SADDLE SOLUTIONS FOR A CLASS OF SYSTEMS OF PERIODIC AND REVERSIBLE SEMILINEAR ELLIPTIC EQUATIONS

. We study systems of elliptic equations − ∆ u ( x )+ F u ( x,u ) = 0 with potentials F ∈ C 2 ( R n , R m ) which are periodic and even in all their variables. We show that if F ( x,u ) has ﬂip symmetry with respect to two of the compo- nents of x and if the minimal periodic solutions are not degenerate then the system has saddle type solutions on R n .

1. Introduction. We consider systems of semilinear elliptic equations − ∆u(x) + F u (x, u) = 0 (PDE) where (F 1 ) F ∈ C 2 (R n × R m ; R) is 1-periodic in all its variables, n, m ≥ 1. When n = 1 and m ≥ 1, (PDE) are particular cases of the dynamical systems considered in the Aubry-Mather Theory ( [9,23,24]). When n > 1 and m = 1 equations like (PDE) were studied by Moser in [25] (indeed in a much more general setting), and then by Bangert [13] and Rabinowitz and Stredulinsky [31], extending some of the results of the Aubry-Mather Theory for partial differential equations. These studies show the presence of a very rich structure of the set of minimal (or locally minimal) entire solutions of (PDE). In particular, when m = 1 the set M 0 of minimal periodic solutions of (PDE) is a non empty ordered set and if M 0 is not a continuum then there exists another ordered family M 1 of minimal entire solutions which are heteroclinic in one space variable to a couple of (extremal) periodic solutions u < v (a gap pair in M 0 ). If M 1 is not a continuum the argument can be iterated to find more complex ordered classes of minimal heteroclinic type solutions and the process continues if the corresponding set of minimal heteroclinics contains gaps. Variational gluing arguments were then employed by Rabinowitz and Stredulinsky to construct various kinds of homoclinic, heteroclinic or more generally multitransition solutions as local minima of renormalized functionals associated to (PDE), see [31]. Other extensions of Moser's results, including changing slope or higher Morse index solutions, have been developed by Bessi [10,11], Bolotin and Rabinowitz [12], de la Llave and Valdinoci [17,33]. Recently, in a symmetric setting and correspondingly to the presence of a gap pair in M 0 symmetric with respect to the origin, entire solutions of saddle type were found by Autuori, Alessio and Montecchiari in [2].
All the above results are based on the ordered structure of the set of minimal solutions of (PDE) in the case m = 1 and a key tool in their proofs is the Maximum Principle, which is no longer available when m > 1.
The study of (PDE) when n, m > 1 was initiated by Rabinowitz in [29,30]. Denoting L(u) = 1 2 |∇u| 2 + F (x, u) and T n = R n /Z n , periodic solutions to (PDE) were found as minima of the functional J 0 (u) = T n L(u)dx on E 0 = W 1,2 (T n , R m ) showing that Paul H. Rabinowitz studied the case of spatially reversible potentials F assuming (F 2 ) F is even in x i for 1 ≤ i ≤ n and proved in [29] that if M 0 is constituted by isolated points then for each v − ∈ M 0 there is a v + ∈ M 0 \ {v − } and a solution u ∈ C 2 (R × T n−1 , R m ) of (PDE) that is heteroclinic in x 1 from v − to v + . These solutions were found by variational methods minimizing the renormalized functional (where T p,0 = [p, p + 1] × [0, 1] n−1 ) on the space Γ(v − , v + ) = {u ∈ W 1,2 (R × T n−1 , R m ) | u − v ± L 2 (Tp,0,R m ) → 0 as p → ±∞}.
In [30] the existence of minimal double heteroclinics was obtained assuming that the elements of M 0 are not degenerate critical points of J 0 and that the set M 1 (v − , v + ) of the minima of J on Γ(v − , v + ) is constituted by isolated points. This research line was continued by Montecchiari and Rabinowitz in [26] where, via variational methods, multitransition solutions of (PDE) were found by glueing different integer phase shifts of minimal heteroclinic connections. The proof of these results does not use the ordering property of the solutions and adapts to the study of (PDE) some of the ideas developed to obtain multi-transition solutions for Hamiltonian systems (see e.g. [3], [28] and the references therein). Aim of the present paper is to show how these methods, in particular a refined study of the concentrating properties of the minimal heteroclinic solutions to (PDE), can be used in a symmetric setting to obtain saddle type solutions to (PDE).
Saddle solutions were first studied by Dang, Fife and Peletier in [16]. In that paper the authors considered Allen-Cahn equations −∆u + W (u) = 0 on R 2 with W an even double well potential. They proved the existence of a (unique) saddle solution v ∈ C 2 (R 2 ) of that equation, i.e., a bounded entire solution having the same sign and symmetry of the product function x 1 x 2 and being asymptotic to the minima of the potential W along any directions not parallel to the coordinate axes. The saddle solution can be seen as a phase transition with cross interface.
We refer to [14,15,6,7,27] for the study of saddle solutions in higher dimensions and to [1,20,8] for the case of systems of autonomous Allen-Cahn equations. Saddle solutions can be moreover viewed as particular k-end solutions (see [4,18,22,19]).
In [5] the existence of saddle type solutions was studied for non autonomous Allen-Cahn type equations and this work motivated the paper [2] where solutions of saddle type for (PDE) were found in the case m = 1.
In the present paper we generalize the setting considered in [2] to the case m > 1. Indeed we consider to have potentials F satisfying (F 1 ) and the symmetry properties (F 2 ) F is even in all its variables; (F 3 ) F has flip symmetry with respect to the first two variables, i.e., By [29] the set M 0 of minimal periodic solution of (PDE) is not empty. The symmetry of F implies that any v ∈ M 0 has components whose sign is constant on R n and if v ∈ M 0 then (ν 1 v 1 , . . . , ν m v m ) ∈ M 0 for every (ν 1 , . . . , ν m ) ∈ {±1} m (see Lemma 2.2 below). In this sense we can say that M 0 is symmetric with respect to the constant function v 0 ≡ 0.
As recalled above, in [2], where m = 1, a saddle solution was found when M 0 has a gap pair symmetric with respect to the origin. In the case m > 1 we generalize this gap condition asking that 0 / ∈ M 0 and, following [30], we look for saddle solutions of (PDE) when any v ∈ M 0 is not degenerate for J 0 . We then assume (N ) 0 / ∈ M 0 and there exists α 0 > 0 such that for every h ∈ W 1,2 ([0, 1] n , R m ) and every v ∈ M 0 .
The assumption (N ) and the symmetries of F allow us to find heteroclinic connections between elements of M 0 which are odd in the variable x 1 . More precisely for v ∈ M 0 these solutions are searched as minima of the functional J (see (1)) on the space Our main result can now be stated as follows and (N ). Then, there exists a classical solution w of (PDE) such that every component w i (for i = 1 . . . , m) satisfies Moreover there exists v 0 ∈ M min 0 such that the solution w satisfies the asymptotic condition where Note that by (i) and (ii) any component of w has the same sign as the product function x 1 x 2 . Moreover by (2), since w is asymptotic as x 2 → +∞ to the compact set M(v 0 ) of odd heteroclinic type solutions, the symmetry of w implies that w is asymptotic to v 0 or −v 0 along any direction not parallel to the planes x 1 = 0, x 2 = 0. In this sense w is a saddle solution, representing a multiple transition between the pure phases v 0 and −v 0 with cross interface.
The proof of Theorem 1.1 uses a variational approach similar to the one already used in previous papers like [5,2]. To adapt this approach to the case m > 1 and so to avoid the use of the Maximum Principle we need a refined analysis of the concentrating properties of the minimizing sequences. For that a series of preliminaries results is given in §2, §3, §4 while the proof of Theorem 1.1 is developed in §5.

2.
The periodic solutions. In this section we recall some results obtained by Rabinowitz in [29], on minimal periodic solutions to (PDE). Moreover, following the argument in [2], we study some symmetry properties related to the assumptions (F 2 ) and (F 3 ). Here and in the following we will work under the not restrictive assumption Let us introduce the set We define the functional J 0 : E 0 → R as and consider the minimizing set Then in [29], [30] it is shown , in particular the even parity of F with respect to the components of u, provides that the elements in M 0 have components with definite sign, thanks to the unique extension property (see [29], Proposition 3).
Proof. It is sufficient to observe that if u = (u 1 , . . . , u m ) ∈ M 0 then, since F is even with respect to the components of u, we have i)ū = (|u 1 |, . . . , |u m |) ∈ M 0 and ii) (ν 1 u 1 , . . . , ν m u m ) ∈ M 0 for every (ν 1 , . . . , ν m ) ∈ {±1} m . Property (ii) gives the second part of the statement while by (i) and the unique extension property proved in [29], we obtain that the components of u do not change sign. If u vanishes on an open set, the unique continuation property gives u ≡ 0, giving a contradiction and concluding the proof.
On the other hand, assumption (F 3 ) gives more structure on the set M 0 : its elements have a flip symmetry property. Indeed, setting Then, we have Proof. Given u ∈ M 0 , without loss of generality, we assume Sinceũ ∈ W 1,2 ([0, 1] n , R m ) by Lemma 2.1-(5) we have J 0 (ũ) ≥ c 0 . By the previous inequality we get
As an immediate consequence, using Lemma 2.1-(5), we have the following.
3 tells us that the elements of M 0 are symmetric with respect to the diagonal iperplane {x ∈ R n | x 1 = x 2 } and by Lemma 2.4 they can be found by minimizing Note that by Lemma 2.1-(1) and the assumption (N) we plainly derive that Note finally that by (N 0 ), setting we have r 0 > 0.
3. The variational setting for Heteroclinic connections. This section is devoted to introduce the variational framework to study solutions of (PDE) which are heteroclinic between minimal periodic solutions. We follow some arguments in [29], [26], introducing the renormalized functional J and studying some of its basic properties.
Let us define the set Denoting briefly u(· + p) the shifting of the function u with respect to the first coordinate (that is, u(· + p) = u(· + p e 1 ) where e 1 = (1, 0, ..., 0)), note that by periodicity we have Then, by Lemma 2.1, we have J p,0 (u) ≥ 0 for any u ∈ E and p ∈ Z, from which J is non-negative on E. Proof. Consider a sequence (u k ) k such that u k → u weakly in E. Then, for every ∈ N, by the weak lower semicontinuity of J 0 , and hence of J p,0 , we have If J(u) = +∞, then we obtain easily lim inf k J(u k ) = +∞. So, let us assume J(u) < +∞, then for any ε > 0 we have that there exists ∈ N such that |p|> J p,0 (u) < ε. We get thus finishing the proof.
Using the notation introduced above, note that if u ∈ E is such that J(u) < +∞, then J p,0 (u) → 0 as |p| → +∞, that is, the sequence (u(· + p)) p∈Z is such that J 0 (u(· + p)) → c 0 as p → ±∞. Hence, by Lemma 2.1-(3), there exist u ± ∈ M 0 such that, up to a subsequence, u(· + p) → u ± as p → ±∞ in E 0 . Using this remark and the local compactness of M 0 given by (N 0 ), we are going to prove some concentration properties of the minimizing sequence of the functional J.
First of all, let us consider the functional J p,0 + J p+1,0 for a certain fixed integer p. Notice that, by Lemma 2.1-(5), min u∈E J p,0 (u) + J p+1,0 (u) = 0 and the set of minima coincide with M 0 . We introduce the following distance Remark 2. Let us fix some constants that will be used in rest of the paper. By Lemma 2.1-(3), we have that for any r > 0 there exists λ(r) > 0 such that if u ∈ E satisfies J p,0 (u) + J p+1,0 (u) ≤ λ(r) for a p ∈ Z, then dist p (u, M 0 ) ≤ r.
(7) It is not restrictive to assume that the function with r → λ(r) is non-decreasing.
By the previous lemmas we obtain that the elements in the sublevels of J satisfy the following boundeness property.
Proof. Let u ∈ E be such that J(u) ≤ Λ. We define J (u) = {k ∈ Z | J k,0 (u) ≥ λ0 2 } and note that the number l(u) of elements of J (u) is at most [ 2Λ λ0 ] + 1, where [·] denotes the integer part. Then, the set Z \ J (u) is constituted byl(u) sets of consecutive elements of Z, I i (u), withl(u) ≤ l(u) + 1. By the triangular inequality, for any p, q ∈ Z, we obtain where the first term in (9) The following lemma states the weak compactness of the sublevels of the functional J.
Lemma 3.5. Given any Λ > 0, let (u k ) k ⊂ E be a sequence such that J(u k ) ≤ Λ for every k ∈ N and let (p k ) k be a sequence of integers. Assume that there exist R < +∞ and v ∈ M 0 such that u k − v W 1,2 (Tp k ,0,R m ) ≤R for all k ∈ N. Then, there exists u ∈ E with J(u) ≤ Λ such that, up to a subsequence, u k → u weakly in E.
Proof. First note that, by Lemma 3.4, there exists R > 0 such that if u ∈ E and J(u) ≤ Λ then u(· + p) − u(· + q) L 2 ([0,1] n ,R m ) ≤ R for any p, q ∈ Z. If u − v W 1,2 (T ,0 ,R m ) ≤R for some ∈ Z and v ∈ M 0 , by triangular inequality for any p ∈ Z we obtain Consider now a sequence as in the statement, setting for any L ∈ N and, by a diagonal argument and the weak lower semicontinuity of J, the statement follows.
By Lemma 3.2 we also deduce the following result concerning the asymptotic behaviour of the functions in the sublevels of J.
Hence the sequence (u(·+p)) p∈N is such that u(·+p)−v + W 1,2 ([0,1] n ,R m ) ≤ r0 4 for every p ≥p and J 0 (u(· + p)) − c 0 = J p,0 (u) → 0 as p → +∞. Then, by Lemma 2.1, By Lemma 3.6, if u ∈ E satisfies J(u) < +∞ we can view it as an heteroclinic or homoclinic connection between two periodic solutions v − and v + belonging to M 0 . Hence, we can consider elements of E belonging to the classes where v ± ∈ M 0 . We note that by Lemma 3.5, every sequence (u k ) k∈N ⊂ Γ(v − , v + ) with J(u k ) ≤ Λ for all k ∈ N, admits a subsequence which converges weakly to some u ∈ E. Indeed, since u k − v + W 1,2 (Tp,0,R m ) → 0 as p → +∞ for every k ∈ N, fixedR > 0 there exists p k ∈ N such that u k − v + W 1,2 (Tp k ,0 ,R m ) ≤R and since J(u k ) ≤ Λ, by Lemma 3.5, there exists u ∈ E such that, up to a subsequence, u k → u weakly as k → +∞.
In particular, given v ± ∈ M 0 and setting as in [29], we obtain that for any v − ∈ M 0 there exist v + ∈ M 0 \ {v − } and u ∈ Γ(v − , v + ) such that c(v − , v + ) = J(u). Moreover, it can be proved that any is a classical solution of (PDE) (see Theorem 3.3 in [29]). Finally, we have that inf v − ≡v + c(v − , v + ) > 0 as a consequence of the following lemma.
In order to prove the second part of the statement, assume the existence of two sequences ) k is bounded, we can find Λ > 0 and a sequence (u k ) k , with u k ∈ Γ(v − k , v + k ), such that J(u k ) ≤ Λ, for every index k. Hence, by Lemma 3.4, there exists R > 0 such that u k (· + p) − u k (· + q) L 2 ([0,1] n ,R m ) ≤ R for every k ∈ N and p, q ∈ Z. Moreover, for every ε > 0 and k ∈ N, since u k ∈ Γ(v − k , v + k ), there exist p k , q k ∈ Z such that u k − v − k W 1,2 (Tp k ,0,R m ) < ε and u k − v + k W 1,2 (Tq k ,0 ,R m ) < ε for every k ∈ N. In

Odd heteroclinic solutions.
We focalize now in the study of heteroclinic solutions which are odd in the first variable, hence we will consider a subset of Γ(−v, v), v ∈ M 0 , so let us introduce the set In what follows, when we will consider functions u ∈ E odd we often present their properties for x 1 ≥ 0, avoiding to write the corresponding ones for x 1 < 0. In particular, for every u ∈ E odd we have J(u) = 2J + (u), where In this setting we can rewrite Lemma 3.6 as follows.
We are going to look for minimizer of J in the set Γ(v). So, for every v ∈ M 0 we set Notice that for any v ∈ M 0 we have c(−v, v) ≤ c(v) < +∞ holds and, by Lemma 3.7 since by (N 0 ), 0 ∈ M 0 , we have the following.
Moreover, note that, by assumption (N 0 ), the intersection between M 0 and a bounded set consists of a finite number of elements. Hence, from the previous lemma, the minimum is well defined and the set is nonempty and consists of a finite number of elements. In particular, we have The following lemma provides a concentration property for u ∈ E odd such that J(u) is close to the value c: the elements of the sequence (u(· + p)) p∈Z remain far from M 0 only for a finite number of indexes p. Moreover, (u(· + p)) p∈Z approaches an element v 0 ∈ M 0 only once. Indeed, recalling the notation introduced in Remark 2, we have Lemma 4.3. For any r ∈ (0, r 1 ] there exists (r) ∈ N, δ(r) ∈ (0, r0 4 ) with δ(r) → 0 as r → 0 + with the following property: if u ∈ E odd is such that J(u) ≤ c + Λ(r) then (i) if dist W 1,2 (Tp,0,R m ) (u, M 0 ) ≥ r for every p in a set I of consecutive integers, then Card(I) ≤ (r), Proof. Note that (i) plainly follows from Lemma 2.1-(3), setting (r) = c+Λ(r) where [·] denotes the integer part.
As a direct consequence of Lemmas 4.3 and 4.4 we obtain the following concentration result.
We are now able to prove the existence of a minimum of J in the set Γ(v) for every v ∈ M min Proof. Let (u k ) k ⊂ Γ(v) be such that J(u k ) → c(v). Without loss of generality we can assume that J(u k ) ≤ c +Λ(r 1 ) for any k ∈ N. By Lemma 4.5, we obtain that for any k ∈ N, By Lemma 3.5, since E odd is weakly closed, there exists u ∈ E odd such that, along a subsequence, u k → u weakly in E odd . Finally, by (14) and the weakly lower semicontinuity of the distance we obtain Therefore, by Lemma 3.6, we conclude that u − v W 1,2 (Tp,0,R m ) → 0 as p → +∞, so that u ∈ Γ(v). Finally, by semicontinuity, J(u) = c(v).
By Theorem 4.6 we know that for every v 0 ∈ M min 0 , M(v 0 ) is nonempty. One can prove that M(v 0 ) consists of weak solutions of (PDE).  The proof can be adapted by the one of Lemma 3.3 of [4] or Lemma 5.2 of [6]. Therefore we get that any u ∈ M(v 0 ) is a classical C 2 (R n , R m ) solution of (PDE) which is 1-periodic in the variables x i , i ≥ 2.
Finally, we now study further compactness properties for the functional J that will be useful in the next section. They will be obtained as consequences of the nondegeneracy property of the elements of M 0 asked in (N ). In particular assumption (N ) asks that, for every v ∈ M 0 , the linearized operator about v has spectrum which does not contain 0. This is the assumption made in [30] and it is indeed equivalent to require as in (N ) that (N 1 ) there exists α 0 > 0 such that for every h ∈ W 1,2 ([0, 1] n , R m ) and every v ∈ M 0 .

Remark 3.
In connection with Remark 1, arguing as in Remark 3.8 of [2], we can prove that (16) holds true also for the functional J σ0 (u) = σ0 L(u) dx − c 0 on Hence, recalling the definition (10), plainly adapting the proof of Lemma 3.10 in [2], we obtain Lemma 4.9. Let v 0 ∈ M min 0 and (u k ) k ⊂ Γ(v 0 ) be such that J(u k ) → c. Then there exists u ∈ M(v 0 ) such that, up to a subsequence, u k − u W 1,2 (R×[0,1] n−1 ,R m ) → 0 as k → +∞. 5. Saddle type solutions. In this section we prove our main theorem. To this aim, following and adapting the argument in [2], we will first prove the existence of a solution of (PDE) on the unbounded triangle T = {x ∈ R n | x 2 ≥ |x 1 |} satisfying Neumann boundary conditions on ∂T , which is odd in the first variable Then, by recursive reflections with respect to the hyperplanes x 2 = ±x 1 , we will recover a solution of (PDE) on the whole R n .
Let us introduce now some notations. We define the squares for every u ∈ E k , where J p,k (u) = T p,k L(u) dx − c 0 .
Remark 4. Notice that J k (u) ≥ 0 for every u ∈ E k , k ∈ N. Indeed, we can view the restriction u| T p,k as a traslation of a function in W 1,2 ([0, 1] n , R m ) and the restriction on u| τ k can be treated similarly using Lemma 2.4, the symmetry of u and Remark 1. Moreover, we note that the functional J k is lower semicontinuous with respect to the weak W 1,2 (T k , R m ) topology for every k ∈ N.
Then, we can set We plainly obtain that M k = ∅ and that the sequence (c k ) k is increasing. Moreover, c k ≤ c, evaluating J k on a function u ∈ M(v 0 ) with v 0 ∈ M min 0 . Moreover, the non degeneracy assumption (N 1 ) permits us to obtain as in [2] (see Lemma 4.2) the following stronger result. Notice that J (u) ≥ 0 for every u ∈ E. Indeed, the restriction u| T k ∈ E k and so J k (u) ≥ c k for any k ∈ N. Moreover, J is lower semicontinuous in the weak topology of W 1,2 loc (T , R m ). By Lemma 5.1 we readily obtain that J is finite for at least one u ∈ E.