数理解析研究所講究録
第 1881 巻 2014 年 1-17

1

A Note on Mountain Pass Solutions for aClass of Allen-Cahn Models
Jaeyoung Byeon
Department of Mathematical Sciences, KAIST
291 Daehak-ro, Yuseong-gu, Daejeon 305-701
Republic of Korea
E-mail: byeon@kaist.ac.kr,
Paul H. Rabinowitz
Department of Mathematics,
University of Wisconsin-Madison,
Madison, Wisconsin, 53706, USA
E-mail: rabinowi@math.wisc.edu
March 16, 2013
Abstract
For an Allen-Cahn model problem having and 1 as solutions, theauthors recently established the existence of a large family of solutions
between and 1 [13]. These solutions were obtained as constrained
minimizers of a functional associated with the model or as limits ofsuch minimizers. In this note, it is shown that corresponding to eachminimizer, , there is a second solution, , of the problem which isof mountain pass type with $0<V<U<1.$
$0$

$0$

$U$

1

$V$

Introduction

In a recent paper [13], the authors studied the following Allen-Cahn model
equation:(1.1)$-\triangle u+A_{\epsilon}(x)G’(u)=0, x\in \mathbb{R}^{n}$

2
, and
$1+A(x)/\epsilon$ with-periodic inis the support. Then $u\equiv 0$ andofand has a smooth boundary, and$u\equiv 1$
are solutions of (1.1), the so-called pure states, and the question ofinterest is whether there are other solutions of (1.1) having $0<u<1$ , i.e.
so called mixed states. There have been many papers written on this topic,
both for Allen-Cahn models: $[1]-[4],$ $[15],$ $[18],$ $[20]-[21]$ , as well as in a moregeneral context motivated by Aubry-Mather Theory and initiated by Moser:[17], $[8]-[12],$ $[22],$ $[24]$ . In particular, it was shown in [13] that there is an
infinitude of such mixed states that shadow and 1 in any prescribed way
on a spatially periodic array of sets. To make this statement more precise,, and set
let

where $G(u)=u^{2}(1-u)^{2}$ is a double well potential,$0\leq A\in C^{1}(\mathbb{R}^{n}),$$1$$x_{1},$$\epsilon>0$$\cdots,$ $x_{n},$$A_{\epsilon}(x)=$$\Omega$$\overline{\Omega}\subset(0,1)^{n}$$A|_{[0,1]^{n}}$$0$$T\subset \mathbb{Z}^{n}$$A^{T}= \bigcup_{i\in T}(i+\Omega)$;$B^{T}= \bigcup_{i\in\{Z^{n}\backslash T\}}(i+\Omega)$.

Then one of the main results in [13] contains
Theorem 1.2. Forandeach
and each$0<U_{\epsilon,T}<1$ . Moreover as
uniformly on
$G$

$A$

as above, there is an, there is a solution
uniformly on$\epsilon_{0}>0$$T\subset \mathbb{Z}^{n}$$\epsilon\in(0, \epsilon_{0})$$\epsilonarrow 0,$$U_{\epsilon,T}$$U_{\epsilon,T}arrow 1$$A^{T}$

such that forof (1.1) with
and
$U_{\epsilon,T}arrow 0$$B^{T}$

is obA more detailed statement can be found in [13]. For finite
tained by a minimization argument where pointwise constraints are imposedon the class of admissible functions. Then, with the aid of further a priori
is obtained from the finite ca.se bybounds, the case of arbitrarypassing to a limit.
The solutions for finite are local minima of a functional correspondingto (1.1) while and 1 are global minima. Possessing a local minimum such as
which is not a global minimum is essentially the geometric hypothesisof the Mountain Pass Theorem. For the current setting, it suggests thepossibility of finding mountain pass solutions for (1.1). The main goal ofthis note is to show that corresponding to each local minimizing solution,, of mountain pass type withof (1.1), there is a second solution,. In \S 2, we will show how to prove this for the case offinite. Then in \S 3, we present a slightly refined result.
There is another way to view the geometric situation just described that
may also be fruitful.
well known result of Matano, Theorem 4.4 of [16],states that for equations like$T,$

$U_{\epsilon,T}$$T\subset Z^{n}$

$T$

$0$

$U_{\epsilon,T}$$V_{\epsilon,T}$$U_{\epsilon,T}$$T$$0<V_{\epsilon,T}<U_{\epsilon,T}$

$A$

(1.3)

$-\triangle u+f(x, u)=0$

3
on bounded domains in , if there is an ordered pair of solutions, $v<w$of (1.3) and andare stable solutions in an appropriate sense for the$\mathbb{R}^{n}$

$w$

$v$

corresponding parabolic equation(1.4)$u_{t}-\triangle u+f(x, u)=0,$

then there exists a solution, of (1.3) with $v<u<w$ . See also Amann [5]$-[7]$ and Sattinger [23], We willcomment further on this point of view in \S 2.
$u$2

The proof for finite$T$

The goal of this section is to show that if
is finite andissufficiently small, there is a mountain pass solution,of (PDE) such that. To do this, first the minimization characterization ofin [13] will be recalled. Let
the closure of
under
$T$

$\epsilon\in(0, \epsilon_{0})$$V_{\epsilon,T}$$0<V_{\epsilon,T}<U_{\epsilon,T}$$U_{\epsilon,T}$$\mathcal{W}$$C_{0}^{\infty}(\mathbb{R}^{n})$$\Vert u\Vert\equiv(\int_{\mathbb{R}^{n}}|\nabla u|^{2}dx+\int_{[-1,1]^{n}}u^{2}dx)^{\frac{1}{2}}.$For$u\in \mathcal{W}$

Let

, let

and
so $d^{*}>0$ . Choose any $d\in(O, d^{*})$ such that if$L_{\epsilon}(u)= \frac{1}{2}|\nabla u|^{2}+A_{\epsilon}G(u)$$d^{*}= \frac{1}{2}|\partial\Omega-\partial[0,1]^{n}|$$J_{\epsilon}(u)= \int_{\mathbb{R}^{n}}L_{\epsilon}(u)dx.$$\Omega_{d}\equiv\{x\in\Omega||x-\partial\Omega|>d\},$thenis diffeomorphic to
. Set
and
Choose constants
such that $0<b< \frac{1}{2}<a<1$ and set$\partial\Omega_{d}$$\partial\Omega$$a,$$A_{T}=\mathbb{Z}^{n}+\Omega_{d}$$B_{T}=(\mathbb{Z}^{n}\backslash T)+\Omega_{d}.$$b$$\Gamma(T)=\{u\in \mathcal{W}|u\geq a$$A_{T}$

and $u\leq b$ on$B_{T}\}.$

Define$c_{\epsilon}(T)= \inf_{u\in\Gamma(T)}J_{\epsilon}(u)$

Let
have:

$\chi_{S}$.

denote the characteristic function of the set$S$. Then from [13], we

Theorem 2.1. Let
and be as above. Then there exists anandindependent of with the property that for eachand each finite$A_{\epsilon}$

$G$

$T$

$T\subset \mathbb{Z}^{n},$$1^{o}\mathcal{M}_{\epsilon}(T)\equiv\{u\in\Gamma(T)|J_{\epsilon}(u)=c_{\epsilon}(T)\}\neq\emptyset.$$\epsilon_{0}>0$$\epsilon\in(0, \epsilon_{0})$4$2^{o}$

Anysatisfies $0<U<1$ and is a classical solution of$U\in \mathcal{M}_{\epsilon}(T)$$(PDE)$ .
$3^{o}\mathcal{M}_{\epsilon}(T)$is an ordered set:

$U,$ $V\in \mathcal{M}_{\epsilon}(T)$implies $U<V,$ $U>V$, or$U\equiv V.$$4^{o}$$5^{o}$

If$T\subset S\subset \mathbb{Z}^{n},$$U_{\epsilon,T}\leq U_{\epsilon,S}$

with strict inequality if $T\neq S.$

There exist constants $C,$ $c>0$ , independent ofsatisfying$T$and of$\epsilon\in(0,\epsilon_{0})$,$|U_{\epsilon,T}(x)-\chi_{T+[0,1]^{n}}(x)|\leq C\exp(-cd(x, T)) , x\in \mathbb{R}^{n}$where $d(x, T)$ denotes the distance frombe finite,Letforward to show that$T\subset \mathbb{Z}^{n}$(2.2)$\epsilon\in(0, \epsilon_{0})$

, and

$x$

to $T.$$U_{\epsilon,T}\in \mathcal{M}_{\epsilon}(T)$. Then, it is straight-$\lim_{\epsilonarrow}\sup_{0}c_{\epsilon}(T)=\lim_{\epsilonarrow}\sup_{0}J_{\epsilon}(U_{\epsilon,T})<\infty.$

To obtain a mountain pass solution of (1.1) corresponding to$\mathcal{G}_{\epsilon,T}\equiv\{g\in C([0,1], W^{1,2}(\mathbb{R}^{n})|0\leq g(\theta)\leq U_{\epsilon,T},$

$T$

, set

$g(O)=0,$ $g(1)=U_{\epsilon,T}\}$and define(2.3)$b_{\epsilon}(T)= \inf_{g\in \mathcal{G}_{\epsilon}\tau},\theta\in[0,1]\max J_{\epsilon}(g(\theta))$.

We will show:
Proposition 2.4.

$\lim\inf_{\epsilonarrow 0}\sqrt{\epsilon}b_{\epsilon}(T)>0.$

Proof: Arguing indirectly, if the result is false, there exists a sequencesuch that$\{\epsilon_{m}\}_{m=1}^{\infty}\subset(0,1)$$\lim_{marrow\infty}\epsilon_{m}=\lim_{marrow\infty}\sqrt{\epsilon_{m}}b_{\epsilon_{m}}(T)=0.$

For notational convenience, we assume that
(2.5)$\lim_{\epsilonarrow 0}\sqrt{\epsilon}b_{\epsilon}(T)-\neg 0.$

5
The definition of

implies there is a$b_{\epsilon}(T)$(2.6)$g\in \mathcal{G}_{\epsilon,T}$

such that$b_{\epsilon}(T) \leq\theta\in[0,1]\max J_{\epsilon}(g(\theta))\leq b_{\epsilon}(T)+1.$

By (2.6) for each $\theta\in[0,1],$(2.7)$\int_{\mathbb{R}^{n}}A_{\epsilon}G(g(\theta))dx\leq b_{\epsilon}(T)+1.$Letdenote an open ball of radius about. Note thatfor any. By shifting by such a if need be, we
can assume $0\in T$ . Suppose. Since there is a constant,$B_{r}(x)$$x\in \mathbb{R}^{n}$$r$$b_{\epsilon}(T)=b_{\epsilon}(T+p)$$T$$p\in \mathbb{Z}^{n}$$p$$B_{r}(z)\subset\Omega_{d}$such that for(2.8)$x\in\Omega_{d},$$A(x)\geq\omega_{d}$$\omega_{d}>0$, it follows that

$A_{\epsilon}(x)\geq\omega_{d}/\epsilon$, and by (2.7),$\int_{B_{r}(z)}G(g(\theta))dx\leq\int_{\Omega_{d}}G(g(\theta))dx\leq\int_{\Omega_{d}}\frac{\epsilon}{\omega_{d}}A_{\epsilon}(x)G(g(\theta))dx$$\leq\frac{\epsilon}{\omega_{d}}(b_{\epsilon}(T)+1)\equiv M(\epsilon)arrow 0$

as$\epsilonarrow 0.$

Set$\mathcal{A}_{\epsilon}(\theta)=\{x\in B_{r}(z)|g(\theta)>a\},$$\mathcal{B}_{\epsilon}(\theta)=\{x\in B_{r}(z)|g(\theta)\leq b\},$and$\mathcal{D}_{\epsilon}(\theta)=\{x\in B_{r}(z)|b<g(\theta)\leq a\}.$The measure of a set,
, and$|\mathcal{B}_{\epsilon}(\theta)|$

(2.9)

$S$

, will be denoted by. Then

$|S|$

. Let

$\alpha(\theta)=|\mathcal{A}_{\epsilon}(\theta)|,$$\beta(\theta)=$$\delta(\theta)=|\mathcal{D}_{\epsilon}(\theta)|$$\alpha(\theta)+\beta(\theta)+\delta(\theta)=|B_{r}(z)|=|B_{r}(0)|.$

There is a constant $\sigma=\sigma(d, a, b)>0$ such that $G(g(\theta)(x))\geq\sigma$ for $x\in$for. Hence by (2.8), for all $\theta\in[0,1],$$\mathcal{D}_{\epsilon}(\theta)\subset\Omega_{d}$$\theta\in[0,1]$$\sigma\delta(\theta)\leq\int_{\mathcal{D}_{\epsilon}(t)}G(g(\theta))dx\leq\int_{B_{r}(z)}G(g(\theta))dx\leq M(\epsilon)$or(2.10)$\delta(\theta)\leq M(\epsilon)/\sigma.$

6Note that by the choice of(2.11)$B_{r}(z)$,$\{\begin{array}{ll}\alpha(0)=0, \beta(0)=|B_{r}(0)|\alpha(1)=|B_{r}(0)|, \beta(1)=0.\end{array}$

We claim that for all$\epsilon\in(0, \epsilon_{0})$, there is a $\rho=\rho(\epsilon)>0$ such that whenever$|t-s|\leq\rho,$(2.12)$|\alpha(t)-\alpha(s)|\leq 2M(\epsilon)/\sigma.$and for each , there is a $t=t(\rho),$ $s=s(\rho)$with $|t-s|\leq\rho$ and $|\alpha(t)-\alpha(s)|>2M(\epsilon)/\sigma$ . Without loss of generality, it. Therefore
can be assumed that

Otherwise for some$\epsilon\in(0, \epsilon_{0})$$\rho$$\alpha(t)\geq\alpha(s)$(2.13)$\alpha(t)>\alpha(s)+2M(\epsilon)/\sigma.$

Set$X=\{x\in B_{r}($ 之$)$’$||g(t)(x)-g(s)(x)|\geq a-b\}.$Then(2.14)$|X|(a-b)^{2} \leq\int_{X}|g(t)(x)-g(s)(x)|^{2}dx\leq\Vert g(t)-g(s)\Vert_{W^{1,2}(\mathbb{R}^{n})}^{2}$and the right hand side of (2.14) goes to(2.13),$0$as$\rhoarrow 0$. On the other hand, by$|(\mathcal{B}_{\epsilon}(s)\cup \mathcal{D}_{\epsilon}(s))\cap \mathcal{A}_{\epsilon}(t)|\geq 2M(\epsilon)/\sigma.$

Therefore by (2.10), we see$|\mathcal{B}_{\epsilon}(s)\cap \mathcal{A}_{\epsilon}(t)|\geq M(\epsilon)/\sigma.$and this implies(2.15)$|X|\geq M(\epsilon)/\sigma.$

For small $(2.15)$ is contrary to (2.14). Thus (2.12) holds.
In a similar fashion, by making still smaller if necessary, it can be,assumed that if$\rho,$$\rho$$|t-s|\leq\rho(\epsilon)$(2.16)$|\beta(t)-\beta(s)|\leq 2M(\epsilon)/\sigma.$

7
Further requiring that is small compared to $|B_{r}(0))|$ , the inequalities (2.12),(2.16), and (2.11) imply there is aandsuch thatbelongto $(|B_{r}(0)|/3,2|B_{r}(0)|/3)$ . By Lemma 3.5 in Chapter 2 of [14], for a constant,, independent of, we have$\epsilon$$t^{*}=t^{*}(\epsilon)$$C$$\epsilon,$(2.17)$r,$ $a,$$\alpha(t^{*})$$\beta(t^{*})$$b$$( \frac{1}{3}|B_{r}(0)|)^{1-\frac{1}{n}}(a-b)\leq\alpha(t^{*})^{1-\frac{1}{n}}(a-b)$$\leq\frac{Cr^{n}}{|B_{r}(0)\backslash \{x\in B_{r}(0)|g(t^{*})(x)>b\}|}\int_{\mathcal{D}_{\epsilon}(t^{*})}|\nabla g(t^{*})|dx$$\leq\frac{Cr^{n}|\delta(t^{*})|^{\frac{1}{2}}}{\beta(t^{*})}(\int_{\mathcal{D}_{\epsilon}(t^{*})}|\nabla g($が$)|^{2}dx)^{\frac{1}{2}}$$\leq\frac{3Cr^{n}(M(\epsilon)/\sigma)^{\frac{1}{2}}}{|B_{r}(0)|}(2J_{\epsilon}(g(t^{*})))^{\frac{1}{2}}\leq\frac{3\sqrt{2}C}{\sqrt{\sigma\omega_{d}}|B_{1}(0)|}\sqrt{\epsilon}(b_{\epsilon}+1)$via (2.10), (2.8) and (2.6). But as, the right hand side of (2.17)
approaches while the left hand side is bounded away from , a contradiction.
This proves Proposition 2.4.
Next we have the an upper estimate for.
$\epsilonarrow 0$

$0$

$0$

$\square$$b_{\epsilon}(T)$Proposition 2.18.

$\lim\sup_{\epsilonarrow 0}\sqrt{\epsilon}b_{\epsilon}(T)<\infty.$

be a smooth function such that $\lim_{sarrow 0}\psi(s)=$$\lim_{sarrow 1}\psi(1)=\infty$ andis monotone increasing. Second, choose a$|\phi’(s)|\leq 2$ for$\phi(s)=0$
functionsuch that$s\geq1$
for $s\leq 0\phi(s)=1$ for. Now define
Proof: First, let$\psi$:$-\infty,$$[0,1]arrow \mathbb{R}$$\psi$$\phi\in C^{1}(\mathbb{R})$$0\leq\emptyset(\mathcal{S})\leq 1,$$s\in \mathbb{R},$$g( \theta)(x_{1}, \cdots, x_{n})=U_{\epsilon,T}(x_{1}, \cdots, x_{n})\phi(\frac{x_{1}+\psi(\theta)}{\sqrt{\epsilon}}) , \theta\in(0,1)$.

Then usingof Theorem 2.1 shows that $\lim_{sarrow 0}g(s)=0,$ $\lim_{sarrow 1}g(s)=$, the limits being taken in. Hence defining $g(O)=0$ and $g(1)=$, showsand$5^{o}$$W^{1,2}(\mathbb{R}^{n})$$U_{\epsilon,T}$$U_{\epsilon,T}$$g\in \mathcal{G}_{\epsilon,T}$(2.19)$b_{\epsilon}(T) \leq\theta\in[0,1]\max J_{\epsilon}(g(\theta))$.

To estimate the right hand side of (2.19), first observe that$\theta\in[0,1]\max\int_{\mathbb{R}^{n}}|\nabla g(\theta)|^{2}dx\leq 2\int_{\mathbb{R}^{n}}(|\nabla U_{\epsilon,T}|^{2}+\frac{1}{\epsilon}U_{\epsilon,T}^{2}|\varphi’(\frac{x_{1}+\psi(\theta)}{\sqrt{\epsilon}})|^{2})dx$

8and$\int_{\mathbb{R}^{n}}U_{\epsilon,T}^{2}|\varphi’(\frac{x_{1}+\psi(\theta)}{\sqrt{\epsilon}})|^{2})dx\leq 4\int_{0\leq x_{1}+\psi(\theta)\leq\sqrt{\epsilon}}U_{\epsilon,T}^{2}dx$

Since, again by$D_{1}>0$ , independent of small$0\leq U_{\epsilon,T}\leq 1$$5^{o}$$\epsilon>0$of Theorem 2.1, there exists a constantsuch that for each$l\in \mathbb{R},$$\int_{x_{1}=l}U_{\epsilon,T}^{2}dx_{2}\cdots dx_{n}\leq D_{1}.$This inequality implies that$\int_{0\leq x_{1}+\psi(\theta)\leq\sqrt{\epsilon}}U_{\epsilon,T}^{2}dx\leq D_{1}\sqrt{\epsilon}.$Hence byof small$5^{o}$of Theorem 2.1 yet again, there is a constant,, so that$D_{2}$

independent$\epsilon>0$(2.20)$\theta\in[0,1]\max\int_{\mathbb{R}^{n}}|\nabla g(\theta)|^{2}dx\leq D_{2}/\sqrt{\epsilon}.$

Next noting that$\int_{\mathbb{R}^{n}}A_{\epsilon}(x)G(g(\theta))dx=\int_{0\leq x_{1}+\psi(\theta)\leq\sqrt{\epsilon}}A_{\epsilon}(x)G(g(\theta))dx$$+ \int_{\sqrt{\epsilon}<x_{1}}A_{\epsilon}(x)G(U_{\epsilon,T})dx$

and estimating as above shows there is a constantsmall
and $\theta\in[0,1]$ such that, independent of$D_{3}>0^{\backslash }$$\epsilon>0$(2.21)$\int_{\mathbb{R}^{n}}A_{\epsilon}(x)G(g(\theta))dx\leq D_{3}/\sqrt{\epsilon}.$Combining (2.19)- (2.21) then yields the Proposition.
Now the main result of this section can be stated.
Theorem 2.22. For each finiteany, there is a solution,.

$T\subset \mathbb{Z}^{n}$$\epsilon\in(0, \epsilon_{1})$$J_{\epsilon}(V_{\epsilon,T})=b_{\epsilon}(T)$, there is anof (1.1) with

$V_{\epsilon,T}$$\square$$\epsilon_{1}(T)>0$

such that for
and$0<V_{\epsilon,T}<U_{\epsilon,T}$

9Before proving Theorem 2.22, some remarks are in order. First to takeadvantage of some results from [11], it is convenient to replace by a new
function,defined by $G^{*}|_{[0,1]}=G|_{[0,1]}$ andis 1-periodic in . Thus
and is a bounded function. Let$G$$G^{*}$$G^{*}(z)$$z$$G^{*}\in C^{2}(\mathbb{R})$$L_{\epsilon}^{*}(u)= \frac{1}{2}|\nabla u|^{2}+A_{\epsilon}G^{*}(u)$and$J_{\epsilon}^{*}(u)= \int_{\mathbb{R}^{n}}L_{\epsilon}^{*}(u)dx.$

When dealing with functions, , where $0\leq u\leq 1,$ $G^{*}(u)=G(u)$ so. In particular any critical point, , ofwith $0\leq u\leq 1$ will be a
critical point of and a solution of (1.1).
Let$u$$J_{\epsilon}^{*}(u)=$$J_{\epsilon}(u)$$u$$J_{\epsilon}^{*}$$J_{\epsilon}$$S_{T}\equiv\{u\in W^{1,2}(\mathbb{R}^{n})|0\leq u\leq U_{\epsilon,T}\}.$

Consider the parabolic initial value problem associated with
(2.23)$G^{*}$:$u_{t}-\triangle u+A_{\epsilon}(x)G_{u}^{*}(u)=0, t>0, x\in \mathbb{R}^{n}$$u(x, 0)=\varphi(x)\in S_{T}.$Let$\Phi^{t}(\varphi)(x)$denote the solution of (2.23). Then we have

Proposition 2.24.
1o For each
(2.23) which exists for all $t>0.$$\varphi\in S_{T}$$2^{o}$For each $t>0,$$3^{0}$For each$4^{o}$, there is a solution,$\Phi^{t}(\varphi)$of$\Phi^{t}:S_{T}\backslash arrow \mathcal{S}_{T}.$$\varphi\in S_{T},$$J_{\epsilon}(\Phi^{t}(\varphi))\leq J_{\epsilon}(\Phi^{s}(\varphi))$

For each, there is a solution,
and a sequencesuch thatas $karrow\infty.$$\varphi\in S_{T}$$(t_{k}^{*})\subset \mathbb{R}$if$\Psi(\varphi)$$t\geq s\geq 0.$, of (1.1), withand$\Psi(\varphi)\in \mathcal{S}_{T}$$\Phi^{t_{k}}(\varphi)arrow\Psi(\varphi)$$t_{k}^{*}arrow\infty$in$W^{1,2}(\mathbb{R}^{n})$Proposition 2.24 was proved in [11] for a more general class of (bounded), but withnonlinearities thanreplaced by an -torus. However
one can use the heat kernel as in [11] (which in turn was based on arguments
in [19] to convert (2.23) to an operator equation in
andthen continue to argue as in [11] to getof the Proposition.
Next we have:$A_{\epsilon}(x)G_{u}^{*}$$\mathbb{R}^{n}$

$n$

$)$

$W^{1,2}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})$$1^{o}-4^{o}$

Proposition 2.25. On
i.e. any sequence
convergent subsequence.

$S_{T},$$(u_{k})\subset S_{T}$$J_{\epsilon}=J_{\epsilon}^{*}$, withsatisfies the Palais-Smale condition,$J_{\epsilon}(u_{k})$bounded and$J_{\epsilon}’(u_{k})arrow 0,$$ha\mathcal{S}a$10
Proof: Suppose
is a Palais-Smale sequence, i.e.
and(recalling of Theorem 2.1),. Due to the form of and of. Hence there is a
is bounded insuch that along a
subsequence,
weakly in.
and in. Sinceas, the tails of are exponentially small so in fact along our
subsequence,$J_{\epsilon}(\varphi_{k})\leq M$$(\varphi_{k})\subset S_{T}$$J_{\epsilon}’(\varphi_{k})arrow 0$$J_{\epsilon}$$5^{o}$$U_{\epsilon,T}$$\varphi\in S_{T}$$W^{1,2}(\mathbb{R}^{n})$$(\varphi_{k})$$W^{1,2}(\mathbb{R}^{n})$$\varphi_{k}arrow\varphi$$0\leq\varphi\leq U_{\epsilon,T}$$L_{loc}^{2}(\mathbb{R}^{n})$$karrow\infty$$\varphi$(2.26)$\Vert\varphi_{k}-\varphi\Vert_{L^{2}(\mathbb{R}^{n})}arrow 0, karrow\infty.$

By (2.26), for any(2.27), as $karrow\infty,$$\psi\in W^{1,2}(\mathbb{R}^{n})$$| \int_{\mathbb{R}^{n}}A_{\epsilon}(x)(G_{u}(x, \varphi_{k})-G_{u}(x, \varphi))\psi dx|$$\leq K\Vert\varphi_{k}-\varphi\Vert_{L^{2}(\mathbb{R}^{n})}\Vert\psi\Vert_{L^{2}(\mathbb{R}^{n})}arrow 0$

where(2.28)$K=\Vert A_{\epsilon}\Vert_{L}\infty\Vert G_{uu}\Vert_{L^{\infty}[0,1]}$. Hence$J_{\epsilon}’(\varphi)=0$

and$J_{\epsilon}’(\varphi_{k})(\varphi_{k}-\varphi)-J_{\epsilon}’(\varphi)(\varphi_{k}-\varphi)$$= \int_{\mathbb{R}^{n}}|\nabla(\varphi_{k}-\varphi)|^{2}+A_{\epsilon}(G_{u}(x, \varphi_{k})-G_{u}(x, \varphi)(\varphi_{k}-\varphi)dxarrow 0.$as$karrow\infty$. Thus by $(2.27)-(2.28)$ ,$\int_{\mathbb{R}^{n}}|\nabla(\varphi_{k}-\varphi)|^{2}dxarrow 0$

as$karrow\infty$. Consequently$\varphi_{k}arrow\varphi$

in$W^{1,2}(\mathbb{R}^{n})$.

$\square$Now we can give theProof of Theorem 2.22: First we note from Proposition 2.4 and (2.2) that.
for smallNow we follow an argument from [12]. Let $\delta>0$ and choose $g\in\Gamma(T)$such that$\epsilon>0,$(2.29)$b_{\epsilon}(T)>J_{\epsilon}(U_{\epsilon,T})$$\max_{\theta\in[0,1]}J_{\epsilon}(g(\theta))\leq b_{\epsilon}(T)+\delta.$

By $1^{o}-2^{o}$ of Proposition 2.24, for each $\tau>0$ and $\theta\in[0,1],$
Therefore there is asuch that$\theta_{\tau}\in[0,1]$(2.30)$J_{\epsilon}(\Phi^{\tau}(g(\theta_{\tau}))\geq b_{\epsilon}(T)$.

$\Phi^{\tau}(g(\theta))\in S_{T}.$11Hence a subsequence of the points,$[0,1]$ . Denote this subsequence by$\{\theta_{\tau}|\tau>0\}$, converges to some
. We claim for all $\tau>0,$$\theta_{\infty}\in$$\{\theta_{\tau_{p}}\equiv\mu_{p}|p\in \mathbb{N}\}$(2.31)$J_{\epsilon}(\Phi^{\tau}(g(\theta_{\infty}))\geq b_{\epsilon}(T)$.

$J_{\epsilon}(\Phi^{\tau}(g(\theta_{\infty}))<b_{\epsilon}(T)$.

Otherwise for some $\tau>0,$(2.32)

Since$g$is continuous, by (2.32), for large(2.33)$p,$$J_{\epsilon}(\Phi^{\tau}(g(\mu_{p}))<b_{\epsilon}(T)$Thus by$3^{o}$of Proposition 2.24, for large(2.34).

$p,$$J_{\epsilon}(\Phi^{\mu_{p}}(g(\mu_{p}))<b_{\epsilon}(T)$,contrary to (2.30). Consequently (2.31) holds.
Next observe that by of Proposition 2.24, there is a sequence
and a solution,of (1.1) insuch thatin
Moreover by (2.31) and (2.29),$4^{o}$$S_{T}$$v_{\delta}$(2.35)$\Phi^{q_{k}}(g(\theta_{\infty}))arrow v_{\delta}$$(q_{k})\subset \mathbb{R}$$W^{1,2}(\mathbb{R}^{n})$.

$b_{\epsilon}(T)\leq J_{\epsilon}(v_{\delta})\leq b_{\epsilon}(T)+\delta.$

Letting, Proposition 2.25 shows there is a solution of (1.1),
such that along a subsequence,inand. This implies thatand completes the proof.
To conclude this section, recall that fromof Theorem 2.1,
isan ordered set. Suppose thatand there are no members oflying between andThen
is called a gap pair of solutionsof (1.1) inand we have:$\deltaarrow 0$$V_{\epsilon,T}\in S_{T}$$W^{1,2}(\mathbb{R}^{n})$$v_{\delta}arrow V_{\epsilon,T}$$c_{\epsilon}(T)$$J_{\epsilon}(V_{\epsilon,T})=b_{\epsilon}(T)>$$0<V_{\epsilon,T}<U_{\epsilon,T}$$\square$$3^{o}$$U,$$U$$\mathcal{M}_{\epsilon}(T)$$U^{*}$$\mathcal{M}_{\epsilon}(T)$$U^{*}\in \mathcal{M}_{\epsilon}(T)$$U,$$U^{*}$$\mathcal{M}_{\epsilon}(T)$Theorem 2.36. Ifexists a solution,
Moreoveris a gap pair of olutions of (1.1) in
there
$V$
(1.1)
of
of mountain pass type with $U<V<U^{*}$
where
$U,$

$U^{*}$$\mathcal{S}$$\mathcal{M}_{\epsilon}(T)_{f}$$J_{\epsilon}(V)=d_{\epsilon}(T)$$d_{\epsilon}(T)= \inf_{h\in \mathcal{H}\tau},\theta\in[0,1]\max J_{\epsilon}(h(\theta))$and$\mathcal{H}_{\epsilon,T}\equiv\{h\in C([0,1], W^{1,2}(\mathbb{R}^{n}))|U\leq h(\theta)\leq U^{*}, h(0)=U, h(1)=U^{*}\}.$

12
Proof. Once it is shown that
(2.37)$d_{\epsilon}(T)>c_{\epsilon}(T)$,the proof is essentially the same as that of Theorem 2.22. Therefore we merely
verify (2.37). Modifying an argument from [11], letand set$\rho\in(0, \Vert U-U^{*}\Vert_{L^{2}(\mathbb{R}^{n})})$$P_{\rho}\equiv\{u\in W^{1,2}(\mathbb{R}^{n})|U\leq u\leq U^{*}and \Vert U-u\Vert_{L^{2}(\mathbb{R}^{n})}=\rho\}.$We claim(2.38)$\inf_{u\in P_{\rho}}J_{\epsilon}(u)\equiv f_{\rho}(T)>c_{\epsilon}(T)$.

Indeed, suppose that (2.38) has been established. Fix, there is a corresponding
Since for any, by (2.38),$\rho\in(0, \Vert U-U^{*}\Vert_{L^{2}(\mathbb{R}^{n})})$$\theta_{h}\in(0,1)$$h\in \mathcal{H}_{\epsilon,T}$

such that.

$h(\theta_{h})\in$$P_{\rho}$$d_{\epsilon}(T)\geq f_{\rho}.(T)>c_{\epsilon}(T)$and (2.37) is established.
such
Finally to verify (2.38), if it is false, there is a sequence.
is bounded in. The sequenceasthatwith $U\leq v\leq U^{*}$ such that, along a
Therefore there is a
. Sinceasand inweakly insubsequence,is weakly lower semicontinuous,for each $R>0,$$(u_{k})\subset P_{\rho}$$J_{\epsilon}(u_{k})arrow c_{\epsilon}(T)$$karrow\infty$$(u_{k})$$W^{1,2}(\mathbb{R}^{n})$$v\in W^{1,2}(\mathbb{R}^{n})$$W^{1,2}(\mathbb{R}^{n})$$u_{k}arrow v$$L_{loc}^{2}(\mathbb{R}^{n})$$karrow\infty$$\int_{B_{R}(0)}L_{\epsilon}(\cdot)dx$$\int_{B_{R}(0)}L_{\epsilon}(v)dx\leq\lim_{karrow}\inf_{\infty}\int_{B_{R}(0)}L_{\epsilon}(u_{k})dx\leq\lim_{karrow}\inf_{\infty}J_{\epsilon}(u_{k})=c_{\epsilon}(T)$.

Since this is true for all $R>0,$(2.39)
But $U,$since(2.40)$J_{\epsilon}(v)\leq c_{\epsilon}(T)$$U^{*}$.

is a gap pair so (2.39) implies $v=U$ or$v=U^{*}\backslash$This is impossible$\Vert U-v\Vert_{L^{2}(\mathbb{R}^{n})}=\rho.$

Indeed, to verify (2.40), observe that

(2.41)$\rho^{2}=\Vert U-u_{k}\Vert_{L^{2}(\mathbb{R}^{n})}^{2}=\int_{B_{R}(0)}|U-u_{k}|^{2}dx+\int_{\mathbb{R}^{n}\backslash B_{R}(0)}|U-u_{k}|^{2}dx$

13
As $karrow\infty$ along our subsequence, the first term on the right in (2.41)
converges to$\int_{B_{R}(0)}|U-v|^{2}dx$

while for each$k\in \mathbb{N},$$\int_{\mathbb{R}^{n}\backslash B_{R}(0)}|U-u_{k}|^{2}dx\leq\int_{\mathbb{R}^{n}\backslash B_{R}(0)}|U-U^{*}|^{2}dx.$

Letting $Rarrow\infty$ and noting that, by of Theorem 2.1, andeach decay$(2.40)$ follows.
exponentially to as
Remark 2.42. Theorem 1.2 provides the existence of a solution,, of(1.1) for any
and any
independent of . If iswithfinite, Theorem 2.22 gives another solution,, for a more restricted setof values of : $0<\epsilon<\epsilon_{1}(T)$ whereis possibly very small. The
size restriction on
is required to get. It is natural to
ask whether the range of values for the existence of a second solution canbe broadened, perhaps even to . One possible approach to this question is
to use the relationship between Theorem 2.22 and the Theorem of Matano[16] mentioned in the Introduction. The latter result is for bounded domains
while the current setting is. Ignoring this fact, the Matano result requires
stability behavior relative to (1.4) for the solutions $u=0$ andof(1.1). The linearized equation of (1.1) about $u=0$ is$U$$5^{O}$$U^{*}$$|x|arrow\infty,$$0$$\square$$U_{\epsilon,T}$$T\in \mathbb{Z}^{n}$$\epsilon\in(0, \epsilon_{0})$$\epsilon_{0}$

$T$

$T$

$V_{\epsilon,T}$$\epsilon_{1}(T)$$\epsilon$$\epsilon_{1}(T)$$b_{\epsilon}(T)>c_{\epsilon}(T)$$\epsilon_{0}$$\mathbb{R}^{n}$$u=U_{\epsilon,T}$(2.43)$-\triangle v+A_{\epsilon}(x)v=0$, showing that will be a stable solution of (1.4). What happens
where
for initial data for (1.4) nearis less clear. However, at the formal level, acalculation shows(where is as in) is a weak subsolutionof (1.1), at least for small , and that may suffice for the stability near
Whether these heuristics can be used to obtain an existence theorem remains
an open question.
$A_{\epsilon}\geq 1$$0$$U_{\epsilon,T}$$\min(U_{\epsilon,T}, a)$$\epsilon$$a$$\Gamma(T)$$U_{\epsilon,T}.$$\backslash$3$A$

generalization of Theorem 2.22 and some

remarks
In this final section we will prove a version of Theorem 2.22 that contains
more qualitative information about solutions of mountain pass type. Let14$S\subset T\subset \mathbb{Z}^{n}wi\dot{t}hT$

finite and $S\neq T$ . Define$G_{\epsilon}(S, T) \equiv \{g\in C([O, 1], W^{1,2}(\mathbb{R}^{n})|U_{\epsilon,S}\leq g(\theta)\leq U_{\epsilon,T}$

and$g(O)=U_{\epsilon,S},$$g(1)=U_{\epsilon,T}\}$and set$b_{\epsilon}(S, T)= \inf_{g\in G_{e}(S,T)\theta}\max_{\in[0,1]}J_{\epsilon}(g(\theta))$.

Then corresponding to Proposition 2.4, we have
Proposition 3.1.

$\lim\inf_{\epsilonarrow 0}\sqrt{\epsilon}b_{\epsilon}(S, T)>0.$

Proof: The proof is quite close to that of the earlier case so we will be brief.
. Then following the previous
so thatandChooseproof yields the Proposition.
Next corresponding to Proposition 2.18, we also have$q\in T\backslash S$$B_{r}(z)\subset q+\Omega_{d}$$r,$ $z$$\square$Proposition 3.2.

$\lim\sup_{\epsilonarrow 0}\sqrt{\epsilon}b_{\epsilon}(S, T)<\infty.$

To continue, replace the set$S_{T}$

of \S 2 by$\Psi(S, T)=\{u\in W^{1,2}(\mathbb{R}^{n})|U_{\epsilon,S}\leq u\leq U_{\epsilon,T}\}.$

Then there are analogues of Proposition 2.24 and Proposition 2.25 corresponding to this replacement. In particular, $\Psi(S, T)$ is invariant underAs earlier, these tools and the proof of Theorem 2.22 yield:$\Phi^{t}.$Theorem 3.3. Let
such that for any
and

$S$

and

$\epsilon\in(0, \epsilon_{2})$$W_{\epsilon,S,T}<U_{\epsilon,T}$be as above. Then there is an, there is a solution,of (1.1) with.

$T$$\epsilon_{2}=\epsilon_{2}(S,T)>0$$W_{\epsilon,S,T}$$U_{\epsilon,S}<$$J_{\epsilon}(W_{\epsilon,S,T})=b_{\epsilon}(S, T)$

We conclude with two remarks and some open questions.
and an infinite set
Remark 3.4. Consider a finite set$S\subset
T$. For each large integer $m>0,$ $S\subset T_{m}\equiv T\cap[-m, m]^{n}.$withandThen by Theorem 2.1, there are corresponding solutions. Theorem 3.3 provides ansuch that for any, there is a solution,of (1.1) with
. In [13], we showed that there is anandand large integer $m>0,$
independent of $m>0$ such that for
converges in
to a
is a solution of (1.1), and that as $marrow\infty,$$S\in \mathbb{Z}^{n}$$T\subset \mathbb{Z}^{n}$$U_{\epsilon,S}\in\Gamma(S)$$\epsilon_{m}=\epsilon_{m}(S, T_{m})$$U_{\epsilon,T_{m}}\in\Gamma(T_{m})$$\epsilon\in(0, \epsilon_{m})$$U_{\epsilon,S}<W_{\epsilon,S,T_{m}}<U_{\epsilon,T_{m}}$$W_{\epsilon,S,T_{m}}$$\epsilon_{0}>0,$$J_{\epsilon}(W_{\epsilon,S,T_{m}})=b_{\epsilon}(S, T_{m})$$\epsilon\in(0, \epsilon_{0})$$U_{\epsilon,T_{m}}$$U_{\epsilon,T_{m}}$$C_{loc}^{2}(\mathbb{R}^{n})$15solutionof (1.1) with the property thatuniformly on
anduniformly on. It is an interesting
open question as to whether. If this is the case, forsmall
converges to a solutionof (1.1) with
. Hence, it is then also an interesting open question to see
whetheror
Acknowledgments This research of the first author was supported by
the Mid-career Researcher Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and$U_{\epsilon,T}$$T+\Omega$$\lim_{\epsilonarrow 0}U_{\epsilon,T}(x)=1$$\lim_{\epsilonarrow 0}U_{\epsilon,T}(x)=0$$(\mathbb{Z}^{n}\backslash T)+\Omega$$\lim\inf_{marrow\infty}\epsilon_{m}>0$$\epsilon>0,$$W_{\epsilon,S,T_{m}}$$W_{\epsilon,S,T}$$U_{\epsilon,S}<$$W_{\epsilon,S,T}\leq U_{\epsilon,T}$$W_{\epsilon,S,T}<U_{\epsilon,T}$$W_{\epsilon,S,T}=U_{\epsilon,T}.$

Technology (MEST) (No. 2010-0014135).

References[1] FRANCESCA ALESSIO, LOUIS JEANJEAN AND PIERO MONTECCHIARI,
Stationary layered solutions infor a class of non autonomous AllenCahn equations, Calc. Var. Partial Differential Equations, 11 (2000),$\mathbb{R}^{2}$

177-202[2] FRANCESCA ALESSIO, LOUIS JEANJEAN AND PIERO MONTECCHIARI,
Existence of infinitely many stationary layered solutions infor a classof periodic Allen-Cahn equations, Comm. Partial Differential Equations,
27 (2002), 1537-1574$\mathbb{R}^{2}$[3] FRANCESCA ALESSIO AND PIERO MONTECCHIARI, Entire solutions infor a class of Allen-Cahn equations, ESAIM Control Optim. Calc.
Var., 11 (2005), 633-672$\mathbb{R}^{2}$[4] FRANCESCA ALESSIO AND PIERO MONTECCHIARI, Multiplicity of entire solutions for a class of almost periodic Allen-Cahn type equations,Adv. Nonlinear Stud., 5 (2005), 515-549[5] HERBERT AMANN, On the existence of positive solutions of nonlinear
elliptic boundary value problems, Indiana Univ. Math. J., 21 (1971),

125-146[6] HERBERT AMANN, Existence of multiple solutions for nonlinear elliptic
boundary value problems, Indiana Univ. Math. J., 21 (1972), 925-935

16[7] HERBERT AMANN, Supersolutions, monotone iterations, and stability,
J. Differential Equations, 21 (1976), 363-377[8] VICTOR BANGERT, On minimal laminations of the torus, Ann. Inst.
Poincare Anal. Non Lineaire, 6 (1989), 95-138

of irrational[9] UGO BESSI, Many solutions of elliptic problems on
slope, Comm. Partial Differential Equations, 30 (2005), 1773-1804$\mathbb{R}^{n}$[10] UGO BESSI, Slope-changing solutions of elliptic problems onlinear Anal., 68 (2008), 3923-3947$\mathbb{R}^{n}$, Non-[11] SERGEY BOLOTIN AND PAUL. H. RABINOWITZ, On the multiplicity of
periodic solutions of mountain pass type for a class of semilinear PDE’s.
J. Fixed Point Theory Appl., 2 $(2007),.313-331.$[12] SERGEY BOLOTIN AND PAUL. H. RABINOWITZ, Hybrid mountain passhomoclinic solutions of a class of semilinear elliptic PDEs, to appear inAnn. Inst. Poincare Anal. Non Lineaire[13] JAEYOUNG BYEON AND PAUL H. RABINOWITZ, On a phase transition
model, to appear in Calc. Var. Partial Differential Equations[14] O. A. LADYZHENSKAYA AND N. N. URAL’TSEVA, Linear elliptic andQuasilinear Elliptic Equations, Academic Press, New York and London,(1968)$[15]$RAFAEL DE LA LLAVE AND ENRICO VALDINOCI, Multiplicity resultsfor interfaces of Ginzburg-Landau Allen-Cahn equations in periodic media, Adv. Math., 215 (2007), 379-426[16] HIROSHI MATANO, Asymptotic Behavior and Stability of Solutions of
Semilinear Diffusion Equations, Publ. RIMS, Kyoto Univ., 15 (1979),
401-454.

[17] J\"URGEN MOSER, Minimal solutions of a variational problems on a torus,
Ann. Inst. Poincare Anal. Non Lineaire, 3 (1986), 229-272[18] MATTEO NOVAGA AND ENRICO VALDINOCI, Bump solutions for themesoscopic Allen-Cahn equation in periodic media. Calc. Var. Partial
Differential Equations, 40 (2011), no. 1-2, 37-49

17[19] AMNON PAZY, Semigroups of Linear Operators and Partial Differential
Equations, Springer Applied Mathematical Sciences series 44, (1983)[20] PAUL H. RABINOWITZ AND ED STREDULINSKY, Mixed states for an
Allen-Cahn type equation, Comm. Pure Appl. Math., 56 (2003), 1078-

1134[21] PAUL H. RABINOWITZ AND ED STREDULINSKY, Mixed states for an
Allen-Cahn type equation. II, Calc. Var. Partial Differential Equations,
21 (2004), 157-207[22] PAUL H. RABINOWITZ AND ED STREDULINSKY, Extensions of MoserBangert Theory: Locally Minimal Solutions, Progress in Nonlinear Differential Equations and Their Applications, 81, Birkhauser, Boston,(2011)[23] Sattinger, D. H., Monotone methods in nonlinear elliptic and parabolic
boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000.
[24] ENRICO VALDINOCI, Plane-like minimizers in periodic media: jet flows
and Ginzburg-Landau-type functionals, J. Reine Angew. Math., 574(2004), 147-185