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Research article

Meaningful secret image sharing for JPEG images with arbitrary quality factors


  • JPEG is the most common format for storing and transmitting photographic images on social network platforms. JPEG image is widely used in people's life because of their low storage space and high visual quality. Secret image sharing (SIS) technology is important to protect image data. Traditional SIS schemes generally focus on spatial images, however there is little research on frequency domain images. In addition, the current tiny research on SIS for JPEG images only focuses on JPEG images with a compression quality factor (QF) of 100. To overcome the limitation of JPEG images in SIS, we propose a meaningful SIS for JPEG images to operate the quantized DCT coefficients of JPEG images. The random elements utilization model is applied to achieve meaningful shadow images. Our proposed scheme has a better quality of the shadow images and the recovered secret image. Experiment results and comparisons indicate the effectiveness of the scheme. The scheme can be used for JPEG images with any compression QF. Besides, the scheme has good characteristics, such as (k,n) threshold, extended shadow images.

    Citation: Yue Jiang, Xuehu Yan, Jia Chen, Jingwen Cheng, Jianguo Zhang. Meaningful secret image sharing for JPEG images with arbitrary quality factors[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 11544-11562. doi: 10.3934/mbe.2022538

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  • JPEG is the most common format for storing and transmitting photographic images on social network platforms. JPEG image is widely used in people's life because of their low storage space and high visual quality. Secret image sharing (SIS) technology is important to protect image data. Traditional SIS schemes generally focus on spatial images, however there is little research on frequency domain images. In addition, the current tiny research on SIS for JPEG images only focuses on JPEG images with a compression quality factor (QF) of 100. To overcome the limitation of JPEG images in SIS, we propose a meaningful SIS for JPEG images to operate the quantized DCT coefficients of JPEG images. The random elements utilization model is applied to achieve meaningful shadow images. Our proposed scheme has a better quality of the shadow images and the recovered secret image. Experiment results and comparisons indicate the effectiveness of the scheme. The scheme can be used for JPEG images with any compression QF. Besides, the scheme has good characteristics, such as (k,n) threshold, extended shadow images.



    We consider systems of semilinear elliptic equations

    Δu(x)+Fu(x,u)=0

    where

    (F1) FC2(Rn×Rm;R) is 1-periodic sin all its variable, n,m1.

    When n=1 and m1, (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 M0 of minimal periodic solutions of (PDE) is a non empty ordered set and if M0 is not a continuum then there exists another ordered family M1 of minimal entire solutions which are heteroclinic in one space variable to a couple of (extremal) periodic solutions u<v (a gap pair in M0). If M1 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 M0 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)=12|u|2+F(x,u) and Tn=Rn/Zn, periodic solutions to (PDE) were found as minima of the functional J0(u)=TnL(u)dx on E0=W1,2(Tn,Rm) showing that

    M0={uE0J0(u)=c0:=infE0J0(u)}.

    Paul H. Rabinowitz studied the case of spatially reversible potentials F assuming (¯F2) F is even in xi for 1in and proved in [29] that if M0 is constituted by isolated points then for each vM0 there is a v+M0{v} and a solution uC2(R×Tn1,Rm) of (PDE) that is heteroclinic in x1 from v to v+. These solutions were found by variational methods minimizing the renormalized functional

    J(u)=pZJp,0(u):=pZ(Tp,0L(u)dxc0), (1)

    (where Tp,0=[p,p+1]×[0,1]n1) on the space

    Γ(v,v+)={uW1,2(R×Tn1,Rm)uv±L2(Tp,0,Rm)0 as p±}.

    In [30] the existence of minimal double heteroclinics was obtained assuming that the elements of M0 are not degenerate critical points of J0 and that the set M1(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 Δu+W(u)=0 on R2 with W an even double well potential. They proved the existence of a (unique) saddle solution vC2(R2) of that equation, i.e., a bounded entire solution having the same sign and symmetry of the product function x1x2 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 (F1) and the symmetry properties

    (F2) F is even in all its variables;

    (F3) F has flip symmetry with respect to the first two variables, i.e.,

    F(x1,x2,x3,...,xn,u)=F(x2,x1,x3,...,xn,u) on Rn×Rm.

    By [29] the set M0 of minimal periodic solution of (PDE) is not empty. The symmetry of F implies that any vM0 has components whose sign is constant on Rn and if vM0 then (ν1v1,,νmvm)M0 for every (ν1,,νm){±1}m (see Lemma 2.2 below). In this sense we can say that M0 is symmetric with respect to the constant function v00.

    As recalled above, in [2], where m=1, a saddle solution was found when M0 has a gap pair symmetric with respect to the origin. In the case m>1 we generalize this gap condition asking that 0M0 and, following [30], we look for saddle solutions of (PDE) when any vM0 is not degenerate for J0. We then assume

    (N) 0M0 and there exists α0>0 such that

    J0(v)hh=[0,1]n|h|2+Fu,u(x,v(x))hhdxα0h2L2([0,1]n,Rm)

    for every hW1,2([0,1]n,Rm) and every vM0.

    The assumption (N) and the symmetries of F allow us to find heteroclinic connections between elements of M0 which are odd in the variable x1. More precisely for vM0 these solutions are searched as minima of the functional J (see (1)) on the space

    Γ(v)={uW1,2(R×Tn1,Rm)u is odd in x1,limp+uvL2([p,p+1]×Tn1,Rm)=0}.

    In §4, setting

    c(v)=infuΓ(v)J(u) for vM0

    we show that

    Mmin0={v0M0c(v0)=minvM0c(v)}

    and that Mmin0 is such that if v0Mmin0 then

    M(v0)={uΓ(v0)J(u)=c(v0)}

    is not empty and compact with respect to the W1,2(R×Tn1,Rn) metric. The elements uM(v0) are classical solutions to (PDE), odd in x1, even and 1-periodic in x2,...,xn and satisfy the asymptotic condition

    uv0W1,2([p,p+1]×Tn1,Rm)0 as p+.

    Our main result can now be stated as follows

    Theorem 1.1. Assume (F1), (F2), (F3) and (N). Then, there exists a classical solution w of (PDE) such that every component wi (for i=1,m) satisfies

    (i) wi0 for x1x2>0;

    (ii) wi is odd in x1 and x2, 1-periodic in x3,...,xn;

    (iii) wi(x1,x2,x3,...,xn)=wi(x2,x1,x3,...,xn) in Rn.

    Moreover there exists v0Mmin0 such that the solution w satisfies the asymptotic condition

    distW1,2(Rk,Rm)(w,M(v0))0,ask+, (2)

    where Rk=[k,k]×[k,k+1]×[0,1]n2.

    Note that by (i) and (ii) any component of w has the same sign as the product function x1x2. Moreover by (2), since w is asymptotic as x2+ to the compact set M(v0) of odd heteroclinic type solutions, the symmetry of w implies that w is asymptotic to v0 or v0 along any direction not parallel to the planes x1=0, x2=0. In this sense w is a saddle solution, representing a multiple transition between the pure phases v0 and v0 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.

    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 (F2) and (F3). Here and in the following we will work under the not restrictive assumption

    (F4) F0 on Rn×Rm.

    Let us introduce the set

    E0=W1,2(Tn,Rm)={uW1,2(Rn,Rm)u is 1-periodic in all its variables}

    with the norm

    uW1,2([0,1]n,Rm)=(mi=1[0,1]n(|ui|2+|ui|2)dx)12.

    We define the functional J0:E0R as

    J0(u)=[0,1]n12|u|2+F(x,u)dx=[0,1]nL(u)dx. (3)

    and consider the minimizing set

    M0={uE0|J0(u)=c0} where c0=infuE0J0(u)

    Then in [29], [30] it is shown

    Lemma 2.1. Assume (F1), then M0. Moreover, setting [u]=[0,1]nudx, we have that

    1. ˆM0={uM0[u][0,1]m} is a compact set in E0;

    2. if (uk)kE0, with [uk][0,1]m, is a minimizing sequence for J0, then there exists uˆM0 such that uku in E0 up to subsequences;

    3. For every ρ>0 there exists β(ρ)>0 such that if uE0 is such that

    distW1,2([0,1]n,Rm)(u,M0):=infvM0uvW1,2([0,1]n,Rm)>ρ,

    then J0(u)c0β(ρ);

    4. If (F2) holds, then any uM0 minimizes also I(u)=[0,12]nL(u)dx on W1,2([0,12]n,Rm). As a consequence, every uM0 is symmetric in xi about xi=0 and xi=12 for every index i and u is even in xi for every index i;

    5. If (F2) holds, there results c0=infuW1,2([0,1]n,Rm)J0(u). Furthermore, if uW1,2([0,1]n,Rm) verifies J0(u)=c0, then for every i=1,2,...,n, u is symmetric in xi about xi=12 and hence uM0.

    Assumption (F2), in particular the even parity of F with respect to the components of u, provides that the elements in M0 have components with definite sign, thanks to the unique extension property (see [29], Proposition 3).

    Lemma 2.2. Assume (F1), (F2) and 0M0. If u=(u1,,um)M0 then, for every i=1,...,m, one has either ui0, or ui0 on [0,1]n and u does not vanish on open sets. Moreover, (ν1u1,,νmum)M0 for every (ν1,,νm){±1}m.

    Proof. It is sufficient to observe that if u=(u1,,um)M0 then, since F is even with respect to the components of u, we have

    ⅰ) ˉu=(|u1|,,|um|)M0 and

    ⅱ) (ν1u1,,νmum)M0 for every (ν1,,νm){±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 u0, giving a contradiction and concluding the proof.

    On the other hand, assumption (F3) gives more structure on the set M0: its elements have a flip symmetry property. Indeed, setting T+={x[0,1]n|x1x2}, for every uW1,2(T+,Rm), let us define ˜uW1,2([0,1]n,Rm) as

    ˜u(x)={u(x),xT+,u(x2,x1,x3,,xn),x[0,1]nT+. (4)

    Then, we have

    Lemma 2.3. If uM0 then, u˜u in [0,1]n.

    Proof. Given uM0, without loss of generality, we assume

    T+L(u)dx[0,1]nT+L(u)dx.

    Since ˜uW1,2([0,1]n,Rm) by Lemma 2.1-(5) we have J0(˜u)c0. By the previous inequality we get

    c0=J0(u)=T+L(u)dx+[0,1]nT+L(u)dx2T+L(u)dx=J0(˜u)c0.

    Hence, again by Lemma 2.1-(5), ˜uM0. By the unique extension property of the solutions of (PDE) (cf. [29], Proposition 3), we have ˜uu in [0,1]n.

    As an immediate consequence, using Lemma 2.1-(5), we have the following.

    Lemma 2.4. There results

    minuW1,2(T+,Rm)T+L(u)dx=c02. (5)

    Moreover, if uW1,2(T+,Rm) verifies T+L(u)dx=c02, then ˜uM0.

    Remark 1. Lemma 2.3 tells us that the elements of M0 are symmetric with respect to the diagonal iperplane {xRnx1=x2} and by Lemma 2.4 they can be found by minimizing T+L(v)dx on W1,2(T+,Rm). Analogously, setting T=[0,1]nT+, we can find the elements of M0 by minimizing TL(v)dx on W1,2(T,Rm) or, by periodicity, by minimizing TL(v)dx on W1,2(T,Rm) whenever T=p+T± with pZn. For future references it is important to note that this property implies in particular that uM0 if and only if u is a minimizer of the functional σ0L(v)dx on W1,2(σ0,Rm) where

    σ0={xR×[0,1]n1|x21x1x2}.

    More precisely we have c0=infvW1,2(σ0,Rm)σ0L(v)dx and uM0 if and only if σ0L(u)dx=c0. From Lemma 2.1-(3) we recover an analogous property in W1,2(σ0,Rm): for any r>0 there exists β(r)>0 such that if uW1,2(σ0,Rm) verifies σ0L(u)dxc0+β(r), then distW1,2(σ0,Rm)(u,M0)r.

    Note that by Lemma 2.1-(1) and the assumption (N) we plainly derive that (N0) ˆM0 is a finite set and 0ˆM0,

    where we recall that ˆM0={uM0[u][0,1]m} and note that M0=ˆM0+Zm.

    Note finally that by (N0), setting

    r0:=min{uvL2(Tn,Rm)u,vM0,u (6)

    we have r_0>0 .

    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

    E = \{u\in W_{loc}^{1, 2}({\mathbb{R}}^n, {\mathbb{R}}^m)\mid u\text{ is $1$-periodic in }x_2, \ldots x_n\}.

    For any u\in E we consider the functional

    J(u) = \sum\limits_{p\in{\mathbb{Z}}}J_{p, 0}(u),

    where, denoting T_{p, 0} = [p, p+1]\times[0, 1]^{n-1} ,

    J_{p, 0}(u) = \int_{T_{p, 0}}L(u)\, dx-c_{0}, \quad \forall p\in{\mathbb{Z}}.

    Denoting briefly u(\cdot+p) the shifting of the function u with respect to the first coordinate (that is, u(\cdot+p) = u(\cdot+p\hskip1pt{{\mathit{\boldsymbol{e}}}}_1) where \hskip1pt{{\mathit{\boldsymbol{e}}}}_1 = (1, 0, ..., 0) ), note that by periodicity we have

    J_{p, 0}(u) = \int_{[0, 1]^n}L(u(\cdot+p))\, dx-c_0 = J_0(u(\cdot+p))-c_0, \quad \forall p\in{\mathbb{Z}}.

    Then, by Lemma 2.1, we have J_{p, 0}(u)\geq 0 for any u\in E and p\in{\mathbb{Z}} , from which J is non-negative on E .

    Lemma 3.1. The functional J: E \to {\mathbb{R}} is weakly lower semicontinuous.

    Proof. Consider a sequence (u_k)_k such that u_k \to u weakly in E . Then, for every \ell\in {\mathbb{N}} , by the weak lower semicontinuity of J_0 , and hence of J_{p, 0} , we have \sum_{p = -\ell}^\ell J_{p, 0}(u) \leq \liminf_{k} \sum_{p = -\ell}^\ell J_{p, 0}(u_k) . If J(u) = +\infty , then we obtain easily \liminf_{k} J(u_k) = +\infty . So, let us assume J(u)<+\infty , then for any \varepsilon>0 we have that there exists \ell\in{\mathbb{N}} such that \sum_{|p|>\ell} J_{p, 0}(u) <\varepsilon . We get

    \liminf\limits_k J(u_k) \geq \liminf\limits_k \sum\limits_{p = -\ell}^\ell J_{p, 0}(u_k) \geq \sum\limits_{p = -\ell}^\ell J_{p, 0}(u) > J(u) -\varepsilon\, ,

    thus finishing the proof.

    Using the notation introduced above, note that if u\in E is such that J(u)<+\infty , then J_{p, 0}(u)\to 0 as |p|\to +\infty , that is, the sequence (u(\cdot+p))_{p\in{\mathbb{Z}}} is such that J_0(u(\cdot+p))\to c_0 as p\to \pm\infty . Hence, by Lemma 2.1-(3), there exist u_\pm \in {\mathcal{M}}_0 such that, up to a subsequence, u(\cdot+p) \to u_\pm as p\to \pm\infty in E_0 . Using this remark and the local compactness of {\mathcal{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\limits_{u\in E} J_{p, 0}(u)+J_{p+1, 0}(u) = 0

    and the set of minima coincide with {\mathcal{M}}_0 . We introduce the following distance

    {\rm dist}_p(u, A) = \inf \{ \| u-v \|_{W^{1, 2}(T_{p, 0}\cup T_{p+1, 0}, {\mathbb{R}}^m)} \mid v\in A \}\, .

    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 \lambda(r)>0 such that

    \begin{equation} \hbox{if }u\in E\hbox{ satisfies }J_{p, 0}(u)+J_{p+1, 0}(u)\leq \lambda(r)\hbox{ for a }p\in{\mathbb{Z}}, \hbox{ then }{\rm dist}_p(u, {\mathcal{M}}_0)\leq r. \end{equation} (7)

    It is not restrictive to assume that the function with r\mapsto \lambda(r) is non-decreasing.

    On the other hand for every \lambda>0 if we set

    \rho(\lambda) = \sup \left\{{\rm dist}_p(u, {\mathcal{M}}_0) \mid u\in E \text{ with } J_{p, 0}(u)+J_{p+1, 0}(u) \leq \lambda, \, p\in{\mathbb{Z}} \right\}\,

    we get \rho(\lambda)>0 and that \lambda \mapsto \rho(\lambda) is non-decreasing. Moreover, for every \varepsilon>0 , since if J_{p, 0}(u)+J_{p+1, 0}(u)\leq \lambda(\varepsilon) for a certain p\in{\mathbb{Z}} , then {\rm dist}_p(u, {\mathcal{M}}_0)\leq \varepsilon , we obtain \rho(\lambda)\le\rho(\lambda(\varepsilon))\leq\varepsilon for every \lambda \in (0, \lambda(\varepsilon)] , so that \lim_{\lambda \to 0^+} \rho(\lambda) = 0 holds. Hence, recalling the definition of r_0 in (6), we can fix \lambda_0>0 satisfying \rho(\lambda_0)\leq \frac{r_0}4 . Finally, we can define

    \begin{equation} \Lambda(r) = \sup \left\{J_{p, 0}(u) \mid u\in E \text{ and } p\in{\mathbb{Z}}\text{ are such that } {\rm dist}_p(u, {\mathcal{M}}_0)\leq 2r \right\} \end{equation} (8)

    which is non-decreasing and \lim_{r\to 0} \Lambda(r) = 0 . Then we fix r_1\in(0, \frac{r_0}4) such that \Lambda(r) \leq \frac{\lambda_0}8 for every r\in(0, r_1] .

    We say that a set \mathcal I\subseteq{\mathbb{Z}} is a set of consecutive integers if it is of the form \{\ell\in{\mathbb{Z}}\, |\, p\le \ell<p+k\} or \{\ell\in{\mathbb{Z}}\, |\, p-k< \ell\le p\} for a p \in {\mathbb{Z}} and k\in {\mathbb{N}} \cup \{+ \infty\} . If u\in E is such that J_{p, 0} is small enough for some consecutive integers p\in \mathcal I , then, using (N_0) , we can prove that, in the corresponding sets T_{p, 0} , u is ``near'' to an element of {\mathcal{M}}_0 , the same for all p\in\mathcal I . Indeed we have

    Lemma 3.2. Given \lambda\in (0, \frac{\lambda_0}2] , u\in E and a set of consecutive integers \mathcal I , if J_{p, 0}(u) \leq \lambda for any p\in\mathcal I , then there exists v\in {\mathcal{M}}_0 such that \|u-v\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq \rho(2\lambda) \leq\frac{ r_0}4 , for every p\in \mathcal I .

    Proof. Let p\in\mathcal I be such that p+1 \in \mathcal I . Then J_{p, 0}(u) +J_{p+1, 0}(u) \leq 2\lambda\leq\lambda_{0} and, by Remark 2 and the definition of \lambda_0 , {\rm dist}_p(u, {\mathcal{M}}_{0})\leq\rho(2\lambda)\leq\rho(\lambda_{0})\leq \frac{r_{0}}{4} . Then, by (N_0) and the choice of r_0 in (6), we can find v_{p}\in{\mathcal{M}}_{0} such that

    \| u-v_p \|_{W^{1, 2}(T_{p, 0}\cup T_{p+1, 0}, {\mathbb{R}}^m)} \leq \tfrac{r_{0}}{4}

    from which \|u-v_p\|_{W^{1, 2}(T_{k, 0}, {\mathbb{R}}^m)} \leq \frac{r_0}4 for k = p, p+1 . If p+2\in\mathcal I , repeating the argument with the couple of indices p+1 and p+2 we find v_{p+1}\in{\mathcal{M}}_{0} such that \|u-v_{p+1}\|_{W^{1, 2}(T_{k, 0}, {\mathbb{R}}^m)} \leq \frac{r_0}4 for k = p+1, p+2 . By the choice of r_0 in (6), we conclude that v_{p+1} = v_{p} and the lemma follows.

    Moreover, using the notations introduced above, we have

    Lemma 3.3. If u\in W^{1, 2}(T_{p, 0}\cup T_{p+1, 0}, {\mathbb{R}}^m) then

    \|u(\cdot+p)-u(\cdot+(p+1))\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}^2\le 2(J_{p, 0}(u)+J_{p+1, 0}(u)+2c_{0}).

    Proof. Setting y = (x_{2}, \ldots, x_{n}) , we have

    \|u(\cdot+p)-u(\cdot+(p+1))\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}^2 = \int_{p}^{p+1}\int_{[0, 1]^{n-1}} |u(x_{1}+1, y)-u(x_{1}, y)|^{2} dy \, dx_{1}

    and so there exists \bar x_{1}\in(p, p+1) such that

    \int_{[0, 1]^{n-1}} |u(\bar x_{1}+1, y) -u(\bar x_{1}, y)|^2 dy \geq \|u(\cdot+p)-u(\cdot+(p+1))\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}^2.

    On the other hand, by Hölder inequality,

    \begin{align*} 2(J_{p, 0}(u)+J_{p+1, 0}(u)+2c_{0})&\geq\int_{p}^{p+2} \int_{[0, 1]^{n-1}} |\partial_{x_{1}}u(x_{1}, y)|^2 dy \, dx_{1}\\ &\geq \int_{[0, 1]^{n-1}} \int_{ \bar x_{1}}^{\bar x_{1} +1} |\partial_{x_{1}}u(x_{1}, y)|^2 dx_{1} \, dy\\ &\geq \int_{[0, 1]^{n-1}} |u(\bar x_{1}+1, y ) -u(\bar x_{1})|^2 dy\\ &\geq \|u(\cdot+p)-u(\cdot+(p+1))\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}^2 \end{align*}

    completing the proof.

    By the previous lemmas we obtain that the elements in the sublevels of J satisfy the following boundeness property.

    Lemma 3.4. For every \Lambda>0 there exists {\mathit{R}}>0 such that for every u\in E satisfying J(u)\leq \Lambda one has \|u(\cdot+p)-u(\cdot+q)\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)} \leq {\mathit{R}} for any p, q\in{\mathbb{Z}} .

    Proof. Let u \in E be such that J(u) \leq \Lambda . We define \mathcal J(u) = \{k\in{\mathbb{Z}} \mid J_{k, 0}(u) \geq \frac{\lambda_{0}}{2} \} and note that the number l(u) of elements of \mathcal J(u) is at most [\frac{2\Lambda}{\lambda_{0}}] +1 , where [\cdot] denotes the integer part. Then, the set {\mathbb{Z}} \setminus \mathcal J(u) is constituted by \bar l(u) sets of consecutive elements of {\mathbb{Z}} , \mathcal I_i(u) , with \bar l (u) \leq l(u) +1 . By the triangular inequality, for any p, q \in {\mathbb{Z}} , we obtain

    \begin{align} \|u(\cdot+p)-u(\cdot+q)\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)} &\leq l(u) \sup\limits_{k \in {\mathcal J}(u)}\|u(\cdot+k)-u(\cdot+k+1)\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}\\ &\, \, \, \, \quad + \sum\limits_{i = 1}^{\bar l (u)} \sup\limits_{p, q \in \mathcal I_i(u)}\|u(\cdot+p)-u(\cdot+q)\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}\\ &\leq l(u)(2(\Lambda +2c_{0}))^{\frac{1}{2}} + \bar l(u) \tfrac{r_{0}}{2}. \end{align} (9)

    where the first term in (9) follows by the application of Lemma 3.3, since

    2(J_{k, 0}(u)+J_{k+1, 0}(u)+2c_{0})\le 2(J(u) +2c_{0})\le 2(\Lambda+2c_0), \quad\forall k\in{\mathbb{Z}},

    while the second one follows by the definition of {\mathcal I}_i(u) and Lemma 3.2.

    Since \bar l(u)\le l(u)+1 and l(u)\le [\frac{2c}{\lambda_{0}}] +1 , the lemma follows by choosing {\mathit{R}} = ( [\frac{2\Lambda}{\lambda_{0}}] +1)(2(\Lambda +2c_{0}))^{\frac{1}{2}} + ( [\frac{2\Lambda}{\lambda_{0}}] +2) \tfrac{r_{0}}{2} .

    The following lemma states the weak compactness of the sublevels of the functional J .

    Lemma 3.5. Given any \Lambda>0 , let (u_k)_k\subset E be a sequence such that J(u_{k})\leq \Lambda for every k\in{\mathbb{N}} and let (p_k)_k be a sequence of integers. Assume that there exist \bar {\mathit{R}} < + \infty and v\in{\mathcal{M}}_{0} such that \|u_{k}-v\|_{W^{1, 2}(T_{p_{k}, 0}, {\mathbb{R}}^m)}\leq \bar {\mathit{R}} for all k\in{\mathbb{N}} . Then, there exists u\in E with J(u)\leq \Lambda such that, up to a subsequence, u_{k}\to u weakly in E .

    Proof. First note that, by Lemma 3.4, there exists {\mathit{R}}>0 such that if u\in E and J(u)\leq \Lambda then \|u(\cdot+p)-u(\cdot+q)\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)} \leq {\mathit{R}} for any p, q\in{\mathbb{Z}} . If \|u-v\|_{W^{1, 2}(T_{\ell, 0}, {\mathbb{R}}^m)}\leq \bar{\mathit{R}} for some \ell\in{\mathbb{Z}} and v\in{\mathcal{M}}_0 , by triangular inequality for any p\in{\mathbb{Z}} we obtain

    \begin{multline*} \|u- v\|_{L^{2}(T_{p, 0}, {\mathbb{R}}^m)} = \|u(\cdot+p)-v\|_{L^2([0, 1]^n, {\mathbb{R}}^{m})}\\ \leq\|u(\cdot+p)-u(\cdot+\ell)\|_{L^2([0, 1]^n, {\mathbb{R}}^{m})}+\|u(\cdot+\ell)-v\|_{L^2([0, 1]^n, {\mathbb{R}}^{m})}\le {\mathit{R}}+\bar{\mathit{R}}. \end{multline*}

    Consider now a sequence as in the statement, setting Q_{L} = [-L, L]\times [0, 1]^{n-1} for L \in {\mathbb{N}} , we get

    \|u_k -v\|^2_{L^2(Q_{L}, {\mathbb{R}}^m)} + \|\nabla u_k\|^2_{L^2(Q_{L}, {\mathbb{R}}^m)} \leq 2L(\bar{\mathit{R}}+{\mathit{R}})^{2} + 4Lc_{0} +2\Lambda.

    Hence, (u_{k}-v)_k is bounded in W^{1, 2}(Q_{L}, {\mathbb{R}}^m) for any L\in{\mathbb{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 .

    Lemma 3.6. If J(u)<+\infty , there exist v^{\pm}\in{\mathcal{M}}_{0} such that

    \|u-v^\pm\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \to 0\quad\mathit{\text{as}}\quad p\to\pm\infty.

    Proof. Since J(u) < + \infty , we have J_{p, 0}(u) \to 0 as |p| \to +\infty and there exists \bar p such that J_{p, 0}(u) \leq \frac{\lambda_{0}}{2} for any |p| \geq \bar p . Thus, by Lemma 3.2, there exists v^{\pm} \in {\mathcal{M}}_{0} such that \|u-v^+\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq \frac{r_{0}}{4} for p \geq \bar p and \|u-v^-\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq \frac{r_{0}}{4} for p \leq -\bar p .

    Hence the sequence (u(\cdot+p))_{p\in{\mathbb{N}}} is such that \|u(\cdot+p)-v^+\|_{W^{1, 2}([0, 1]^n, {\mathbb{R}}^{m})}\le \frac{r_0}4 for every p\ge \bar p and J_0(u(\cdot +p))-c_0 = J_{p, 0}(u)\to 0 as p\to +\infty . Then, by Lemma 2.1, \|u-v^+\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} = \|u(\cdot+p)-v^+\|_{W^{1, 2}([0, 1]^n, {\mathbb{R}}^m)}\to 0 as p\to +\infty . Analogously we obtain that \|u-v^-\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)}\to 0 as p\to -\infty

    By Lemma 3.6, if u\in E satisfies J(u)<+\infty we can view it as an heteroclinic or homoclinic connection between two periodic solutions v^- and v^+ belonging to {\mathcal{M}}_0 . Hence, we can consider elements of E belonging to the classes

    \begin{align*} \Gamma(v^-, v^+) = \big\{ u\in E \mid \|u-v^\pm\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \to 0\, \text{as}\, p\to\pm\infty \big\} \end{align*}

    where v^\pm\in{\mathcal{M}}_0 .

    We note that by Lemma 3.5, every sequence (u_k)_{k\in {\mathbb{N}}}\subset\Gamma(v^-, v^+) with J(u_k)\leq \Lambda for all k\in{\mathbb{N}} , admits a subsequence which converges weakly to some u\in E . Indeed, since \|u_k-v^+\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \to 0 as p\to +\infty for every k\in{\mathbb{N}} , fixed \bar{\mathit{R}}>0 there exists p_k\in{\mathbb{N}} such that \|u_k-v^+\|_{W^{1, 2}(T_{p_k, 0}, {\mathbb{R}}^m)}\le \bar{\mathit{R}} and since J(u_k)\le \Lambda , by Lemma 3.5, there exists u\in E such that, up to a subsequence, u_k\to u weakly as k\to +\infty .

    In particular, given v^\pm\in{\mathcal{M}}_0 and setting

    c(v^-, v^+) = \inf\limits_{u\in\Gamma(v^-, v^+)}J(u)\, ,

    as in [29], we obtain that for any v^-\in{\mathcal{M}}_0 there exist v^+\in{\mathcal{M}}_0\setminus\{v^-\} and u\in\Gamma(v^-, v^+) such that c(v^-, v^+) = J(u) . Moreover, it can be proved that any u\in\Gamma(v^-, v^+) such that c(v^-, v^+) = J(u) is a classical solution of (PDE) (see Theorem 3.3 in [29]).

    Finally, we have that \inf_{v^-\not\equiv v^+}c(v^-, v^+)>0 as a consequence of the following lemma.

    Lemma 3.7. For every v^\pm\in{\mathcal{M}}_{0} with v^-\not\equiv v^+ , we have c(v^-, v^+)\geq \frac{\lambda_{0}}{2} . Moreover, c(v^-, v^+)\to +\infty as \|v^+-v^-\|_{W^{1, 2}([0, 1]^n, {\mathbb{R}}^m)}\to+\infty .

    Proof. Assume that there exists u\in \Gamma(v^-, v^+) satisfying J(u)<\frac{\lambda_0}2 . Then J_{p, 0}(u)<\frac{\lambda_0}2 for every p\in {\mathbb{Z}} , so that by Lemma 3.2 there exists v\in{\mathcal{M}}_0 such that \|u- v\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq \frac{r_0}4 for every p\in{\mathbb{Z}} . Since u\in\Gamma(v^-, v^+) we know that \|u- v^-\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \to 0 as p\to-\infty and \|u- v^+\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \to 0 as p\to+\infty , so that by (6) we would have v^- = v = v^+ giving a contradiction.

    In order to prove the second part of the statement, assume the existence of two sequences (v^-_k)_k and (v^+_k)_k in {\mathcal{M}}_0 such that (c(v^-_k, v^+_k))_k is bounded while \|v^+_k-v^-_k\|_{W^{1, 2}({\mathbb{T}}^n, {\mathbb{R}}^m)}\to+\infty as k\to +\infty . Since (c(v^-_k, v^+_k))_k is bounded, we can find \Lambda>0 and a sequence (u_k)_k , with u_k\in \Gamma(v^-_k, v^+_k) , such that J(u_k)\leq \Lambda , for every index k . Hence, by Lemma 3.4, there exists {\mathit{R}}>0 such that \|u_k(\cdot+p)-u_k(\cdot+q)\|_{L^2([0, 1]^n , {\mathbb{R}}^{m})}\le {\mathit{R}} for every k\in{\mathbb{N}} and p, q\in{\mathbb{Z}} . Moreover, for every \varepsilon>0 and k\in{\mathbb{N}} , since u_k\in \Gamma(v^-_k, v^+_k) , there exist p_k, q_k\in{\mathbb{Z}} such that \|u_k-v_k^-\|_{W^{1, 2}(T_{ p_k, 0}, {\mathbb{R}}^m)}<\varepsilon and \|u_k-v_k^+\|_{W^{1, 2}(T_{ q_k, 0}, {\mathbb{R}}^m)}<\varepsilon for every k\in{\mathbb{N}} . In particular we get

    \begin{align*} \|v^+_k-v^-_k\|_{L^2([0, 1]^n, {\mathbb{R}}^m)} &\leq \|v^-_k-u_k(\cdot+ p_k)\|_{L^2([0, 1]^n, {\mathbb{R}}^m)} \\ &\phantom{\leq}+\|u_k(\cdot+ p_k)-u_k(\cdot+ q_k)\|_{L^2([0, 1]^n, {\mathbb{R}}^m)} \\ &\phantom{\leq}+ \|v^+_k-u_k(\cdot+ q_k)\|_{L^2([0, 1]^n, {\mathbb{R}}^m)} \\ &\leq \varepsilon + {\mathit{R}} +\varepsilon \end{align*}

    since, by periodicity, \|v_k^\pm-u_k(\cdot+p)\|_{L^2([0, 1]^n, {\mathbb{R}}^m)} = \|v_k^\pm-u_k\|_{W^{1, 2}(L^2, {\mathbb{R}}^m)} for any k\in{\mathbb{N}} , p\in{\mathbb{Z}} . Finally, since \|\nabla v\|^2_{L^2([0, 1]^n, {\mathbb{R}}^m)}\leq 2c_0 for every v\in{\mathcal{M}}_0 , we recover \| v^+ - v^- \|_{W^{1, 2}([0, 1]^n, {\mathbb{R}}^m)} \leq 2\sqrt{2c_0}+2\varepsilon+R in contradiction with \|v^+_k-v^-_k\|_{W^{1, 2}({\mathbb{T}}^n, {\mathbb{R}}^m)}\to+\infty .

    We focalize now in the study of heteroclinic solutions which are odd in the first variable, hence we will consider a subset of \Gamma(-v, v) , v\in{\mathcal{M}}_0 , so let us introduce the set

    E^{odd} = \{u\in E \mid \text{$u$ is odd with respect to $x_1$} \},

    In what follows, when we will consider functions u\in E^{odd} we often present their properties for x_1\geq 0 , avoiding to write the corresponding ones for x_1< 0 . In particular, for every u\in E^{odd} we have J(u) = 2J^+(u) , where

    J^+(u) = \sum\limits_{p\geq 0} J_{p, 0}(u)\, .

    For any v\in{\mathcal{M}}_0 let

    \Gamma(v) = \{ u\in E^{odd} \mid \|u - v\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \to 0 \text{ as } p\to +\infty \} \subseteq \Gamma(-v, v)\, .

    In this setting we can rewrite Lemma 3.6 as follows.

    Lemma 4.1. For every u\in E^{odd} for which J(u)<+\infty there exists v\in{\mathcal{M}}_0 such that \|u - v\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \to 0 as p\to +\infty , that is u\in\Gamma(v) .

    We are going to look for minimizer of J in the set \Gamma(v) . So, for every v\in{\mathcal{M}}_0 we set

    \begin{equation} c(v) = \inf\limits_{u\in \Gamma(v)} J(u)\quad \text{ and }\quad{\mathcal{M}}(v) = \{u\in \Gamma(v) \mid J(u) = c(v)\}\, . \end{equation} (10)

    Notice that for any v\in{\mathcal{M}}_0 we have c(-v, v)\leq c(v) <+\infty holds and, by Lemma 3.7 since by (N_0) , 0\not\in{\mathcal{M}}_0 , we have the following.

    Lemma 4.2. For any v\in{\mathcal{M}}_{0} , c(v)\geq \frac{\lambda_{0}}{2} , and c(v)\to +\infty as \|v\|_{W^{1, 2}([0, 1]^n, {\mathbb{R}}^m)}\to+\infty .

    Moreover, note that, by assumption ( N_0 ), the intersection between {\mathcal{M}}_0 and a bounded set consists of a finite number of elements. Hence, from the previous lemma, the minimum

    \begin{equation} c = \min\limits_{v\in{\mathcal{M}}_0} c(v) \end{equation} (11)

    is well defined and the set

    \begin{equation} {\mathcal{M}}_0^{min} = \{v \in{\mathcal{M}}_0 \mid c(v) = c \} \end{equation} (12)

    is nonempty and consists of a finite number of elements. In particular, we have

    \begin{equation} \min\limits_{ v\in {\mathcal{M}}_0\setminus {\mathcal{M}}_0^{min}}c(v) > c\, . \end{equation} (13)

    The following lemma provides a concentration property for u\in E^{odd} such that J(u) is close to the value c : the elements of the sequence (u(\cdot +p))_{p\in {\mathbb{Z}}} remain far from {\mathcal{M}}_0 only for a finite number of indexes p . Moreover, (u(\cdot +p))_{p\in {\mathbb{Z}}} approaches an element v_0\in{\mathcal{M}}_0 only once. Indeed, recalling the notation introduced in Remark 2, we have

    Lemma 4.3. For any r\in(0, r_{1}] there exists \ell(r) \in {\mathbb{N}} , \delta(r)\in(0, \frac{r_{0}}4) with \delta(r)\to 0 as r\to 0^{+} with the following property: if u\in E^{odd} is such that J(u)\leq c+\Lambda(r) then

    (i) if {\rm dist}_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)}(u, {\mathcal{M}}_0) \geq r for every p in a set \mathcal I of consecutive integers, then {\rm Card}(\mathcal I) \leq \ell(r) ,

    (ii) if \|u-v_0 \|_{W^{1, 2}(T_{p_0, 0}, {\mathbb{R}}^m)} \leq r for some index p_0\geq 0 and v_0\in{\mathcal{M}}_0 , then \|u-v_0 \|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq \delta(r) for every p \geq p_0 , and \sum_{p = p_0}^{+ \infty} J_{p, 0}(u) \leq 2\Lambda(r) .

    Proof. Note that (i) plainly follows from Lemma 2.1-(3), setting \ell(r) = \left[\frac{c + \Lambda(r)}{\beta(r)}\right]+1 , where [\cdot] denotes the integer part.

    To prove (ii) , we consider \tilde u \in E^{odd} defined for x_1\ge 0 as

    \tilde u(x_{1}, y) = \begin{cases} u(x_{1}, y)&\hbox{if $x_{1}\in[0, p_0]$, }\\ u(x_{1}, y)(p_0+1-x_{1})+v_0(x_{1}, y)(x_{1}-p_0)& \hbox{if $x_{1}\in (p_0, p_0+1)$, }\\ v_0(x_{1}, y) &\hbox{if $x_{1}\in[p_0+1, +\infty)$} \end{cases}

    Hence, \tilde u \in \Gamma(v_0) and since \tilde u\equiv u in [-p_0, p_0]\times {\mathbb{R}}^{n-1} , while \tilde u = v_0 in [p_0+1, +\infty)\times {\mathbb{R}}^{n-1} , we obtain

    \tfrac12 c\le\tfrac12c(v_0)\leq \tfrac12 J(\tilde u) = J^+(\tilde u) = J^+(u) -\sum\limits_{p = p_0}^{+ \infty} J_{p, 0}(u) + J_{p_{0}, 0}(\tilde u).

    By definition, on T_{p_{0}, 0} we have \tilde u(x_{1}, y) - v_0(x_{1}, y) = (p_0+1-x_{1})(u(x_{1}, y)-v_0(x_{1}, y)) and so \| \tilde u-v_0\|_{W^{1, 2}(T_{p_0, 0}, {\mathbb{R}}^m)} \leq 2\|u-v_0\|_{W^{1, 2}(T_{p_0, 0}, {\mathbb{R}}^m)} \leq 2r . Since \tilde u = v_0 in [p_0+1, p_0+2]\times {\mathbb{R}}^{n-1} , we have {\rm dist}_p(\tilde u, {\mathcal{M}}_0) = \| \tilde u-v_0\|_{W^{1, 2}(T_{p_0, 0}, {\mathbb{R}}^m)}\leq 2r , so that, by Remark 2, we obtain J_{p_{0}, 0}(\tilde u) \leq \Lambda(r) \leq \frac{\lambda_{0}}8 and therefore

    \tfrac 12 c\leq \tfrac12 J(\tilde u) \leq J^+(u) -{\sum}_{p = p_0}^{+ \infty} J_{p, 0}(u) + \Lambda(r) \leq \tfrac 12 c -{\sum}_{p = p_0}^{+ \infty} J_{p, 0}(u) + \tfrac32 \Lambda(r).

    Then {\sum}_{p = p_0}^{+ \infty} J_{p, 0}(u) \leq \frac32 \Lambda(r) and in particular J_{p, 0}(u) \leq \frac32 \Lambda(r)\leq \frac{\lambda_0}{2} for any p \geq p_0 . Hence, by Lemma 3.2, \| u-v_0\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq \rho(3\Lambda(r))<r_0 for any p \geq p_0 . Hence (ii) follows setting \delta(r) = \rho(3\Lambda(r)) . Indeed, by Remark 2, we have \lim_{r\to 0^+} \delta(r) = 0 and, since \Lambda(r)\le\frac{\lambda_0}8 for all r\in (0, r_1] , we get \delta(r)\leq\rho(\lambda_0)\le \frac{r_{0}}4 for every r\in (0, r_1) .

    By the previous lemma we get

    Lemma 4.4. For any r\in(0, r_{1}] , if u\in E^{odd} satisfies J(u)\leq c+\Lambda(r) , then there exists v_0\in{\mathcal{M}}_0 such that u\in\Gamma(v_0) and

    (i) if \|u-v_0 \|_{W^{1, 2}(T_{p_0, 0}, {\mathbb{R}}^m)} \leq r for a certain index p_0\geq 0 , then we have \|u-v_0 \|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq \delta(r) for every p \geq p_0 , and \sum_{p = p_0}^{+ \infty} J_{p, 0}(u) \leq 2\Lambda(r) .

    (ii) if w \in {\mathcal{M}}_{0} \setminus \{ v_{0}\} , then \|u-w \|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} > r_1 for every p \in{\mathbb{Z}}, \, p\ge 0 .

    Proof. Note that the existence of v_0 such that u\in\Gamma(v_0) is ensured by Lemma 4.1 and (i) plainly follows from Lemma 4.3- (ii) . To prove (ii) we argue by contradiction assuming that there exist \bar p_{0}\in{\mathbb{Z}} , \bar p_0\ge 0 and w\in{\mathcal{M}}_{0}\setminus\{v_{0}\} such that \|u-w \|_{W^{1, 2}(T_{\bar p_0, 0}, {\mathbb{R}}^m)} \leq r_1 . Again, by Lemma 4.3- (ii) we get \|u-w \|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq \delta(r_1) \leq \frac{r_0}{4} for every p\geq \bar p_0 which is in contradiction with u\in\Gamma(v_0) , recalling the definition of r_0 in (6).

    As a direct consequence of Lemmas 4.3 and 4.4 we obtain the following concentration result.

    Lemma 4.5. For any \rho\in (0, {r_{1}}] there exists \tilde \Lambda(\rho) , with \tilde \Lambda(\rho)\to 0 as \rho\to 0^+ , and \tilde\ell(\rho)\in{\mathbb{N}} such that if u\in E^{odd} satisfies J(u)\leq c+\tilde \Lambda(\rho) , then there exists v_0\in{\mathcal{M}}_0 such that u\in\Gamma(v_0) and

    (i) \| u -v_0\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq \rho \mathit{\text{for every}} p\geq \tilde\ell(\rho) ;

    (ii) \sum_{p = \tilde\ell(\rho)}^{+\infty}J_{p, 0}(u) \leq 2\tilde \Lambda(\rho) ;

    (iii) \| u - w \|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \geq r_1 for every p\in{\mathbb{Z}}, \, p\ge 0 and w\in {\mathcal{M}}_0\setminus\{v_0\} .

    Proof. The existence of v_0 such that u\in\Gamma(v_0) is again ensured by Lemma 4.1. By Lemma 4.3, given any \rho\in (0, {r_{1}}] , there exists r\in(0, \rho) such that \delta(r)\le\rho . Then, if u\in \Gamma(v_0) is such that J(u)\leq c+\Lambda(r) , by Lemma 4.3- (i) , there exists p_0\in[0, \ell(r)+1] such that {\rm dist}_{W^{1, 2}(T_{p_0, 0}, {\mathbb{R}}^m)}(u, {\mathcal{M}}_0) < r and hence a v\in{\mathcal{M}}_0 such that \|u-v\|_{W^{1, 2}(T_{p_0, 0}, {\mathbb{R}}^m)} < r. Therefore, by Lemma 4.3- (ii) , we obtain \|u-v\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} < \delta(r) for all p\geq p_0 and since \delta(r)<\rho<r_1<\frac{r_0}4 , we can conclude that v\equiv v_0 and hence that \|u -v_0\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq \rho for every p\geq p_0 . Moreover, again by Lemma 4.3- (ii) , we have \sum_{p = p_0}^{+ \infty} J_{p, 0}(u) \leq 2\Lambda(r) . Hence (i) and (ii) follows setting \tilde\ell(\rho) = \ell(r)+1 and \tilde \Lambda(\rho) = \Lambda(r) .

    Finally, \| u - w \|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \geq r_1 for every p\in{\mathbb{Z}}, \, p\ge 0 , and w\in {\mathcal{M}}_0\setminus\{v_0\} follows directly by Lemma 4.4 - (ii) .

    We are now able to prove the existence of a minimum of J in the set \Gamma(v) for every v\in{\mathcal{M}}_0^{min} , i.e., that {\mathcal{M}}(v)\ne\varnothing for all v\in{\mathcal{M}}_0^{min} .

    Theorem 4.6. Let v\in{\mathcal{M}}_0^{min} , then there exists u\in \Gamma(v) such that J(u) = c(v) = c .

    Proof. Let (u_{k})_k\subset \Gamma(v) be such that J (u_{k})\to c(v) . Without loss of generality we can assume that J(u_{k})\leq c+\tilde \Lambda(r_{1}) for any k\in{\mathbb{N}} . By Lemma 4.5, we obtain that for any k\in{\mathbb{N}} ,

    \begin{equation} \|u_k -v\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq r_{1} \text{ for every } p\geq \tilde\ell(r_{1}). \end{equation} (14)

    By Lemma 3.5, since E^{odd} is weakly closed, there exists u\in E^{odd} such that, along a subsequence, u_{k}\to u weakly in E^{odd} . Finally, by (14) and the weakly lower semicontinuity of the distance we obtain

    \begin{equation} \|u -v \|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq r_{1} \text{ for every } p\geq \tilde\ell(r_{1}). \end{equation} (15)

    Therefore, by Lemma 3.6, we conclude that \|u -v \|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)}\to 0 as p\to +\infty , so that u\in \Gamma(v) . Finally, by semicontinuity, J(u) = c(v) .

    By Theorem 4.6 we know that for every v_0\in{\mathcal{M}}_0^{min} , {\mathcal{M}}(v_0) is nonempty. One can prove that {\mathcal{M}}(v_0) consists of weak solutions of (PDE).

    Lemma 4.7. Given \bar u \in {\mathcal{M}}(v_0) , with v_0\in{\mathcal{M}}_0^{min} , then for any \psi\in {\mathcal{C}}_0^\infty({\mathbb{R}}\times{\mathbb{T}}^{n-1}, {\mathbb{R}}^m) we have

    \int_{{\mathbb{R}}\times[0, 1]^{n-1}} \nabla \bar u \cdot \nabla \psi + F_u(x, \bar u) \psi \, dx = 0\, .

    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\in {\mathcal{M}}(v_{0}) is a classical {\mathcal{C}}^2({\mathbb{R}}^{n}, {\mathbb{R}}^m) solution of (PDE) which is 1 -periodic in the variables x_{i} , i\ge 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 {\mathcal{M}}_{0} asked in ( N ). In particular assumption ( N ) asks that, for every v\in{\mathcal{M}}_0 , the linearized operator about v

    L_{v}: W^{2, 2}([0, 1]^n, {\mathbb{R}}^m)\subset L^{2}([0, 1]^n, {\mathbb{R}}^m)\to L^{2}([0, 1]^n, {\mathbb{R}}^m)\, ,
    L_{v}h = -\Delta h+F_{u, u}(\cdot, v(\cdot))h

    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 \alpha_{0}>0 such that

    J_{0}''(v)h\cdot h = \int_{[0, 1]^n}|\nabla h(x)|^{2}+F_{u, u}(x, v(x))|h(x)|^2\, dx\geq \alpha_{0}\| h\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}^{2}

    for every h\in W^{1, 2}([0, 1]^n, {\mathbb{R}}^m) and every v\in{\mathcal{M}}_{0} .

    As a consequence of (N_1) we obtain the following (see also Lemma 3.6 in [2]).

    Lemma 4.8. There exist r_{2}\!\in \!(0, r_1) and \omega_1\!>\!\omega_0\!>\!0 such that if u\in W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m) , p\in{\mathbb{Z}} , verifies \|u-v\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq r_{2} for some v\in{\mathcal{M}}_0 then

    \begin{equation} \omega_{0}\|u-v\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)}^{2} \leq J_{p, 0}(u) \leq \omega_1\|u-v\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)}^{2}\, . \end{equation} (16)

    Proof. Notice that, by ( N_{1} ), if h\in W^{1, 2}([0, 1]^n, {\mathbb{R}}^m) and v\in{\mathcal{M}}_{0} then

    \begin{align*} \int_{[0, 1]^n} |\nabla h(x)|^{2}+F_{u, u}(x, v(x))|h(x)|^2\, dx& \geq \alpha_{0}\| h\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}^{2}\\&\geq -\alpha_{0}f_{0} \int_{[0, 1]^n}F_{u, u}(x, v(x))|h(x)|^2\, dx, \end{align*}

    where f_{0} = 1/\|F_{uu}\|_{\infty} , and so

    \int_{[0, 1]^n}\frac 1{1+\alpha_{0}f_{0}}|\nabla h(x)|^{2}\, dx+ \int_{[0, 1]^n} F_{u, u}(x, v(x))|h(x)|^2 \, dx \geq 0\, .

    We conclude that

    J_0''(v)h\cdot h = \int_{[0, 1]^n} |\nabla h(x)|^{2}+ F_{u, u}(x, v(x))|h(x)|^2 \, dx \geq \frac{\alpha_{0}f_{0}}{1+\alpha_{0}f_{0}}\|\nabla h\|^{2}_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}

    and so, using (N_{1}) and setting \omega_{0} = \frac{\alpha_0}6\min\{1, \frac{f_{0}}{1+\alpha_0 f_{0}}\} , we obtain

    \begin{equation*} \label{eq:**bis} J_{0}''(v)h\cdot h\geq 3\omega_{0}\|h\|_{W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2}, \quad\forall\, h\in W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m). \end{equation*}

    Since by Taylor's formula we have J_{0}(u)-c_{0} = \frac12J_{0}''(v)(u-v)\cdot(u-v)+o(\|u-v\|_{W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2}) for all v\in{\mathcal{M}}_0 and u\in W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m) , we obtain that there exists r_{2}\in (0, \frac{r_{1}}4) such that if u\in W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m) verifies \|u-v\|_{W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)}\leq r_{2} for some v\in{\mathcal{M}}_0 , then

    \begin{equation} J_{0}(u)-c_{0}\geq \omega_{0} \|u-v\|_{W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2}. \end{equation} (17)

    On the other hand, again Taylor's expansion gives us

    \begin{align*} J_{0}(u)-c_{0}& = \tfrac12J_{0}''(v)(u-v)\cdot(u-v)+o(\|u-v\|_{W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2})\\ & = \tfrac12 \|\nabla(u-v)\|_{L^{2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2}+\tfrac12\int_{[0, 1]^{n}}F_{u, u}(x, v(x))|u(x)-v(x)|^{2}\, dx\\ &\phantom{ = } + o(\|u-v\|_{W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2})\\ &\le \tfrac12 \|\nabla(u-v)\|_{L^{2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2}+\tfrac1{2f_{0}}\|u-v\|_{L^{2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2}\\ &\phantom{\le} +o(\|u-v\|_{W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2}) \end{align*}

    and we deduce that there exists \omega_{1}>\omega_{0} such that, taking r_{2} smaller if necessary, if u\in W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m) verifies \|u-v\|_{W^{1, 2}([0, 1]^{n})}\leq r_{2} , v\in{\mathcal{M}}_0 , then

    \begin{equation} J_{0}(u)-c_{0}\le\omega_{1}\|u-v\|_{W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2}. \end{equation} (18)

    The lemma follows by periodicity from (17) and (18) recalling that T_{p, 0} = [p, p+1]\times[0, 1]^{n-1} and that J_{p, 0}(u) = J_0(u(\cdot+p))-c_0 for all u\in W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m) .

    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_{\sigma_0}(u) = \int_{\sigma_{0}}L(u)\, dx-c_{0} on W^{1, 2}(\sigma_{0}, {\mathbb{R}}^m) , that is, if \|u-v\|_{W^{1, 2}(\sigma_{0}, {\mathbb{R}}^m)}\le r_{1} for some v\in{\mathcal{M}}_0 then

    \begin{equation} \omega_{0}\|u-v\|_{W^{1, 2}(\sigma_{0}, {\mathbb{R}}^m)}^{2}\le J_{\sigma_0}(u)\le \omega_{1}\|u-v\|_{W^{1, 2}(\sigma_{0}, {\mathbb{R}}^m)}^{2}. \end{equation} (19)

    Hence, recalling the definition (10), plainly adapting the proof of Lemma 3.10 in [2], we obtain

    Lemma 4.9. Let v_0\in{\mathcal{M}}_0^{min} and (u_{k})_k\subset \Gamma(v_0) be such that J(u_{k})\to c . Then there exists u\in{\mathcal{M}}(v_0) such that, up to a subsequence, \|u_{k}- u\|_{W^{1, 2}({\mathbb{R}}\times[0, 1]^{n-1}, {\mathbb{R}}^m)}\to 0 as k\to+\infty .

    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

    {\mathcal{T}} = \{ x\in {\mathbb{R}}^n \mid x_2 \geq |x_1| \}

    satisfying Neumann boundary conditions on \partial {\mathcal{T}} , which is odd in the first variable x_1 , asymptotic as x_2 \to +\infty to a certain heterocline v_0\in{\mathcal{M}} where

    {\mathcal{M}}: = \bigcup\limits_{v\in{\mathcal{M}}_0^{min}}{\mathcal{M}}(v).

    Then, by recursive reflections with respect to the hyperplanes x_2 = \pm x_1 , we will recover a solution of (PDE) on the whole {\mathbb{R}}^n .

    Let us introduce now some notations. We define the squares

    T_{p, k}: = [p, p+1] \times [k, k+1] \times [0, 1]^{n-2}\, , \quad p\in{\mathbb{Z}}, \, k\in{\mathbb{N}}

    and the horizontal strips

    {\mathcal{S}}_k : = {\mathbb{R}} \times [k, k+1] \times [0, 1]^{n-2} = \bigcup\limits_{p\in{\mathbb{Z}}} T_{p, k}\, , \quad k\in{\mathbb{N}}

    The intersection between the strip {\mathcal{S}}_k and the triangle {\mathcal{T}} consists of a bounded strip

    {\mathcal{T}}_k : = {\mathcal{S}}_k\cap {\mathcal{T}} = \left(\bigcup\limits_{p = -k}^{k-1} T_{p, k}\right) \cup \tau_k

    where \tau_k = \{x\in T_{k, k} \cup T_{-k-1, k} \mid x_2 \geq |x_1| \} .

    Figure 1. 

    The decomposition of the triangular set {\mathcal{T}}

    .

    For every k\in{\mathbb{N}} we define the sets of functions

    \begin{align*} E_{k} = \{u\in W^{1, 2}({\mathcal{T}}_k, {\mathbb{R}}^m) \mid u& \text{ is odd in } x_1, \, \text{1-periodic in }x_3, ..., x_n \} \end{align*}

    and the normalized functionals on the bounded strips {\mathcal{T}}_k as

    J_{k}(u) = \int_{{\mathcal{T}}_{k}} L(u) \, dx -(2k+1)c_0 = \sum\limits_{p = -k}^{k-1} J_{p, k}(u) + \int_{\tau_k} L(u) \, dx-c_0\, , \quad k\in{\mathbb{N}}\, ,

    for every u\in E_{k} , where J_{p, k}(u) = \int_{T_{p, k}} L(u) \, dx -c_0 .

    Remark 4. Notice that J_{k}(u)\geq0 for every u\in E_{k} , k\in{\mathbb{N}} . Indeed, we can view the restriction u|_{T_{p, k}} as a traslation of a function in W^{1, 2}([0, 1]^n, {\mathbb{R}}^m) and the restriction on u|_{\tau_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}({\mathcal{T}}_k, {\mathbb{R}}^m) topology for every k\in{\mathbb{N}} .

    Then, we can set

    c_{k} = \inf\limits_{E_{k}} J_{k}(u) \quad\text{ and }\quad {\mathcal{M}}_{k} = \{u\in E_{k} \mid J_{k}(u) = c_{k}\}\, .

    We plainly obtain that {\mathcal{M}}_{k} \neq \varnothing and that the sequence (c_{k})_k is increasing. Moreover, c_{k} \leq c , evaluating J_{k} on a function u\in {\mathcal{M}}(v_0) with v_0\in{\mathcal{M}}_0^{min} . Moreover, the non degeneracy assumption ( N_1 ) permits us to obtain as in [2] (see Lemma 4.2) the following stronger result.

    Lemma 5.1. We have \sum_{k = 0}^\infty \left(c-c_{k}\right) < +\infty , in particular c_k \to c as {k\to+\infty} .

    We can now introduce on the set

    {\mathcal{E}} = \{u\in W^{1, 2}_{loc}({\mathcal{T}}, {\mathbb{R}}^m) \mid u \text{ is odd in } x_1, \, u_i(x)\ge0 \text{ for } x_1\geq 0 \, , \forall i = 1, ..., m \}.

    the functional

    {\mathcal{J}}(u) = \sum\limits_{k = 0}^{+\infty} \left( J_{k}(u) - c_{k}\right)\, .

    Notice that {\mathcal{J}}(u)\geq 0 for every u\in {\mathcal{E}} . Indeed, the restriction u|_{{\mathcal{T}}_k}\in E_{k} and so J_{k}(u)\geq c_{k} for any k\in{\mathbb{N}} . Moreover, {\mathcal{J}} is lower semicontinuous in the weak topology of W^{1, 2}_{loc}({\mathcal{T}}, {\mathbb{R}}^m) . By Lemma 5.1 we readily obtain that {\mathcal{J}} is finite for at least one u\in {\mathcal{E}} .

    Lemma 5.2. If u\in{\mathcal{M}}(v_0) for some v_0\in{\mathcal{M}}_0^{min} , then {\mathcal{J}}(u)<+\infty .

    We now look for a minimum of the functional {\mathcal{J}} on {\mathcal{E}} , thus we set

    \tilde c = \inf\limits_{{\mathcal{E}}} {\mathcal{J}}(u) \quad \text{and} \quad \widetilde{{\mathcal{M}}} = \{u\in {\mathcal{E}} \mid {\mathcal{J}}(u) = \tilde c \, \}\, .

    Lemma 5.2, gives that \tilde c\in {\mathbb{R}} and we can prove the existence of the minimum applying the direct method of the Calculus of Variations (see e.g. the proof of Proposition 4.4 in [2]).

    Proposition 1. We have \widetilde{\mathcal{M}}\neq \varnothing .

    Arguing as in [2,4,6] (see e.g. the argument in Lemma 3.3 of [4] or Lemma 5.2 of [6]), we can prove that if u\in\widetilde{\mathcal{M}} then it is a weak solution of (PDE) on {\mathcal{T}} with Neumann boundary condition on \partial {\mathcal{T}} . Then we can conclude that every u\in\widetilde{\mathcal{M}} is indeed a classical {\mathcal{C}}^2 solution of (PDE). Finally, using ( F_3 ), we can recursively reflect w with respect to the hyperplanes x_2 = \pm x_1 , obtaining an entire solution w of (PDE) (see e.g. [2]). By construction, it is odd both in x_1 and x_2 , symmetric with respect to the hyperplanes x_1 = \pm x_2 and it is 1–periodic in x_{3}, ..., x_{n} . Hence, it satisfies hypotheses (ii) - (iii) of Theorem 1.1.

    In the next lemma we finally characterise the asymptotic behavior of the solution w .

    Lemma 5.3. Let w\in W^{1, 2}_{loc}({\mathbb{R}}^n, {\mathbb{R}}^m) be the function obtained by recursive reflection of a given w_0 \in\widetilde{\mathcal{M}} . Then there exists \bar v \in {\mathcal{M}}_0^{min} such that

    \lim\limits_{k\to +\infty} {\rm dist}_{W^{1, 2}({\mathcal{T}}_k, {\mathbb{R}}^m)} (w, {\mathcal{M}}(\bar v)) = 0.

    Proof. Let w be as in the statement, we start proving that there exists \bar v\in {\mathcal{M}}_0^{min} such that

    \begin{equation} \lim\limits_{k\to+\infty} \|w-\bar v\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)} = 0\, . \end{equation} (20)

    We have {\mathcal{J}}(w) = {\mathcal{J}}(w_0) = \tilde c<+\infty . Hence, J_{k}(w)- c_{k}\to 0 as k\to +\infty so that, by Lemma 5.1, J_{k}(w)\to c as k\to+\infty . Therefore, we can find a sequence (p_k)_{k\in{\mathbb{N}}} , with p_k\in[0, k-1]\cap{\mathbb{N}} such that J_{p_k, k}(w) \to 0 as k\to+\infty , and in particular J_{0}(w(\cdot +p_k \hskip1pt{{\mathit{\boldsymbol{e}}}}_1 + k \hskip1pt{{\mathit{\boldsymbol{e}}}}_2 ))\to c_0 . By Lemma 2.1-(3), we get {\rm dist}_{W^{1, 2}(T_{p_k, k}, {\mathbb{R}}^m)}(w, {\mathcal{M}}_0) = {\rm dist}_{W^{1, 2}([0, 1]^n, {\mathbb{R}}^m)}(w(\cdot +p_k \hskip1pt{{\mathit{\boldsymbol{e}}}}_1 + k \hskip1pt{{\mathit{\boldsymbol{e}}}}_2 ), {\mathcal{M}}_0) \to 0 as k\to +\infty thus giving the existence of v_k\in{\mathcal{M}}_0 such that

    \|w-v_k\|_{W^{1, 2}(T_{p_k, k}, {\mathbb{R}}^m)} \to 0\, , \text{ as } k\to +\infty.

    Now, for every k\in{\mathbb{N}} , we define in the horizontal strip {\mathcal{S}}_k the following interpolation between w and v_k :

    w_k(x_1, x_2, y) = \begin{cases} w(x_1, x_2, y) & \text{if } 0\leq x_1 \leq p_k\\ w(x_1, x_2, y)(p_k-x_1+1)\\ \phantom{w(x_1, x_2, y)} +v_k(x_1, x_2, y)(x_1-p_k) & \text{if } p_k < x_1 \leq p_k+1\\ v_k(x_1, x_2, y) & \text{if } x_1 > p_k+1\\ \text{odd extended for } x_1 < 0 \end{cases}

    A computation gives \|w_k-v_k\|_{W^{1, 2}(T_{p_k, k}, {\mathbb{R}}^m)} \leq 2 \|w-v_k\|_{W^{1, 2}(T_{p_k, k}, {\mathbb{R}}^m)} \to 0 so that

    \lim\limits_{k\to+\infty} J_{p_k, k}(w_k) = 0\, .

    Now, consider w_k^{\downarrow}(x) = w_k(x+k\hskip1pt{{\mathit{\boldsymbol{e}}}}_2) defined on {\mathcal{S}}_0 . We have w_k^{\downarrow}\in\Gamma(v_k) , therefore

    c \leq J(w_k^{\downarrow}) = 2 \sum\limits_{p = 0}^{p_k} J_{p, 0}(w_k^{\downarrow}) = 2 \sum\limits_{p = 0}^{p_k-1} J_{p, k}(w) + 2J_{p_k, k}(w_k) \leq J_{k}(w) + 2J_{p_k, k}(w_k)\, .

    and hence, since J_{k}(w) \to c and J_{p_k, k}(w_k)\to 0 , we obtain J(w_k^{\downarrow})\to c as k\to +\infty . As a consequence, since w_k^{\downarrow}\in \Gamma(v_k) , by (13), we can conclude that v_k\in{\mathcal{M}}_0^{min} . Moreover we have

    J_{k}(w) -J_{k}(w_k) = 2 \sum\limits_{p = p_k}^{k-1} J_{p, k}(w) + \int_{\tau_k} L(w) \, dx - c_0 - 2 J_{p_k, k}(w_k)

    and since J_{k}(w) \to c , J_{k}(w_k) = J(w_k^{\downarrow}) \to c and J_{p_k, k}(w_k) \to 0 , we obtain

    \begin{equation} 2 \sum\limits_{p = p_k}^{k-1} J_{p, k}(w) + \int_{\tau_k} L(w) \, dx - c_0 \to 0\, , \text{ as } k\to +\infty\, . \end{equation} (21)

    In particular \int_{\tau_k} L(w) \, dx - c_0\to 0 , so that \lim_{k\to+\infty} J_{k, k}(w) = 0 , by the symmetry of w with respect to x_2 = \pm x_1 . Summing up, using (21), we get \sum_{p = p_k}^{k} J_{p, 0}(w(\cdot + k \hskip1pt{{\mathit{\boldsymbol{e}}}}_2)) = \sum_{p = p_k}^{k} J_{p, k}(w) \to 0 , so we can apply Lemma 3.2 and conclude that

    \begin{equation} \|w-v_k\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)}\to 0\, . \end{equation} (22)

    Let us now consider, for every k\in{\mathbb{N}} , a different interpolation in the horizontal strip {\mathcal{S}}_k between w and the periodic solution v_k\in{\mathcal{M}}_0^{min} previously introduced:

    \omega_k(x_1, x_2, y) = \begin{cases} w(x_1, x_2, y) & \text{if } 0\leq x_1 \leq k\\ w(x_1, x_2, y)(k-x_1+1)\\ \phantom{w(x_1, x_2, y)}+v_k(x_1, x_2, y)(x_1-k) & \text{if } k < x_1 \leq k+1\\ v_k(x_1, x_2, y) & \text{if } x_1 > k+1\\ \text{odd extended for } x_1 < 0 \end{cases}

    Arguing as above \|\omega_k-v_k\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)} \leq 2 \|w-v_k\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)} , so that, defining \omega_k^{\downarrow}(x) = \omega_k(x+k \hskip1pt{{\mathit{\boldsymbol{e}}}}_2) in {\mathcal{S}}_0 we find \|\omega_k^{\downarrow}-v_k\|_{W^{1, 2}(T_{k, 0}, {\mathbb{R}}^m)} \to 0 {and hence, } J_{k, 0}(\omega_k^{\downarrow}) \to 0 . Since \omega_k^{\downarrow}\in \Gamma(v_k) and v_k\in{\mathcal{M}}_0^{min} we obtain, reasoning as above,

    c \leq J(\omega_k^{\downarrow}) \leq J_{k}(w) + 2 J_{k, 0}(\omega_k^{\downarrow}) = c + o(1) \, ,

    thus giving J(\omega_k^{\downarrow})\to c .

    We now prove that the sequence (v_k)_k\in{\mathcal{M}}_0^{min} is indeed a (definitively) constant sequence, i.e. v_k = \bar v for every k sufficiently large. Being J(\omega_k^{\downarrow})\to c , we can assume J(\omega_k^{\downarrow}) \leq c + \tilde\Lambda(r_1) and since \omega_k^{\downarrow}\in \Gamma(v_k) and v_k\in{\mathcal{M}}_0^{min} , we can apply Lemma 4.5 obtaining that

    (i) \| \omega_k^{\downarrow} -v_k\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq r_1 \text{ for every } p\geq \tilde\ell(r_1) ;

    (ii) \sum_{p = \tilde\ell(\rho)}^{+\infty}J_{p, 0}(\omega_k^{\downarrow}) \leq 2\tilde \Lambda(r_1)< \tfrac{\lambda_0}{4} ;

    As a consequence, by definition of \omega_k^{\downarrow} and recalling that \omega_k = w when 0\le x_1\leq k we obtain J_{p, k}(w) < \tfrac{\lambda_0}{4} and

    \begin{align} &\|w-v_k\|_{W^{1, 2}(T_{p, k}, {\mathbb{R}}^m)} \leq r_1 < \tfrac{r_0}{4} \end{align} (23)

    provided that p_0\leq p\leq k-1 where p_0 = \tilde \ell(r_1) . Consider now the vertical rectangle [ p_0, p_0+1]\times[p_0+1, +\infty)\times{[0, 1]^{n-2}} = \cup_{k\geq p_0+1} T_{p_0, k} . We have J_{p_0, k}(w) \leq \tfrac{\lambda_0}{4} for any k in the set of consecutive integers \mathcal I = \{k\in{\mathbb{Z}} \mid k\geq p_0+1\} , so that we can argue as in Lemma 3.2 and conclude that there exists \bar v \in {\mathcal{M}}_0 such that

    \begin{equation} \|w-\bar v\|_{W^{1, 2}(T_{ p_0, k}, {\mathbb{R}}^m)} \leq \tfrac{r_0}{4} \text{ for every } k\geq p_0+1\, . \end{equation} (24)

    Finally, recalling (6), since both (23) and (24) holds, we must have \bar v = v_k\in{\mathcal{M}}_0^{min} for every k \geq p_0+1 . In particular, (22) gives the claim in (20).

    Moreover, we have proved that (\omega_k^{\downarrow})_{k\ge p_0+1} \subset \Gamma(\bar v) with \bar v\in{\mathcal{M}}_0^{min} and since J(\omega_k^{\downarrow})\to c , we can apply Lemma 4.9 to get that there exists \bar u\in{\mathcal{M}}(\bar v) for which, up to a subsequence,

    \begin{equation*} \lim\limits_{k\to +\infty} \| \omega_k^{\downarrow} - \bar u \|_{W^{1, 2}({\mathcal{S}}_0, {\mathbb{R}}^m)} = 0\, . \end{equation*}

    Hence we obtain that

    \begin{equation} {\rm dist}_{W^{1, 2}({\mathcal{S}}_0, {\mathbb{R}}^m)}( \omega_k^{\downarrow} , {\mathcal{M}}(\bar v))\to 0\quad\text{as }k\to +\infty. \end{equation} (25)

    Finally, for every u\in {\mathcal{M}}(\bar v) we have

    \begin{align*} \|w-u\|_{W^{1, 2}({\mathcal{T}}_k, {\mathbb{R}}^m)}^2 & = 2\|w-u\|_{W^{1, 2}(\cup_{p = 0}^{k-1}T_{p, k}, {\mathbb{R}}^m)}^2 + \|w- u\|_{W^{1, 2}(\tau_k, {\mathbb{R}}^m)}^2\\ & = \|\omega_k^{\downarrow}- u\|_{W^{1, 2}({\mathcal{S}}_0, {\mathbb{R}}^m)}^2 - 2 \|\omega_k^{\downarrow}-u\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)}^2\\ &\phantom{ = } + \|w- u\|_{W^{1, 2}(\tau_k, {\mathbb{R}}^m)}^2\\ &\leq \|\omega_k^{\downarrow}- u\|_{W^{1, 2}({\mathcal{S}}_0, {\mathbb{R}}^m)}^2 + \|w- u\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)}^2\, . \end{align*}

    Notice that since u\in\Gamma(\bar v) and using (20), we have

    \begin{multline*} \|w- u\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)}^2 \leq \|w-\bar v\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)}^2 \\ + \| u-\bar v\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)}^2 \to 0\, , \text{ as } k\to +\infty\, . \end{multline*}

    Hence, by (25), we conclude

    \lim\limits_{k\to +\infty} {\rm dist}_{W^{1, 2}({\mathcal{T}}_k, {\mathbb{R}}^m)} (w, {\mathcal{M}}(\bar v)) = 0 \, .

    The previous lemma gives the asymptotic estimate in Theorem 1.1 since {\mathcal{R}}_k\subset{\mathcal{T}}_k .

    We can conclude now the proof of Theorem 1.1 proving the sign property (i) . By Lemma 2.2, for any periodic solution v = (v_1, \ldots, v_m)\in{\mathcal{M}}_0^{min} we can define v^a = (|v_1|, \ldots, |v_m|) belonging to {\mathcal{M}}_0^{min} too, being J_0(v^a) = J_0(v) = c_0 easily verified. Now, by Theorem 4.6, there exists a heteroclinic solution u = (u_1, \ldots, u_m)\in{\mathcal{M}}(v) . We can define the function u^a\in E^{odd} , such that u^a = (|u_1|, \ldots, |u_m|) when x_1\geq 0 , and verify that u^a\in{\mathcal{M}}(v^a) being J(u^a) = J(u) = c .

    Finally, for any w = (w_1, \ldots, w_m)\in\widetilde{\mathcal{M}} we can find v\in{\mathcal{M}}_0^{min} as in Lemma 5.3. Similarly as above, we can define w^a\in{\mathcal{E}} such that w^a = (|w_1|, \ldots, |w_m|) when x_1\geq 0 . Then, we can verify that w^a\in \widetilde{\mathcal{M}} verifies Lemma 5.3 with the choice v^a\in{\mathcal{M}}_0^{min} . By reflecting w^a with rispect to the hyperplanes x_2 = \pm x_1 , we obtain the saddle-type solution satisfying (i) in Theorem 1.1, thus completing the proof.



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