (x, u) = 0 on R ⁿ , it is odd in x ₁ , even and 1-periodic in x ₂ , ..., x _n .

The elements in M ⁺ have moreover well characterized asymptotic behav- ior as x ₁ → ±∞ being asymptotic to the gap pair elements v ^± .

Lemma 3.5. If u ∈ M ⁺ , then ku − v ⁺ k W

(T

) → 0 as p → +∞.

Proof. Take u ∈ M ⁺ . Since J(u) < + ∞, we have that J p,0 (u) → 0 as

|p| → +∞. Setting u p = u( · + pe 1 ), we get 0 ≤ u p ≤ v ⁺ a.e. in [0, 1] ⁿ and I 0 (u p ) → c 0 as p → +∞. Hence, by Lemma 2.9, we find that

ku − v ⁺ k W

(T

) = ku p − v ⁺ k W

([0,1]

) → 0 as p → +∞. 

It is important for our argument to derive further compactness proper-

ties of the functional J. To this end, we make use of the non degeneracy

assumption on v ^± :

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(N 1 ) there exists ω > 0 such that for all h ∈ W ^1,2 ([0, 1] ⁿ ) I ₀ ⁰⁰ (v ⁺ )h · h =

Z

[0,1]

|∇h(x)| ² + F _u,u (x, v ⁺ (x)) |h(x)| ² dx ≥ 4ωkhk ² _L

_([0,1]

₎ . As a first consequence of (N 1 ), the following Lemma displays coerciveness properties of the functional I ₀ in small neighborhoods of v ^± .

Lemma 3.6. There exist 0 < ω 0 < ω 1 and r 1 ∈ (0, ^r ₄

) such that if u ∈ W ^1,2 ([0, 1] ⁿ ) verifies ku − v ⁺ k W

([0,1]

) ≤ r 1 , then

ω ₀ ku − v ⁺ k ² W

([0,1]

) ≤ I 0 (u) − c 0 ≤ ω 1 ku − v ⁺ k ² W

([0,1]

) . (3.1) Proof. We note that, by (N ₁ ), if h ∈ W ^1,2 ([0, 1] ⁿ ), then

Z

[0,1]

|∇h(x)| ² + F _u,u (x, v ⁺ (x)) |h(x)| ² dx ≥ 4ωkhk ² _L

_([0,1]

₎

≥ −4ωf 0

Z

[0,1]

F _u,u (x, v ⁺ (x)) |h(x)| ² dx, where f ₀ = 1/ kF uu ( ·, v ⁺ ( ·))k L

([0,1]

) , and so

Z

[0,1]

1

1 + 4ωf ₀ |∇h(x)| ² dx + Z

[0,1]

F _u,u (x, v ⁺ (x)) |h(x)| ² dx ≥ 0.

We conclude that Z

[0,1]

|∇h(x)| ² + F _u,u (x, v ⁺ (x)) |h(x)| ² dx ≥ 4ωf ₀

1 + 4ωf 0 k∇hk ² _L

_([0,1]

₎ , so that, using (N ₁ ) and setting

ω 0 = ω min {1, f 0

1 + 4ωf ₀ }, we obtain

I ₀ ⁰⁰ (v ⁺ )h · h ≥ 4ω 0 khk ² W

([0,1]

) , ∀ h ∈ W ^1,2 ([0, 1] ⁿ ).

Since by Taylor’s formula we have I ₀ (u) − c 0 = 1

2 I ₀ ⁰⁰ (v ⁺ )(u − v ⁺ ) · (u − v ⁺ ) + o( ku − v ⁺ k ² _W

_([0,1]

₎ ), we obtain that there exists r 1 ∈ (0, ^r ₄

) such that if u ∈ W ^1,2 ([0, 1] ⁿ ) verifies

ku − v ⁺ k W

([0,1]

) ≤ r 1 , then

I 0 (u) − c 0 ≥ ω 0 ku − v ⁺ k ² _W

_([0,1]

₎ . (3.2)

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On the other hand, again Taylor’s expansion gives us that I ₀ (u) − c 0 = 1

2 I ₀ ⁰⁰ (v ⁺ )(u − v ⁺ ) · (u − v ⁺ ) + o( ku − v ⁺ k ² W

([0,1]

) )

= 1

2 k∇(u − v ⁺ ) k ² _L

_([0,1]

₎ + 1 2

Z

[0,1]

F _u,u (x, v ⁺ (x)) |u(x) − v ⁺ (x) | ² dx + o( ku − v ⁺ k ² _W

_([0,1]

₎ )

≤ 1

2 k∇(u − v ⁺ ) k ² _L

_([0,1]

₎ + 1

2f ₀ ku − v ⁺ k ² _L

_([0,1]

₎ + o( ku − v ⁺ k ² _W

_([0,1]

₎ ) and we deduce that there exists ω ₁ > ω ₀ such that, taking r ₁ smaller if necessary, if u ∈ W ^1,2 ([0, 1] ⁿ ) verifies ku − v ⁺ k W

([0,1]

) ≤ r 1 , then

I ₀ (u) − c 0 ≤ ω 1 ku − v ⁺ k ² W

([0,1]

) . (3.3)

The lemma follows by (3.2) and (3.3). 

Remark 3.7. By definition of J _p,0 and the periodicity of the problem, Lemma 3.6 gives information about the behavior of the functional J p,0 nearby v ⁺ . Precisely, we have that if u ∈ W ^1,2 (T p,0 ) verifies

ku − v ⁺ k W

(T

) ≤ r 1 , for a p ∈ Z, then

ω 0 ku − v ⁺ k ² _W

_(T

₎ ≤ J p,0 (u) ≤ ω 1 ku − v ⁺ k ² _W

_(T

₎ , with ω ₀ , ω ₁ and r ₁ as in Lemma 3.6.

Remark 3.8. In connection with Remark 2.6, we have that (3.1) holds true also for the functional

I ˜ 0 (u) = Z

σ

L(u) dx − c 0 , on W ^1,2 (σ ₀ ): if ku − v ⁺ k W

(σ

) ≤ r 1 , then

ω 0 ku − v ⁺ k ² _W

_(σ

₎ ≤ ˜ I 0 (u) ≤ ω 1 ku − v ⁺ k ² _W

_(σ

₎ . (3.4) Indeed, let

I ˜ _0,+ (u) = Z

T

L(u) dx − c ₀ 2 ,

for every u ∈ W ^1,2 (T ⁺ ), where we recall that T ⁺ = {x ∈ [0, 1] ⁿ | x 1 ≤ x 2 }.

Take any h ∈ W ^1,2 (T ⁺ ) and let ˜ h be defined as in (2.1). Hence, by (N ₁ ),

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Lemma 2.4 and Remark 2.6, we have ∀h ∈ W ^1,2 (T ⁺ ), I ˜ _0,+ ⁰⁰ (v ⁺ )h · h = 1

2 I ₀ ⁰⁰ (v ⁺ )˜ h · ˜h ≥ 2ωk˜hk ² _L

_([0,1]

₎ = 4ω khk ² _L

_(T

₎ . In a similar way, letting

I ˜ _0,− (u) = Z

T

L(u) dx − c ₀ 2 , for every u ∈ W ^1,2 (T ⁻ ), where

T ⁻ = {x ∈ [−1, 0] × [0, 1] ^{n −1} | x 2 ≤ x 1 + 1 }, we obtain that

I ˜ _0, ⁰⁰ ₋ (v ⁺ )h · h ≥ 4ωkhk ² _L

_(T

) , ∀h ∈ W ^1,2 (T ⁻ ).

Finally, since

I ˜ 0 (u) = ˜ I 0,+ (u) + ˜ I 0, − (u), for all u ∈ W ^1,2 (σ ₀ ), from the above inequalities, we get

I ˜ ₀ ⁰⁰ (v ⁺ )h · h ≥ 4ωkhk ² _L

_(σ

₎ , ∀ h ∈ W ^1,2 (σ ₀ ),

from which, arguing as in the previous lemma and using Remark 2.6, we deduce (3.4).

We now study uniform concentration properties of the elements in E ⁺ on which J is sufficiently small. To this aim, using Lemma 2.9, we fix λ ₀ > 0 such that

if u ∈ E ⁺ , p ≥ 0 are such that J p,0 (u) ≤ λ 0 , then ku − v ⁺ k W

(T

) ≤ r 1 , (3.5) where r ₁ is given in Lemma 3.6. Then, we have

Lemma 3.9. For any ε > 0 there exist λ(ε) ∈ (0, ^λ ₂

) and p(ε) ∈ N for which if u ∈ E ⁺ is such that J(u) ≤ c + λ(ε), then

X

p ≥p(ε)

J _p,0 (u) ≤ ε and ku − v ⁺ k W

(T

) ≤ r 1 , for all p ≥ p(ε).

Proof. Given r > 0, we set

Λ(r) = sup {J p,0 (u) | u ∈ E ⁺ , ku − v ⁺ k W

(T

) ≤ 2r},

noting that Λ(r) is not decreasing for r > 0 and Λ(r) → 0 as r → 0 ⁺ . We fix r ₂ ∈ (0, r 1 ) such that

Λ(r) < λ ₀

2 for r ∈ (0, r 2 ].

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We claim that if r ∈ (0, r 2 ), J(u) ≤ c + Λ(r) and ku − v ⁺ k W

(T

) ≤ r for some p ₀ ≥ 0, then

P

p ≥p

J _p,0 (u) ≤ 2Λ(r) and ku − v ⁺ k W

(T

) ≤ r 1 , for all p ≥ p 0 . (3.6) To prove (3.6), let u ∈ E ⁺ and p ₀ ≥ 0 be such that J(u) ≤ c + Λ(r) and ku − v ⁺ k W

(T

) ≤ r. We define for x 1 ≥ 0 and y ∈ [0, 1] ⁿ⁻¹

˜

u(x ₁ , y) =

 



 

u(x ₁ , y) if x ₁ ∈ [0, p 0 ],

u(x ₁ , y)(p ₀ + 1 − x 1 )+ v ⁺ (x ₁ , y)(x ₁ − p 0 ) if x ₁ ∈ (p 0 , p ₀ + 1),

v ⁺ (x ₁ , y) if x ₁ ∈ [p 0 + 1, + ∞),

and consider the odd extension for x 1 ≤ 0. Noting that ˜u ∈ E ⁺ , we obtain c ≤ J(˜u) = J(u) − 2

+∞ X

p=p

J p,0 (u) + 2J p

,0 (˜ u).

By definition

˜

u(x ₁ , y) − v ⁺ (x ₁ , y) = (p ₀ + 1 − x 1 )(u(x ₁ , y) − v ⁺ (x ₁ , y)), on T p

,0 and so

k˜u − v ⁺ k W

(T

) ≤ 2ku − v ⁺ k W

(T

) ≤ 2r.

Then, by definition of Λ(r), we obtain J _p

_,0 (˜ u) ≤ Λ(r) and hence c ≤ J(u) − 2

X +∞

p=p

J p,0 (u) + 2Λ(r) ≤ c − 2

+∞ X

p=p

J p,0 (u) + 3Λ(r) from which

+ ∞

X

p=p

J p,0 (u) ≤ 2Λ(r).

In particular, J p,0 (u) ≤ 2Λ(r) for all p ≥ p 0 and since Λ(r) ≤ ^λ ₂

, by (3.5), we have ku − v ⁺ k W

(T

) ≤ r 1 for all p ≥ p 0 , and (3.6) is proved.

It is now simple to obtain our statement. Let ε > 0 and r ε ∈ (0, r 2 ) be such that 2Λ(r _ε ) ≤ ε. By Lemma 2.2-(i), there exists p(ε) ∈ N such that if ku − v ⁺ k W

(T

) > r ε for 0 ≤ p ≤ p(ε), then J(u) > c + Λ(r ε ). Then, if u ∈ E ⁺ is such that J(u) ≤ c + Λ(r ε ), there exists p ₀ ∈ [0, p(ε)] for which ku − v ⁺ k W

(T

) ≤ r ε . Then (3.6) implies that

+ ∞

X

p=p(ε)

J _p,0 (u) ≤ 2Λ(r ε ) ≤ ε,

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and ku − v ⁺ k W

(T

) ≤ r 1 for all p ≥ p(ε), concluding the proof of the

Lemma. 

The concentration property described in Lemma 3.9 is sufficient to obtain the following compactness property of J.

Lemma 3.10. Let (u _k ) ⊂ E ⁺ be such that J(u _k ) → c. Then, there exists u ∈ M ⁺ such that, up to a subsequence, ku k − uk W

(S

) → 0 as k → +∞, where S ₀ = R × [0, 1] ⁿ⁻¹ .

Proof. Let (u _k ) be as in the statement. By Lemma 3.2 there exist a subse- quence of (u _k ), still denoted (u _k ), and u ∈ M ⁺ such that u _k → u weakly in E. We claim that

J _p,0 (u _k ) → J p,0 (u) as k → +∞ for all p ∈ Z. (3.7) Indeed, by semicontinuity, J p,0 (u) ≤ lim inf J p,0 (u _k ) for all p ∈ Z. By con- tradiction, assume that there exists p ₀ ∈ Z such that

lim sup(J _p

_,0 (u _k ) − J p

,0 (u)) = ε > 0.

In this case there exists a subsequence (u _k

) of (u _k ) such that J _p

_,0 (u _k

) − J p

,0 (u) → ε as j → +∞.

Since X

|p|≥q

(J p,0 (u _k ) − J p,0 (u)) ≥ − X

|p|≥q

J p,0 (u) → 0 as q → +∞, there exists q 0 > |p 0 | such that

X

|p|≥q

(J _p,0 (u _k ) − J p,0 (u)) ≥ − ε 2 . Then, setting

b _k = J(u _k ) − J(u) = X

p ∈Z

(J p,0 (u _k ) − J p,0 (u)), we have b _k → 0 as k → +∞ (since J(u k ) → c = J(u)), while

b _k

= J _p

_,0 (u _k

) − J p

,0 (u) + X

|p|<q

,p6=p

(J _p,0 (u _n

) − J p,0 (u))

+ X

|p|≥q

(J p,0 (u _k

) − J p,0 (u))

≥ J p

,0 (u _k

) − J p

,0 (u) + X

|p|<q

,p 6=p

(J _p,0 (u _k

) − J p,0 (u)) − ε

2 .

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Moreover, lim inf(J p,0 (u _k ) −J p,0 (u)) ≥ 0 for all p ∈ Z, and hence lim inf b k

≥

ε

2 , a contradiction which implies (3.7).

Note now that, since J _p,0 (u _k ) → J p,0 (u) and since k∇u k k ² L

(T

) − k∇uk ² L

(T

)

= 2(J _p,0 (u _k ) − J p,0 (u)) − 2 Z

T

F (x, u _k ) − F (x, u) dx, by semicontinuity, we obtain that for any p ∈ Z

0 ≤ lim inf

k→+∞

 k∇u k k ² _L

_(T

₎ − k∇uk ² _L

_(T

₎ 

≤ lim sup

k→+∞

 k∇u k k ² _L

_(T

₎ − k∇uk ² _L

_(T

₎ 

= −2 lim inf

k →+∞

Z

T

F (x, u _k ) − F (x, u) dx ≤ 0.

Since we already know that u _k → u weakly in E, we deduce that

ku k − uk W

(T

) → 0 as n → +∞ for any p ∈ Z. (3.8) To conclude the proof of the Lemma it is then sufficient to show that

∀ε > 0 ∃ k(ε), p(ε) ∈ N such that X

|p|≥p(ε)

ku k − uk ² _W

_(T

₎ ≤ ε for k ≥ k(ε).

(3.9) To this end observe that, by Lemma 3.9, for all ε > 0 there exist λ(ε) > 0 and p(ε) ∈ N for which if ¯u ∈ E ⁺ is such that J(¯ u) ≤ c + λ(ε), then

X

p ≥p(ε)

J _p,0 (¯ u) ≤ ε and k¯u − v ⁺ k W

(T

) ≤ r 1 , ∀p ≥ p(ε). (3.10) By the fact that J(u _k ) → J(u) = c, we derive the existence of k(ε) ∈ N such that (3.10) is satisfied with ¯ u = u or ¯ u = u _k whenever k ≥ k(ε). Then, by Remark 3.7 and (3.10), we deduce that for k ≥ k(ε), we have

X

p ≥p(ε)

ku k − v ⁺ k ² _W

_(T

₎ ≤ ε

ω ₀ and X

p ≥p(ε)

ku − v ⁺ k ² _W

_(T

₎ ≤ ε ω ₀ . Hence,

X

p ≥p()

ku k − uk ² _W

_(T

₎ ≤ 2ε ω 0

,

for k ≥ k(ε) and by symmetry (3.9) follows. 

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4. Saddle type solutions

In this section, we prove our main theorem, constructing saddle type solu- tions. To this aim, we first study an auxiliary problem proving the existence of a solution u of (PDE) on the triangular set

T = {x ∈ R ⁿ | x 2 ≥ |x 1 |},

which satisfies Neumann conditions on ∂ T , is odd in the variable x 1 , positive for x ₁ > 0 and asymptotic to M + as x ₂ → +∞. The saddle solution will be then recovered by recursive reflections of u about the faces of T .

To study the auxiliary problem, we make use of a doubly renormalized variational approach inspired, even if technically different, by the one intro- duced in [4].

First, we look at the set T as the union of the strips T _k = {x ∈ T | k ≤ x 2 ≤ k + 1}, k ∈ Z.

For any k ≥ 0, the domain T k is itself considered decomposed in the union of the squares

T _p,k = [p, p + 1] × [k, k + 1] × [0, 1] ⁿ⁻² = T _p,0 + ke ₂ , p = −k, . . . , k − 1, and the ending set

τ _k = {x ∈ T k,k ∪ T −k−1,k | |x 1 | ≤ x 2 }.

SADDLE TYPE SOLUTIONS FOR REVERSIBLE ELLIPTIC EQUATIONS 15

For any k 0, the domain T k is itself considered decomposed in the union of the squares T p,k = [p, p + 1] ⇥ [k, k + 1] ⇥ [0, 1] ^{n 2} = T p,0 + ke 2 , p = k, . . . , k 1

and the ending set

⌧ k = {x 2 T k,k [ T k 1,k | |x 1 |  x 2 }.

k+1 k

k k+1

⌧

x

x

T

k k 1 T

T

k+1 k

p p+1 x

x

T

Figure 1. The decomposition of the triangular set T For any k 2 N [ {0} we consider the space

E k = {u 2 W ^1,2 (T k ) | u odd in x 1 , 0  u(x)  v ⁺ (x) for x 1 0 }, and, setting J p,k (u) = R

T

L(u) dx c 0 , the renormalized functional J k (u) =

k 1 X

p= k

J p,k (u) + Z

⌧

L(u) dx c 0 , u 2 E k .

Note that J 0 (u) = R

⌧

L(u) dx c 0 and that for k 2 N

(4.1) J k (u) =

Z

T

L(u) dx (2k + 1)c 0 .

Remark 4.1. As first important property we note that J k (u) 0 for all u 2 E k and k 0.

Indeed, if u 2 E 0 by symmetry and Lemma 2.5, we have that J 0 (u) = R

⌧

L(u) dx c 0 dx = 2 R

T

L(u) dx c 0 0. If k 1 and u 2 E k , Lemma 2.2-(iii) and the periodicity of F give that J p,k (u) = I 0 (u(· + pe 1 + ke 2 )) c 0 0 for any p 2 [ k, k 1]. Moreover setting

⌧ _k ⁺ = {x 2 ⌧ k | x 1 > 0 }, we have by symmetry that R

⌧

L(u) dx c 0 = 2 R

⌧

L(u) dx c 0

and since by periodicity R

⌧

L(u) dx = R

T

L(u( · + k(e 1 + e 2 )) dx, again by Lemma 2.5 we derive R

⌧

L(u) dx c 0 0.

Note that the functional J k is lower semicontinuous with respect to the weak W ^1,2 (T k ) topology for every k 2 N [ {0}. Moreover, setting

c k = inf

E

J k (u) and M k = {u 2 E k | J k (u) = c k },

we plainly obtain that M k 6= ;, c k  c k+1 < c for any k 2 N [ {0} and that c k ! c as k ! +1. Thanks to the non degeneracy assumption (N 1 ) we can say more.

Figure 1. The decomposition of the triangular set T For any k ∈ N ∪ {0}, we consider the space

E _k = {u ∈ W ^1,2 (T _k ) | u odd in x 1 , 0 ≤ u(x) ≤ v ⁺ (x) for x ₁ ≥ 0},

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and, setting

J _p,k (u) = Z

T

L(u) dx − c 0 , the renormalized functional

J _k (u) =

k −1

X

p= −k

J _p,k (u) + Z

τ

L(u) dx − c 0 , u ∈ E k . Note that

J ₀ (u) = Z

τ

L(u) dx − c 0

and that for k ∈ N

J _k (u) = Z

T

L(u) dx − (2k + 1)c 0 . (4.1) Remark 4.1. As first important property, we note that J _k (u) ≥ 0 for all u ∈ E k and k ≥ 0. Indeed, if u ∈ E 0 by symmetry and Lemma 2.5, we have that

J 0 (u) = Z

τ

L(u) dx − c 0 dx = 2 Z

T

L(u) dx − c 0 ≥ 0.

If k ≥ 1 and u ∈ E k , Lemma 2.2-(iii) and the periodicity of F give that J _p,k (u) = I ₀ (u( · + pe 1 + ke ₂ )) − c 0 ≥ 0,

for any p ∈ [−k, k − 1]. Moreover, setting

τ _k ⁺ = {x ∈ τ k | x 1 > 0 }, we have by symmetry that

Z

τ

L(u) dx − c 0 = 2 Z

τ

L(u) dx − c 0 , and since by periodicity

Z

τ

L(u) dx = Z

T

L(u( · + k(e 1 + e ₂ )) dx, again by Lemma 2.5, we derive

Z

τ

L(u) dx − c 0 ≥ 0.

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Note that the functional J _k is lower semicontinuous with respect to the weak W ^1,2 (T _k ) topology for every k ∈ N ∪ {0}. Moreover, setting

c _k = inf

E

J _k (u) and M k = {u ∈ E k | J k (u) = c _k },

we plainly obtain that M k 6= ∅, c k ≤ c k+1 < c for any k ∈ N ∪ {0} and that c _k → c as k → +∞. Thanks to the non degeneracy assumption (N 1 ), we can say more.

Lemma 4.2. There results P _∞

k=1 (c − c k ) < + ∞.

Proof. Fixed k ∈ N, let u k ∈ M k . Noting that u _k ( · + e 2 ) | T

∈ E k −1 , we obtain

c _{k −1} ≤ J k −1 (u _k ( · + e 2 ))

= 2

k −2

X

p=0

J _{p,k −1} (u _k ( · + e 2 )) + Z

τ

L(u _k ( · + e 2 )) dx − c 0

= 2

k −2

X

p=0

J _p,k (u _k ) + Z

τ

+e

L(u _k ) dx − c 0

= J _k (u _k ) − 2 Z

σ

L(u _k ) dx + 2c ₀ = c _k − 2  Z

σ

L(u _k ) dx − c 0

 ,

where for all k ∈ N

σ _k = {x ∈ R × [k, k + 1] × [0, 1] ⁿ⁻² | x 2 − 1 ≤ x 1 ≤ x 2 } = σ 0 + k(e ₁ + e ₂ ).

Setting

I ˜ _k (u) = Z

σ

L(u) dx − c 0 ,

the above inequality says that c _k ≥ c k −1 + 2 ˜ I _k (u _k ) and so, we deduce c ≥ c k ≥ c k −1 + 2 ˜ I _k (u _k ) ≥ ... ≥ c 1 + 2

X k h=2

I ˜ _h (u _h ).

Since σ _k = σ ₀ + k(e ₁ + e ₂ ), by periodicity and Remark 2.6, we have I ˜ _k (u) =

Z

σ

L(u(x + k(e ₁ + e ₂ )) dx − c 0 ≥ 0, for any u ∈ W ^1,2 (σ _k ), and we can conclude that

+ ∞

X

h=2

I ˜ _h (u _h ) < + ∞.

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In particular, ˜ I _k (u _k ) → 0 as k → +∞ and hence, using again Remark 2.6 and arguing as in Lemma 2.9, we obtain that there exists k ₀ ∈ N such that for all k ≥ k 0 , we have ku k − v ⁺ k W

(σ

) ≤ ^r ₄

, where r 1 is given by Lemma 3.6.

For k ≥ k 0 + 1, x ₁ ≥ 0, x 2 ∈ [k, k + 1] and y ∈ [0, 1] ⁿ⁻² , let

˜

u _k (x 1 , x 2 , y) =

 

 



 

 

u _k (x ₁ , x ₂ , y) if 0 ≤ x 1 ≤ x 2 − 1, u _k (x 1 , x 2 , y)(x 2 − x 1 )+

+v ⁺ (x ₁ , x ₂ , y)(x ₁ − x 2 + 1) if x ₂ − 1 ≤ x 1 ≤ x 2 , v ⁺ (x ₁ , x ₂ , y) if x ₂ ≤ x 1 ,

and consider the odd extension for x ₁ < 0. As noted above, since k ≥ k 0 , we have that ku k − v ⁺ k W

(σ

) ≤ ^r ₄

and hence, since

k˜u k − v ⁺ k ² _W

_(σ

₎ ≤ 2ku k − v ⁺ k ² _W

_(σ

₎ , we have also k˜u k − v ⁺ k W

(σ

) ≤ r 1 . Then, by (3.4), we have

I ˜ _k (˜ u _k ) ≤ ω 1 k˜u k − v ⁺ k ² W

(σ

) ≤ 2ω 1 ku k − v ⁺ k ² W

(σ

) ≤ 2ω ₁ ω ₀ I ˜ _k (u _k ) from which we can conclude that also

+ ∞

X

h=2

I ˜ _h (˜ u _h ) < + ∞, being

X +∞

h=2

I ˜ _h (u _h ) < + ∞.

Then, since ˜ u _k ( · + ke 2 ) ∈ E, we finally obtain

c ≤ J(˜u k ( · + ke 2 )) = J _k (˜ u _k ) = c _k + 2 ˜ I _k (˜ u _k ) − 2 ˜ I _k (u _k ) ≤ c k + 2 ˜ I _k (˜ u _k ), so that 0 ≤ c − c k ≤ 2 ˜ I _k (˜ u _k ) and the Lemma follows. 

Using the renormalized functionals J _k , we now define on the function space

E = {u ∈ W _loc ^1,2 ( T ) | u odd in x 1 , 0 ≤ u(x) ≤ v ⁺ (x) for x ₁ ≥ 0}, the doubly renormalized functional

J (u) =

+ ∞

X

k=0

(J _k (u) − c k ).

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Note that J is non negative on E since if u ∈ E, then u _|

∈ E k and so J _k (u) − c k ≥ 0 for any k ∈ N ∪ {0}. One also recognizes that J is weakly lower semicontinuous with respect to the W _loc ^1,2 ( T ) topology. Thanks to Lemma 4.2, we can easily show that J is finite on M ⁺ .

Lemma 4.3. If u ∈ M ⁺ , then J (u) < +∞.

Proof. If u ∈ M ⁺ , then u | T ∈ E and for every k ∈ N, we have J k (u) − c k ≤ J(u) − c k = c − c k . By Lemma 4.2, we obtain

J (u) = X

k≥0

(J _k (u) − c k ) ≤ X

k≥0

(c − c k ) < + ∞. 

We look for a minimum of J on E and we set m = inf

E J and K = {u ∈ E | J (u) = m}.

We can apply standard arguments of the Calculus of Variations to show Proposition 4.4. There exists u ₀ ∈ E such that J (u 0 ) = m, with 0 ≤ u 0 ≤ v ⁺ in T for x 1 ≥ 0.

Proof. Let (u j ) ⊂ E be a minimizing sequence for J . Fixed r ∈ N and denoted T r = T ∩ {x 2 ≤ r}, we have that (u j ) is bounded in W ^1,2 ( T r ).

Indeed, since 0 ≤ u j (x) ≤ v ⁺ (x) a.e. in T with x 1 ≥ 0, we have ku j k L

( T

) ≤ kv ⁺ k L

( T

) for every j ∈ N. Moreover, for each k ∈ N, by (4.1)

k∇u j k ² _L

_(T

₎ ≤ 2J k (u _j ) + 2(2k + 1)c ₀ , and so

k∇u j k ² L

( T

) =

r −1

X

k=0

k∇u j k ² L

(T

) ≤ 2

r −1

X

k=0

{J k (u _j ) + (2k + 1)c ₀ }

= 2

r−1 X

k=0

(J _k (u _j ) − c k ) + 2 X r−1 k=0

{(2k + 1)c 0 + c _k }

≤ 2J (u j ) + 2r((2r − 1)c 0 + c).

Then, by a diagonal argument, there exist u ₀ ∈ W _loc ^1,2 ( T ) and a subsequence of (u _j ), still denoted by (u _j ), which converges weakly to u ₀ in W _loc ^1,2 ( T ) and a.e. in T . By the pointwise convergence, we recover that u 0 ∈ E and by the weak lower semicontinuity of J , we obtain J (u 0 ) = m. This concludes the

proof. 

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By Proposition 4.4, we have that K 6= ∅. Applying the argument of Lemma 3.1, one shows that K is in fact the minimal set of J on the larger set X = {u ∈ W _loc ^1,2 ( T ) | u is odd in x 1 }, showing that the constraint condition defining E, 0 ≤ u ≤ v ⁺ on T when x 1 ≥ 0, is natural in the minimal problem. Then, slightly varying the argument in the proof of Lemma 3.3 of [3], or Lemma 5.2 of [10], we obtain that any u ∈ K verifies

Z

T ∇u(x) · ∇ψ(x) + F u (x, u(x))ψ(x) dx = 0 for any ψ ∈ C 0 ^∞ ( R ⁿ ), (4.2) from which in particular, we obtain that any u ∈ K is a weak solution of

−∆u + F u (x, u) = 0 on T ,

which is odd in x ₁ and such that 0 ≤ u(x) ≤ v ⁺ (x) on T for x 1 ≥ 0. Again by reversibility and periodicity, we also obtain that u is 1-periodic in the variables x 3 , ..., x n .

By recursive reflection of the function u ₀ : T → R given by Proposition 4.4 it is possible to construct a function w : R ⁿ → R solution to (PDE) satisfying the asymptotic condition (S), concluding the proof of Theorem 1.1.

To this aim, setting T 0 = T , we consider the rotation matrix

A =

0 1 0 ... 0

−1 0 0 ... 0 0 0 1 ... 0 ... ... ... ... ...

0 0 0 ... 1 ,

and for k = 0, ..., 3, we denote T k = A ^k T . Note that we have R ⁿ = ∪ ³ _k=0 T k , and that if k ₁ 6= k 2 , then int( T k

) ∩ int(T k

) = ∅. By definition A ^−k T k = T and we define

w(x) = ( −1) ^k u 0 (A ^−k x), ∀x ∈ T k . (4.3) Since u ₀ ∈ W _loc ^1,2 ( T 0 ), we have that w ∈ W _loc ^1,2 ( R ⁿ ). Moreover, if ψ ∈ C ₀ ^∞ ( R ⁿ ) and k ∈ {0, . . . , 3}, then trivially, ψ ◦ A ^k ∈ C ₀ ^∞ ( R ⁿ ) while, by (F ₂ ) (F 3) and the definition of w, we have

F _u (A ^k x, w(A ^k x)) = ( −1) ^k F _u (x, u ₀ (x)), for any x ∈ T 0 . Thus, putting y = A ^k x, by (4.2), we obtain

Z

T

∇w(y) · ∇ψ(y) + F u (y, w(y))ψ(y) dy

= ( −1) ^k Z

T

∇u 0 (x) · ∇(ψ ◦ A ^k )(x) + F _u (x, ¯ u(x))(ψ ◦ A ^k )(x) dx = 0.