1 Introduction
In recent years, a great deal of attention has been devoted to the study of nonlinear elliptic equations involving the nonlocal operators. Nonlocal operators such as the fractional Laplacian (-Δ)s appear in continuum mechanics, phase transition phenomena, population dynamics and game theory (e.g., [18, 20]). The fractional Laplacian (-Δ)s is a pseudo-differential operator defined by
(
-
Δ
)
s
u
(
x
)
=
C
N
,
s
lim
ϵ
→
0
∫
ℝ
N
∖
B
ϵ
(
x
)
u
(
x
)
-
u
(
y
)
|
x
-
y
|
N
+
2
s
𝑑
y
,
x
∈
ℝ
N
,
where s∈(0,1) and CN,s>0 is a constant given by
C
N
,
s
=
[
∫
ℝ
N
1
-
cos
ζ
1
|
ζ
|
N
+
2
s
𝑑
ζ
]
-
1
.
For any u∈𝒮, the Schwartz space of rapidly decaying C∞ functions in ℝN, the fractional Laplacian (-Δ)s can also be defined as follows:
(
-
Δ
)
s
u
(
x
)
=
ℱ
-
1
(
|
ξ
|
2
s
(
ℱ
u
)
)
(
x
)
for all
ξ
∈
ℝ
N
,
where
ℱ
u
=
u
^
=
(
2
π
)
-
N
/
2
∫
ℝ
N
u
(
x
)
e
-
i
x
⋅
ξ
𝑑
x
denotes the Fourier transform of u.
In this paper, we consider the following problem:
(1.1)
(
-
Δ
)
s
u
(
x
)
+
F
′
(
u
(
x
)
)
=
0
,
x
∈
ℝ
.
This nonlocal equation has been studied in many works for different types of potentials F. Cabré and Sire studied layer solutions of this equation with some special potential F (see [5, 6]). Under the general Berestycki–Lions-type assumptions on -F′(u), Chang and Wang [10] obtained a ground state positive solution of (1.1). Frank and Lenzmann [16] gave the uniqueness of the ground state solution of (1.1) with F′(u)=u-|u|p-2u. And the regularity of solutions was examined in [7, 8, 22, 23].
In this paper, we focus on the existence of periodic solutions of (1.1). As far as we know, not much study of the periodic solution to elliptic equations involving (-Δ)s was yet conducted. Only [12, 15, 17] considered the periodic solution of (1.1), and [13] considered the periodic solution of an elliptic system involving (-Δ)s. We refer the readers to [3, 2, 4] and the references therein, where periodic solutions have been studied for nonlocal problems involving a spectral fractional Laplacian. In [12, 17], the existence of periodic solutions was studied by using the variational method. Our study is motivated by these works, in particular by the works [12, 17]. Our goal here is to study the multiplicity of periodic solutions, which was not considered in these works. We make use of Clark’s theorem and Morse theory to establish multiple solutions. These are classical tools in the study of periodic solutions for local operators (e.g.,[9, 19, 21]). We develop a framework so these machinery can be applied to fractional Laplacian equations. This may be of independent interest.
To study our multiplicity results of periodic solutions of (1.1), we make the following assumptions:
F is a smooth enough double-well potential with the well bottom at ±1, i.e. it satisfies
F
(
±
1
)
=
0
<
F
(
t
)
,
F
′
(
±
1
)
=
0
,
t
∈
(
-
1
,
1
)
,
F is non-decreasing in (-1,0) and is non-increasing in (0,1).
Now our results are stated as follows.
Theorem 1.1.
Assume F satisfies (F1), (F2) and is even. Then for any k≥1 there exists a Tk such that for any T>Tk problem (1.1) has k pairs of odd periodic solutions ±ui,T (i=1,…,k) of period T and |ui,T|∞∈(0,1).
Theorem 1.2.
Assume F is even and satisfies (F1). If F′′(0)<0, then for any k≥1 there exists a T~k such that for any T>T~k, problem (1.1) has k pairs of odd periodic solutions ±u~i,T of period T and |u~i,T|∞∈(0,1), i=1,…,k.
Theorem 1.3.
Assume F satisfies (F1) and F′′(±1)>0. Suppose F also satisfies the following conditions:
F
is increasing in
(
-
1
,
0
)
and is decreasing in
(
0
,
1
)
.
If
T
(
-
F
′′
(
0
)
)
1
2
s
2
π
is a positive integer, then either
F
′
(
t
)
t
≥
F
′′
(
0
)
or
F
′
(
t
)
t
≤
F
′′
(
0
)
holds for
t
≠
0
in a neighborhood of 0.
Then there exists a T such that for any T>T problem (1.1) has two periodic solutions v1,T and v2,T of period T, and |v1,T|∞,|v2,T|∞∈(0,1).
Remark 1.4.
We note that Theorem 1.1 and Theorem 1.2 require F to be even and we establish multiple solutions, while Theorem 1.3 does not require the evenness of F and we give two solutions. Our work is motivated by [17] and is the further study of [17]. Gui, Zhang and Du [17] obtained one nonconstant periodic solution of (1.1) under the assumptions of Theorem 1.1 and Theorem 1.3 without (F3). In Theorem 1.2, we do not assume (F2) and there may exist some other constant solutions than 0 and ±1.
We shall use a localization argument developed by Caffarelli–Silvestre [8] which transforms the nonlocal problem to a local one in a domain in ℝ2 for which we will develop necessary estimates so that some classical minimax arguments and Morse theory may be applied. These will enable us to establish the multiplicity of periodic solutions for the original nonlocal problem.
This paper is organized as follows. In Section 2, we will introduce some notation and give some preliminary results. In Section 3, we consider a truncated problem proving the k pairs solutions exploiting Clark’s theorem, and use the maximum principle proving that these k pairs solutions are actually solutions of (1.1). Section 4 is devoted to the proof of Theorem 1.3, mainly using a truncated argument, the mountain pass theorem and the Morse index.
2 Preliminaries
In this section, for the convenience of the readers, we collect some basic results that will be used in the forthcoming sections. As we mention in Section 1, we will use a localization argument developed by Caffarelli–Silvestre [8]. For a given function u, the solution of the boundary value problem
{
-
div
(
y
1
-
2
s
∇
w
)
=
0
,
(
x
,
y
)
∈
ℝ
+
2
=
{
(
x
,
y
)
∈
ℝ
2
:
y
>
0
}
,
w
(
x
,
0
)
=
u
(
x
)
,
x
∈
∂
ℝ
+
2
=
{
(
x
,
0
)
∈
ℝ
2
}
.
is called s-harmonic extension of u, denoted by w=hs(u). In particular, w=Ps(⋅,y)∗u, where
P
s
(
x
,
y
)
=
p
s
y
2
s
(
x
2
+
y
2
)
1
+
2
s
2
,
is the s-Poisson kernel and ps is a constant so that ∫ℝPs(x,y)𝑑x=1. From [8] we know that
lim
y
→
0
y
1
-
2
s
∂
w
∂
y
(
x
,
y
)
=
-
k
s
(
-
Δ
)
s
u
(
x
)
,
where
k
s
=
2
1
-
2
s
Γ
(
1
-
s
)
Γ
(
s
)
.
By w=Ps(⋅,y)∗u, we can conclude that w(x,y) is odd with respect to x if u(x) is odd, and w(x,y) is periodic with respect to x if u(x) is periodic. Hence, we can reformulate problem (1.1) in a relatively local way as follows:
(2.1)
{
-
div
(
y
1
-
2
s
∇
w
)
=
0
in
ℝ
+
2
,
-
1
k
s
∂
w
∂
ν
=
-
F
′
(
w
(
x
,
0
)
)
in
ℝ
,
where
∂
w
∂
ν
=
lim
y
→
0
y
1
-
2
s
∂
w
∂
y
(
x
,
y
)
=
-
k
s
(
-
Δ
)
s
u
(
x
)
.
We will omit the constant ks in the following.
2.1 Variational Framework
We first introduce some function spaces:
ℋ
τ
=
{
w
(
x
,
y
)
:
w
(
x
+
T
,
y
)
=
w
(
x
,
y
)
for all
y
≥
0
,
∥
w
∥
2
=
∫
0
∞
∫
-
T
/
2
T
/
2
y
1
-
2
s
|
∇
w
(
x
,
y
)
|
2
d
x
d
y
+
∫
-
T
/
2
T
/
2
|
w
(
x
,
0
)
|
2
d
x
<
∞
}
,
H
τ
s
=
{
u
(
x
)
:
u
(
x
+
T
)
=
u
(
x
)
,
∥
u
∥
s
2
=
∫
-
T
/
2
T
/
2
∫
-
T
/
2
T
/
2
|
u
(
x
)
-
u
(
z
)
|
2
|
x
-
z
|
1
+
2
s
𝑑
x
𝑑
z
+
∫
-
T
/
2
T
/
2
|
u
(
x
)
|
2
𝑑
x
<
∞
}
,
H
^
τ
s
=
{
u
∈
H
τ
s
:
∥
u
∥
H
^
τ
s
2
=
∫
-
T
/
2
T
/
2
∫
-
T
/
2
T
/
2
∑
j
=
-
∞
,
j
∈
Z
+
∞
|
u
(
x
)
-
u
(
z
)
|
2
|
x
-
z
+
j
T
|
1
+
2
s
d
x
d
z
+
∫
-
T
/
2
T
/
2
|
u
(
x
)
|
2
𝑑
x
<
∞
}
,
L
τ
p
=
{
u
(
x
)
:
u
(
x
+
T
)
=
u
(
x
)
,
∥
u
∥
p
p
=
∫
-
T
/
2
T
/
2
|
u
(
x
)
|
p
𝑑
x
<
∞
}
.
Clearly, H^τs continuously imbeds into Hτs. The following results are well-known.
Lemma 2.1 (see [11]).
Let s∈(0,1). The following imbeddings are continuous:
H
τ
s
↪
L
τ
p
,
p
∈
[
2
,
2
s
*
]
,
s
>
1
2
,
H
τ
s
↪
L
τ
p
,
p
∈
[
2
,
2
s
*
)
,
s
=
1
2
,
H
τ
s
↪
L
τ
p
,
p
∈
[
2
,
2
s
*
]
,
s
<
1
2
,
where 2s*=21-2s if s<12, and 2s*=∞ if s≥12. Moreover, Hτs is compactly imbedded in Lτp for all 2≤p<2s*.
Lemma 2.2 (see [12]).
Let s∈(0,1). There exists a constant C=C(s)>0 such that
∥
Tr
(
w
)
∥
H
^
τ
s
≤
C
∥
w
∥
,
where Tr(w) is the trace of w.
We shall say that w∈ℋτ is a weak periodic solution of (2.1) if
∫
Ω
T
y
1
-
2
s
∇
w
(
x
,
y
)
∇
v
(
x
,
y
)
𝑑
x
𝑑
y
+
∫
-
T
/
2
T
/
2
F
′
(
w
(
x
,
0
)
)
v
(
x
,
0
)
𝑑
x
=
0
for all v∈ℋτ. For all w∈ℋτ we set
(2.2)
J
(
w
)
=
1
2
∫
Ω
T
y
1
-
2
s
|
∇
w
(
x
,
y
)
|
2
𝑑
x
𝑑
y
+
∫
-
T
/
2
T
/
2
F
(
w
(
x
,
0
)
)
𝑑
x
,
where ΩT=[-T2,T2]×(0,∞), and for any w,v∈ℋτ it holds that
〈
J
′
(
w
)
,
v
〉
=
∫
Ω
T
y
1
-
2
s
∇
w
(
x
,
y
)
∇
v
(
x
,
y
)
𝑑
x
𝑑
y
+
∫
-
T
2
T
2
F
′
(
w
(
x
,
0
)
)
v
(
x
,
0
)
𝑑
x
.
Therefore, weak solutions to (2.1) correspond to critical points of the functional (2.2).
For c∈ℝ, we say that J satisfies the (PS)c condition if for any sequence {wn}n≥1⊂ℋτ with J(wn)→c, the sequence J′(wn)→0 in ℋτ* has a convergent subsequence in ℋτ.
The standard version of the mountain pass theorem is due to Ambrosetti–Rabinowitz [1] and goes as follows.
Lemma 2.3.
Let X be a real Banach space and let I∈C1(X,R) satisfy the (PS)c condition for any c∈R. Suppose there exist u0,u1∈X and r>0 such that u1∉Br(u0) and I|∂Br(u0)>max{I(u0),I(u1)}. Then
c
=
inf
g
∈
Γ
max
t
∈
[
0
,
1
]
I
(
g
(
t
)
)
is a critical value of I, where
Γ
=
{
g
∈
C
(
[
0
,
1
]
)
:
g
(
0
)
=
u
0
,
g
(
1
)
=
u
1
}
.
The following version of Clark’s theorem can be found in [21].
Lemma 2.4.
Let X be a real Banach space and let I∈C1(X,R) with I even, bounded from below, and satisfying the (PS)c condition for any c∈R. Suppose I(0)=0. There exists a set K⊂X such that K is homeomorphic to ∂Bk by an odd map where Bk is the k-dimensional unit ball, and supKI<0. Then I possesses at least k distinct pairs of critical points.
2.2 Critical Groups
We now recall some results about critical group and Morse theory, for which we refer the readers to [9, 19] for further information. Let X be a Hilbert space with Y⊂X and let Hq(X,Y,G) be the q-th singular relative homology group with the abelian coefficient group G. Assume u0 is an isolated critical point of f∈C2(X,ℝ) with f(u0)=c. Let U be a neighborhood of u0 such that U∩K∩fc={u0}, where K is the critical set of f and fc={u∈X:f(u)≤c}. The group
C
q
(
f
,
u
0
)
=
H
q
(
f
c
∩
U
,
f
c
∩
U
∖
{
u
0
}
,
G
)
,
q
∈
ℕ
,
is said to be the q-th critical group of f at u0. If f′′(u0) has a bounded inverse, then u0 is called a nondegenerate critical point. And we call the dimension of the negative subspace of f′′(u0) the Morse index of u0, denoted by i(f,u0).
We recall some special cases in which the computation of critical groups is immediate. Let δh,k be the Kronecker symbol.
Lemma 2.5.
Assume f∈C2(X,R) satisfies the (PS)c condition for any c∈R. Let u be an isolated critical point of f. Then the following assertions hold:
If
u
is a local minimum of
f
, then
C
q
(
f
,
u
)
=
δ
q
,
0
G
for all
q
∈
ℕ
.
If
u
is a nondegenerate critical point of
f
with Morse index
j
, then
C
q
(
f
,
u
)
=
δ
q
,
j
G
for all
q
∈
ℕ
.
If
u
is a mountain-pass critical point of
f
with Morse index 1 , then
C
q
(
f
,
u
)
=
δ
q
,
1
G
for all
q
∈
ℕ
.
If
dim
X
=
n
<
∞
and
u
is a local maximum of
f
, then
C
q
(
f
,
u
)
=
δ
q
,
n
G
for all
q
∈
ℕ
.
Lemma 2.6 (Shifting Theorem).
Assume f∈C2(X,R) satisfies the (PS)c condition for any c∈R. Let u be an isolated critical point of f with Morse index j. Then
C
q
(
f
,
u
)
=
C
q
-
j
(
f
ˇ
,
u
)
for all
q
∈
ℕ
,
where fˇ=f|N and N=kerf′′(u).
We now recall the following Morse identity.
Lemma 2.7.
Assume that f∈C2(X,R) satisfies the (PS)c condition for any c∈R and that f has only m critical points {ui}1≤i≤m. Set
β
q
=
rank
H
q
(
X
,
f
η
,
G
)
for all
q
∈
ℕ
,
where η<infi∈[1,m]f(ui), and
M
q
=
∑
i
=
1
m
rank
C
q
(
f
,
u
i
)
for all
q
∈
ℕ
.
Then there exists a formal power series Q(t)=∑k=0∞qktk (qk≥0 for all k∈N) such that for all t∈R,
∑
q
=
0
∞
M
q
t
q
=
∑
q
=
0
∞
β
q
t
q
+
(
1
+
t
)
Q
(
t
)
.
2.3 Some Regularity Results
This section is devoted to give some regularity results about the fractional equation. We first introduce
B
R
=
{
(
x
,
y
)
∈
ℝ
2
:
|
(
x
,
y
)
|
<
R
}
,
B
R
+
=
{
(
x
,
y
)
∈
ℝ
2
:
y
>
0
,
|
(
x
,
y
)
|
<
R
}
,
Γ
R
0
=
{
(
x
,
0
)
∈
ℝ
2
:
|
(
x
,
0
)
|
<
R
}
,
Γ
R
+
=
{
(
x
,
y
)
∈
ℝ
2
:
y
≥
0
,
|
(
x
,
y
)
|
=
R
}
,
and for a smooth domain Ω⊂ℝ+2 and a∈(-1,1) we set
H
1
(
Ω
,
y
a
d
x
d
y
)
=
{
w
∈
L
2
(
Ω
,
y
a
d
x
d
y
)
:
|
∇
w
|
∈
L
2
(
Ω
,
y
a
d
x
d
y
)
}
.
We consider the following equation:
(2.3)
{
-
div
(
y
1
-
2
s
∇
w
)
=
0
,
(
x
,
y
)
∈
B
R
+
,
∂
w
∂
ν
=
-
d
(
x
)
w
(
x
,
0
)
,
(
x
,
0
)
∈
Γ
R
0
,
where d(x)∈L∞(ΓR0).
We say w is a weak solution of (2.3) if
∫
B
R
+
y
1
-
2
s
|
∇
w
|
2
𝑑
x
𝑑
y
+
∫
Γ
R
0
w
2
(
x
,
0
)
𝑑
x
<
+
∞
and
∫
B
R
+
y
1
-
2
s
∇
w
(
x
,
y
)
∇
v
(
x
,
y
)
𝑑
x
𝑑
y
+
∫
Γ
R
0
d
(
x
)
w
(
x
,
0
)
v
(
x
,
0
)
𝑑
x
=
0
for all v∈C1(BR+¯) such that v≡0 on ΓR+.
We now recall a maximum principle.
Lemma 2.8 ([5, Corollary 4.12]).
Let R>0, let d be a Hölder continuous function in ΓR0 and let
w
∈
L
∞
(
B
R
+
)
∩
H
1
(
B
R
+
,
y
1
-
2
s
)
with w≥0 in BR+ be a weak solution of (2.3). Then w>0 in BR+∪ΓR0 unless w≡0 in BR+.
The following estimate is from [7, Corollary 2.5] and [14, Theorem 2.3.12].
Lemma 2.9.
Let w∈H1(B2R,y1-2sdxdy) be a weak solution of
-
div
(
|
y
|
1
-
2
s
∇
w
)
=
0
,
(
x
,
y
)
∈
B
2
R
.
Then there exists small α>0 depending on s such that
sup
B
R
|
D
x
k
w
(
x
,
y
)
|
≤
C
R
k
(
sup
B
2
R
w
-
inf
B
2
R
w
)
,
≤
C
α
(
B
R
)
C
R
k
+
α
(
sup
B
2
R
w
-
inf
B
2
R
w
)
,
Adopting some ideas of [5], we have the following lemma.
Lemma 2.10.
Let f∈C1,γ(R) with γ>max{0,1-2s} and let
w
∈
L
∞
(
B
4
R
+
)
∩
H
1
(
B
4
R
+
,
y
1
-
2
s
d
x
d
y
)
with w(x,0)∈L∞(Γ4R0) be a weak solution of
{
-
div
(
y
1
-
2
s
∇
w
)
=
0
,
(
x
,
y
)
∈
B
4
R
+
,
∂
w
∂
ν
=
f
(
w
(
x
,
0
)
)
,
(
x
,
0
)
∈
Γ
4
R
0
.
Then w(x,0)∈C2,α(ΓR0¯) for some 0<α<1 depending only on s and γ.
Proof.
Fix η∈C0∞(ℝ) such that η=1 on Γ3R0, 0≤η≤1 on ℝ and suppη⊂Γ4R0. Multiply f(w(x,0)) by η and call it f~(x,0). Let v(x,y)=Γs(⋅,y)∗f~(x,0). Then v(x,y) is a bounded weak solution of
{
-
div
(
y
1
-
2
s
∇
v
)
=
0
,
(
x
,
y
)
∈
ℝ
+
2
,
∂
v
∂
ν
=
f
~
(
x
,
0
)
,
(
x
,
0
)
∈
∂
ℝ
+
2
,
where we used that Γs(x,y)=cs|(x,y)|2s-1 (cs>0 is a constant) is the solution of
{
-
div
(
y
1
-
2
s
∇
Γ
s
)
=
0
,
(
x
,
y
)
∈
ℝ
+
2
,
∂
Γ
s
∂
ν
=
δ
0
,
(
x
,
0
)
∈
∂
ℝ
+
2
,
where δ0 is the Dirac delta distribution at the origin. And the function v(x,0) solves
(2.4)
(
-
Δ
)
s
v
=
f
~
(
x
,
0
)
for all
(
x
,
0
)
∈
ℝ
.
Applying [23, Proposition 2.9], we have the following assertions:
If s≤12, then v(x,0)∈C0,β(ℝ) for any β<2s.
If s>12, then v(x,0)∈C1,β(ℝ) for any β<2s-1.
Let U=w-v. Then U is a bounded weak solution of
{
-
div
(
y
1
-
2
s
∇
U
)
=
0
,
(
x
,
y
)
∈
B
3
R
+
,
∂
U
∂
ν
=
0
,
(
x
,
0
)
∈
Γ
3
R
0
.
By [8, Lemma 4.1], the extension function
U
~
(
x
,
y
)
=
{
U
(
x
,
y
)
,
y
≥
0
,
U
(
x
,
-
y
)
,
y
<
0
,
is the solution of
-
div
(
|
y
|
1
-
2
s
∇
U
~
)
=
0
,
(
x
,
y
)
∈
B
3
R
.
By Lemma 2.9, there exist two constants μ=μ(s)>0 and C=C(R,s,|f|∞,|w|∞)>0 such that
∥
U
~
∥
C
μ
(
B
5
R
/
2
)
+
∥
D
x
U
~
∥
C
μ
(
B
5
R
/
2
)
+
∥
D
x
2
U
~
∥
C
μ
(
B
5
R
/
2
)
≤
C
.
Hence w(x,0)∈C1,ν(Γ2R0¯) with ν<2s-1 for s>12. If 14<s≤12, fix η1∈C0∞(ℝ) such that η1=1 on Γ9R/40, 0≤η1≤1, on ℝ and suppη1⊂Γ5R/20. Multiply f(w(x,0)) by η1 and still call it f~(x,0). Moreover, still set
v
(
x
,
y
)
=
Γ
s
(
⋅
,
y
)
∗
f
~
(
x
,
0
)
.
Then, using the above procedure again, we can see that w∈C1,ν(Γ2R0¯) with ν<4s-1. The only difference is that in equation (2.4), for f~(x,0)∈Cβ, we give the following estimates by applying [23, Proposition 2.8]:
If β+2s≤1, then v(x,0)∈C0,β+2s(ℝ).
If β+2s>1, then v(x,0)∈C1,β+2s-1(ℝ).
And for any s∈(0,14], we can iterate the above procedure a finite number of times, until β+2ks>1 for some integer k. Then we have
w
(
x
,
0
)
∈
C
1
,
ν
(
Γ
2
R
0
¯
)
for some ν>0.
For the C2,α regularity, we can differentiate equation (2.4) to obtain that
(
-
Δ
)
s
D
x
v
=
D
x
f
~
(
x
,
0
)
for all
(
x
,
0
)
∈
ℝ
,
with Dxf~(x,0) belonging to Cσ (σ<min{γ,ν}). Therefore, by [23, Proposition 2.8], we can see that Dxv(x,0) belongs to C0,σ+2s(ℝ) if σ+2s<1, and it belongs to C1,σ+2s-1(ℝ) if σ+2s>1. Adding this estimate to the procedure of obtaining the C1,ν regularity, we iterate this new procedure a finite number of times. Since γ+2s>1, we finally arrive at Dxv(x,0)∈C1,α(ℝ). Then
w
(
x
,
0
)
∈
C
2
,
α
(
Γ
R
0
¯
)
for some α depending only on s, γ. ∎
2.4 The Eigenvalue Problem
In this subsection, we consider the following equation:
(2.5)
(
-
Δ
)
s
v
=
μ
v
,
v
(
x
)
=
v
(
x
+
2
π
)
.
Lemma 2.11 ([12, Lemma 3.1]).
The only eigenvalues of problem (2.5) are μ0=0, with a constant eigenfunction, and μk=k2s, k=1,2,…, with an eigenspace spanned by cos(kx) and sin(kx).
Corollary 2.12.
The only eigenvalues of
(2.6)
(
-
Δ
)
s
v
=
μ
v
,
v
(
x
)
=
v
(
x
+
2
T
)
,
are μ0=0, with a constant eigenfunction, and μk=(kπT)2s, k=1,2,…, with an eigenspace spanned by cos(kπxT) and sin(kπxT).
Corollary 2.13.
The only eigenvalues of
(2.7)
{
-
div
(
y
1
-
2
s
∇
w
)
=
0
,
(
x
,
y
)
∈
[
-
T
,
T
]
×
ℝ
+
,
∂
w
∂
ν
=
-
μ
w
(
x
,
0
)
,
(
x
,
0
)
∈
[
-
T
,
T
]
,
w
(
x
,
y
)
=
w
(
x
+
2
T
,
y
)
,
(
x
,
y
)
∈
[
-
T
,
T
]
×
ℝ
+
¯
,
are μ0=0, with a constant eigenfunction, and μk=(kπT)2s, k=1,2,…, with an eigenspace spanned by Ps(⋅,y)∗cos(kπxT) and Ps(⋅,y)∗sin(kπxT).
Proof.
The conclusion can be easily verified by the one-to-one correspondence between the eigenvalue of (2.7) and the eigenvalue of (2.6). ∎
3 Existence of k Solutions
In this section, we will work in the following Hilbert space:
X
T
=
{
w
(
x
,
y
)
:
w
(
T
2
,
y
)
=
w
(
0
,
y
)
=
0
for all
y
≥
0
,
w
∈
ℋ
τ
}
.
We choose F^ such that
{
F
^
(
t
)
=
(
t
-
1
)
2
2
,
t
>
1
,
F
^
(
t
)
=
F
(
t
)
,
|
t
|
≤
1
,
F
^
(
t
)
=
(
t
+
1
)
2
2
,
t
<
-
1
.
Let DT=[0,T2]×(0,∞) and consider the following functional:
J
^
(
w
)
=
1
2
∫
D
T
y
1
-
2
s
|
∇
w
|
2
𝑑
x
𝑑
y
+
∫
0
T
/
2
F
^
(
w
(
x
,
0
)
)
𝑑
x
.
3.1 Proof of Theorem 1.1
Now we give the proof of Theorem 1.1. It is standard to check J^∈C1(XT,ℝ) by Lemma 2.1, Lemma 2.2 and the assumption of F^. Clearly, J^ is even and J^(w)≥0 for all w∈XT. It is easily to see that there exists a constant C>0 independent of w∈XT such that
(3.1)
J
^
(
w
)
≥
1
2
∫
D
T
y
1
-
2
s
|
∇
w
|
2
𝑑
x
𝑑
y
+
1
4
∫
0
T
/
2
w
2
(
x
,
0
)
𝑑
x
-
C
.
Then J^ is bounded below and coercive. Now let {wn}n≥1 be a (PS)c sequence, that is, J^(wn)→c and J^′(wn)→0. Then, by (3.1), we can easily see that {wn}n≥1 is bounded. Assume without loss of generality that wn converges to w weakly in XT and strongly in L2([0,T2]) by Lemma 2.1 and Lemma 2.2. Then J^′(w)=0. Note that
∫
D
T
y
1
-
2
s
|
∇
(
w
n
(
x
,
y
)
-
w
(
x
,
y
)
)
|
2
𝑑
x
𝑑
y
=
〈
J
^
′
(
w
n
)
-
J
^
′
(
w
)
,
w
n
-
w
〉
-
∫
0
T
/
2
[
F
^
′
(
w
n
(
x
,
0
)
)
-
F
^
′
(
w
(
x
,
0
)
)
]
(
w
n
(
x
,
0
)
-
w
(
x
,
0
)
)
𝑑
x
.
Clearly,
〈
J
^
′
(
w
n
)
-
J
^
′
(
w
)
,
w
n
-
w
〉
→
0
,
and by the choice of F^ we have
∫
0
T
/
2
[
F
^
′
(
w
n
(
x
,
0
)
)
-
F
^
′
(
w
(
x
,
0
)
)
]
(
w
n
(
x
,
0
)
-
w
(
x
,
0
)
)
𝑑
x
≤
C
(
1
+
∥
w
n
(
x
,
0
)
∥
L
2
(
[
0
,
T
2
]
)
+
∥
w
(
x
,
0
)
∥
L
2
(
[
0
,
T
2
]
)
)
∥
w
n
(
x
,
0
)
-
w
(
x
,
0
)
∥
L
2
(
[
0
,
T
2
]
)
.
Therefore, wn converges to w strongly in XT and the (PS)c condition holds for J^.
For σ∈(0,12), we define k functions as follows:
h
i
(
x
)
=
(
-
1
)
j
g
(
x
)
,
x
∈
[
0
,
T
2
i
]
+
j
T
2
i
,
j
∈
{
0
,
…
,
2
i
-
1
-
1
}
,
i
=
1
,
…
,
k
,
where g satisfies g(x)=g(x+T2k) and
g
(
x
)
=
{
2
k
+
1
σ
T
x
,
x
∈
[
0
,
σ
T
2
k
+
1
]
,
1
,
x
∈
[
σ
T
2
k
+
1
,
T
2
k
-
σ
T
2
k
+
1
]
,
2
σ
-
2
k
+
1
σ
T
x
,
x
∈
[
T
2
k
-
σ
T
2
k
+
1
,
T
2
k
]
.
Then we can construct k functions ψi(x,y)=exp{-y2b+1}hi(x), i=1,…,k, for some b>1. Set
K
=
{
∑
i
=
1
k
α
i
ψ
i
(
x
,
y
)
:
∑
i
=
1
k
|
α
i
|
=
1
}
.
It is clear that K is homeomorphic to ∂Bk by an odd map for any b. We now claim supKJ^<J^(0) if b is sufficiently large. Let
ψ
(
x
,
y
)
=
∑
i
=
1
k
α
i
ψ
i
(
x
,
y
)
∈
K
.
After a direct checking, we can see that
|
∑
i
=
1
k
α
i
h
i
(
x
)
|
≤
1
,
|
∑
i
=
1
k
α
i
D
x
h
i
(
x
)
|
≤
2
k
+
1
σ
T
for any
x
∈
[
0
,
T
2
]
,
|
∑
i
=
1
k
α
i
D
x
h
i
(
x
)
|
=
0
,
x
∈
[
σ
T
2
k
+
1
,
T
2
k
-
σ
T
2
k
+
1
]
+
m
T
2
k
,
m
=
0
,
…
,
2
k
-
1
-
1
,
|
∑
i
=
1
k
α
i
h
i
(
x
)
|
=
1
,
x
∈
[
σ
T
2
k
+
1
,
T
2
k
-
σ
T
2
k
+
1
]
+
m
1
T
2
k
,
for a suitable
m
1
∈
{
0
,
…
,
2
k
-
1
-
1
}
.
By the assumptions (F1) and (F2), for any ψ∈K,
(3.2)
∫
0
T
/
2
F
^
(
ψ
(
x
,
0
)
)
𝑑
x
=
∫
0
T
/
2
F
(
∑
i
=
1
k
α
i
ψ
i
(
x
,
0
)
)
𝑑
x
<
F
(
0
)
T
2
k
(
2
k
-
1
-
1
+
σ
)
.
We also have
∫
D
T
y
1
-
2
s
|
∇
ψ
(
x
,
y
)
|
2
𝑑
x
𝑑
y
=
∫
0
∞
∫
0
T
/
2
y
1
-
2
s
|
∇
(
∑
i
=
1
k
α
i
ψ
i
(
x
,
y
)
)
|
2
𝑑
x
𝑑
y
=
∫
0
∞
y
1
-
2
s
exp
{
-
y
2
b
}
𝑑
y
∫
0
T
/
2
|
∑
i
=
1
k
α
i
h
i
(
x
)
|
2
2
2
b
+
2
+
|
∑
i
=
1
k
α
i
D
x
h
i
(
x
)
|
2
d
x
≤
(
1
2
2
b
+
2
⋅
T
2
+
(
2
k
+
1
σ
T
)
2
⋅
σ
T
2
)
∫
0
∞
y
1
-
2
s
exp
{
-
y
2
b
}
𝑑
y
=
2
b
(
2
-
2
s
)
(
1
2
2
b
⋅
T
8
+
2
2
k
+
1
σ
T
)
∫
0
∞
z
1
-
2
s
exp
{
-
z
}
𝑑
z
=
Γ
(
2
-
2
s
)
2
-
2
s
b
(
T
8
+
2
2
b
⋅
2
2
k
+
1
σ
T
)
.
For sufficiently large b, we have that Γ(2-2s)2-2sb is small, and 22b⋅22k+1σT is also small provided that T is large enough. Therefore, there exists Tk>0 such that for any T>Tk the following estimate holds true:
∫
D
T
y
1
-
2
s
|
∇
ψ
(
x
,
y
)
|
2
𝑑
x
𝑑
y
<
F
(
0
)
(
1
-
σ
)
T
2
k
.
Hence, by (3.2), we can get that J^(ψ)<F(0)T2=J^(0) for any ψ∈K. Then all the conditions of Clark’s theorem are satisfied, and hence we can get k pairs solutions ±Wi,T∈XT (i=1,…,k) of
{
-
div
(
y
1
-
2
s
∇
w
)
=
0
in
D
T
,
∂
w
∂
ν
=
F
^
′
(
w
(
x
,
0
)
)
in
[
0
,
T
2
]
.
We now show that Wi,T∈[-1,1] for any i∈{1,…,k}. Let W±=max{±W,0}. Then we have
0
=
-
∫
D
T
y
1
-
2
s
∇
W
i
,
T
∇
(
W
i
,
T
+
1
)
-
d
x
d
y
-
∫
0
T
/
2
F
^
′
(
W
i
,
T
(
x
,
0
)
)
(
W
i
(
x
,
0
)
+
1
)
-
d
x
=
∫
D
T
y
1
-
2
s
|
∇
(
W
i
,
T
+
1
)
-
|
2
d
x
d
y
-
∫
[
0
,
T
/
2
]
∩
{
W
i
,
T
≤
-
1
}
F
^
′
(
W
i
,
T
(
x
,
0
)
)
(
W
i
,
T
(
x
,
0
)
+
1
)
-
d
x
=
∫
D
T
y
1
-
2
s
|
∇
(
W
i
,
T
+
1
)
-
|
2
d
x
d
y
+
∫
0
T
/
2
|
(
W
i
,
T
(
x
,
0
)
+
1
)
-
|
2
d
x
,
where we have used that F^′(t)=t+1 for t≤-1. We can conclude that (Wi,T+1)-=0, and therefore Wi,T≥-1. Similarly, we can prove that Wi,T≤1 by using that F^′(t)=t-1 for t≥1. Therefore, Wi,T (i=1,…,k) are the solutions of
{
-
div
(
y
1
-
2
s
∇
w
)
=
0
in
D
T
,
∂
w
∂
ν
=
F
′
(
w
(
x
,
0
)
)
in
[
0
,
T
2
]
.
Now, we make odd extensions of Wi,T (i=1,…,k) (with respect to x) from D¯T to Ω¯T=[-T2,T2]×[0,∞). Furthermore, we extend it periodically (with respect to x again) from Ω¯T to ℝ+2¯, and we still denote it by Wi,T. Then Wi,T (i=1,…,k) are the weak solutions of (2.1). And ui,T(x)=Wi,T(x,0) (i=1,…,k) are the odd periodic solutions of (1.1). By Lemma 2.8, we can conclude that ui,T(x)∈(-1,1), where we used that F′(±1)=0.
3.2 Proof of Theorem 1.2
We now prove Theorem 1.2. Recall from the proof of Theorem 1.1 that we only need to find a set K homeomorphic to ∂Bk such that supKJ^<J^(0). By F′(0)=0 and Taylor’s formula, we can see that
F
^
(
u
)
=
F
^
(
0
)
+
F
^
′′
(
θ
u
)
2
u
2
≤
F
^
(
0
)
+
(
F
^
′′
(
0
)
+
ϵ
)
2
u
2
for all
u
∈
B
ϱ
(
0
)
,
where θ∈(0,1), ϱ>0 and ϵ=ϵ(ϱ)>0 are small. From Corollary 2.13,
μ
k
=
(
2
k
π
T
)
2
s
and
φ
k
=
P
s
(
⋅
,
y
)
∗
sin
(
2
k
π
x
T
)
are the k-th eigenvalue and eigenfunction, respectively, of equation (2.7) with φ(0,y)=φ(T2,y)=0. Define
K
=
{
φ
∈
∑
i
=
1
k
α
i
φ
i
:
∑
i
=
1
k
α
i
2
=
r
2
}
,
r
<
k
ϱ
.
Then K is homeomorphic to ∂Bk. It is clear that
∫
D
T
y
1
-
2
s
∇
φ
i
∇
φ
j
d
x
d
y
=
μ
i
∫
0
T
/
2
φ
i
(
x
,
0
)
φ
j
(
x
,
0
)
𝑑
x
for all
i
,
j
∈
{
1
,
…
,
k
}
.
Then for any φ∈K we have |φ|≤ϱ and
J
^
(
φ
)
=
1
2
∫
D
T
y
1
-
2
s
|
∇
φ
|
2
𝑑
x
𝑑
y
+
∫
0
T
/
2
F
^
(
φ
(
x
,
0
)
)
𝑑
x
≤
1
2
∫
D
T
y
1
-
2
s
|
∇
φ
|
2
𝑑
x
𝑑
y
+
(
F
^
′′
(
0
)
+
ϵ
)
2
∫
0
T
/
2
φ
2
(
x
,
0
)
𝑑
x
+
F
^
(
0
)
T
2
≤
1
2
(
(
2
k
π
T
)
2
s
+
F
^
′′
(
0
)
+
ϵ
)
∫
0
T
/
2
φ
2
(
x
,
0
)
𝑑
x
+
F
^
(
0
)
T
2
.
For T large, we have J^(φ)<J^(0) for any φ∈K since F^′′(0)=F′′(0)<0. Then, as in the proof of Theorem 1.1, we can complete the proof of Theorem 1.2.
4 Proof of Theorem 1.3
First, we define the extension function F~ as follows:
{
F
~
(
t
)
=
F
′′
(
1
)
2
(
t
-
1
)
2
,
t
≥
1
,
F
~
(
t
)
=
F
(
t
)
,
|
t
|
<
1
,
F
~
(
t
)
=
F
′′
(
-
1
)
2
(
t
+
1
)
2
,
t
≤
-
1
.
Clearly,
{
F
~
′
(
t
)
=
F
′′
(
1
)
t
-
F
′′
(
1
)
,
t
≥
1
,
F
~
′
(
t
)
=
F
′′
(
-
1
)
t
+
F
′′
(
-
1
)
,
t
≤
-
1
,
F
~
′′
(
t
)
=
F
′′
(
1
)
,
t
≥
1
,
F
~
′′
(
t
)
=
F
′′
(
-
1
)
,
t
≤
-
1
.
Recall
Ω
T
=
[
-
T
2
,
T
2
]
×
(
0
,
∞
)
.
We now consider the following functional:
J
~
(
w
)
=
1
2
∫
Ω
T
y
1
-
2
s
|
∇
w
|
2
𝑑
x
𝑑
y
+
∫
-
T
/
2
T
/
2
F
~
(
w
(
x
,
0
)
)
𝑑
x
.
It is easily to see that there exists a constant C>0 such that
J
~
(
w
)
≥
1
2
∫
Ω
T
y
1
-
2
s
|
∇
w
|
2
𝑑
x
𝑑
y
+
min
{
F
′′
(
1
)
,
F
′′
(
-
1
)
}
4
∫
-
T
/
2
T
/
2
w
2
(
x
,
0
)
𝑑
x
-
C
,
that is, J~ is coercive and bounded below. And as in the proof of Theorem 1.1, we can see that J~ satisfies the (PS)c condition for any c∈ℝ. By the assumptions (F1) and (F2’), we also have
J
~
(
w
)
≥
0
=
J
~
(
±
1
)
for all
w
∈
ℋ
τ
.
Note that F′′(±1)>0. Then
J
~
′′
(
±
1
)
φ
2
=
∫
Ω
T
y
1
-
2
s
|
∇
φ
|
2
𝑑
x
𝑑
y
+
∫
-
T
/
2
T
/
2
F
~
′′
(
±
1
)
φ
2
(
x
,
0
)
𝑑
x
>
0
for all
φ
∈
ℋ
τ
∖
{
0
}
,
and one can deduce that ±1 are both local minimizers for J~. Then, by the mountain pass theorem (Lemma 2.3), we can get a V1,T∈ℋτ such that
J
~
(
V
1
,
T
)
=
inf
γ
∈
Γ
sup
t
∈
[
0
,
1
]
J
~
(
γ
(
t
)
)
>
0
,
where the class of a path Γ is defined by
Γ
=
{
γ
∈
C
(
[
0
,
1
]
,
ℋ
τ
)
:
γ
(
0
)
=
-
1
,
γ
(
1
)
=
1
}
.
It is known that there exists 𝒯>0 such that for T>𝒯,
J
~
(
V
1
,
T
)
<
J
~
(
0
)
,
holds by a construction from [17].
It is easy to see that V1,T∈[-1,1]. Indeed, by
0
=
∫
Ω
T
y
1
-
2
s
∇
V
1
,
T
∇
(
V
1
,
T
-
1
)
+
d
x
d
y
+
∫
-
T
/
2
T
/
2
F
~
′
(
V
1
,
T
(
x
,
0
)
)
(
V
1
,
T
(
x
,
0
)
-
1
)
+
d
x
=
∫
Ω
T
y
1
-
2
s
|
∇
(
V
1
,
T
-
1
)
+
|
2
d
x
d
y
+
∫
Γ
T
/
2
0
∩
{
V
1
,
T
≥
1
}
F
~
′
(
V
1
,
T
(
x
,
0
)
)
(
V
1
,
T
(
x
,
0
)
-
1
)
+
d
x
=
∫
Ω
T
y
1
-
2
s
|
∇
(
V
1
,
T
-
1
)
+
|
2
d
x
d
y
+
F
′′
(
1
)
∫
-
T
/
2
T
/
2
|
(
V
1
,
T
(
x
,
0
)
-
1
)
+
|
2
d
x
,
we can conclude that V1,T≤1. Also, we can prove that V1,T≥-1.
Now we extend V1,T periodically (with respect to x) from Ω¯T to the whole half space ℝ+2¯, and we still denote it by V1,T . Then we know that V1,T is a weak solution of (2.1).
Claim 1: i(J~,V1,T)=1, that is, the Morse index of V1,T is equal to 1. By Lemma 2.10, we have
V
1
,
T
(
x
,
0
)
∈
C
2
,
α
(
Γ
3
T
/
8
0
¯
)
for some α∈(0,1). We now show that
(4.1)
V
1
,
T
(
x
,
0
)
∈
C
2
,
α
(
±
D
1
¯
)
,
D
1
=
[
T
8
,
7
T
8
]
.
For simplicity, we assume T=2π. By Corollary 2.13, for any φ(x,y)∈ℋ2π, we have
φ
=
∑
k
=
0
∞
a
k
P
s
(
⋅
,
y
)
∗
sin
(
k
x
)
+
b
k
P
s
(
⋅
,
y
)
∗
cos
(
k
x
)
,
with
a
k
=
∫
-
π
π
φ
(
x
,
0
)
sin
(
k
x
)
𝑑
x
and
b
k
=
∫
-
π
π
φ
(
x
,
0
)
cos
(
k
x
)
𝑑
x
.
Let
φ
m
=
∑
k
=
0
m
a
k
P
s
(
⋅
,
y
)
∗
sin
(
k
x
)
+
b
k
P
s
(
⋅
,
y
)
∗
cos
(
k
x
)
.
Then
φ
(
x
+
π
,
y
)
=
lim
m
→
∞
φ
m
(
x
+
π
,
y
)
=
∑
k
=
0
∞
(
-
1
)
k
a
k
P
s
(
⋅
,
y
)
∗
sin
(
k
x
)
+
(
-
1
)
k
b
k
P
s
(
⋅
,
y
)
∗
cos
(
k
x
)
.
Hence, again by Corollary 2.13, we can get that
∫
0
∞
∫
-
π
π
y
1
-
2
s
|
∇
φ
|
2
𝑑
x
𝑑
y
=
lim
m
→
∞
∫
0
∞
∫
-
π
π
y
1
-
2
s
|
∇
φ
m
|
2
𝑑
x
𝑑
y
=
lim
m
→
∞
∑
k
=
0
m
k
2
s
(
a
k
2
+
b
k
2
)
=
lim
m
→
∞
∑
k
=
0
m
k
2
s
(
(
(
-
1
)
k
a
k
)
2
+
(
(
-
1
)
k
b
k
)
2
)
=
∫
0
∞
∫
0
2
π
y
1
-
2
s
|
∇
φ
|
2
𝑑
x
𝑑
y
.
This implies that for any U,V∈ℋ2π,
∫
0
∞
∫
0
2
π
y
1
-
2
s
∇
U
∇
V
d
x
d
y
=
∫
0
∞
∫
-
π
π
y
1
-
2
s
∇
U
∇
V
d
x
d
y
.
Then fix φ∈C∞([0,T]×[0,∞)) with φ(x,y)=φ(x+T,y) and
∫
0
∞
∫
0
T
y
1
-
2
s
|
∇
φ
|
2
𝑑
x
𝑑
y
+
∫
0
T
|
φ
(
x
,
0
)
|
2
𝑑
x
<
+
∞
.
By the periodicity of V1,T, we have
∫
0
∞
∫
0
T
y
1
-
2
s
∇
V
1
,
T
∇
φ
d
x
d
y
+
∫
0
T
F
~
′
(
V
1
,
T
(
x
,
0
)
)
φ
(
x
,
0
)
𝑑
x
=
∫
Ω
T
y
1
-
2
s
∇
V
1
,
T
∇
φ
d
x
d
y
+
∫
0
T
/
2
F
~
′
(
V
1
,
T
(
x
,
0
)
)
φ
(
x
,
0
)
𝑑
x
+
∫
T
/
2
T
F
~
′
(
V
1
,
T
(
x
,
0
)
)
φ
(
x
,
0
)
𝑑
x
=
∫
Ω
T
y
1
-
2
s
∇
V
1
,
T
∇
φ
d
x
d
y
+
∫
-
T
/
2
T
/
2
F
~
′
(
V
1
,
T
(
x
,
0
)
)
φ
(
x
,
0
)
𝑑
x
=
0
.
Therefore, V1,T satisfies
{
-
div
(
y
1
-
2
s
∇
v
)
=
0
,
(
x
,
y
)
∈
[
0
,
T
]
×
[
0
,
∞
)
,
∂
v
∂
ν
=
F
~
′
(
v
(
x
,
0
)
)
,
(
x
,
0
)
∈
[
0
,
T
]
,
and (4.1) follows by Lemma 2.10.
Then, by (4.1), we have V1,T(x,0)∈C2,α(ℝ), and ψ(x,y)=Ps(⋅,y)∗DxV1,T(x,0) is a bounded weak solution of the following equation:
(4.2)
{
-
div
(
y
1
-
2
s
∇
v
)
=
0
,
(
x
,
y
)
∈
Ω
T
,
∂
v
∂
ν
=
F
~
′′
(
V
1
,
T
(
x
,
0
)
)
v
,
(
x
,
0
)
∈
Γ
T
/
2
0
.
It follows that V1,T is a degenerate critical point.
One can easily verify i(J~,V1,T)≤1 because V1,T is a mountain-pass type critical point (e.g., [9, (1.9) in Chapter 2]). We now prove i(J~,V1,T)=1, and argue by contradiction. Assume i(J~,V1,T)=0, that is,
(4.3)
J
~
′′
(
V
1
,
T
)
v
2
=
∫
Ω
T
y
1
-
2
s
|
∇
v
|
2
𝑑
x
𝑑
y
+
∫
-
T
/
2
T
/
2
F
~
′′
(
V
1
,
T
(
x
,
0
)
)
v
2
(
x
,
0
)
𝑑
x
≥
0
for all
v
∈
ℋ
τ
.
Then we can prove that 1 is the smallest positive eigenvalue of
(4.4)
{
-
div
(
y
1
-
2
s
∇
v
)
=
0
,
(
x
,
y
)
∈
Ω
T
,
∂
v
∂
ν
-
v
=
-
λ
[
1
-
F
~
′′
(
V
1
,
T
(
x
,
0
)
)
]
v
,
(
x
,
0
)
∈
Γ
T
/
2
0
.
Indeed, provided by (4.3), for all positive eigenvalues λ of (4.4), we have
λ
=
∫
Ω
T
y
1
-
2
s
|
∇
v
|
2
𝑑
x
𝑑
y
+
∫
-
T
/
2
T
/
2
v
2
(
x
,
0
)
𝑑
x
∫
-
T
/
2
T
/
2
(
1
-
F
~
′′
(
V
1
,
T
)
)
v
2
(
x
,
0
)
𝑑
x
≥
1
.
We also know that
λ
1
=
inf
v
∈
ℋ
τ
∖
{
0
}
∫
Ω
T
y
1
-
2
s
|
∇
v
|
2
𝑑
x
𝑑
y
+
∫
-
T
/
2
T
/
2
v
2
(
x
,
0
)
𝑑
x
∫
-
T
/
2
T
/
2
(
1
-
F
~
′′
(
V
1
,
T
)
)
v
2
(
x
,
0
)
𝑑
x
≤
∫
Ω
T
y
1
-
2
s
|
∇
ψ
|
2
𝑑
x
𝑑
y
+
∫
-
T
/
2
T
/
2
ψ
2
(
x
,
0
)
𝑑
x
∫
-
T
/
2
T
/
2
(
1
-
F
~
′′
(
V
1
,
T
)
)
ψ
2
(
x
,
0
)
𝑑
x
=
1
,
where we used that ψ(x,y)=Ps(⋅,y)∗DxV1,T(x,0) is the solution of (4.2). Hence 1 is the smallest positive eigenvalue of (4.4), and ψ(x,y) is the corresponding eigenfunction.
We now claim that ψ(x,y) does not change sign in x∈BT/2+∪ΓT/20. Otherwise, let
ψ
±
(
x
,
y
)
=
max
{
±
ψ
(
x
,
y
)
,
0
}
.
Since they are both not zero, we have
∫
-
T
/
2
T
/
2
(
1
-
F
~
′′
(
V
1
,
T
)
)
ψ
2
(
x
,
0
)
𝑑
x
=
∫
-
T
/
2
T
/
2
(
1
-
F
~
′′
(
V
1
,
T
)
)
ψ
+
2
(
x
,
0
)
𝑑
x
+
∫
-
T
/
2
T
/
2
(
1
-
F
~
′′
(
V
1
,
T
)
)
ψ
-
2
(
x
,
0
)
𝑑
x
=
a
+
+
a
-
and
∫
Ω
T
y
1
-
2
s
|
∇
ψ
|
2
𝑑
x
𝑑
y
+
∫
-
T
/
2
T
/
2
ψ
2
(
x
,
0
)
𝑑
x
=
∫
Ω
T
y
1
-
2
s
|
∇
ψ
+
|
2
𝑑
x
𝑑
y
+
∫
-
T
/
2
T
/
2
ψ
+
2
(
x
,
0
)
𝑑
x
+
∫
Ω
T
y
1
-
2
s
|
∇
ψ
-
|
2
𝑑
x
𝑑
y
+
∫
-
T
/
2
T
/
2
ψ
-
2
(
x
,
0
)
𝑑
x
=
b
+
+
b
-
.
It is easily to see that a±>0. Otherwise, if a-<0, then
b
+
a
+
<
b
+
a
+
+
a
-
<
b
+
+
b
-
a
+
+
a
-
=
1
.
This contradicts the fact that 1 is the smallest eigenvalue.
We have either
b
+
+
b
-
a
+
+
a
-
=
b
+
a
+
=
b
-
a
-
or
b
+
+
b
-
a
+
+
a
-
>
min
{
b
+
a
+
,
b
-
a
-
}
.
However, the latter case cannot occur due to the fact the 1=b++b-a++a- is the smallest positive eigenvalue. In the former case, both ψ±(x,y) are eigenfunctions with respect to 1. Then by Lemma 2.8, we have ψ±(x,y)>0 for a.e. x∈BT/2+∪ΓT/20. This is impossible.
However, by the periodicity of V1,T(x,0), we have that DxV1,T(x,0) must change sign in ΓT/20. This is a contradiction. Then we deduce that i(J~,V1,T)=1.
We now come back to find the another nonconstant periodic solution.
By Lemma 2.5, we can get that
C
q
(
J
~
,
V
1
,
T
)
=
{
G
,
q
=
1
,
0
,
q
≠
1
,
due to i(J~,V1,T)=1. By F′′(±1)>0, we can claim that ±1 are local minimizers of J~. Again applying Lemma 2.5, we have
C
q
(
J
~
,
±
1
)
=
{
G
,
q
=
0
,
0
,
q
≠
0
.
We now consider the trivial solution θ. If F′′(0)=0, then 𝒩=ker(J~′′(θ))=ℝ. We recall assumption (F2’): F(t)<F(0) for t∈(-1,1). We can conclude that θ is a local maximum of J~|𝒩. Then, by Lemmas 2.5 and 2.6,
C
q
(
J
~
,
θ
)
=
{
G
,
q
=
1
,
0
,
q
≠
1
.
If there exists a positive integer k such that
T
(
-
F
′′
(
0
)
)
1
2
s
2
π
=
k
,
then
𝒩
=
ker
(
J
~
′′
(
θ
)
)
=
span
{
P
s
(
⋅
,
y
)
∗
cos
(
2
π
k
x
T
)
,
P
s
(
⋅
,
y
)
∗
sin
(
2
π
k
x
T
)
}
.
And if F′(t)t≤F′′(0) for t≠0 in a neighborhood U of 0, then θ is a local maximum of J~|𝒩. Let i(J~,θ) be the Morse index of θ. By Lemma 2.5 and Lemma 2.6,
C
q
(
J
~
,
θ
)
=
{
G
,
q
=
i
(
J
~
,
θ
)
+
2
,
0
,
q
≠
i
(
J
~
,
θ
)
+
2
.
Similarly, in the cases that either T(-F′′(0))1/2s2π is not equal to a positive integer or T(-F′′(0))1/2s2π is a positive integer but F′(t)t≥F′′(0) for t≠0 in a neighborhood U of 0, then θ is a local minimum of J~|𝒩. Thus
C
q
(
J
~
,
θ
)
=
{
G
,
q
=
i
(
J
~
,
θ
)
,
0
,
q
≠
i
(
J
~
,
θ
)
.
If there were no other critical points, then a contradiction would occur due to the Morse inequalities as follows. We have βq=rankHq(X,J~-c,G)=δ0,q (q∈ℕ) since J~ is bounded below. Now for k>2+i(J~,θ) we would have
M
k
-
M
k
-
1
+
⋯
+
(
-
1
)
k
M
0
=
β
k
-
β
k
-
1
+
⋯
+
(
-
1
)
k
β
0
.
The left-hand side is even, but the right-hand side is odd, a contradiction. Therefore, there must be another solution V2,T.
Arguing as in the proof of V1,T∈[-1,1], we have V2,T∈[-1,1].
Now we extend V2,T periodically (with respect to x) from Ω¯T to the whole half space ℝ+2¯, and we still denote it by V2,T. Then we know that V2,T is a weak solution of (2.1). Further, vi,T(x)=Vi,T(x,0) (i=1,2) are periodic solutions of (1.1). By Lemma 2.8, we can conclude that vi,T(x)∈(-1,1), for which we use F′(±1)=0.
and
Zhi-Qiang Wang
