Feasibility of in-situ evaluation of soil void ratio in clean sands using high resolution measurements of Vp and Vs from DPCH testing

1.   Introduction

We consider systems of semilinear elliptic equations

where

($F_1$) $F\in C^{2}({\mathbb{R}}^{n}\times {\mathbb{R}}^m;{\mathbb{R}})$ is $1$-periodic sin all its variable, $n, m\geq 1$.

When $n = 1$ and $m\geq 1$, (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 $\mathcal{M}_{0}$ of minimal periodic solutions of (PDE) is a non empty ordered set and if ${\mathcal{M}}_0$ is not a continuum then there exists another ordered family $\mathcal{M}_{1}$ of minimal entire solutions which are heteroclinic in one space variable to a couple of (extremal) periodic solutions $u<v$ (a gap pair in ${\mathcal{M}}_{0}$). If $\mathcal{M}_{1}$ 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 ${\mathcal{M}}_{0}$ 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 $L(u) = \frac 12|\nabla u|^{2} + F (x, u)$ and ${\mathbb{T}}^{n} = {\mathbb{R}}^{n}/{\mathbb{Z}}^{n}$, periodic solutions to (PDE) were found as minima of the functional $J_{0}(u) = \int_{{\mathbb{T}}^{n}}L(u) dx$ on $E_{0} = W^{1, 2}({\mathbb{T}}^{n}, {\mathbb{R}}^{m})$ showing that

Paul H. Rabinowitz studied the case of spatially reversible potentials $F$ assuming ($\overline F_2$) $F$ is even in $x_{i}$ for $1\leq i\leq n$ and proved in [29] that if ${\mathcal{M}}_{0}$ is constituted by isolated points then for each $v_{-}\in {\mathcal{M}}_{0}$ there is a $v_{+}\in {\mathcal{M}}_{0} \setminus\{v_{-}\}$ and a solution $u\in C^{2}({\mathbb{R}}\times{\mathbb{T}}^{n-1}, {\mathbb{R}}^{m})$ of (PDE) that is heteroclinic in $x_{1}$ from $v_{-}$ to $v_{+}$. These solutions were found by variational methods minimizing the renormalized functional

(where $T_{p, 0} = [p, p+1]\times[0, 1]^{n-1}$) on the space

In [30] the existence of minimal double heteroclinics was obtained assuming that the elements of ${\mathcal{M}}_{0}$ are not degenerate critical points of $J_{0}$ and that the set ${\mathcal{M}}_{1}(v_{-}, v_{+})$ of the minima of $J$ on $\Gamma(v_{-}, v_{+})$ 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 $-\Delta u+W'(u) = 0$ on ${\mathbb{R}}^{2}$ with $W$ an even double well potential. They proved the existence of a (unique) saddle solution $v\in C^{2}({\mathbb{R}}^{2})$ 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 ${\mathcal{M}}_{0}$ of minimal periodic solution of (PDE) is not empty. The symmetry of $F$ implies that any $v\in{\mathcal{M}}_{0}$ has components whose sign is constant on ${\mathbb{R}}^{n}$ and if $v\in{\mathcal{M}}_{0}$ then $(\nu_1 v_1, \ldots, \nu_m v_m)\in{\mathcal{M}}_{0}$ for every $(\nu_1, \ldots, \nu_m)\in\{\pm1\}^m$ (see Lemma 2.2 below). In this sense we can say that ${\mathcal{M}}_{0}$ is symmetric with respect to the constant function $v_{0}\equiv0$.

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

($N$) $0\notin{\mathcal{M}}_{0}$ and there exists $\alpha_{0}>0$ such that

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

The assumption ($N$) and the symmetries of $F$ allow us to find heteroclinic connections between elements of ${\mathcal{M}}_{0}$ which are odd in the variable $x_{1}$. More precisely for $v\in{\mathcal{M}}_{0}$ these solutions are searched as minima of the functional $J$ (see (1)) on the space

In §4, setting

we show that

and that ${\mathcal{M}}_{0}^{min}$ is such that if $v_{0}\in{\mathcal{M}}_{0}^{min}$ then

is not empty and compact with respect to the $W^{1, 2}({\mathbb{R}}\times{\mathbb{T}}^{n-1}, {\mathbb{R}}^{n})$ metric. The elements $u\in{\mathcal{M}}(v_{0})$ are classical solutions to (PDE), odd in $x_{1}$, even and $1$-periodic in $x_{2}, ..., x_{n}$ and satisfy the asymptotic condition

Our main result can now be stated as follows

Theorem 1.1. Assume $(F_1)$, $(F_{2})$, $(F_3)$ and $(N)$. Then, there exists a classical solution $w$ of (PDE) such that every component $w_i$ (for $i = 1\ldots, m$) satisfies

$(i)$ $w_i\geq 0$ for $x_{1}x_{2}>0$;

$(ii)$ $w_i$ is odd in $x_{1}$ and $x_{2}$, $1$-periodic in $x_{3}, ..., x_{n}$;

$(iii)$ $w_i(x_{1}, x_{2}, x_{3}, ..., x_{n}) = w_i(x_{2}, x_{1}, x_{3}, ..., x_{n})$ in ${\mathbb{R}}^{n}$.

Moreover there exists $v_{0}\in{\mathcal{M}}_{0}^{min}$ such that the solution $w$ satisfies the asymptotic condition

where ${\mathcal{R}}_k = [-k, k]\times[k, k+1]\times[0, 1]^{n-2}$.

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 $x_{2}\to+\infty$ to the compact set ${\mathcal{M}}(v_{0})$ 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

($F_4$) $F\geq 0$ on ${\mathbb{R}}^{n}\times{\mathbb{R}}^{m}$.

Let us introduce the set

with the norm

We define the functional $J_0 : E_0 \to {\mathbb{R}}$ as

and consider the minimizing set

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

Lemma 2.1. Assume $(F_1)$, then ${\mathcal{M}}_{0}\neq \varnothing$. Moreover, setting $[u] = \int_{[0, 1]^n} u \, dx$, we have that

1. $\hat {\mathcal{M}}_0 = \{u\in {\mathcal{M}}_0 \mid [u]\in [0, 1]^m \}$ is a compact set in $E_0$;

2. if $(u_k)_k \subset E_0$, with $[u_k]\in[0, 1]^m$, is a minimizing sequence for $J_0$, then there exists $u\in \hat{\mathcal{M}}_0$ such that $u_k\to u$ in $E_0$ up to subsequences;

3. For every $\rho>0$ there exists $\beta(\rho)>0$ such that if $u\in E_0$ is such that

then $J_0(u) - c_0 \geq \beta(\rho)$;

4. If $(F_2)$ holds, then any $u\in{\mathcal{M}}_0$ minimizes also $I(u) = \int_{[0, \frac{1}{2}]^n} L(u) \, dx$ on $W^{1, 2}([0, \frac{1}{2}]^n, {\mathbb{R}}^m)$. As a consequence, every $u\in{\mathcal{M}}_0$ is symmetric in $x_i$ about $x_i = 0$ and $x_i = \frac12$ for every index $i$ and $u$ is even in $x_i$ for every index $i$;

5. If $(F_2)$ holds, there results $c_0 = \inf_{u\in W^{1, 2}([0, 1]^n, {\mathbb{R}}^m)} J_0(u)$. Furthermore, if $u\in W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^{m})$ verifies $J_{0}(u) = c_{0}$, then for every $i = 1, 2, ..., n$, $u$ is symmetric in $x_i$ about $x_i = \frac12$ and hence $u\in{\mathcal{M}}_0$.

Assumption $(F_2)$, in particular the even parity of $F$ with respect to the components of $u$, provides that the elements in ${\mathcal{M}}_{0}$ have components with definite sign, thanks to the unique extension property (see [29], Proposition 3).

Lemma 2.2. Assume $(F_1)$, $(F_2)$ and $0\notin{\mathcal{M}}_0$. If $u = (u_1, \ldots, u_m)\in{\mathcal{M}}_{0}$ then, for every $i = 1, ..., m$, one has either $u_i\ge 0$, or $u_i\le 0$ on $[0, 1]^n$ and $u$ does not vanish on open sets. Moreover, $(\nu_1 u_1, \ldots, \nu_m u_m)\in{\mathcal{M}}_{0}$ for every $(\nu_1, \ldots, \nu_m)\in\{\pm1\}^m$.

Proof. It is sufficient to observe that if $u = (u_{1}, \ldots, u_{m})\in{\mathcal{M}}_{0}$ then, since $F$ is even with respect to the components of $u$, we have

ⅰ) $\bar u = (|u_{1}|, \ldots, |u_{m}|)\in{\mathcal{M}}_{0}$ and

ⅱ) $(\nu_1 u_1, \ldots, \nu_m u_m)\in{\mathcal{M}}_{0}$ for every $(\nu_1, \ldots, \nu_m)\in\{\pm1\}^m$.

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 $u\equiv 0$, giving a contradiction and concluding the proof.

On the other hand, assumption $(F_{3})$ gives more structure on the set ${\mathcal{M}}_0$: its elements have a flip symmetry property. Indeed, setting $T^+ = \{x\in [0, 1]^n\, | \; x_1\le x_2\}$, for every $u\in W^{1, 2}(T^+, {\mathbb{R}}^m)$, let us define $\tilde u\in W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)$ as

Then, we have

Lemma 2.3. If $u\in{\mathcal{M}}_{0}$ then, $u\equiv \tilde u$ in $[0, 1]^{n}$.

Proof. Given $u\in {\mathcal{M}}_0$, without loss of generality, we assume

Since $\tilde u \in W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)$ by Lemma 2.1-(5) we have $J_{0}(\tilde u)\geq c_0$. By the previous inequality we get

Hence, again by Lemma 2.1-(5), $\tilde u\in {\mathcal{M}}_0$. By the unique extension property of the solutions of (PDE) (cf. [29], Proposition 3), we have $\tilde u\equiv u$ in $[0, 1]^n$.

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

Lemma 2.4. There results

Moreover, if $u\in W^{1, 2}(T^{+}, {\mathbb{R}}^m)$ verifies $\int_{T^{+}}L(u)\, dx = \tfrac{c_{0}}2$, then $\tilde u\in{\mathcal{M}}_{0}$.

Remark 1. Lemma 2.3 tells us that the elements of ${\mathcal{M}}_{0}$ are symmetric with respect to the diagonal iperplane $\{x\in{\mathbb{R}}^{n}\mid x_{1} = x_{2}\}$ and by Lemma 2.4 they can be found by minimizing $\int_{T^{+}} L(v)\, dx$ on $W^{1, 2}(T^+, {\mathbb{R}}^m)$. Analogously, setting $T^{-} = [0, 1]^{n}\setminus T^{+}$, we can find the elements of ${\mathcal{M}}_{0}$ by minimizing $\int_{T^{-}} L(v)\, dx$ on $W^{1, 2}(T^-, {\mathbb{R}}^m)$ or, by periodicity, by minimizing $\int_{T} L(v)\, dx$ on $W^{1, 2}(T, {\mathbb{R}}^m)$ whenever $T = p+T^{\pm}$ with $p\in{\mathbb{Z}}^{n}$. For future references it is important to note that this property implies in particular that $u\in{\mathcal{M}}_{0}$ if and only if $u$ is a minimizer of the functional $\int_{\sigma_{0}} L(v)\, dx$ on $W^{1, 2}(\sigma_{0}, {\mathbb{R}}^m)$ where

More precisely we have $c_{0} = \inf_{v\in W^{1, 2}(\sigma_{0}, {\mathbb{R}}^m)}\int_{\sigma_{0}}L(v)\, dx$ and $u\in{\mathcal{M}}_{0}$ if and only if $\int_{\sigma_{0}}L(u)\, dx = c_0$. From Lemma 2.1-(3) we recover an analogous property in $W^{1, 2}(\sigma_{0}, {\mathbb{R}}^m)$: for any $r>0$ there exists $\beta(r)>0$ such that if $u\in W^{1, 2}(\sigma_{0}, {\mathbb{R}}^m)$ verifies $\int_{\sigma_{0}}L(u)\, dx \leq c_0 + \beta(r)$, then ${\rm dist}_{W^{1, 2}(\sigma_{0}, {\mathbb{R}}^m)}(u, {\mathcal{M}}_0)\leq r$.

Note that by Lemma 2.1-(1) and the assumption (N) we plainly derive that ($N_0$) $\hat {\mathcal{M}}_0$ is a finite set and $0\notin \hat {\mathcal{M}}_0$,

where we recall that $\hat{\mathcal{M}}_0 = \{u\in {\mathcal{M}}_0 \mid [u]\in [0, 1]^m \}$ and note that ${\mathcal{M}}_0 = \hat{\mathcal{M}}_0+{\mathbb{Z}}^m$.

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

For any $u\in E$ we consider the functional

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

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

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

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),

and the set of minima coincide with ${\mathcal{M}}_0$. 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 $\lambda(r)>0$ such that

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

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

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

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

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

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

On the other hand, by Hölder inequality,

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

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

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

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

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

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

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

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

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 $\|v^+_k-v^-_k\|_{W^{1, 2}({\mathbb{T}}^n, {\mathbb{R}}^m)}\to+\infty$.

4.   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 $\Gamma(-v, v)$, $v\in{\mathcal{M}}_0$, so let us introduce the set

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

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

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

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

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 $u\in E^{odd}$ 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

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

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

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 $k\in{\mathbb{N}}$,

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

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

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 $v$

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

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

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

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

We conclude that

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

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

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

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

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

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 $k\to+\infty$.

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

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

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

and the horizontal strips

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

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

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

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

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

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

the functional

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

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

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

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

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

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

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

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

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

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

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:

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,

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

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

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,

Hence we obtain that

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

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

Hence, by (25), we conclude

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.