Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional Laplacian

Zhenping Feng; Zhuoran Du

doi:10.1515/ans-2020-2075

Artikel Open Access

Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional Laplacian

Zhenping Feng und Zhuoran Du

Veröffentlicht/Copyright: 19. März 2020

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Abstract

We consider periodic solutions of the following problem associated with the fractional Laplacian: (-∂x⁢x)s⁢u⁢(x)+∂u⁡F⁢(x,u⁢(x))=0 in ℝ. The smooth function F⁢(x,u) is periodic about x and is a double-well potential with respect to u with wells at +1 and -1 for any x∈ℝ. We prove the existence of periodic solutions whose periods are large integer multiples of the period of F about the variable x by using variational methods. An estimate of the energy functional, Hamiltonian identity and Modica-type inequality for periodic solutions are also established.

Keywords: Fractional Laplacian; Periodic Solutions; Non-autonomous; Variational Method

MSC 2010: 35J61; 35B10

1 Introduction

We consider periodic solutions of the non-autonomous Allen–Cahn equation

(1.1) ( - ∂ x ⁢ x ) s ⁢ u ⁢ ( x ) + ∂ u ⁡ F ⁢ ( x , u ⁢ ( x ) ) = 0 , x ∈ ℝ ,

where (-∂x⁢x)s, 0<s<1, denotes the usual fractional Laplace operator. The function F⁢(x,u) is periodic about x; for simplicity of notation, we assume the period is 1, namely, for any given u∈ℝ,

(1.2) F ⁢ ( x + 1 , u ) = F ⁢ ( x , u ) for all ⁢ x ∈ ℝ .

F is also a smooth double-well potential with respect to u with wells at +1 and -1, namely, for any given x∈ℝ, it satisfies

(1.3) { F ⁢ ( x , 1 ) = F ⁢ ( x , - 1 ) = 0 < F ⁢ ( x , u ) for all - 1 < u < 1 , ∂ u ⁡ F ⁢ ( x , 1 ) = ∂ u ⁡ F ⁢ ( x , - 1 ) = 0 .

We also assume that, for any x,

(1.4) F ⁢ is nondecreasing in ⁢ ( - 1 , 0 ) ⁢ and nonincreasing in ⁢ ( 0 , 1 ) ⁢ with respect to ⁢ u .

Note that conditions (1.3), (1.4) mean that, for fixed x,

F ⁢ ( x , 0 ) = max - 1 ≤ u ≤ 1 ⁡ F ⁢ ( x , u ) > 0 .

We may assume that F is even with respect to u and x respectively, that is to say

(1.5) { for any given ⁢ x , F ⁢ ( x , u ) = F ⁢ ( x , - u ) , u ∈ ℝ , for any given ⁢ u , F ⁢ ( x , u ) = F ⁢ ( - x , u ) , x ∈ ℝ .

For the corresponding autonomous equation of (1.1),

(1.6) ( - ∂ x ⁢ x ) s ⁢ u + F ′ ⁢ ( u ) = 0 , x ∈ ℝ ,

while conditions (1.3), (1.4) turn into

(1.7) { F ⁢ ( 1 ) = F ⁢ ( - 1 ) = 0 < F ⁢ ( u ) for all - 1 < u < 1 , F ′ ⁢ ( 1 ) = F ′ ⁢ ( - 1 ) = 0 ,

and

(1.8) F ⁢ is nondecreasing in ⁢ ( - 1 , 0 ) ⁢ and nonincreasing in ⁢ ( 0 , 1 ) .

Under conditions (1.7), (1.8), the authors in [14] proved that if F is even about u, then (1.6) admits an odd periodic solution, which is a minimizer of the corresponding energy functional J. If F is not necessarily even, the existence of mountain passing periodic solutions are also established, and the solutions are not necessarily odd. In [10], an upper bound of the least positive period is given, and Hamiltonian identity, Modica-type inequality and an estimate of the energy functional for periodic solutions are also established. Recently, the existence of periodic solutions to an Allen–Cahn system with the fractional Laplacian is obtained in [11]. Existence and multiplicity of periodic solutions to so-called pesudo-relativistic Schrödinger equations are also established in [1, 2, 3]. The authors in [4] obtain some existence results of periodic solutions to some fractional Laplace equations by using variational method and bifurcation theory. In [18], the authors establish interior and boundary Harnack inequalities for nonnegative solutions to (-Δ)s⁢u=0 with periodic boundary conditions, and they also obtain regularity properties of the fractional Laplacian with periodic boundary conditions and the pointwise integro-differential formula for the operator.

The authors in [6, 7] proved that conditions (1.7) are both necessary and sufficient for the existence of a layer solution to problem (1.6). Moreover, they prove that such layer solution is unique and establish asymptotic behavior of the layer solution. These results also have been proven with different techniques in [17]. The author in [16] studied layer solutions of the non-autonomous Allen–Cahn-type fractional Laplace equation (-∂x⁢x)s⁢u=b⁢(x)⁢F′⁢(u), x∈ℝ, where F satisfies (1.7), (1.8) and b:ℝ→ℝ is a positive even periodic function, and obtained the existence of layer solutions for s≥12.

A natural question is that whether there exist periodic solutions of such non-autonomous Allen–Cahn-type fractional Laplace equation. Our interest in the present work is to find periodic solutions of the more general non-autonomous Allen–Cahn equation (1.1).

Plainly, equation (1.1) possesses three constant periodic solutions u=1,-1,0, under conditions (1.3) and (1.4). We will try to find nonconstant periodic solutions. We investigate the existence of odd periodic solutions to equation (1.1) when F satisfies conditions (1.2)–(1.5). If F⁢(x,u) only satisfies condition (1.2)–(1.4), namely, it is not necessarily an even function with respect to u and x, in this case, we will obtain the existence of periodic solutions (not necessarily odd) to equation (1.1). We will also establish Hamiltonian identity and Modica-type inequality for periodic solutions of (1.1).

Our main results are the followings.

Theorem 1.1.

Let s∈(0,1). Assume F⁢(x,u) satisfies conditions (1.2)–(1.5). Then there exists T1>0, for any integer T with T>T1, equation (1.1) admits an odd periodic solution uT with period T, and uT⁢(x)∈(0,1) for x∈(0,T2). Moreover, we have

(1.9) J ⁢ ( U T , Ω T ) ≤ { C ⁢ T 1 - 2 ⁢ s , s ∈ ( 0 , 1 2 ) , C ⁢ ln ⁡ T , s = 1 2 , C , s ∈ ( 1 2 , 1 ) .

Here UT is the s-harmonic extension of uT, and J⁢(UT,ΩT) is the energy functional of equation (3.1), which will be given in Section 3.

Theorem 1.2.

Let s∈(0,1). Assume F⁢(x,u) satisfies conditions (1.2)–(1.4) and satisfies ∂u⁢u⁡F⁢(x,±1)>0 for all x∈R. Then there exists T2>0, for any integer with T>T2, equation (1.1) admits a periodic solution uT with period T, |uT⁢(x)|<1 in R, and it changes its sign at least once in a period. Moreover, for any σ∈(0,12), there exists Tσ≥T2 such that, for any integer T with T>Tσ, we have

(1.10) J ⁢ ( U T , Ω T ) < σ ⁢ max x ∈ [ - T 2 , T 2 ] ⁡ F ⁢ ( x , 0 ) ⁢ T .

Theorem 1.3 (Hamiltonian Identity).

Assume U⁢(x,y) is the s-harmonic extension of a periodic solution u⁢(x) of (1.1). Then, for all x∈R, we have

∫ 0 ∞ y a 2 ⁢ { U x 2 - U y 2 } ⁢ d ⁢ y + ∫ 0 x ∂ x ⁡ F ⁢ ( τ , U ⁢ ( τ , 0 ) ) ⁢ d ⁢ τ - F ⁢ ( x , U ⁢ ( x , 0 ) ) ≡ C T ,

where a=1-2⁢s.

Theorem 1.4 (Modica-Type Inequality).

Assume U is the s-harmonic extension of an even periodic solution u of (1.1). Then, for all y≥0 and x∈R, we have

∫ 0 y τ a 2 ⁢ { U x 2 ⁢ ( x , τ ) - U y 2 ⁢ ( x , τ ) } ⁢ d ⁢ τ + ∫ 0 x ∂ x ⁡ F ⁢ ( τ , U ⁢ ( τ , 0 ) ) ⁢ d ⁢ τ - F ⁢ ( x , U ⁢ ( x , 0 ) ) - C T ≤ C ^ ,

where C^=supx∈R⁡{∫0x∂x⁡F⁢(τ,U⁢(τ,0))⁢d⁢τ-F⁢(x,U⁢(x,0))-CT}>0 and CT is the constant given in Theorem 1.3.

A similar Hamiltonian identity and Modica-type inequality can be found in [6, 15]. We want to point out that the Hamiltonian identity and Modica-type inequality established in [6] correspond to our particular case, namely the case that ∂x⁡F⁢(x,u)≡0.

2 Some Basic Properties

We introduce some useful lemmas which will be used in the sequel.

Lemma 2.1 (Weighted Poincaré Inequality [13]).

Assume D belongs to the class S which is defined as

S = { D ⊂ ℝ n : D ⁢ is an open bounded set, and there exists ⁢ α , ρ 0 > 0 such that, for all x ^ ∈ ∂ D , ρ < ρ 0 , | B ρ ( x ^ ) ∖ D | ≥ α | B ρ ( x ^ ) | } .

Then, for all x^∈∂⁡D, 0<ρ<ρ0, and all u∈C1⁢(Bρ⁢(x^)∩D)¯ vanishing on Bρ⁢(x^)∩∂⁡D,

∫ B ρ ⁢ ( x ^ ) ∩ D x n a ⁢ u 2 ⁢ ( x ) ⁢ d ⁢ x ≤ C ⁢ ( ρ , n , a ) ⁢ ∫ B ρ ⁢ ( x ^ ) ∩ D x n a ⁢ | ∇ ⁡ u ⁢ ( x ) | 2 ⁢ d ⁢ x ,

where a∈(-1,1).

Lemma 2.2 (Hopf Principle [6]).

Assume that a∈(-1,1), and consider the cylinder

C R , 1 : = Γ R 0 × ( 0 , 1 ) ⊂ ℝ + n + 1 ,

where ΓR0={(x,0)∈∂⁡R+n+1:|x|<R}. Let u∈C⁢(CR,1¯)∩H1⁢(CR,1,ya) satisfy

{ L a u : = div ( y a ∇ w ) ≤ 0 𝑖𝑛 ⁢ C R , 1 , u > 0 𝑖𝑛 ⁢ C R , 1 , u ⁢ ( 0 , 0 ) = 0 .

Then

lim sup y → 0 + - y a ⁢ u ⁢ ( 0 , y ) y < 0 .

In addition, if ya⁢uy∈C⁢(CR,1¯), then

- lim y → 0 + ⁡ y a ⁢ ∂ ⁡ u ∂ ⁡ y < 0 .

Lemma 2.3 (Strong Maximum Principles [6]).

Assume that a∈(-1,1). Let v∈H1⁢(ya,CR,1)∩C⁢(CR,1¯) satisfy

{ div ⁡ ( y a ⁢ ∇ ⁡ v ) ≤ 0 , ( x , y ) ∈ C R , 1 , - lim y → 0 + ⁡ y a ⁢ ∂ ⁡ v ∂ ⁡ y ≥ 0 , ( x , y ) ∈ Γ R 0 , v ≥ 0 , ( x , y ) ∈ C R , 1 .

Then either v>0 or v≡0 in CR,1∪ΓR0.

Equation (1.1) is related to a degenerate elliptic problem (3.1) on ℝ+2 (see [8]). For more properties of fractional Laplacian and solutions of degenerate elliptic equations, readers can see [9, 5, 12, 13, 21, 20].

3 Proof of Existence

We follow the methods in [14, 10] to prove Theorem 1.1.

Proof of Theorem 1.1.

For given positive integer T, we denote ΩT:=[0,T2]×[0,+∞). The solution of equation (1.1) is related to the following equation (see [8]):

(3.1) { div ⁡ ( y a ⁢ ∇ ⁡ U ) = 0 in ⁢ ℝ + 2 = { ( x , y ) : x ∈ ℝ , y > 0 } , ∂ ⁡ U ∂ ⁡ ν a = - ∂ U ⁡ F ⁢ ( x , U ⁢ ( x , 0 ) ) on ⁢ ℝ ,

where a=1-2⁢s, ∂⁡U∂⁡νa=-limy→0⁡ya⁢∂y⁡U. In other words, if U is a solution of (3.1), then a positive constant multiple of u(x):=U(x,0) satisfies (1.1). Problem (3.1) corresponds to an energy functional

J ⁢ ( U , Ω T ) = 1 2 ⁢ ∫ Ω T y a ⁢ | ∇ ⁡ U ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y + ∫ 0 T 2 F ⁢ ( x , U ⁢ ( x , 0 ) ) ⁢ d ⁢ x .

We denote the admissible set of the energy J as

Λ T : = { U : U ≥ 0 , U ( 0 , y ) = 0 = U ( T 2 , y ) for all y ≥ 0 , U ∈ H 1 ( Ω T , y a ) } .

Here

H 1 ( Ω T , y a ) : = { U : y a ( U 2 + | ∇ U | 2 ) ∈ L 1 ( Ω T ) } .

Note that J⁢(U,ΩT)≥0. On the other hand, we have that 0∈ΛT and J⁢(0,ΩT)<+∞. Hence there exists a minimizing sequence Uk∈ΛT of J, namely

lim k → ∞ ⁡ J ⁢ ( U k , Ω T ) = m T = inf U ∈ Λ T ⁡ J ⁢ ( U , Ω T ) .

Clearly, we have 0≤Uk≤1 on ΩT since F⁢(x,u) achieves its minimum value 0 at u=1 for any x. Then, for sufficiently large k, we have

(3.2) ∫ Ω T y a ⁢ | ∇ ⁡ U k ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y ≤ 2 ⁢ m T + 1 .

From the weighted Poincaré inequality, we obtain

(3.3) ∫ Ω T y a ⁢ U k 2 ⁢ ( x , y ) ⁢ d ⁢ x ⁢ d ⁢ y ≤ C < + ∞ .

From (3.2) and (3.3), we deduce that there exists a subsequence of {Uk} still denoted as {Uk}, converging weakly in H1⁢(ΩT,ya) to a function UT∈H1⁢(ΩT,ya). Due to the weak lower semi-continuity of the norm, we obtain

∫ Ω T y a ⁢ | ∇ ⁡ U T ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y ≤ lim inf k → + ∞ ⁡ ∫ Ω T y a ⁢ | ∇ ⁡ U k ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y .

By Fatou’s lemma, we also have

∫ 0 T 2 F ⁢ ( x , U T ⁢ ( x , 0 ) ) ⁢ d ⁢ x ≤ lim inf k → + ∞ ⁡ ∫ 0 T 2 F ⁢ ( x , U k ⁢ ( x , 0 ) ) ⁢ d ⁢ x .

Hence J⁢(UT,ΩT)≤mT. Note that ΛT is weakly closed; then J⁢(UT,ΩT)=mT. For any given η∈ΛT, τ>0, then UT+τ⁢η∈ΛT. We construct a real-valued function i(τ):=J(UT+τη,ΩT). Then

0 ≤ i ′ ⁢ ( 0 + ) = ∫ Ω T y a ⁢ ∇ ⁡ U T ⁢ ( x , y ) ⁢ ∇ ⁡ η ⁢ ( x , y ) ⁢ d ⁢ x ⁢ d ⁢ y + ∫ 0 T 2 ∂ u ⁡ F ⁢ ( x , U T ⁢ ( x , 0 ) ) ⁢ η ⁢ ( x , 0 ) ⁢ d ⁢ x = - ∫ Ω T div ⁡ ( y a ⁢ ∇ ⁡ U T ⁢ ( x , y ) ) ⁢ η ⁢ ( x , y ) ⁢ d ⁢ x ⁢ d ⁢ y + ∫ 0 T 2 [ ∂ ⁡ U T ∂ ⁡ ν a + ∂ u ⁡ F ⁢ ( x , U T ⁢ ( x , 0 ) ) ] ⁢ η ⁢ ( x , 0 ) ⁢ d ⁢ x .

Hence, by the arbitrariness of η, we obtain

{ div ⁡ ( y a ⁢ ∇ ⁡ U T ) ≤ 0 in ⁢ Ω T , ∂ ⁡ U T ∂ ⁡ ν a ≥ - ∂ u ⁡ F ⁢ ( x , U T ) on ⁢ ( ∂ ⁡ Ω ) 0 = [ 0 , T 2 ] .

Next, our task is to prove that UT≢0. For σ∈(0,1), we define the continuous function

h ⁢ ( x ) = { 4 σ ⁢ T ⁢ x if ⁢ x ∈ [ 0 , σ ⁢ T 4 ] , 1 if ⁢ x ∈ [ σ ⁢ T 4 , T 2 - σ ⁢ T 4 ] , 2 σ - 4 σ ⁢ T ⁢ x if ⁢ x ∈ [ T 2 - σ ⁢ T 4 , T 2 ] .

Note that 0≤h≤1; then we construct a function ψ∈ΛT as follows:

ψ ⁢ ( x , y ) = exp ⁡ { - y 2 b + 1 } ⁢ h ⁢ ( x ) ,

where the parameter b will be determined later. We next compute the energy J⁢(ψ,ΩT). From conditions (1.3) and (1.4) of F, we have

(3.4) ∫ 0 T 2 F ⁢ ( x , ψ ⁢ ( x , 0 ) ) ⁢ d ⁢ x = ∫ 0 T 2 F ⁢ ( x , h ⁢ ( x ) ) ⁢ d ⁢ x < ∫ 0 σ ⁢ T 4 F ⁢ ( x , 0 ) ⁢ d ⁢ x + ∫ T 2 - σ ⁢ T 4 T 2 F ⁢ ( x , 0 ) ⁢ d ⁢ x .

For the other part of energy, we have

∫ 0 T 2 ∫ 0 + ∞ y a ⁢ | ∇ ⁡ ψ ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y = ∫ 0 ∞ y a ⁢ exp ⁡ { - y 2 b } ⁢ d ⁢ y ⁢ ∫ 0 T 2 [ h 2 ⁢ ( x ) 2 2 ⁢ b + 2 + ( h ′ ⁢ ( x ) ) 2 ] ⁢ d ⁢ x

≤ [ 1 2 2 ⁢ b + 8 σ ⁢ T ] ⁢ ∫ 0 ∞ y a ⁢ exp ⁡ { - y 2 b } ⁢ d ⁢ y

= 2 b ⁢ ( a + 1 ) ⁢ [ 1 2 2 ⁢ b + 8 σ ⁢ T ] ⁢ ∫ 0 ∞ z a ⁢ e - z ⁢ d ⁢ z

≤ Γ ⁢ ( a + 1 ) ⁢ 2 b ⁢ ( a - 1 ) ⁢ ( T 8 + 2 2 ⁢ b ⁢ 8 σ ⁢ T ) .

Note that a-1<0. For the purpose that the term 2b⁢(a-1)⁢Γ⁢(a+1) is small, we can choose sufficiently large b. The other term 22⁢b⁢8σ⁢T is also small provided that T is large enough. Hence there exists T1>0 such that, for any T>T1, the following estimate holds true:

(3.5) 1 2 ⁢ ∫ 0 T 2 ∫ 0 + ∞ y a ⁢ | ∇ ⁡ ψ ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y < ∫ σ ⁢ T 4 T 2 - σ ⁢ T 4 F ⁢ ( x , 0 ) ⁢ d ⁢ x .

From (3.4) and (3.5), we have

J ⁢ ( 0 , Ω T ) > J ⁢ ( ψ , Ω T ) ≥ J ⁢ ( U T , Ω T ) ,

which shows that UT≢0.

In view of UT≥0 and UT≢0, by the strong maximum principle of a strictly elliptical operator and the Hopf principle (Lemma 2.2), we have that UT>0 in (0,T2)×[0,+∞). Choosing any function

η 0 ∈ C c ∞ ⁢ ( ( 0 , T 2 ) × [ 0 , + ∞ ) ) ,

if |τ| is sufficiently small, then one has UT+τ⁢η0∈ΛT. Then

0 = i ′ ⁢ ( 0 ) = - ∫ Ω T div ⁡ ( y a ⁢ ∇ ⁡ U T ) ⁢ η 0 ⁢ d ⁢ x ⁢ d ⁢ y + ∫ 0 T 2 ∂ ⁡ U T ∂ ⁡ ν a ⁢ η 0 + ∂ u ⁡ F ⁢ ( x , U T ) ⁢ η 0 ⁢ d ⁢ x ,

which yields

{ div ⁡ ( y a ⁢ ∇ ⁡ U T ) = 0 in ⁢ Ω T , ∂ ⁡ U T ∂ ⁡ ν a = - ∂ u ⁡ F ⁢ ( x , U T ) on ⁢ ( ∂ ⁡ Ω ) 0 = [ 0 , T 2 ] .

Now we extend UT oddly (in x) from ΩT to [-T2,T2]×[0,+∞). Furthermore, we extend it periodically (in x again) from [-T2,T2]×[0,+∞) to the whole half space ℝ+2¯, and we still denote it as UT. For any integer T>T1, we claim that UT is a weak solution of (3.1).

We first prove that the odd function UT in x is a weak solution of

{ div ⁡ ( y a ⁢ ∇ ⁡ U T ) = 0 in ⁢ [ - T 2 , T 2 ] × [ 0 , + ∞ ) , ∂ ⁡ U T ∂ ⁡ ν a = - ∂ u ⁡ F ⁢ ( x , U T ) on ⁢ [ - T 2 , T 2 ] .

Namely, for any η∈Cc∞⁢((-T2,T2)×[0,+∞)), the following equality holds:

(3.6) ∫ - T 2 T 2 ∫ 0 ∞ y a ⁢ ∇ ⁡ U T ⁢ ( x , y ) ⁢ ∇ ⁡ η ⁢ ( x , y ) ⁢ d ⁢ x ⁢ d ⁢ y + ∫ - T 2 T 2 ∂ u ⁡ F ⁢ ( x , U T ⁢ ( x , 0 ) ) ⁢ η ⁢ ( x , 0 ) ⁢ d ⁢ x = 0 .

We compute

∫ - T 2 0 ∫ 0 ∞ y a ⁢ ∇ ⁡ U T ⁢ ( x , y ) ⁢ ∇ ⁡ η ⁢ ( x , y ) ⁢ d ⁢ x ⁢ d ⁢ y + ∫ - T 2 0 ∂ u ⁡ F ⁢ ( x , U T ⁢ ( x , 0 ) ) ⁢ η ⁢ ( x , 0 ) ⁢ d ⁢ x = ∫ 0 T 2 ∫ 0 ∞ y a ⁢ ∇ ⁡ U T ⁢ ( - x , y ) ⁢ ∇ ⁡ η ⁢ ( - x , y ) ⁢ d ⁢ x ⁢ d ⁢ y + ∫ 0 T 2 ∂ u ⁡ F ⁢ ( - x , U T ⁢ ( - x , 0 ) ) ⁢ η ⁢ ( - x , 0 ) ⁢ d ⁢ x = ∫ 0 T 2 ∫ 0 ∞ y a ⁢ ∇ ⁡ U T ⁢ ( x , y ) ⁢ ( - ∂ x ⁡ η ⁢ ( - x , y ) , - ∂ y ⁡ η ⁢ ( - x , y ) ) ⁢ d ⁢ x ⁢ d ⁢ y - ∫ 0 T 2 ∂ u ⁡ F ⁢ ( x , U T ⁢ ( x , 0 ) ) ⁢ η ⁢ ( - x , 0 ) ⁢ d ⁢ x ,

where we used the facts that F⁢(x,u) is even in x and u, respectively, and UT is odd in x. Hence we have

∫ - T 2 T 2 ∫ 0 ∞ y a ⁢ ∇ ⁡ U T ⁢ ( x , y ) ⁢ ∇ ⁡ η ⁢ ( x , y ) ⁢ d ⁢ x ⁢ d ⁢ y + ∫ - T 2 T 2 ∂ u ⁡ F ⁢ ( x , U T ⁢ ( x , 0 ) ) ⁢ η ⁢ ( x , 0 ) ⁢ d ⁢ x = ∫ 0 T 2 ∫ 0 ∞ y a ⁢ ∇ ⁡ U T ⁢ ( x , y ) ⁢ ∇ ⁡ φ ⁢ ( x , y ) ⁢ d ⁢ x ⁢ d ⁢ y + ∫ 0 T 2 ∂ u ⁡ F ⁢ ( x , U T ⁢ ( x , 0 ) ) ⁢ φ ⁢ ( x , 0 ) ⁢ d ⁢ x ,

where we have set φ⁢(x,y)=η⁢(x,y)-η⁢(-x,y). Clearly, this function is admissible since it vanishes on x=0, x=T2 for any y≥0. Hence (3.6) holds true. A similar argument shows that, for any integer T>T1, the periodic function UT in x is a weak solution of (3.1), where we need to use the periodic condition (1.2) and the assumption that T is an integer.

Now we set uT⁢(x)=UT⁢(x,0). Then uT is an odd periodic solution of (1.1) with period T. Using the Hopf principle, we get UT⁢(x,0)=uT⁢(x)∈(0,1) in (0,T2), where we used the fact that ∂u⁡F⁢(x,1)=∂u⁡F⁢(x,-1)=0. So uT⁢(x)∈(0,1) for x∈(0,T2).

Next let us calculate the estimate of (1.9). From [10], we know that

∫ - T 2 T 2 ∫ ℝ + y a ⁢ | ∇ ⁡ U ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y = C ⁢ ( s ) 2 ⁢ d s ⁢ ∫ - T 2 T 2 ∫ ℝ | u ⁢ ( x ) - u ⁢ ( x ¯ ) | 2 | x - x ¯ | 1 + 2 ⁢ s ⁢ d ⁢ x ¯ ⁢ d ⁢ x .

We construct the continuous function

ϕ ⁢ ( x ) = { x d , x ∈ [ 0 , d ] , 1 , x ∈ [ d , T 2 - d ] , - 1 d ⁢ ( x - T 2 ) , x ∈ [ T 2 - d , T 2 ]

for some constant d. We extend ϕ oddly from [0,T2] to [-T2,T2]. Further, we extend it periodically with period T. We still denote it as ϕ. It is easy to verify that there exists a function Φ⁢(x,y)∈ΛT such that ϕ⁢(x)=Φ⁢(x,0). Therefore, to prove (1.9), it is enough to show that

(3.7) ∫ - T 2 T 2 ∫ ℝ | ϕ ⁢ ( x ) - ϕ ⁢ ( x ¯ ) | 2 | x - x ¯ | 1 + 2 ⁢ s ⁢ d ⁢ x ¯ ⁢ d ⁢ x + ∫ - T 2 T 2 F ⁢ ( x , Φ ⁢ ( x , 0 ) ) ⁢ d ⁢ x ≤ { C ⁢ T 1 - 2 ⁢ s , s ∈ ( 0 , 1 2 ) , C ⁢ ln ⁡ T , s = 1 2 , C , s ∈ ( 1 2 , 1 ) .

To obtain (3.7), we only need to prove that

(3.8) ∫ - T 2 T 2 ∫ | x - x ¯ | ≥ T 2 | ϕ ⁢ ( x ) - ϕ ⁢ ( x ¯ ) | 2 | x - x ¯ | 1 + 2 ⁢ s ⁢ d ⁢ x ¯ ⁢ d ⁢ x + ∫ - T 2 + d - d ∫ d T 2 - d | ϕ ⁢ ( x ) - ϕ ⁢ ( x ¯ ) | 2 | x - x ¯ | 1 + 2 ⁢ s ⁢ d ⁢ x ¯ ⁢ d ⁢ x + ∫ - T 2 + d - d ∫ - d d | ϕ ⁢ ( x ) - ϕ ⁢ ( x ¯ ) | 2 | x - x ¯ | 1 + 2 ⁢ s ⁢ d ⁢ x ¯ ⁢ d ⁢ x + ∫ - d d ∫ - d d | ϕ ⁢ ( x ) - ϕ ⁢ ( x ¯ ) | 2 | x - x ¯ | 1 + 2 ⁢ s ⁢ d ⁢ x ¯ ⁢ d ⁢ x ≤ { C ⁢ T 1 - 2 ⁢ s , s ∈ ( 0 , 1 2 ) , C ⁢ ln ⁡ T , s = 1 2 , C , s ∈ ( 1 2 , 1 ) .

For the first integral, we have

(3.9) ∫ - T 2 T 2 ∫ | x - x ¯ | ≥ T 2 | ϕ ⁢ ( x ) - ϕ ⁢ ( x ¯ ) | 2 | x - x ¯ | 1 + 2 ⁢ s ⁢ d ⁢ x ¯ ⁢ d ⁢ x ≤ C ⁢ T 1 - 2 ⁢ s .

For the second integral, we have

(3.10) ∫ - T 2 + d - d ∫ d T 2 - d | ϕ ⁢ ( x ) - ϕ ⁢ ( x ¯ ) | 2 | x - x ¯ | 1 + 2 ⁢ s ⁢ d ⁢ x ¯ ⁢ d ⁢ x ≤ 4 2 ⁢ s ⁢ ∫ - T 2 + d - d | d - x | - 2 ⁢ s ⁢ d ⁢ x ≤ { C ⁢ T 1 - 2 ⁢ s , s ∈ ( 0 , 1 2 ) , C ⁢ ln ⁡ T , s = 1 2 , C , s ∈ ( 1 2 , 1 ) .

For the third integral, we have

(3.11) ∫ - T 2 + d - d ∫ - d d | ϕ ⁢ ( x ) - ϕ ⁢ ( x ¯ ) | 2 | x - x ¯ | 1 + 2 ⁢ s ⁢ d ⁢ x ¯ ⁢ d ⁢ x = d - 2 ⁢ ∫ - T 2 + d - d ∫ - d d | d + x ¯ | 2 | x - x ¯ | 1 + 2 ⁢ s ⁢ d ⁢ x ¯ ⁢ d ⁢ x = d - 2 ⁢ ∫ - d d ( ∫ - T 2 + d - d | d + x ¯ | 2 | x - x ¯ | 1 + 2 ⁢ s ⁢ d ⁢ x ) ⁢ d ⁢ x ¯ ≤ C ⁢ d - 2 ⁢ ∫ - d d | d + x ¯ | 2 ⁢ | d + x ¯ | - 2 ⁢ s ⁢ d ⁢ x ¯ ≤ C .

For the last integral, we have

(3.12) ∫ - d d ∫ - d d | ϕ ⁢ ( x ) - ϕ ⁢ ( x ¯ ) | 2 | x - x ¯ | 1 + 2 ⁢ s ⁢ d ⁢ x ¯ ⁢ d ⁢ x ≤ d - 2 ⁢ ∫ - d d ∫ - d d | x - x ¯ | 1 - 2 ⁢ s ⁢ d ⁢ x ⁢ d ⁢ x ¯ ≤ C ⁢ d 1 - 2 ⁢ s ≤ C .

From inequalities (3.8)–(3.12), we obtain (3.7). ∎

Remark 3.1.

By adjusting the admissible set and using an argument similar to Theorem 1.1, we can obtain the existence of even periodic solutions of (1.1). Precisely, we consider the energy functional J⁢(U,ΩT) in the admissible set

Λ T = { U ∈ H 1 ⁢ ( Ω T , y a ) : U ⁢ ( - x , y ) = U ⁢ ( x , y ) , U ⁢ ( 0 , y ) ≤ 0 ≤ U ⁢ ( T 2 , y ) } ,

where ΩT=[-T2,T2]×[0,+∞). We can find a minimizer UT of the energy J in ΛT and prove that UT≢0. From the even symmetry of UT (in x) and F, we know that UT is also a minimizer of the energy J⁢(U,Ω^T), where Ω^T=[0,T2]×[0,+∞). Hence UT satisfies

{ div ⁡ ( y a ⁢ ∇ ⁡ U T ) = 0 in ⁢ Ω ^ T , ∂ ⁡ U T ∂ ⁡ ν a = - ∂ u ⁡ F ⁢ ( x , U T ) on ⁢ [ 0 , T 2 ] .

From this and the facts that ∂x⁡(UT)=0 on x=0 and φ(T2,y):=η(T2,y)+η(-T2,y)=0 on x=T2 for any y≥0, we obtain (3.6) for any η∈Cc∞⁢((-T2,T2)×[0,+∞)). We extend UT periodically (in x) from [-T2,T2]×[0,+∞) to the whole half space ℝ+2¯ (still denoted as UT). An argument similar to that of (3.6) shows that UT is a weak solution of (3.1), where we need to use the assumptions (1.2) and that T is an integer.

Note that similar energy estimates are obtained in [17] for minimizers of the functional in a finite interval [a,b] with a homogeneous condition outside the interval instead of a periodic condition, and higher-dimensional estimates have been subsequently obtained in [19].

Proof of Theorem 1.2.

We borrow the idea in [14] to prove this theorem. Now define the Hilbert space as

ℋ : = { U ( x , y ) : | U ⁢ ( x , y ) | ≤ 1 , U ⁢ ( - T 2 , y ) = U ⁢ ( T 2 , y ) ⁢ for all ⁢ y ≥ 0 , ∥ U ∥ ℋ 2 = ∫ Ω T ¯ y a | ∇ U ( x , y ) | 2 d x d y + ∫ - T 2 T 2 U 2 ( x , 0 ) d x < + ∞ } ,

where ΩT¯:=[-T2,T2]×[0,+∞). We consider the corresponding energy functional

J ⁢ ( U , Ω T ¯ ) = 1 2 ⁢ ∫ Ω T ¯ y a ⁢ | ∇ ⁡ U ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y + ∫ - T 2 T 2 F ⁢ ( x , U ⁢ ( x , 0 ) ) ⁢ d ⁢ x .

Since F⁢(x,u) is a smooth function, we can obtain J∈C1⁢(ℋ,ℝ). Next we verify the Palais–Smale condition. Namely, for any sequence {Uk}⊂ℋ with J⁢(Uk,ΩT¯) bounded and J′⁢(Uk,ΩT¯)→0 in ℋ, it contains a convergent subsequence of {Uk}. Estimates similar to (3.2) and (3.3) yield that there exists a subsequence, still denoted as {Uk}, converging weakly to a function U¯ in ℋ. In view of ℋ(ΩT¯)↪Hs(-T2,T2)↪↪L2(-T2,T2), we have

(3.13) U k ⁢ ( x , 0 ) → U ¯ ⁢ ( x , 0 ) in ⁢ L 2 ⁢ ( - T 2 , T 2 ) .

Note that

∫ Ω T ¯ y a ⁢ | ∇ ⁡ U k ⁢ ( x , y ) - ∇ ⁡ U ¯ ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y = 〈 J ′ ⁢ ( U k , Ω T ¯ ) - J ′ ⁢ ( U ¯ , Ω T ¯ ) , U k - U ¯ 〉 - ∫ - T 2 T 2 [ ∂ u ⁡ F ⁢ ( x , U k ⁢ ( x , 0 ) ) - ∂ u ⁡ F ⁢ ( x , U ¯ ⁢ ( x , 0 ) ) ] ⁢ ( U k ⁢ ( x , 0 ) - U ¯ ⁢ ( x , 0 ) ) ⁢ d ⁢ x .

Clearly, 〈J′⁢(Uk,ΩT¯)-J′⁢(U¯,ΩT¯),Uk-U¯〉→0. We also have

| ∫ - T 2 T 2 [ ∂ u ⁡ F ⁢ ( x , U k ⁢ ( x , 0 ) ) - ∂ u ⁡ F ⁢ ( x , U ¯ ⁢ ( x , 0 ) ) ] ⁢ ( U k ⁢ ( x , 0 ) - U ¯ ⁢ ( x , 0 ) ) ⁢ d ⁢ x | ≤ C ⁢ ∫ - T 2 T 2 | U k ⁢ ( x , 0 ) - U ¯ ⁢ ( x , 0 ) | 2 ⁢ d ⁢ x → 0 ,

where the convergence result follows from (3.13). Hence

∫ Ω T ¯ y a ⁢ | ∇ ⁡ U k ⁢ ( x , y ) - ∇ ⁡ U ¯ ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y → 0 .

This and (3.13) give that Uk→U¯ in ℋ. We have obtained the Palais–Smale condition.

We set Γ:={g∈C([0,1];ℋ):g(0)=-1,g(1)=1}. Note that

J ⁢ ( 1 , Ω T ¯ ) = J ⁢ ( - 1 , Ω T ¯ ) = ∫ - T 2 T 2 F ⁢ ( x , ± 1 ) ⁢ d ⁢ x = 0 ≤ J ⁢ ( v , Ω T ¯ ) for all ⁢ v ∈ ℋ .

and J is stable at 1 and -1, namely

∫ Ω T ¯ y a ⁢ | ∇ ⁡ φ ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y + ∫ - T 2 T 2 ∂ u ⁢ u ⁡ F ⁢ ( x , ± 1 ) ⁢ φ 2 ⁢ ( x , 0 ) ⁢ d ⁢ x > 0 for all ⁢ φ ≠ 0 ∈ ℋ .

Hence we have

δ T = inf g ∈ Γ ⁢ sup t ∈ [ 0 , 1 ] ⁡ J ⁢ ( g ⁢ ( t ) , Ω ¯ T ) > 0 .

We set J⁢(UT,Ω¯T)=δT, where UT=g⁢(t0) for some g∈Γ and some t0∈(0,1). Choose T as an integer. We extend UT periodically (in x) to the whole half space ℝ+2¯ (still denoted it as UT). An argument similar to that of (3.6) shows that UT is a solution of (3.1).

Next we show that UT≢0. It is enough to prove that UT≢0 on ΩT¯. We choose a function ψ∈ℋ similar to the above section,

ψ ⁢ ( x , y ) = exp ⁡ { - y 2 b + 1 } ⁢ h ~ ⁢ ( x ) ,

where h~⁢(x) is the odd extension of h from (0,T2) onto (-T2,T2). We construct a path as

g ¯ ⁢ ( t ) = { 2 ⁢ t ⁢ ψ + ( 1 - 2 ⁢ t ) × ( - 1 ) for ⁢ 0 ≤ t ≤ 1 2 , ( 2 - 2 ⁢ t ) ⁢ ψ + ( 2 ⁢ t - 1 ) for ⁢ 1 2 ≤ t ≤ 1 .

Clearly, g¯∈Γ, and we denote g¯⁢(t) as gt¯⁢(x,y). Then

∫ Ω T ¯ y a ⁢ | ∇ ⁡ g ¯ t ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y ≤ ∫ Ω T ¯ y a ⁢ | ∇ ⁡ ψ ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y .

Then, for 0≤t≤12, we have

∫ - T 2 T 2 F ⁢ ( x , g t ¯ ⁢ ( x , 0 ) ) ⁢ d ⁢ x = ∫ - T 2 0 F ⁢ ( x , g t ¯ ⁢ ( x , 0 ) ) ⁢ d ⁢ x + ∫ 0 T 2 F ⁢ ( x , g t ¯ ⁢ ( x , 0 ) ) ⁢ d ⁢ x ≤ ∫ - T 2 0 F ⁢ ( x , ψ ⁢ ( x , 0 ) ) ⁢ d ⁢ x + ∫ 0 T 2 F ⁢ ( x , 0 ) ⁢ d ⁢ x .

Similarly, for 12≤t≤1, we have

∫ - T 2 T 2 F ⁢ ( x , g t ¯ ⁢ ( x , 0 ) ) ⁢ d ⁢ x ≤ ∫ 0 T 2 F ⁢ ( x , ψ ⁢ ( x , 0 ) ) ⁢ d ⁢ x + ∫ - T 2 0 F ⁢ ( x , 0 ) ⁢ d ⁢ x .

Then a computation similar to (3.4) and (3.5) shows that there exists T2>0 such that, for any T>T2, we have J⁢(gt¯,ΩT¯)<J⁢(0,ΩT¯) for all t∈[0,1]. Hence

J ⁢ ( U T , Ω T ¯ ) = δ T ≤ max t ∈ [ 0 , 1 ] ⁡ J ⁢ ( g t ¯ , Ω T ¯ ) < J ⁢ ( 0 , Ω T ¯ ) ,

which gives that UT≢0 on ΩT¯.

For large enough integer T, uT(x):=UT(x,0) is a periodic solution of equation (1.1). Plainly, uT⁢(x) changes its sign at least once in a period. The Hopf principle gives again that |uT⁢(x)|=|UT⁢(x,0)|<1.

Finally, we show estimate (1.10). To this end, for any given integer m>1, we define 2⁢m-1 continuous functions hi (1≤i≤2⁢m-1) as follows:

h i ⁢ ( x ) = { - 8 ⁢ m T ⁢ x + 4 ⁢ m for ⁢ x ∈ [ T 2 - T 8 ⁢ m , T 2 ] , 1 for ⁢ x ∈ [ T 2 - i ⁢ T 2 ⁢ m + T 8 ⁢ m , T 2 - T 8 ⁢ m ] , 8 ⁢ m T ⁢ x - 4 ⁢ ( m - i ) for ⁢ x ∈ [ T 2 - i ⁢ T 2 ⁢ m - T 8 ⁢ m , T 2 - i ⁢ T 2 ⁢ m + T 8 ⁢ m ] , - 1 for ⁢ x ∈ [ - T 2 + T 8 ⁢ m , T 2 - i ⁢ T 2 ⁢ m - T 8 ⁢ m ] , - 8 ⁢ m T ⁢ x - 4 ⁢ m for ⁢ x ∈ [ - T 2 , - T 2 + T 8 ⁢ m ] .

Note that hi∈[-1,1]. Similarly, we define ψi∈ℋ (1≤i≤(2⁢m-1)) by

ψ i ⁢ ( x , y ) = exp ⁡ { - y 2 b + 1 } ⁢ h i ⁢ ( x ) .

Now we construct a path as

g ^ ⁢ ( t ) = { 2 ⁢ m ⁢ t ⁢ ψ 1 + ( 1 - 2 ⁢ m ⁢ t ) ⁢ ( - 1 ) for ⁢ 0 ≤ t ≤ 1 2 ⁢ m , ( ( i + 1 ) - 2 ⁢ m ⁢ t ) ⁢ ψ i + ( 2 ⁢ m ⁢ t - i ) ⁢ ψ i + 1 for ⁢ i 2 ⁢ m ≤ t ≤ i + 1 2 ⁢ m , 1 ≤ i ≤ 2 ⁢ m - 2 , ( 2 ⁢ m - 2 ⁢ m ⁢ t ) ⁢ ψ 2 ⁢ m - 1 + ( 2 ⁢ m ⁢ t - ( 2 ⁢ m - 1 ) ) for ⁢ 2 ⁢ m - 1 2 ⁢ m ≤ t ≤ 1 .

Clearly, g^∈Γ. For 0≤t≤12⁢m, from the definition of g^, we have

∫ - T 2 T 2 F ⁢ ( x , g t ^ ⁢ ( x , 0 ) ) ⁢ d ⁢ x = ∫ - T 2 - T 2 + T 8 ⁢ m F ⁢ ( x , g t ^ ⁢ ( x , 0 ) ) ⁢ d ⁢ x + ∫ T 2 - 5 ⁢ T 8 ⁢ m T 2 F ⁢ ( x , g t ^ ⁢ ( x , 0 ) ) ⁢ d ⁢ x ≤ max x ∈ [ - T 2 , T 2 ] ⁡ F ⁢ ( x , 0 ) ⁢ 6 ⁢ T 8 ⁢ m .

Similarly, for 2⁢m-12⁢m≤t≤1, we have

∫ - T 2 T 2 F ⁢ ( x , g t ^ ⁢ ( x , 0 ) ) ⁢ d ⁢ x = ∫ - T 2 T 2 - ( 2 ⁢ m - 1 ) 2 ⁢ m ⁢ T + T 8 ⁢ m F ⁢ ( x , g t ^ ⁢ ( x , 0 ) ) ⁢ d ⁢ x + ∫ T 2 - T 8 ⁢ m T 2 F ⁢ ( x , g t ^ ⁢ ( x , 0 ) ) ⁢ d ⁢ x ≤ max x ∈ [ - T 2 , T 2 ] ⁡ F ⁢ ( x , 0 ) ⁢ 6 ⁢ T 8 ⁢ m .

For the case i2⁢m≤t≤i+12⁢m (1≤i≤2⁢m-2), we have

∫ - T 2 T 2 F ⁢ ( x , g t ^ ⁢ ( x , 0 ) ) ⁢ d ⁢ x = ∫ - T 2 - T 2 + T 8 ⁢ m F ⁢ ( x , g t ^ ⁢ ( x , 0 ) ) ⁢ d ⁢ x + ∫ T 2 - ( i + 1 ) 2 ⁢ m ⁢ T - T 8 ⁢ m T 2 - ( i ) 2 ⁢ m ⁢ T + T 8 ⁢ m F ⁢ ( x , g t ^ ⁢ ( x , 0 ) ) ⁢ d ⁢ x + ∫ T 2 - T 8 ⁢ m T 2 F ⁢ ( x , g t ^ ⁢ ( x , 0 ) ) ⁢ d ⁢ x ≤ max x ∈ [ - T 2 , T 2 ] ⁡ F ⁢ ( x , 0 ) ⁢ T m .

Therefore,

(3.14) ∫ - T 2 T 2 F ⁢ ( x , g t ^ ⁢ ( x , 0 ) ) ⁢ d ⁢ x ≤ max x ∈ [ - T 2 , T 2 ] ⁡ F ⁢ ( x , 0 ) ⁢ T m for all ⁢ t ∈ [ 0 , 1 ] .

For the other part of the energy, we have

∫ Ω T ¯ y a ⁢ | ∇ ⁡ g ^ ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y ≤ max 1 ≤ i ≤ 2 ⁢ m - 1 ⁡ 2 ⁢ ∫ Ω T ¯ y a ⁢ | ∇ ⁡ ψ i ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y .

Similarly, by choosing large enough b and T, we obtain

(3.15) max 1 ≤ i ≤ 2 ⁢ m - 1 ⁡ 2 ⁢ ∫ Ω T ¯ y a ⁢ | ∇ ⁡ ψ i ⁢ ( x , y ) | 2 ⁢ d ⁢ x ⁢ d ⁢ y ≤ max x ∈ [ - T 2 , T 2 ] ⁡ F ⁢ ( x , 0 ) ⁢ T m for all ⁢ T > T m ,

where Tm≥T2 and limm→0⁡Tm→+∞. Inequalities (3.14) and (3.15) give that

max t ∈ [ 0 , 1 ] ⁡ J ⁢ ( g t ^ , Ω T ¯ ) ≤ max x ∈ [ - T 2 , T 2 ] ⁡ F ⁢ ( x , 0 ) ⁢ 2 ⁢ T m for all ⁢ T > T m .

Hence, for any 0<σ<12, we can take large m=m⁢(σ) such that, for any T>Tσ,

J ⁢ ( U T , Ω T ¯ ) ≤ max t ∈ [ 0 , 1 ] ⁡ J ⁢ ( g t ^ , Ω T ¯ ) ≤ max x ∈ [ - T 2 , T 2 ] ⁡ F ⁢ ( x , 0 ) ⁢ 2 ⁢ T m < σ ⁢ max x ∈ [ - T 2 , T 2 ] ⁡ F ⁢ ( x , 0 ) ⁢ T ,

which is the desired estimate (1.10). Here Tσ:=Tm⁢(σ)→∞ as σ→0. ∎

4 Hamiltonian Identity and Modica-Type Inequality

We will first prove the Hamiltonian identity for periodic solutions of (3.1).

Proof of Theorem 1.3.

Similarly to [6, Lemma 5.1], we have ∫0∞ya⁢|∇⁡U⁢(x,y)|2⁢d⁢y<∞. Hence

lim y → + ∞ ⁡ y a ⁢ U y ⁢ ( x , y ) ⁢ U x ⁢ ( x , y ) = 0 .

We introduce the function

w ( x ) : = 1 2 ∫ 0 ∞ y a [ U x 2 ( x , y ) - U y 2 ( x , y ) ] d y .

The regularity result allows us to differentiate within the integral in the above equality to get

w ′ ⁢ ( x ) = ∫ 0 ∞ y a ⁢ [ U x ⁢ U x ⁢ x - U y ⁢ U x ⁢ y ] ⁢ ( x , y ) ⁢ d ⁢ y .

Note that (ya⁢Uy)y+ya⁢Ux⁢x=0. Using integration by parts, we have

w ′ ⁢ ( x ) = - [ y a ⁢ U y ⁢ ( x , y ) ⁢ U x ⁢ ( x , y ) ] | y = 0 + ∞ = lim y → 0 + ⁡ y a ⁢ U y ⁢ ( x , y ) ⁢ U x ⁢ ( x , y ) = ∂ u ⁡ F ⁢ ( x , U ⁢ ( x , 0 ) ) ⁢ U x ⁢ ( x , 0 ) .

Owing to

d d ⁢ x ⁢ { F ⁢ ( x , U ⁢ ( x , 0 ) ) - ∫ 0 x ∂ x ⁡ F ⁢ ( σ , U ⁢ ( σ , 0 ) ) ⁢ d ⁢ σ } = ∂ x ⁡ F ⁢ ( x , U ⁢ ( x , 0 ) ) + ∂ u ⁡ F ⁢ ( x , U ⁢ ( x , 0 ) ) ⋅ U x ⁢ ( x , 0 ) - ∂ x ⁡ F ⁢ ( x , U ⁢ ( x , 0 ) ) = ∂ u ⁡ F ⁢ ( x , U ⁢ ( x , 0 ) ) ⁢ U x ⁢ ( x , 0 ) ,

we obtain

∫ 0 ∞ y a 2 ⁢ { U x 2 - U y 2 } ⁢ d ⁢ y + ∫ 0 x ∂ x ⁡ F ⁢ ( σ , U ⁢ ( σ , 0 ) ) ⁢ d ⁢ σ - F ⁢ ( x , U ⁢ ( x , 0 ) ) ≡ C T ,

where CT is a constant depending on T. ∎

Proof of Theorem 1.4.

We introduce the function

v ( x , y ) : = 1 2 ∫ 0 y [ U x 2 ( x , τ ) - U y 2 ( x , τ ) ] τ a d τ

and define

v ^ : = 1 2 ∫ 0 y [ U x 2 ( x , τ ) - U y 2 ( x , τ ) ] τ a d τ - F ( x , U ( x , 0 ) ) + ∫ 0 x ∂ x F ( σ , U ( σ , 0 ) ) d σ - C T .

By the periodicity and even symmetry of U⁢(x,y) (in x), it suffices to prove the Modica-type inequality for every y≥0 and all x∈[0,T2]. Note that

(4.1) lim y → + ∞ ⁡ v ^ ⁢ ( x , y ) = 0

and Ux⁢(0,y)=0=Ux⁢(T2,y) for all y≥0. Then we have

(4.2) v ^ ⁢ ( 0 , y ) < v ^ ⁢ ( 0 , 0 ) , v ^ ⁢ ( T 2 , y ) < v ^ ⁢ ( T 2 , 0 ) .

Hence v^ is not identically constant.

Elementary calculation shows v^x=-ya⁢Ux⁢Uy and

(4.3) div ⁡ ( y - a ⁢ ∇ ⁡ v ^ ) = a ⁢ y - 1 ⁢ u y 2 .

Without loss of generality, we may assume that Ux≤0 in (0,T2)×(0,+∞). The strong maximum principle yields that Ux is strictly negative in this domain. Equation (4.3) can be written as

div ⁡ ( y - a ⁢ ∇ ⁡ v ^ ) + a ⁢ y - 1 - a ⁢ U y U x ⁢ v ^ x = 0 .

Note that the operator in the left-hand side is uniformly elliptic with continuous coefficients in compact sets of (0,T2)×(0,+∞). Since v^ is not identically constant, v^ cannot achieve its maximum in any interior point of (0,T2)×(0,+∞). This fact and (4.1), (4.2) show that v^ achieves its maximum in [0,T2]×[0,+∞) at [0,T2]×{0}, and we denote the maximum as C^. Then

C ^ = sup x ∈ [ 0 , T 2 ] ⁡ { ∫ 0 x ∂ x ⁡ F ⁢ ( σ , U ⁢ ( σ , 0 ) ) ⁢ d ⁢ σ - F ⁢ ( x , U ⁢ ( x , 0 ) ) - C T } ≥ ∫ 0 T 2 ∂ x ⁡ F ⁢ ( σ , U ⁢ ( σ , 0 ) ) ⁢ d ⁢ σ - F ⁢ ( T 2 , U ⁢ ( T 2 , 0 ) ) - C T = 1 2 ⁢ ∫ 0 + ∞ U y 2 ⁢ ( T 2 , τ ) ⁢ d ⁢ τ > 0 . ∎

Communicated by Changfeng Gui

Funding source: Natural Science Foundation ofÂ Hunan Province

Award Identifier / Grant number: 2016JJ2018

Funding statement: The second author is supported by the Natural Science Foundation of Hunan Province, China (Grant No. 2016JJ2018).

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Received: 2019-12-05

Accepted: 2020-02-07

Published Online: 2020-03-19

Published in Print: 2020-08-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/ans-2020-2075

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Fractional Laplacian; Periodic Solutions; Non-autonomous; Variational Method

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