Layered solutions for a nonlocal Ginzburg-Landau model with periodic modulation

1.   Introduction

In this paper, we consider minimization of the following nonlocal energy functional:

in the set

Here $\alpha, \beta$ are positive constants, $g\left(x\right)$ is a $1$-periodic (a general period $T > 0$ can be treated similarly) nonnegative function whose reciprocal satisfies the following integrability assumption

The fixed function $\eta \in C^{\infty }\left(\mathbb{R}\right)$ satisfies $\left\vert \eta \right\vert \leq 1,$ $\eta \left(x\right) = 1$ for $x\geq 1,$ $\eta(x) = -1$ for $x\leq -1,$ $W\left(u\right)$ is a double well potential satisfying

Formally, for $u \in C^{2}_{loc}(\mathbb{R}) \cap L^\infty(\mathbb{R})$ the Euler-Lagrange equation associated with (1.1) is

where

We are mainly interested in solutions of (1.4) that satisfy

we call such solutions layered solutions.

Equation (1.4) is a special case of the more general equation

where $0 < s < 1$. For $u\in C_{loc}^{2}\left(\mathbb{R}^{n}\right) \cap L^{\infty }\left(\mathbb{R}^{n}\right)$ the operator $\left(-\Delta \right) ^{s}$ is the fractional Laplacian defined by

where $C_{n, s}$ is a normalization constant to guarantee that the Fourier symbol of the resulting operator is $\left\vert \xi \right\vert ^{2s},$ see e.g., ^[25], Section 3 for more details.

When $g\left(x\right) = \gamma$ is a constant, (1.7) reduces to

This type of equation has attracted a lot of attention over the last twenty years (see e.g., ^{[17,25,26,27,28,47,49,53,64]}). In particular, the structure of layered solutions in the case $\beta = 0$ (Allen-Cahn) or $\alpha = 0$ (fractional Allen-Cahn) is well understood at present. Here a layered solution of (1.8) is a bounded solution which is monotone in one direction. When $\beta = 0,$ De Giorgi conjecture posits that the level sets of such a layered solution are hyperplanes for $n\leq 8.$ De Giorgi's initial conjecture was for $W\left(u\right) = \frac14(1 - u^2)^2.$ This conjecture was proved for any $C^{2}$ function $W(u)$ satisfying (1.3) by Ghoussoub and Gui ^[42] when $n = 2$. When $n = 3,$ Ambrosio and Cabré ^[10] proved the conjecture for a large class of $W(u)$ which includes the original De Giorgi's choice. Later, Alberti, Ambrosio and Cabré ^[2] extended their results to cover all $C^{2}$ function $W(u)$ with the properties specified in (1.3). Under the additional assumption of anti-symmetry of solutions, Ghoussoub and Gui ^[43] established the De Giorgi conjecture for $n = 4, 5.$ Further developments on the conjecture can be found in ^[12]. De Giorgi conjecture was completely solved by Savin ^[60,61] for $4\leq n\leq 8$ under the additional assumption $\lim_{x_{n}\rightarrow \pm \infty }u(x) = \pm 1.$ For dimensions $n\geq 9,$ a counter-example was constructed by Del Pino, Kowalczyk and Wei ^[31]. A weaker version of the De Gigorgi conjecture, known as Gibbons conjecture, replaced monotonicity assumption by the stronger condition

Gibbons conjecture was proved in all dimensions ^[12,13,40].

De Giorgi's conjecture has also been extended to the fractional Allen-Cahn case. The fractional De Giorgi conjecture was proved in ^{[25,26,27,63]} for the case $n = 2, s\in (0, 1)$, and in ^[22,23] for $n = 3$ and $s\geq \frac{1}{2}.$ Under additional limit conditions, fractional De Giorgi conjecture was proved for $n = 3$ and $s\in \left(0, \frac{1}{2}\right)$ by Dipierro, Serra and Valdinoci in ^[37] and by Savin in ^[62] for $4\leq n\leq 8$ and $s\in \lbrack \frac{1}{2}, 1)$. The limit condition is removed in ^[34] for $n = 3$ and $s\in \left(0, \frac{1}{2}\right)$. Recently, Figalli and Serra ^[41] solved the De Giorgi conjecture for half-Laplacian when $n = 4$ (such a result is not known for the classical case $s = 1$). Based on all these results, when $g\left(x\right)$ is a constant, solutions to (1.8) satisfying (1.6) reduces to the unique one-dimensional solution (modulo translation) which is monotone and the problem is essentially one-dimensional.

When $g\left(x\right)$ is not constant, but rather periodic, the continuous translational symmetry of layered solutions of (1.7) is broken and the structure of the set of solutions is much more complex. When $\beta = 0,$ the nonautonomous Allen-Cahn equation

with

has been studied extensively over the last three decades. Equation (1.10) is a special case of a model problem initiated by Moser ^[48] for developing a PDE version of Aubry-Mather theory of monotone twist maps (see ^{[11,14,15,57,58]} for related work). A different motivation is to view (1.10) as a model for phase transitions. When $n = 1$ and subject to homogeneous Neumann boundary conditions on the interval of $x \in \left(0, 1\right),$ the following results have been proved for (1.10) for various choices of the potential term $W(x, u).$ Angenent, Mallet-Paret and Peletier ^[1] gave a complete classification of all stable equilibrium solutions to (1.10) for $W_{u}(x, u) = - u\left(1-u\right) (u - a(x)).$ Existence and stability of equilibrium solutions with a single transiton layer is proved in ^[44] for a general class of $W_{u}\left(x, u\right) = - f\left(x, u\right)$, with $f$ satisfying $f\left(x, 0\right) = f\left(x, 1\right) = 0$ and $f\left(x+k, u\right) = f\left(x, u\right)$ for some $k > 0$. Nakashima ^[50] proved existence of stable solutions with multiple transition layers for the case $W_{u}(x, u) = \left(u-a\left(x\right) \right) \left(u-b\left(x\right) \right) (u-c\left(x\right)).$ Existence and stability of multilayered solutions were provided in ^[51] for $W_{u}(x, u) = h^{2}\left(x\right) f\left(u\right)$. Nakashima and Tanaka ^[52] studied the one-dimensional case with a general potential $W(x, u)$ and obtained existence of solutions with clustering layers. For higher dimensions and the special case of

Alessio, Jeanjean and Montecchiari ^[5] proved existence of infinitely many solutions which are distinct up to periodic translations and satisfy $\lim_{x_{1}\rightarrow \pm \infty }u\left(x_{1}, x_{2}\right) = \pm b$ uniformly in $x_{2}$ for the case $n = 2$ when $a\left(x_{1}, x_{2}\right)$ is a positive, even, periodic function in $x_1, x_2$, and $W(u)$ is a double well potential vanishing at $u = \pm b$ for some $b > 0$. For the same equation, Alessio and Montecchiari ^[7] showed existence of brake orbits type solutions, and Alessio, Gui and Montecchiari ^[3] proved existence and asymptotic behavior of saddle solutions. When $a (x_1, x_2)$ depends only on one variable, existence of two-dimensional solutions was proved by Alessio and Montecchiari ^[4], and existence of infinitely many solutions can be found in ^[6,66]. Alessio and Montecchiari ^[8] proved existence of infinitely many solutions verifying $\lim_{x_{1}\rightarrow \pm \infty }u\left(x_{1}, x_{2}, x_{3}\right) = \pm 1$ uniformly in $\left(x_{2}, x_{3}\right)$ for $n = 3$ and $a = a(x_1)$. For results on solutions to (1.10) for $n = 2$ with general potentials, see the papers by Rabinowitz and Stredulinsky ^[55,56]. Existence of various (multi-layer, mountain pass or higher topological complexity) solutions to (1.11) for general $n$ was obtained in a series of papers by Byeon and Rabinowitz ^{[18,19,20,21]}. A review on existence results for (1.10) is given in ^[54] (see book ^[59] for a more thorough review on extensions of Moser-Bangert theory).

An extensive discussion on moving front solutions for time-dependent inhomogeneous Allen-Cahn equation can also be found in the literature. For example, Xin ^[65] considered propagating front solutions (which include stationary layered solutions) for

when $a(x)$, $b(x)$ are periodic and $f(u)$ is bistable. Keener ^[46] studied propagation of waves in periodic media for the following model:

where $g(x)$ is a 1-periodic function and obtained a nearly complete picture of propagation in periodic medium. In particular, his results show how wave front shape changes when the medium becomes more and more nonuniform, and how propagation failure occurs when the medium becomes sufficiently nonuniform. Pinning and de-pinning phenomena for front propagation in heterogenous media was discussed in ^[38]. Existence and qualitative properties of pulsating travelling wave solutions is proved in ^[33] for equation

where $a(x)$ and $f(x, u)$ are 1-periodic in $x$.

Studies of layered solution in the fractional case when $g\left(x\right)$ is not constant caught less attention. Existence of layered solutions to

was obtained in ^[45] for $s\in [\frac{1}{2}, 1)$ when $g > 0$ is an even, periodic function and $W^{\prime }\left(u\right)$ is odd. Another related work is ^[39], where the authors studied existence of multi-layered solution to the following equation:

when $g\left(x\right)$ is not constant and $s \in (\frac{1}{2}, 1).$ Existence of heteroclinic orbits was proved in ^[9] for equation

in each of the three cases of $g$, namely, $g$ (asymptotically) periodic, $g$ coercive and $g$ satisfying Rabinowitz's condition. For a more general nonlocal operator in the form

where

heteroclinic orbits were constructed for $s\in \left(\frac{1}{2}, 1\right]$ in ^[35] (see also ^[30]) and for $s\in \left(\frac{1}{4}, \frac{1}{2}\right]$ in a later work ^[36] for the nonlocal equation

with an oscillatory $g.$

The mixed case, with local and nonlocal operators ($\alpha \neq 0$ and $\beta \neq 0$), has gained attention in the last years at least in the autonomous case. For example, layered solutions of Allen-Cahn type equations in the form of a sum of fractional Laplacians of different orders was addressed in ^[24]. Systematic study on regularity and maximum principles for mixed local and nonlocal operators in the form $\mathcal{L} = - \Delta+\left(-\Delta \right) ^{s}$ has recently been developed in ^[16]. The work in the current paper is partly motivated by a recent work by the authors ^[29] where we considered the following renormalized nonlocal Ginzburg-Landau energy

We proved existence, regularity, monotonicity and uniqueness (up to translation) of the minimizer of $E_{\varepsilon }\left(u\right)$ in $\mathcal{A}.$ Moreover, as $\varepsilon \rightarrow 0$ we recovered the solution in ^[53] as the global minimizer (unique up to translations) of

The proof of existence and uniqueness of minimizers in ^[29] relies on an essential observation that a minimizer of $E_{\varepsilon }$ among all functions satisfying $u-\eta \in W_{0}^{1, 2}$ $\left(I\right)$ on any sufficiently large fixed interval $I$ is monotone. Such conclusion follows from the key assumption that (1.21) is translation invariant. For model (1.1) with only discrete translation invariance, this argument fails and we need to seek a new method. The main difficulty to prove the existence of minimizer of (1.1) lies in two parts. Firstly, since $\eta \notin \mathring{H}^{\frac{1}{2}}\left(\mathbb{R}\right),$ it is not a priori clear that $J\left(u\right)$ is bounded from below on $\mathcal{A}.$ Secondly, the energy bound does not necessarily imply the boundedness of $u$ in a suitable Sobolev space in general. Therefore, we cannot a priori apply the direct method of calculus of variations to obtain a minimizer. To show that $J$ is bounded from below on $\mathcal{A},$ we divide the real line into the regions where $u$ is close to $\pm 1$ and where $u$ is away from $\pm 1.$ By carefully matching the contributions from each region, all negative parts of the potential infinite energy are cancelled out.

To prove the existence of a minimizer, our main idea is as follows. Given an arbitrary minimizing sequence $\left\{ u_{n}\right\}$, we replace this sequence by another sequence $\left\{ \overline{u}_{n}\right\}$ constructed via reflecting the negative parts of $u_{n}$ outside suitable regions. Taking into account our energy estimates from the lower bound argument, we can carefully choose the region where we apply the reflection to $u_{n}$ so that the energy $J\left(\overline{u}_{n}\right)$ differs only slightly from $J\left(u_{n}\right).$ The sequence $\left\{ \overline{u} _{n}\right\}$ satisfies $\left\vert \overline{u}_{n}\left(x\right) +sgn(x)\right\vert \geq c > 0$ outside a uniformly bounded interval. For such a sequence, boundedness of energy implies boundedness of $\overline{u} _{n}-\eta$ in $H^{1}\left(\mathbb{R}\right).$ From this and a lower semicontinuity argument, we obtain a limit function which attains a minimum of $J\left(u\right)$ in $\mathcal{A}.$

Our main result is the following existence and regularity theorem.

Theorem 1.1. Let $\alpha, \beta$ be positive constants. Assume $g \in C^\infty(\mathbb{R})$ is a nonnegative 1-periodic function which satisfies (1.2), $\eta \in C^{\infty }\left(\mathbb{R}\right)$ is a given function satisfying $\left\vert \eta \right\vert \leq 1,$ $\eta \left(x\right) = 1$ for $x\geq 1,$ $\eta(x) = -1$ for $x\leq -1$, $W\left(u\right)$ is a double well potential satisfying (1.3). Then there exists a minimizer $u_{0}$ of $J\left(u\right)$ over $\mathcal{A}$. Moreover, $u_{0}\in C^{2, \frac{1}{ 2}}\left(\mathbb{R}\right) \cap L^\infty(\mathbb{R})$ and satisfies the Euler-Lagrange equation

and the condition at infinity

Here the fractional operator $\left(-\frac{d^{2}}{dx^{2}}\right) ^{\frac{1}{ 2}}$ is defined by (1.5).

Remark 1.2. Part of the motivation for our choice of the assumptions on $g$ is to study layer solutions in excitable media. Keener ^[46] introduced a model for calcium release in cardiac cells where the release sites are discrete, resulting in a production term that may degenerate in space. Here we consider a medium exhibiting bistability locally away from a suitable measure zero set, with one possible choice of $g$ satisfying (1.2) being $g(x) = (1+\cos(2\pi x))^{1/4}$.

Remark 1.3. When $g(x) = 1$, the authors proved in ^[29] the monotonicity and uniqueness (up to translation) of minimizers $u_{\alpha}$ of $J(u)$ for each $\alpha > 0$. In addition, when $\alpha \rightarrow 0$, the unique miminzer with $u_{\alpha}(0) = 0$ converges to a global minimizer of $J(u)$ with $\alpha = 0$. The proof relies crucially on the translational invariance of $J(u)$ when $g$ is constant and does not extend to the periodic case studied here. In contrast, for $g$ periodic it is not at all clear whether a layer solution exhibits monotonicity or uniqueness (up to discrete translation).

Remark 1.4. When $\alpha = 0$, our argument can still be used to establish that the energy $J(u)$ is bounded from below. It is not clear, though, which function space is a suitable choice for carrying out the minimization in this case. Indeed, for $\alpha > 0$ our argument uses the continuous representative of an $H^1(\mathbb{R})$ function. If $\alpha = 0$, however, one can no longer conclude a priori that the weak limit of the minimizing sequence remains in $H^1(\mathbb{R})$. When $\alpha \rightarrow 0$, it would be interesting to know whether the layer minimizers $u_{\alpha}$ we constructed for $\alpha > 0$ converge to a function that minimizes $J(u)$ with $\alpha = 0$ in some sense. In contrast to our earlier results in ^[29], due to the lack of monotonicity we were not able to derive uniform $L^2(\mathbb R)$ bounds on $u_{\alpha} - \eta$, and it is not clear how to pass to the limit.

We prove Theorem 1.1 in three steps. We first check that $J\left(u\right)$ is bounded from below. Let

We show that $F$ is bounded from below in Section 2. In the second step, we construct a global minimizer of $J$ in $\mathcal{A}$ in Section 3. Regularity is treated in Section 4 and follows from a bootstrap argument, since $u_{0} = v_{0}+\eta$ with $v_{0}\in H^{1}\left(\mathbb{R}\right).$ However, a priori it is not clear whether $\left(-\frac{d^{2}}{dx^{2}}\right) ^{ \frac{1}{2}}u_{0}\in L^{2}(\mathbb{R}),$ and we handle this term separately when deriving the Euler-Lagrange equation.

We note that our result and the variational approach are also closely related to those in ^[36]. There, Dipierro, Patrizi and Valdinoci prove existence of heteroclinic orbits for (1.20) and (1.21) in the case $\alpha = 0$ as follows. Given $\alpha, \beta \in \left(0, 1\right],$ they considered a constrained double obstacle minimization of a perturbed renormalized energy with an additional penalization term in the following form

The authors showed that the constrained minimizer becomes an unconstrained minimizer when the obstacles are far apart [36,Proposition 10.1]. The existence of heteoclinic orbits for (1.20) is obtained by first letting $\alpha \rightarrow 0$ and then sending $\beta$ to zero. Our renormalized energy in this paper corresponds to $\beta = 0$ in their setting and requires a much more delicate analysis of the tail behavior of the members of minimizing sequences.

2.   Lower bound on $J\left(u\right)$

Let $J\left(u\right),$ $F\left(u\right)$ be defined by (1.1) and (1.25). We shall prove the following lower bound in this section.

Proposition 2.1. There exists a positive constant $C$ independent of $u$ such that $F\left(u\right) > -C$ for any $u$ $\in \mathcal{A}.$

Remark 2.2. The proof of the lower bound of $F$ can be simplified if $g(x)$ is a positive periodic function. In this case, we have $g(x) \geq a_0 > 0$ for all $x$. Thus for any $u \in \mathcal{A}$,

and lower bound on $F$ follows directly from the results in ^[29] since

where $u_0$ is the global minimizer of $F_0$ obtained in ^[29]. The main difference in the argument employed in this paper is, therefore, to allow for a degeneracy in the energy occurring on a set of measure zero.

The lower bound on $J\left(u\right)$ follows directly from Proposition 2.1.

2.1.   Overview

First we observe that replacing $u$ by $\pm 1$ whenever $\left\vert u\right\vert > 1,$ the energy is only getting smaller. Here for the nonlocal term, a direct calculation shows $(u(x)-u(y))^2\geq (\tilde{u}(x)-\tilde{u} (y))^2$ where $\tilde{u}(x) = \max\{\min(u(x), 1), -1\}$). Without loss of generality, we shall assume $\left\vert u\left(x\right) \right\vert$ $\leq 1$ on $\mathbb{R}$ throughout the paper. Moreover, the following Lemma can be proved by the same argument as in the proof of Lemma 2.2 from ^[29].

Lemma 2.3. Given $u\in \mathcal{A},$ there exists a sequence $u_{n}\in \mathcal{A}$ such that $u_{n}-\eta \in C_{0}^{\infty }\left(\mathbb{R} \right)$ and $J\left(u_{n}\right) \rightarrow J\left(u\right)$ as $n\rightarrow \infty$.

We introduce the following subset of $\mathcal{A}$:

Letting

we can write $F\left(u\right)$ as

Direct calculation shows that the integrals

are all bounded. To show $F\left(u\right)$ is bounded from below, the question reduces to showing that

for some $C > 0$ independent of $u \in \mathcal{A}_{0}.$ Since $\eta \notin \mathring{H}^{1/2}\left(\mathbb{R}\right),$ the term $\int_{1}^{\infty }\int_{-\infty }^{-1}f\left(x, y\right) dydx$ could potentially be negative infinity. In particular, if we choose a sequence $u_{n}\left(x\right)$ which oscillates between $1$ and $-1$ on intervals which get larger and larger, it is not clear that we can have a uniform lower bound on the left-hand side in (2.2). Our idea is the following: if $\left\vert u\right\vert$ stays away from $1$ on a big portion of $\mathbb{R},$ the term

would dominate

On the other hand, if $\left\vert u\right\vert \sim 1$ on $\mathbb{R}$ and $u$ oscillates between $1$ and $-1$, the sum

would approach infinity at the same order as

thus eventually cancelling out the potential negative infinite energy. In both cases, we obtain a finite lower bound on $F\left(u\right).$

To explain our ideas more precisely, recall that $u-\eta \in H^{1}\left(\mathbb{R}\right)$ and has compact support for every $u \in \mathcal{A}_0$. By Sobolev embedding theorem, $u-\eta$ and, therefore, $u$ are continuous. Given any $\delta > 0,$ we define the following decomposition of $\left(-\infty, -1\right] \cup \left[ 1, \infty \right)$ with respect to $u$:

and

Under these notations, we observe $I_{\delta }^{+}$, $II_{\delta }^{-}$, $III_{\delta }^{+}$ and $III_{\delta }^{-}$ are all bounded sets. We show that there exists a constant $C = C(\delta, g, \beta, W) > 0$ and independent of $u \in \mathcal{A}_0$ such that

and

Throughout the paper, we will use $C$ to represent a generic constant independent of $u \in \mathcal{A}_{0}$, and depending only on $\delta$, $g$, $\beta$ and $W$, which might change from line to line. A lower bound in (2.2) follows from (2.5) and (2.6).

Since

the proof of (2.6) reduces to the following main technical inequalities:

The proof of (2.7) and (2.8) uses a contradiction argument. We prove one bound, the other one can be proved similarly. Assume $J_{\delta }^{+}\left(u_{n}\right) \rightarrow -\infty$ for some sequence $(u_{n}).$ Representing the decomposition of $\left(-\infty, -1\right] \cup \left[ 1, \infty \right)$ with respect to $u_{n}$ by adding index $n$ in (2.3) and (2.4), decompose $I_{\delta, n}^{+},$ $I_{\delta, n}^{-}$ and $II_{\delta, n}^{+}$ into union of disjoint intervals. We can estimate

and

in terms of summation of integral over those intervals. In particular, $J_{\delta }^{+}\left(u_{n}\right) \rightarrow -\infty$ implies $I_{\delta, n}^{+}\subset \left[ 1, R_{n}\right]$ where $R_{n}\rightarrow \infty$ and (2.10) goes to infinity at most logarithmically in $R_{n}$. If $\left\vert u_{n}\right\vert$ is bounded away from 1 on a large portion of $\left[ 1, \infty \right),$ then the term $\int_{1}^{\infty }W\left(u_{n}\right)$ dominates (2.10). If $u_{n}\approx -1$ on a large portion of $\left[ 1, \infty \right)$, then (2.9) would approach infinity at the same logarithmic order as (2.10). In either case, we can always conclude that $J_{\delta }^{+}\left(u_{n}\right)$ is bounded from below$,$ a contradiction.

2.2.   $F\left(u\right)$ is bounded from below on $\mathcal{A}_0$

We prove Proposition 2.1 in several steps.

2.2.1.   Preliminaries

We first state the following lemma.

Lemma 2.4. Given $u\in \mathcal{A}_{0},$, the following bounds hold:

Proof. The bounds in (2.11) are straightforward. By the definition of $\eta,$ we have

The second inequality in (2.12) follows from a similar argument. □

Lemma 2.4 implies that the terms involving integration on $\left[ -1, 1\right]$ in (2.1) are all bounded from below$.$ The boundedness of $F\left(u\right)$ from below would then follow from the following lemma.

Lemma 2.5. There exists a constant $C = C(\delta, g, \beta, W) > 0$ such that for all $u\in \mathcal{A}_0$, the following lower bound holds:

Lemma 2.5 is proved in two steps. Under decompositions (2.3) and (2.4), we can write

The following lower bound will be used in the proof of Lemma 2.5.

Lemma 2.6. Let $A\subset \lbrack 1, \infty)$ and $B\subset (-\infty, -1]$. Assume either $A$ or $B$ is bounded, then there exists a constant $C = C(\beta, g) > 0$ such that for all $u \in \mathcal{A}_0$, the following bounds hold:

and

for any $\varepsilon > 0$.

Proof. We only show how to obtain (2.14), as the other inequality follows by a similar argument. Since $(\eta (x)-\eta (y))^{2} = 4$ for $x\in A$ and $y\in B$, we have

By (1.2) and periodicity of $g(x)$, we conclude

Hence we deduce

where we applied Hölder's inequality and (2.16) in the last line. Notice that when either $A$ or $B$ is bounded, the integral $\int_{A}\int_{B}\frac{\left(u\left(x\right) -u\left(y\right) -2\right) ^{2}}{\left(x-y\right) ^{2}} dydx$ is finite for any $u \in \mathcal{A}_0$, justifying the spliting of the integrals in the calculation above. □

The first step to prove Lemma 2.5 is the following Lemma.

Lemma 2.7. For any $0 < \delta \ll 1,$ there exists a constant $C = C(\delta, g, \beta, W)$ such that for all $u\in \mathcal{A}_0$

Proof. Recall that $I_{\delta}^{+}$, $II_{\delta}^{-}$, $III_{\delta}^{+}$ and $III_{\delta}^{+}$ are all bounded sets. By estimate (2.14) in Lemma 2.6, we have

and

Here for (2.17) we used the fact that the integral $\int_{II_{\delta }^{+}}\int_{I_{\delta }^{-}}f\left(x, y\right) dydx$ can be written as a sum of integrals of the form $\int_A\int_Bf\left(x, y\right) dydx$, where either $A$ or $B$ is bounded when $u \in \mathcal{A}_0$.

By estimate (2.15) from Lemma 2.6, we have

and

Summing up (2.17)−(2.23) yields

Recall that $W\left(\pm 1\right) = W^{\prime }\left(\pm 1\right) = 0,$ $W\left(u\right) > 0$ for $\left\vert u\right\vert < 1$ and $W^{\prime \prime }\left(\pm 1\right) > 0.$ Picking $\delta \ll 1,$ we have the following estimates

Since

the conclusion follows by taking $\varepsilon = \min \left(\frac{1 }{48} \min_{\left\vert u\right\vert \leq 1-\delta }W\left(u\right), \frac{1 }{32}W^{\prime \prime }\left(-1\right), \frac{1 }{32}W^{\prime \prime }\left(1\right) \right).$ □

The second step to prove Lemma 2.5 is the following Lemma.

Lemma 2.8. There exists a constant $C = C(\delta, g, \beta, W) > 0$ such that for all $u\in \mathcal{A}_0$ we have

Lemma 2.7 and Lemma 2.8 imply Lemma 2.5.

2.2.2.   Proof of Lemma 2.8

Decompositions and some basic estimates The proof of Lemma 2.8 is rather long and technical. First we decompose each set into intervals. By our assumption, for any given $\delta \in \left(0, 1\right),$ there exists $R_{1}\left(u\right) > 0$ and $R_{2}\left(u\right) > 0$ such that

and

It follows that $III_{\delta }^{+}$ and $III_{\delta }^{-}$ are open subsets of $\left(1, R_{1}\left(u\right) \right)$ and $\left(-R_{2}\left(u\right), -1\right)$ respectively. By the structure theorem of open sets in $\mathbb{R}$ and choosing $u$ to be the continuous representative, there exist indices $N^{\pm },$ and positive numbers $\alpha _{i}^{\pm },$ $\beta _{i}^{\pm }$ such that we can write $III_{\delta }^{+}$ and $III_{\delta }^{-}$ as unions of disjoint open intervals in the following form.

Without loss of generality, we can assume $N^{\pm }$ are finite and $\alpha _{i} < \beta _{i} < \alpha _{i+1} < \beta _{i+1}$ for all $i.$ In fact, recall that $u-\eta$ in $H^1_0(\mathbb{R})$, write $R = \max(R_1(u), R_2(u))$, we can obtain an approximation $u_1$ in $C^\infty(\mathbb{R})$ that is equal to $\eta$ for $|x| > R + 1$ and arbitrarily close to $u$ in $H^1(\mathbb{R})$. Taking a linear interpolant $u_2$ of $u_1$ over a sufficiently fine partition $X$ of $[-1-R, 1 + R]$, we get a function that is arbitrarily close to $u_1$ in $W^{1, \infty}(\mathbb{R})$. Finally, shifting the values of the function $u_2$ at the (finitely many) points of $X$ by arbitrarily small amounts if necessary, we get a function $u_3$ that is arbitrarily close to $u_2$ in $W^{1, \infty}(\mathbb{R})$ and $u_3^{\prime }$ is non-zero a.e. in $\mathbb{R}$. Hence on every interval of the partition $X$ there is at most one point at which $|u_3| = 1 - \delta.$ From this, we conclude that each interval of the partition $X$ intersects $III_{\delta }^{+}$ (or $III_{\delta }^{-}$) at most once. Relabling if necessary, we thus find finitely many disjoint intervals $(\alpha_k, \beta_k)$ (merge $(\alpha_k, \beta_k)\cup(\alpha_{k+1}, \beta_{k+1})$ into $(\alpha_k, \beta_{k+1})$ if $\beta_k = \alpha_{k+1}$) where

Renaming our endpoints we find disjoint intervals $\left[ a_{i, }b_{i}\right]$ $,$ $\left[ c_{j}, d_{j}\right] \subset [1, R_1(u)]$ and indices $K, L$ such that

and

with

and

Here by a slight abuse of notation, we denote $\left[ c_{l, }d_{l}\right] = \left[ c_{l}, \infty \right)$, and if $a_{1} = 1$ we replace $\left[ a_{1}, b_{1}\right]$ by $\left(1, b_{1}\right]$ in (2.29), or if $c_{1} = 1,$ replace $\left[ c_{1}, d_{1} \right]$ by $\left(1, d_{1}\right]$ in (2.30). Similarly we write $II_{\delta }^{-}$ and $I_{\delta }^{-}$ as unions of disjoint intervals. For the rest of the paper, we write

with the understanding that $\left[ -\widetilde{d}_{\widetilde{L}}, - \widetilde{c}_{\widetilde{L}}\right] = \left(-\infty, -\widetilde{c}_{ \widetilde{L}}\right]$ and $\left[ -\widetilde{b}_{1}, -\widetilde{a}_{1} \right] = \left[ -\widetilde{b}_{1}, -1\right)$ if $\widetilde{a}_{1} = 1$, or $\left[ -\widetilde{d}_{1}, -\widetilde{c}_{1}\right] = \left[ -\widetilde{d} _{1}, -1\right)$ if $\widetilde{c}_{1} = 1$. Note that in this form, all $a_{i}, b_{i, }$ $c_{j}, d_{j},$ $\widetilde{a}_{i}, \widetilde{b}_{i, }$ $\widetilde{c}_{j}, \widetilde{d}_{j}$ are greater or equal to 1.

We first state some basic estimates.

Lemma 2.9. The following estimates hold:

Proof. By (2.31) and (2.32), we have

(2.3), (2.35) and (2.36) are proved similarly. □

An immediate corollary of Lemmas 2.9 is the following.

Corollary 2.10. We have the following bounds.

Proof.

If $\widetilde{c}_{1} > b_{K},$

if $\widetilde{c}_{1} < b_{K},$

so (2.37) follows. The estimate (2.38) follows from a similar argument. □

The following estimate on the potential term is important for the lower bound estimate.

Lemma 2.11. Given any $\varepsilon \in (0, 1)$, there exists a positive constant $\tau = \tau(\varepsilon, g)$ such that

Proof. By (2.16). the zero level set of $g$ $E_0 = \{x\in [0, 1]:g(x) = 0\}$ has measure zero. In particular, given any $\varepsilon > 0$, we can find an open set $U_{\varepsilon}$ of $(0, 1)$ which covers $E_0$ and $|U_{\varepsilon}| < \varepsilon$. Let $O_{\varepsilon} = \cup_{n = 0}^{\infty}\left\{U_{\varepsilon}+n\right\}$, then since $g(x)$ is continuous and positive on $[1, \infty)\backslash O_{\varepsilon}$, there exists $\tau = \tau(g, \varepsilon) > 0$ such that $g(x) \geq \tau$ on $[1, \infty)\backslash O_{\varepsilon}$, therefore

(2.40) can be proved similarly. □

Remark 2.12. Fix an $\varepsilon = \varepsilon_0$ and write $\gamma = \tau(\varepsilon_0, g)(1-\varepsilon_0)$, we can write the lower bounds (2.39)–(2.40) as

Asume $I_{\delta }^{\pm },$ $II_{\delta }^{\pm }$ are written as unions of intervals in the form (2.31) and (2.32). Let $\cup _{i = 1}^{n}\left[ a_{i}, b_{i}\right] \subset I_{\delta }^{+}$ and $\cup _{j = 1}^{m}\left[ c_{j}, d_{j}\right] \subset II_{\delta }^{+},$ $\cup _{i = 1}^{\widetilde{n}}\left[ -\widetilde{b} _{i}, -\widetilde{a}_{i}\right] \subset I_{\delta }^{-}$ and $\cup _{j = 1}^{ \widetilde{m}}\left[ -\widetilde{d}_{j}, -\widetilde{c}_{j}\right] \subset II_{\delta }^{-}$. If $d_{j} = \infty,$ we write $\left[ c_{j}, d_{j}\right] = \left[ c_{j}, \infty \right)$ and $\left[ -\widetilde{d}_{j}, -\widetilde{c} _{j}\right] = \left(-\infty, \widetilde{c}_{j}\right]$ if $\widetilde{d} _{j} = \infty.$ Assume also

and

We have the following estimates.

Lemma 2.13. Assume $\left\vert c_{1}-b_{n}\right\vert \geq 1,$ $\left\vert \widetilde{c}_{1}-\widetilde{b}_{\widetilde{n}}\right\vert \geq 1$ for all $i$ and $j.$ For $\delta \ll 1,$ there exist positive constants $C = C(\delta, g, \beta, W)$ and $\gamma = \gamma(g)$ such that the following estimates hold for any $u\in \mathcal{A}_{0}.$

Proof. We prove (2.43), (2.44) follows from a similar argument. By Corollary 2.10, Remark 2.12 and (2.24−2.25),

The last two steps follow from

and

□

Proof of the main technical lemma Observe that

Lemma 2.8 would follow from (2.45) and the following Lemma.

Lemma 2.14. There exists a constant $C = C(\delta, \gamma, \beta, W) > 0$ such that

Lemma 2.14 is a direct corollary of the following Lemma.

Lemma 2.15. There exists a constant $C = C(\delta, \gamma, \beta, W) > 0$ such that for all $u\in \mathcal{A}_{0}$ the following bounds hold:

Proof. We prove (2.46), as the proof of (2.47) is similar. We argue by contradiction and let

We show that

implies

On the other hand, (2.48) and a diagonal argument would imply $J_{\delta }^{+}\left(u_{n}\right)$ is bounded from below$,$ a contradiction. We prove (2.48) in a series of steps. We will only explain the first three steps in detail. The remaining steps can be proved similarly.

Assume (2.46) fails. Then there would exist a sequence $\left\{ u_{n}\right\}$ such that $J_{\delta }^{+}\left(u_{n}\right) < -n.$ We denote the decomposition of $I_{\delta }^{\pm }\left(u_{n}\right)$ and $II_{\delta }^{\pm }\left(u_{n}\right)$ as follows.

Here we have $d_{L_{n}}^{n} = \infty$ and $c_{L_{n}}^{n} > b_{K_{n}}^{n}.$ We assume

Step 1: $J_{\delta }^{+}\left(u_{n}\right) \rightarrow -\infty$ implies there exists $i_{1}^{n}$ satisfying $j_{1}^{n}\leq i_{1}^{n}\leq L_{n}$ such that

First we observe

Therefore, $J_{\delta }^{+}\left(u_{n}\right) \rightarrow -\infty$ implies

Case Ⅰ: Assume that for a subsequence, $c_{L_{n}}^{n}-b_{K_{n}}^{n} < 1$ (without relabeling for simplicity of notations). Then by Lemma 2.13,

Taking liminf on both sides, we have

For the remaining cases, we shall always assume $c_{L_{n}}^{n}-b_{K_{n}}^{n} \geq 1.$ Also whenever we need to work on a subsequence, we always use the original sequence for simplicity of notations.

Case Ⅱ: Assume for a subsequence that there exists $i_{1}^{n}\in \left\{ \text{ }j_{1}^{n}, \text{ }j_{1}^{n}+1, \text{ }\cdots, L_{n}-1\right\}$ such that $c_{i_{1}^{n}}^{n}-b_{K_{n}}^{n} < 1.$ We also assume $b_{K_{n}}^{n}-a_{K_{n}}^{n} > 1$ (otherwise (2.49) follows directly). Then by

and Lemma 2.13, we bound $J_{\delta }^{+}\left(u_{n}\right)$ as follows.

Taking liminf on both sides, we must have

Case Ⅲ: Assume no such $i_{1}^{n}$ from case Ⅱ exists, then we must have $c_{j_{1}^{n}}^{n}-b_{K_{n}}^{n}\geq 1$. Thus

The last step used (2.51) and $\ \left(b_{K_{n}}^{n}, c_{j_{1}^{n}}^{n}\right) \subset \left(b_{K_{n}}^{n}, c_{L_{n}}^{n}\right) \cap III_{\delta, n}^{+}.$ Taking liminf on both sides of the equation above, we must have

Step 2: $J_{\delta }^{+}\left(u_{n}\right) \rightarrow -\infty$ implies there exists $i_{2}^{n}$ satisfying $j_{2}^{n}\leq i_{2}^{n}\leq i_{1}^{n}$ such that

First we observe

Therefore, by Step 1 we have that $J_{\delta }^{+}\left(u_{n}\right) \rightarrow -\infty$ implies

Case Ⅰ: If $\lim \inf_{n\rightarrow \infty }\frac{ a_{K_{n}}^{n}-b_{K_{n}-1}^{n}}{c_{i_{1}^{n}}^{n}-b_{K_{n}-1}^{n}} = 0,$ then $\lim \sup_{n\rightarrow \infty }\frac{ c_{i_{1}^{n}}^{n}-a_{K_{n}}^{n}}{c_{i_{1}^{n}}^{n}-b_{K_{n}-1}^{n}} = 1.$ In this case, then we can replace $\left(a_{K_{n}-1}^{n}, b_{K_{n}-1}^{n}\right) \cup \left(a_{K_{n}}^{n}, b_{K_{n}}^{n}\right)$ by $\left(a_{K_{n}-1}^{n}, b_{K_{n}}^{n}\right)$ and repeat our argument in Step 1 as follows.

Taking liminf on both sides, we get

Case Ⅱ: If $\lim \inf_{n\rightarrow \infty }\frac{ a_{K_{n}}^{n}-b_{K_{n}-1}^{n}}{c_{i_{1}^{n}}^{n}-b_{K_{n}-1}^{n}} > 0,$ There are three cases.

Case Ⅱ-ⅰ: If $\left(b_{K_{n}-1}^{n}, a_{K_{n}}^{n}\right) \cap II_{\delta, n}^{+} = \emptyset,$ then $\left(b_{K_{n}-1}^{n}, a_{K_{n}}^{n}\right) \subset III_{\delta, n}^{+},$

where we used (2.51) and the facts

Taking liminf on both sides of (2.53), we conclude

Case Ⅱ-ⅱ: If $\left(b_{K_{n}-1}^{n}, a_{K_{n}}^{n}\right) \cap II_{\delta, n}^{+}\neq \emptyset,$ there are two cases. If $\lim \inf_{n\rightarrow \infty }\frac{c_{i_{1}^{n}}^{n}}{a_{K_{n}-1}^{n}} = 1,$ then (2.52) follows directly. We therefore assume $\lim \inf_{n\rightarrow \infty }\frac{c_{i_{1}^{n}}^{n}}{a_{K_{n}-1}^{n}} > 1$ for the remaining two cases.

Case Ⅱ-ⅱ-a: There exists $i_{2}^{n}\in \left\{ j_{2}^{n}, \text{ } j_{2}^{n}+1, \text{ }\cdots, \text{ }i_{1}^{n}\right\}$ such that $c_{i_{2}^{n}}^{n}-b_{K_{n}-1}^{n} < 1.$ We assume $b_{K_{n}-1}^{n}-a_{K_{n}-1}^{n} > 1$ without loss of generality. Applying Corollary 2.10, Lemma 2.13 and (2.51), we bound $J_{\delta }^{+}\left(u_{n}\right)$ as follows.

Taking liminf on both sides, by Step 1 and $\lim \inf_{n\rightarrow \infty } \frac{a_{K_{n}}^{n}-b_{K_{n}-1}^{n}}{a_{K_{n}}^{n}-a_{K_{n}-1}^{n}} > 0,$ we must have

Case Ⅱ-ⅰ-b: No such $i_{2}^{n}$ exists, then we must have $c_{j_{2}^{n}}^{n}-b_{K_{n}}^{n}\geq 1.$ Applying Corollary 2.10, Lemma 2.13 and (2.51), we bound $J_{\delta }^{+}\left(u_{n}\right)$ as follows.

Taking liminf on both sides, we must have

Step 3: $J_{\delta }^{+}\left(u_{n}\right) \rightarrow -\infty$ implies there exists $i_{3}^{n}$ satisfying $j_{3}^{n}\leq i_{3}^{n}\leq i_{2}^{n}$ such that

First we observe

Therefore

Case Ⅰ: $\lim \inf \frac{a_{K_{n}}^{n}-b_{K_{n}-2}^{n}}{ c_{i_{1}^{n}}^{n}-b_{K_{n}-2}^{n}} = 0.$ This implies

In this case, we can replace $\left(a_{K_{n}-2}^{n}, b_{K_{n}-2}^{n}\right) \cup \left(a_{K_{n}-1}^{n}, b_{K_{n}-1}^{n}\right) \cup \left(a_{K_{n}}^{n}, b_{K_{n}}^{n}\right)$ by $\left(a_{K_{n}-2}^{n}, b_{K_{n}}^{n}\right),$ and repeat our argument in step 1.

We estimate $J_{\delta }^{+}\left(u_{n}\right)$ as follows.

By (2.55), we are back to the situation in Step 1 with $\left(a_{K_{n}}^{n}, b_{K_{n}}^{n}\right)$ replaced by $\left(a_{K_{n}-2}^{n}, b_{K_{n}}^{n}\right)$ and we conclude

Case Ⅱ: $\lim \inf \frac{a_{K_{n}}^{n}-b_{K_{n}-2}^{n}}{ c_{i_{1}^{n}}^{n}-b_{K_{n}-2}^{n}} > 0$ and $\lim \inf \frac{ a_{K_{n}-1}^{n}-b_{K_{n}-2}^{n}}{c_{i_{2}^{n}}^{n}-b_{K_{n}-2}^{n}} = 0.$ This implies

In this case, we replace $\left(a_{K_{n}-2}^{n}, b_{K_{n}-2}^{n}\right) \cup \left(a_{K_{n}-1}^{n}, b_{K_{n}-1}^{n}\right)$ by $\left(a_{K_{n}-2}^{n}, b_{K_{n}-1}^{n}\right)$ and repeat our argument in step 2.

We bound $J_{\delta }^{+}\left(u_{n}\right)$ as follows.

By (2.56), we are back to the case of step 2 with $\left(a_{K_{n}-1}^{n}, b_{K_{n}-1}^{n}\right)$ replaced by $\left(a_{K_{n}-2}^{n}, b_{K_{n}-1}^{n}\right)$ and we can conclude

Case Ⅲ: $\lim \inf \frac{a_{K_{n}}^{n}-b_{K_{n}-2}^{n}}{ c_{i_{1}^{n}}^{n}-b_{K_{n}-2}^{n}} > 0$ and $\lim \inf \frac{ a_{K_{n}-1}^{n}-b_{K_{n}-2}^{n}}{c_{i_{2}^{n}}^{n}-b_{K_{n}-2}^{n}} > 0.$ We discuss several cases.

Case Ⅲ-ⅰ: $\left(b_{K_{n}-2}^{n}, a_{K_{n}-1}^{n}\right) \cap II_{\delta, n}^{+} = \emptyset.$ Then $\left(b_{K_{n}-2}^{n}, a_{K_{n}-1}^{n}\right) \subset III_{\delta, n}^{+}.$ We also assume $\lim \inf_{n\rightarrow \infty }\frac{a_{K_{n}}^{n}-b_{K_{n}-1}^{n}+1 }{c_{i_{1}^{n}}^{n}-b_{K_{n}-1}^{n}} > 0.$

We have

Taking liminf on both sides, we get

If $\lim \inf_{n\rightarrow \infty }\frac{a_{K_{n}}^{n}-b_{K_{n}-1}^{n}+1}{ c_{i_{1}^{n}}^{n}-b_{K_{n}-1}^{n}} = 0,$ then $\lim \sup_{n\rightarrow \infty } \frac{c_{i_{1}^{n}}^{n}-a_{K_{n}}^{n}}{c_{i_{1}^{n}}^{n}-b_{K_{n}-1}^{n}} = 1.$ We can modify the argument above by replacing $\left(a_{K_{n}-1, }^{n}b_{K_{n}-1}^{n}\right) \cup \left(a_{K_{n}}^{n}, b_{K_{n}}^{n}\right)$ by $\left(a_{K_{n}-1, }^{n}b_{K_{n}}^{n}\right)$ and same conclusion follows.

Case Ⅲ-ⅱ: $\left(b_{K_{n}-2}^{n}, a_{K_{n}-1}^{n}\right) \cap II_{\delta, n}^{+}\neq \emptyset.$ There are two cases.

Case Ⅲ-ⅱ-a: There exists $i_{3}^{n}\in \left\{ j_{3}^{n}, \text{ } j_{3}^{n}+1, \text{ }\cdots, j_{2}^{n}-1\right\}$ such that

We bound $J_{\delta }^{+}\left(u_{n}\right)$ as follows.

Taking liminf on both sides, we conclude

Case Ⅲ-ⅱ-b: no such $i_{3}^{n}$ satisfying (2.57) exists. Then we must have

In this case, we can estimate $J_{\delta }^{+}\left(u_{n}\right)$ in the same way as case Ⅲ-ⅰ with $c_{i_{3}^{n}}^{n}$ replaced by $c_{j_{3}^{n}}^{n}$ and conclude

Continuing this way, we conclude that $J_{\delta }^{+}\left(u_{n}\right) \rightarrow -\infty$ implies

We now pick our subsequence as follows. Pick our first subsequence $\left\{ u_{n}\right\}$ such that for its decomposition

Next we pick a subsequence of the chosen subsequence such that

Continuing this way, we pick our final subsequence using a diagonal argument. For simplicity of notations, we use the original sequence. For the final subsequence, we have for all $n,$

it then follows that

a contradiction to the assumption that $J_{\delta}^+\left(u_{n}\right) \rightarrow -\infty.$ □

3.   Existence of minimizer of $J\left(u\right)$

3.1.   Overview

In this section, we prove the existence of a minimizer of $J\left(u\right).$ Observe that in general the boundedness of $J\left(u_{n}\right)$ does not imply the boundedness of $v_{n} = u_{n}-\eta$ in $H^{1}\left(\mathbb{R} \right).$ A priori it is not clear whether we can obtain a suitable limit function from a minimizing sequence, using the direct method of calculus of variations. On the other hand, for a sequence $\left\{ u_{n}\right\}$ satisfying $\left\vert u_{n}\left(x\right) +sgn(x)\right\vert \geq c_{0} > 0$ outside a uniformly bounded interval, the boundedness of $J\left(u_{n}\right)$ implies the boundedness of $\left\{ u_{n}-\eta \right\}$ in $H^{1}\left(\mathbb{R}\right).$ Our main idea, therefore, is to show that we can replace a minimizing sequence $\left\{ u_{n}\right\}$ by another one $\{\overline{u}_{n}\}$ which satisfies $\left\vert \overline{u}_{n}\left(x\right) +sgn(x)\right\vert \geq c_{0} > 0$ for some constant $c_{0}$ outside a uniformly bounded interval$.$ The new sequence $\left\{ \overline{u} _{n}\right\}$ has uniformly bounded energy $J\left(\overline{u}_{n}\right)$, and $\left\{ \overline{u} _{n}-\eta \right\}$ is bounded in $H^{1}\left(\mathbb{R}\right).$ From this, we obtain a subsequence which weakly converges to a limit function $\overline{v}\in \mathcal{A}$ that achieves the minimum energy in $\mathcal{A}$. To construct the replacement sequence $\left\{ \overline{u}_{n}\right\}$, we use the interval decompositions of $u_{n}$ from the previous section. We first construct $\widetilde{u}_{n}$ by reflecting $u_{n}$ over suitable regions. Keeping track of the energy contributions from each interval in the decomposition, we show that we can choose our regions of reflection so that the energy difference between $J\left(\widetilde{u}_{n}\right)$ and $J\left(u_{n}\right)$ is approaching zero as $n\rightarrow \infty.$ Lastly, we define $\overline{u} _{n}$ by a suitable translation of $\widetilde{u}_{n}$ $\ $so that $\left\vert \overline{u}_{n}\left(x\right) +sgn(x)\right\vert \geq c_{0} > 0$ for some constant $c_{0}$ outside a uniformly bounded interval$.$ By periodic translation invariance of $J$ and the above property of $J\left(\widetilde{u}_{n}\right),$ we conclude that $\left\{ \overline{u} _{n}\right\}$ is another minimizing sequence.

We first state a translation invariant lemma.

Lemma 3.1. Given any $c\in \mathbb{Z},$ let $u_{c}\left(x\right) = u\left(x+c\right),$ then $J\left(u_{c}\left(x\right) \right) = J\left(u\left(x\right) \right).$

Proof. Since the first two terms are translation invariant for $c\in \mathbb{Z}$, $J(u_{c}) = J(u)+D(\eta _{c}, \eta)$, where

By Lemma 2.1 in ^[29], we have $D\left(\eta _{c}, \eta \right) = 0$ for any constant $c.$ The conclusion of the Lemma then follows. □

3.2.   Existence of a minimizer

Let $\left\{ u_{n}\right\}$ be a minimizing sequence. By Lemma 2.3, we may assume that $u_n-\eta$ is compactly supported in $\mathbb{R}$. By Lemma 3.1 and our assumption on the behavior of $\left\{ u_{n}\right\}$ at infinity, after a suitable translation by an integer there exists $c_{n}\in [ 0, 1)$ such that $u_{n}\left(1+c_{n}\right)$ $= 0$ and

Throughout this section, we assume that on $\left[ 1+c_{n}, \infty \right),$ $u_{n}$ has a decomposition

and

Here we understand $d_{L_{n}}^{n} = \infty.$ Throughout this section, we fix $\delta \ll 1$ such that $W^{\prime \prime }\left(u\right) \geq \frac{1}{4} W^{\prime \prime }\left(1\right)$ when $\left(1-u\right) \leq \delta$ and $W^{\prime \prime }\left(u\right) \geq \frac{1}{4}W^{\prime \prime }\left(-1\right)$ when $1+u\geq \delta.$ Since $W\left(u\right) > 0$ for $u\in \left(-1, 1\right),$ there exists $C_{\delta } > 0$ such that

and

3.2.1.   Case Ⅰ: $\lim \sup b_{K_{n}}^{n} < \infty$

In this case, we prove the following proposition.

Proposition 3.2. Let $\delta > 0$ be such that $W\left(u\right)$ satisfies (3.3) and (3.4). Let $u_n$ be a minimizing sequence for $J(u)$ in $\mathcal{A}_0$ with decompositions (3.1) and (3.2). If there exists a constant $M > 1$ such that $b_{K_{n}}^{n} < M$ for all $n$, then a subsequence of $\left\{ u_{n}\right\}$ converges weakly to a minimizer $u_{0}$ of $J\left(u\right)$ in $\mathcal{A}$.

Proof. Without loss of generality, we can assume $\left\vert u_{n}\right\vert \leq 1.$ Consider $v_{n}: = u_{n}-\eta.$ Decomposition (3.1) and our assumption imply

and

We write $J\left(u_{n}\right)$ in terms of $v_{n}$ as follows:

We have

The last inequality in (3.8) follows from Hö lder inequality, (1.2) and bounds on

By (3.3), (3.4), (3.7) and (3.8), we have

From this, we conclude that $v_{n}$ is bounded in $H^{1}\left(\mathbb{R} \right)$, and, hence, there exists a subsequence $v_{n_{k}}\rightharpoonup v\in H^{1}\left(\mathbb{R}\right)$ and

Here the convergence above follows from the identity

the fact that

lies in $L^2(\mathbb{R})$ (see the discussion around (4.5) in Section 4) and the weak convergence of $v_{n_k}$ in $L^2(\mathbb{R})$.

Let $u_{0} = v+\eta,$ then $u_{0}\in \mathcal{A}$ and

□

3.2.2.   Case Ⅱ: $\lim \sup b_{K_{n}}^{n}=\infty$

We prove the following proposition.

Proposition 3.3. If $\lim \sup b_{K_{n}}^{n} = \infty,$ we can find a new minimizing sequence $\left\{ \overline{u}_{n}\right\}$ and $\ \overline{c} _{n}\in \left[ 0, 1\right)$ such that

and

with $\lim \sup \overline{b}_{\overline{K}_{n}}^{n} < \infty$.

Proof. Let $\rho _{i}^{n} < a_{i}^{n}$ be the biggest zero point of $u_{n}$ that lies to the left of $a_{i}^{n}$ and $\sigma _{i}^{n} > b_{i}^{n}$ be the smallest zero of $u_{n}$ lying to the right of $b_{i}^{n}.$ Set

and

We construct our replacement minimizing sequence $\left\{ \widetilde{u} _{n}\right\}$ in two cases.

Case Ⅰ: There exists $0\leq l\leq K_{n},$ such that $\lim \sup_{n}\frac{a_{K_{n}-l}^{n}}{b_{K_{n}-l}^{n}} = \nu _{l} < 1$ and $\lim \sup_{n}\frac{a_{K_{n}-i}^{n}}{b_{K_{n}-i}^{n}} = 1$ for all $i = 0, 1, \cdots, l-1.$

The main idea to construct a replacement minimizing sequence in this case is to show that we must have $\lim \sup_{n}\left(a_{K_{n}-l}^{n}-b_{1}^{n}\right) < \infty$ and $\lim \sup_{n}\frac{a_{1}^{n} }{b_{K_{n}-l}^{n}} = 0.$ We then reflect the positive part of $u_{n}$ defined on $\left[ 1+c_{n}, \rho _{K_{n}-l}^{n}\right]$ to $-u_{n}$. It can be shown that the energy of the resulting function differs from initial minimizing sequence by a small amount, a suitable translation of this reflected minimizing sequence satisfies the assumption in Proposition 3.2 and we can obtain a limit function from this replacement minimizing sequence. To illustrate our main idea, we first assume $l = 0.$

Case Ⅰ-ⅰ: $\lim \sup_{n}\frac{a_{K_{n}}^{n}}{b_{K_{n}}^{n}} = 0.$ By definition of $\rho _{i}^{n},$ $\sigma _{i}^{n},$ we have $\lim \sup_{n} \frac{\rho _{K_{n}}^{n}}{\sigma _{K_{n}}^{n}} = 0.$ In this case, we consider the sequence $\left\{ \widetilde{u}_{n}\right\}$ defined by

Then

Since

(3.10) implies the subsequence of $\left\{ \widetilde{u}_{n}\right\}$ is also a minimizing sequence. Let $s_{n} = \left[ \sigma _{K_{n}}^{n}\right]$ be the largest integer smaller than $\sigma _{K_{n}}^{n}.$ By periodic translation invariance of the energy, we define $\overline{u}_{n}\left(x\right) = \widetilde{u}_{n}\left(x+s_{n}-1\right)$. Then $J\left(\overline{u}_{n}\right) = J\left(\widetilde{u}_{n}\right)$ and $\overline{u}_{n}$ satisfies

and

We conclude from Proposition 3.2 that $\overline{u}_{n}$ is bounded in $H^{1}\left(\mathbb{R}\right),$ and $\overline{u}_{n}$ converges weakly to a minimizer in $\mathcal{A}$.

Case Ⅰ-ⅱ: $\lim \sup_{n}\frac{a_{K_{n}}^{n}}{b_{K_{n}}^{n}} = \nu < 1.$ Then we must have $\lim \sup_{n}\left(a_{K_{n}}^{n}-b_{1}^{n}\right) < \infty$ and $\lim \sup_{n}\frac{ a_{1}^{n}}{b_{K_{n}}^{n}} = 0.$ In this case, we prove that we essentially get back to the same situation as case Ⅰ-ⅰ, with $a_{K_{n}}^{n}$replaced by $a_{1}^{n}.$ By estimates in Section 2 (we use the same notations in this section)

Assuming $\lim \sup_{n}\frac{a_{K_{n}}^{n}}{b_{K_{n}}^{n}} = \nu,$ (3.11) and the boundedness of $J_{\delta }^{+}\left(u_{n}\right)$ imply $\left\vert III_{\delta, n}^{+}\right\vert \leq C.$ We construct $\widetilde{u}_{n}$ as follows.

We first show that there does not exist an $s$ such that $\lim \sup_{n}\left(a_{K_{n}}^{n}-a_{s+1}^{n}\right) = A_{s} < \infty,$ $\lim \sup_{n}\left(a_{K_{n}}^{n}-b_{s}\right) = \infty$.

Otherwise, letting $t_{n} = \left\vert \left[ 1+c_{n}, \rho _{K_{n}}^{n}\right] \cap A_{n}^{+}\right\vert,$ $t_{s}^{n} = \left\vert \left[ b_{s}^{n}, a_{s+1}^{n}\right] \cap II_{\delta, n}^{+}\right\vert,$ we can write

Recall that

and

Taking liminf on both sides of (3.13), we have

contradicting the assumption that $\left\{ u_{n}\right\}$ is a minimizing sequence and the fact that $J$ is bounded from below. Therefore we must have $\lim \sup_{n}\left(a_{K_{n}}^{n}-b_{1}^{n}\right) = a_{0} < \infty$.

Next we show that $\lim \sup_{n}\frac{a_{1}^{n}}{b_{K_{n}}^{n}} = 0.$ If $\lim \sup_{n}\frac{a_{1}^{n}}{b_{K_{n}}^{n}} > 0$ and $\kappa _{n} : = \left\vert \left[ 1+c_{n}, \rho _{K_{n}}^{n}\right] \cap II_{\delta, n}^{+}\right\vert$, then

and

Therefore, we have

Since

and

taking liminf on both sides of (3.17), it follows from (3.18), (3.19) and (3.20) that

a contradiction.

Lastly we estimate the energy difference between $u_{n}$ and $\widetilde{u} _{n}.$ We have

Recalling that

and

we conclude that

Thus by (3.21) we have that $\left\{ \widetilde{u }_{n}\right\}$ is also a minimizing sequence. Defining

where $s_{n} = \left[ \sigma _{K_{n}}^{n}\right]$ is the largest integer smaller than $\sigma _{K_{n}}^{n},$ then $\left\{ \overline{u}_{n}\right\}$ is a minimizing sequence satisfying

Proposition 3.2 applies to $\overline{u}_{n}$, from which we can extract a converging subsequence to a minimizer $u_0 \in \mathcal{A}_0$.

Case Ⅰ-ⅲ: There exists $l > 1$ such that $\lim \sup_{n} \frac{a_{K_{n}-j}^{n}}{b_{K_{n}-j}^{n}} = 1$ for $j = 0, 1, \cdots, l-1$ and $\lim \sup_{n}\frac{a_{K_{n}-l}^{n}}{b_{K_{n}-l}^{n}} < 1.$ Then we must have $\lim \sup_{n}\left(a_{K_{n}-l}^{n}-b_{1}^{n}\right) < \infty$ and $\lim \sup_{n}\frac{a_{1}^{n}}{b_{K_{n}-l}^{n}} = 0.$ We construct $\widetilde{u}_{n}$ as follows.

We also need

We can follow a similar argument in Case Ⅰ-ⅰ and Case Ⅰ-ⅱ by replacing any estimates on $\left[ a_{K_{n}}^{n}, b_{K_{n}}^{n}\right]$ in (3.13), (3.21) and (3.17) by estimates on $\left[ a_{K_{n}-l}^{n}, b_{K_{n}-l}^{n}\right]$, using $\overline{u}_{n} =$ $\widetilde{u}_{n}\left(x+s_{n, l}-1\right),$ where $s_{n, l} = \left[ \sigma _{l}^{n}\right]$ is the largest integer less than or equal to $\sigma _{l}^{n}.$

Case Ⅱ: No such $l$ exists, i.e., $\lim \sup_{n} \frac{a_{j}^{n}}{b_{j}^{n}} = 1$ for all $j$ where $\lim \sup_{n}b_{j}^{n} = \infty.$ Let $l$ be such that $\lim \sup_{n}b_{K_{n}-j}^{n} = \infty$ for all $j < l$ and $\lim \sup_{n}b_{K_{n}-l}^{n} < \infty.$ In this case, we will reflect the negative part of $u_{n}$ to $-u_{n}$ outside a big portion of $\left(b_{K_{n}-l}^{n}, a_{K_{n}-l+1}^{n}\right) \cap II_{\delta, n}^{+}.$ Following notations in section 2, we write

First by

we conclude that

Case Ⅱ-ⅰ: There exists $j\left(l\right) \in \left\{ j_{l+1}^{n}, \cdots, j_{l}^{n}-1\right\}$ such that

Let $T_{n, l} = \left[ d_{j\left(l\right) }^{n}, c_{L_{n}}^{n}\right] \cap A_{n}^{-},$ $M_{n, l} = \left\vert T_{n, l}\cap III_{\delta, n}^{+}\right\vert.$ We define

Since

for $x\in T_{n, l},$ $y_{1}\in \left[ c_{j\left(l\right) }^{n}, d_{j\left(l\right) }^{n}\right]$ and $y_{2}\in \left[ b_{K_{n}-l}^{n}, \text{ } c_{j\left(l\right) }^{n}\right] \cap A_{n}^{-},$ together with (3.24) and the fact that

we conclude that

It then follows that

i.e., $\left\{ \overline{u}_{n}\right\}$ is also a minimizing sequence with $\overline{b}_{\overline{K}_{n}}^{n} = b_{K_{n}-l}^{n}$ satisfying $\lim \sup \overline{b}_{\overline{K}_{n}}^{n} < \infty.$

Case Ⅱ-ⅱ:

Let $S_{0} = \left\{ \text{ }j_{l+1}^{n}+1, \text{ }\cdots, \text{ } j_{l}^{n}-1\right\}.$ We define $S_{0}^{+}$ as follows.

There exists $k = k\left(n\right)$ and indices $p\left(1\right), \cdots,$ $p\left(k\right) \in S_{0}$, $q\left(1\right), \cdots, q\left(k\right) \in S_{0}$ such that $i\in S_{0}^{+}$ if $p\left(s\right) \leq i\leq q\left(s\right)$ and $i\notin S_{0}^{+}$ if $q\left(s\right) \leq i\leq p\left(s+1\right)$ for $s = 1, \cdots, k.$ We write $S_{1} = \left\{ 1, \cdots, k\right\}.$

Case Ⅱ-ⅱ-1: There exists $s\leq k$ such that

Let

We define

Since

and

together with the observation that

for $x\in T_{n, l, s},$ $y_{1}\in \left(c_{j}^{n}, d_{j}^{n}\right),$ $y_{2}\in \left(d_{j-1}^{n}, c_{j}^{n}\right),$ $p\left(s\right) \leq j\leq q\left(s\right)$, and

for $x\in T_{n, l, s},$ $y_{1}\in \cup _{j = p\left(s\right) }^{q\left(s\right) }\left(c_{j}^{n}, d_{j}^{n}\right),$ $y_{2}\in III_{\delta, n}^{+}\cap \left(b_{K_{n}-l}^{n}, c_{p\left(s\right) }^{n}\right),$ we conclude that

Therefore

Case Ⅱ-ⅱ-2:

We consider

There exists $m = m\left(n\right) \leq k$ and $p_{1}\left(\tau \right), q_{1}\left(\tau \right) \in S_{1}$ for each $\tau \leq m$ and $\tau \in \mathbb{N}$ such that $\gamma \in S_{1}^{+}$ if $p_{1}\left(\tau \right) \leq \gamma \leq q_{1}\left(\tau \right)$ and $\gamma \notin S_{1}^{+}$ if $q_{1}\left(\tau \right) \leq \gamma \leq p_{1}\left(\tau +1\right).$ Let $S_{2} = \left\{ 1, \cdots, m\right\}$

Case Ⅱ-ⅱ-2-a: There exists $\tau = \tau \left(n\right)$ such that

Then we consider

Let $T_{n, l, q_{1}\left(\tau \right) } = \left[ d_{q\left(q_{1}\left(\tau \right) \right) }^{n}, \sigma _{K_{n}}^{n}\right] \cap A^{-}.$ $M_{n, l, q_{1}\left(\tau \right) }^{{}} = \left\vert \left[ d_{q\left(q_{1}\left(\tau \right) \right) }^{n}, \sigma _{K_{n}}^{n}\right] \cap A^{-}\cap III_{\delta, n}^{+}\right\vert.$ Observe

for $x\in T_{n, l, q_{1}\left(\tau \right) },$ $y_{1}\in \left(c_{j}^{n}, d_{j}^{n}\right),$ $y_{2}\in \left(d_{j-1}^{n}, c_{j}^{n}\right)$ when $p\left(p_{1}\left(\tau \right) \right) \leq j\leq q\left(q_{1}\left(\tau \right) \right).$ The same inequality also holds for $x\in T_{n, l, q_{1}\left(\tau \right) },$ $y_{1}\in \cup _{j = p\left(p_{1}\left(\tau \right) \right) }^{q\left(q_{1}\left(\tau \right) \right) }\left(c_{j}^{n}, d_{j}^{n}\right),$ $y_{2}\in \left[ b_{K_{n}-l}^{n}, \text{ } c_{p\left(p_{1}\left(\tau \right) \right) }^{n}\right] \cap A_{n}^{-}.$ Moreover, by (3.25),

and

we conclude that

Therefore

Continuing this way if necessary, we can define the set $S_{i}$ inductively by each $m\in S_{i},$ $p_{i}\left(m\right), q_{i}\left(m\right) \in S_{i-1}$ such that any $p_{i}\left(m\right) \leq p\leq q_{i}\left(m\right),$ $p\in S_{i-1}^{+},$ if $q_{i}\left(m\right) \leq p\leq p_{i}\left(m+1\right),$ $p\in S_{i-1}\backslash S_{i-1}^{+}.$ Here

By the definition of $S_{i},$ we have $\left\vert S_{i}\right\vert \leq \left\vert S_{i-1}\right\vert \leq \cdots \leq \left\vert S_{0}\right\vert.$ Since $\left\vert III_{\delta, n}^{+}\right\vert$ is uniformly bounded and $\lim \sup \left(a_{K_{n}-l+1}^{n}-b_{K_{n}-l}^{n}\right) = \infty,$ we would be able to find $r_{n}$ such that $S_{r_{n}} = S_{r_{n}}^{+}.$ Let $\mu = \left\vert S_{r_{n}}\right\vert$ and define

By a similar argument, we can show that $\{\overline{u}_{n}\}$ is a minimizing sequence which is close to $\pm 1$ away from a uniformly bounded interval. □

Proof of the first half of Theorem 1.1.Given a minimizing sequence $\left\{ u_{n}\right\},$ if $\lim \sup_{n}b_{K_{n}}^{n} < \infty$ we obtain a minimizer by Proposition 3.2. If $\lim \sup b_{K_{n}}^{n} = \infty,$ we obtain a new minimizing sequence $\left\{ \overline{u}_{n}\right\}$ which satisfies $\lim \sup \overline{b}_{\overline{K}_{n}}^{n} < \infty$ by Proposition 3.3, existence then follows from Proposition 3.2.

4.   Regularity of the minimizers

Proof of the second half of Theorem 1.1.

Proposition 4.1. Any minimizer $u_{0}$ of $J$ over $\mathcal{A}$ is a $C^{2, \frac{1}{2} }\left(\mathbb{R}\right)$ solution of

where we understand the fractional operator in the sense of (1.5).

Proof. Let $v_{0} = u_{0}-\eta$. We write $J\left(u_{0}\right)$ in terms of $v_{0}$ as

Consider now variations $v_{\varepsilon } = v_{0}+\varepsilon \varphi$, where $\varphi$ is any smooth compactly supported function. Since $u_{0}$ is a minimizer, we must have

Since $v_{0}\in H^{1}\left(\mathbb{R}\right),$ we can define $\left(- \frac{d^{2}}{dx^{2}}\right) ^{\frac{1}{2}}v_{0}$ via Fourier transform as (see e.g., ^[32] Proposition 3.3)

and write the second term in (4.1) (see ^[32] Remark 3.7) as

Since $\eta \in C^{\infty }\left(\mathbb{R}\right),$ for $x > 1$ take $\varepsilon \ll 1$ such that

and for $x < -1$ take $\varepsilon \ll 1$ such that

For $-1\leq x\leq 1,$ we can write

where the last step follows from the fact that $\eta \in C^{\infty }\left(\mathbb{\ R}\right).$

For each $x\in \mathbb{R}$ we have

Combining (4.2), (4.3) and (4.4), we conclude that the function

belongs to $L^2(\mathbb{R})$. Thus the third term in (4.1) can be written as

We now introduce the notation

where the fractional operator in the right-hand side is understood via Fourier transform. Since $\varphi$ is arbritrary, we conclude from (4.1) that $u_{0}$ satisfies the following equation in the distributional sense:

Since $\left\vert u_{0}\right\vert \leq 1$ and $v_{0} = u_{0}-\eta \in H^{1}\left(\mathbb{R}\right),$ we have $W^{\prime }\left(u_{0}\right) \in L^{2}\left(\mathbb{R}\right)$ and $\left(-\frac{d^{2}}{dx^{2}}\right) ^{ \frac{1}{2}}v_{0}$ $\in L^{2}\left(\mathbb{R}\right).$ Thus (4.5) implies that $\left(-\frac{d^{2}}{dx^{2}}\right) ^{\frac{1 }{2}}u_{0}\in L^{2}\left(\mathbb{R}\right).$ By elliptic estimates, we then conclude that $u_{0}\in W^{2, 2}\left(\mathbb{R}\right).$

Weakly differentiating (4.6) yields

in the sense of distributions. Here we used the facts that

and

which follow from the properties of Fourier transform of Sobolev functions and the following calculation:

The same arguments as in the case of (4.5) can be used to prove that the function

belongs to $L^2(\mathbb{R})$ as well. Define

Since $W\in C^{2, 1}\left(\mathbb{R}\right),$ we have $g^{\prime } W^{\prime }(u_{0})+gW^{^{\prime \prime }}\left(u_{0}\right) u_{0x}\in L^{2}\left(\mathbb{R}\right),$ $\left(-\frac{d^{2}}{dx^{2}}\right) ^{ \frac{1}{2}}v_{0x}\in L^{2}\left(\mathbb{R}\right),$ we have $\left(- \frac{d^{2}}{dx^{2}}\right) ^{\frac{1}{2}}u_{0x}\in L^{2}\left(\mathbb{R} \right).$ Thus elliptic estimates imply $u_{0x}\in W^{2, 2}\left(\mathbb{R} \right),$ i.e., $u_{0}\in W^{3, 2}\left(\mathbb{R}\right) \subset C^{2, \frac{1}{2}}\left(\mathbb{R}\right).$ Thus $u_{0}$ is a classical solution of (4.6). Moreover, since $u_{0}\in C^{2, \frac{1 }{2} }\left(\mathbb{R}\right),$ we can write

The second half of Theorem 1.1 follows. □

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We thank the referee for careful reading of our manuscript and many helpful suggestions for improving this paper. In particular, the referee pointed out Remark 2.2 when $g(x)$ is nondegenerate. The work of CM was supported, in part, by NSF via grants DMS-1614948 and DMS-1908709. XY's research was sponsored by a Research Excellence Grant from University of Connecticut, CLAS Dean's summer research grant and Simons Collaboration Grant #947054.

Conflict of interest

The authors declare no conflict of interest.