Existence of Multiple Periodic Solutions for a Semilinear Wave Equation in an n-Dimensional Ball

Hui Wei; Shuguan Ji

doi:10.1515/ans-2018-2036

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Existence of Multiple Periodic Solutions for a Semilinear Wave Equation in an n-Dimensional Ball

Hui Wei and Shuguan Ji

Published/Copyright: December 21, 2018

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From the journal Advanced Nonlinear Studies Volume 19 Issue 3

Abstract

This paper is devoted to the study of periodic solutions for a radially symmetric semilinear wave equation in an n-dimensional ball. By combining the variational methods and saddle point reduction technique, we obtain the existence of at least three periodic solutions for arbitrary space dimension n. The structure of the spectrum of the linearized problem plays an essential role in the proof, and the construction of a suitable working space is devised to overcome the restriction of space dimension.

Keywords: Existence; Periodic Solutions; Wave Equation

MSC 2010: 35L71; 35B10

1 Introduction

In this paper, we consider the radially symmetric periodic solutions for a semilinear wave equation

(1.1) u t ⁢ t - Δ ⁢ u = μ ⁢ u + f ⁢ ( t , x , u ) , t ∈ ℝ , x ∈ 𝔹 R n ,

with Dirichlet boundary conditions

(1.2) u ⁢ ( t , x ) = 0 , t ∈ ℝ , x ∈ ∂ ⁡ 𝔹 R n ,

and periodic conditions

(1.3) u ⁢ ( t + T , x ) = u ⁢ ( t , x ) , t ∈ ℝ , x ∈ 𝔹 R n ,

where Δ is the classical Laplacian operator, μ is a positive constant, 𝔹Rn denotes the open ball of radius R centered at 0 in ℝn, ∂⁡𝔹Rn denotes the boundary of 𝔹Rn, f is a radically symmetric (in x) T-periodic (in t) function, and R and T satisfy 8⁢RT=ab for some relative prime positive integers a and b.

Starting from the pioneering work of Rabinowitz [20], the problem of finding periodic solutions to nonlinear wave equations has been extensively studied. For example, see [6, 8, 12, 13, 15, 16, 18, 14, 17, 21, 23, 22, 24, 25] and references therein. It is known that, in general, the spectral structure of the wave operator plays an important role for the study of periodic solution. For the one-dimensional case, if the time period T was a rational multiple of the length of the spatial interval, the spectrum of the wave operator with Dirichlet boundary condition possesses the zero eigenvalue with infinite multiplicity, but the other eigenvalues are well separated. The variational technique is a sophisticated approach to treat this case, and most known results rely on the separation properties. On the other hand, if the time period T was an irrational multiple of the length of the spatial interval, the wave operator does not possess the zero eigenvalue, but its spectrum accumulates to zero for almost every frequency. This is the notorious small divisor problem. For this case, a quite different approach which used the Kolmogorov–Arnold–Moser theory was developed from the viewpoint of infinite-dimensional dynamical systems by Kuksin [19] and Wayne [26]. However, for the higher-dimensional case (i.e., the space dimension n>1), the spectral structure of the wave operator becomes more complicated, which makes it more difficult to investigate the periodic solutions. In recent years, many authors (see [3, 4, 11, 10, 9, 22]) considered the periodic solution of the radially symmetric wave equation in an n-dimensional ball. For this case, it is not difficult to realize that the study of periodic solutions is closely related to the arithmetical properties of radius R and period T. Here it is worth mentioning that Ben-Naoum and Mawhin [4] studied the wave equation with a general nonlinear term and obtained at least one 2⁢π-periodic solution in an n-dimensional ball with radius R=π2 when n=3 or n is even.

As regards the multiplicity problem, Chen and Zhang [9] considered the wave equation

u t ⁢ t - Δ ⁢ u = μ ⁢ u + | u | p - 1 ⁢ u

in a ball in ℝn and obtained infinitely many weak solutions for the case that n-3 is an integer multiple of (4,a) and 8⁢RT=ab, where a, b are relative prime positive integers and (4,a) denotes the greatest common divisor of 4 and a. Later, they also dealt with the wave equation ut⁢t-Δ⁢u=μ⁢u+a⁢(t,x)⁢|u|p-1⁢u in a ball with R=π2 and obtained infinitely many 2⁢π-periodic solutions for the cases that n is an even integer or n>3 is an odd integer [10]. Recently, they also investigated the wave equation ut⁢t-Δ⁢u=g⁢(t,x,u) in a ball in ℝn and proved that there exist at least three radially symmetric periodic solutions under some certain suitable conditions for the case that n=2 or n>3 is odd and RT=d4 with d∈ℕ+ (see [11]). In this case, it is proved that 0 is not in the spectral set of the wave operator, which plays an important role in the work.

In this paper, we shall investigate the existence of multiple periodic solutions of problem (1.1)–(1.3) for arbitrary space dimension n and 8⁢RT=ab. By constructing a suitable working space, we can overcome the restriction on the space dimension n and prove that problem (1.1)–(1.3) possesses at least three periodic solutions.

Throughout this paper we make the following assumption:

f ⁢ ( t , x , u ) ∈ C 1 ⁢ ( ℝ × 𝔹 R n × ℝ ) is radially symmetric in x, T-periodic in t (i.e., f⁢(t+T,x,u)=f⁢(t,x,u)), satisfies

(1.4) | f ⁢ ( t , x , u ) | = o ⁢ ( | u | ) as ⁢ | u | → 0 ⁢ uniformly in ⁢ ( t , x ) ,

and f⁢(t,x,u) is asymptotically linear in u at ∞, which means that there exists a constant β>0 such that

(1.5) | f ⁢ ( t , x , u ) - β ⁢ u | = o ⁢ ( | u | ) as ⁢ | u | → ∞ ⁢ uniformly in ⁢ ( t , x ) .

The rest of this paper is organized as follows: In Section 2, we give the definition of a weak solution of problem (1.1)–(1.3) and transform it into the critical point of the corresponding functional. Meanwhile, we also give the spectral analysis of the wave operator and the statement of the main result as well as some preliminaries. In Section 3, we reduce the critical point problem on an infinite-dimensional space to a finite-dimensional subspace via the saddle point reduction argument. Section 4 and Section 5 are respectively dedicated to the verification of the (PS)c condition and the bounds of reduction functional. Finally, in Section 6, we complete the proof of the main result.

2 Definition of Weak Solution and some Preliminaries

Let r=∥x∥. In virtue of the assumption of radial symmetry, problem (1.1)–(1.3) can be transformed into

(2.1) u t ⁢ t - u r ⁢ r - n - 1 r ⁢ u r = μ ⁢ u + f ⁢ ( t , r , u ) , ( t , r ) ∈ Ω ≡ [ 0 , T ] × [ 0 , R ] ,

with Dirichlet boundary conditions

(2.2) u ⁢ ( t , R ) = 0 , t ∈ [ 0 , T ] ,

and periodic conditions

(2.3) u ⁢ ( 0 , r ) = u ⁢ ( T , r ) , u t ⁢ ( 0 , r ) = u t ⁢ ( T , r ) , r ∈ [ 0 , R ] .

Consequently, in order to look for radially symmetric periodic solutions of problem (1.1)–(1.3), it suffices to deal with problem (2.1)–(2.3).

Set

Ψ = { φ ∈ C ∞ ⁢ ( Ω ) : φ ⁢ ( t , R ) = 0 , φ ⁢ ( 0 , r ) = φ ⁢ ( T , r ) , φ t ⁢ ( 0 , r ) = φ t ⁢ ( T , r ) }

and define

∥ φ ∥ L q ⁢ ( Ω ) q = ∬ Ω | φ ⁢ ( t , r ) | q ⁢ r n - 1 ⁢ d t ⁢ d r < ∞ for all ⁢ φ ∈ Ψ

for q≥1. It is well known that Ψ is dense in Lq⁢(Ω), and L2⁢(Ω) is a Hilbert space with the inner product

〈 u , v 〉 = ∬ Ω u ⁢ ( t , r ) ⁢ v ⁢ ( t , r ) ¯ ⁢ r n - 1 ⁢ d t ⁢ d r for all ⁢ u , v ∈ L 2 ⁢ ( Ω ) .

Definition 2.1.

A function u∈Lq⁢(Ω) is called a weak solution of problem (2.1)–(2.3) if it satisfies

∬ Ω ( u ⁢ ( φ t ⁢ t - φ r ⁢ r - n - 1 r ⁢ φ r ) - ( μ ⁢ u + f ⁢ ( t , r , u ) ) ⁢ φ ) ⁢ r n - 1 ⁢ d t ⁢ d r = 0 for all ⁢ φ ∈ Ψ .

Define the linear operator L0 by

L 0 ⁢ φ = φ t ⁢ t - φ r ⁢ r - n - 1 r ⁢ φ r for all ⁢ φ ∈ Ψ ,

and denote its extension in L2⁢(Ω) by L. It is known (see [3, 11, 22]) that L is a selfadjoint operator on L2⁢(Ω) and the spectrum of L is made of the eigenvalues

λ j ⁢ k = ( γ j R ) 2 - ( 2 ⁢ k ⁢ π T ) 2 , j ∈ ℕ + , k ∈ ℤ ,

where γj is the j-th positive zero point of Jν⁢(x) which is the Bessel function of the first kind of order ν=n-22. The corresponding eigenfunctions are

ψ j ⁢ k ⁢ ( t , r ) = 1 R ⁢ 2 T ⁢ 1 J ν + 1 ⁢ ( γ j ) ⁢ 1 r ν ⁢ J ν ⁢ ( γ j ⁢ r R ) ⁢ e 2 ⁢ k ⁢ π ⁢ i T ⁢ t , j ∈ ℕ + , k ∈ ℤ .

Furthermore, the eigenfunctions {ψj⁢k⁢(t,r):j∈ℕ+,k∈ℤ} form a complete orthonormal sequence in L2⁢(Ω).

It is well known that, in the case R=π2 and T=2⁢π, Bessel functions have the form J-1/2⁢(x)=(2⁢xπ)-1/2⁢cos⁡x for n=1 and J1/2⁢(x)=(2⁢xπ)-1/2⁢sin⁡x for n=3. Thus, we obtain the eigenvalues

λ j ⁢ k = ( 2 ⁢ j - 1 ) 2 - k 2 , j ∈ ℕ + , k ∈ ℤ ⁢ for ⁢ n = 1 ,

and

λ j ⁢ k = 4 ⁢ j 2 - k 2 , j ∈ ℕ + , k ∈ ℤ ⁢ for ⁢ n = 3 ,

which implies that, in both cases, the eigenvalues are isolated and 0 is the only eigenvalue of infinite multiplicity. Moreover, Ben-Naoum and Mawhin [4] deduced that, if R=π2, T=2⁢π and n=2, then 0 is not an eigenvalue of L and the eigenvalues of L are isolated with finite multiplicity. For the more general case that n is a positive integer and the ratio RT is a rational number, Ben-Naoum and Berkovits [3] and Schechter [22] obtained the following lemma.

Lemma 2.2 ([3, 22]).

Assume that 8⁢RT=ab, where a, b are relatively prime integers. Let βj=(4⁢j+n-3)⁢π4, τk=2⁢|k|⁢π⁢R/T for j∈N+, k∈Z, and let (4,a) denote the greatest common divisor of 4 and a.

If n - 3 is not an integer multiple of ( 4 , a ) , then L has no essential spectrum and | β j - τ k | ≥ π 4 ⁢ b for every j and k.
If n - 3 is an integer multiple of ( 4 , a ) , then the essential spectrum of L is precisely the point

λ 0 = - ( n - 3 ) ⁢ ( n - 1 ) 4 ⁢ R 2 .

Assume λ is in the spectrum of L and λ∉[2⁢π⁢λ0,λ0]. Then λ is isolated and the multiplicity of λ is finite.

Moreover, for every j, k, either βj=τk or |βj-τk|≥π4⁢b holds. When j, k satisfy βj=τk, the eigenvalues λj⁢k of L accumulate to λ0=-(n-3)⁢(n-1)/4⁢R2 as j,k→∞, while λj′⁢k′→∞ as j′,k′→∞ for j′,k′ satisfying βj′≠τk′.

Denote the spectrum of L by σ⁢(L), and assume that β∉σ⁢(L) and R, T satisfy 8⁢RT=ab for some relative prime positive integers a and b. Then, by Lemma 2.2,

β + = min { λ ∈ σ ( L ) : λ > β } and β - = max { λ ∈ σ ( L ) : λ < β }

can be well defined, where β is present in (1.5). Obviously, β-<β<β+.

Now we state the main result as follows.

Theorem 2.3.

Assume that n is an arbitrary positive integer, 8⁢RT=ab for some relative prime positive integers a and b, f satisfies (A), and μ,β∉σ⁢(L) satisfy μ∈(0,β+-β) and (μ,β)∩σ⁢(L)≠∅. Set μ0=β+-μ. If there exists 0<η<μ0 such that

0 ≤ ∂ ⁡ f ∂ ⁡ u ⁢ ( t , x , u ) ≤ μ 0 - η for all ⁢ ( t , x , u ) ∈ ℝ × 𝔹 R n × ℝ ,

then problem (2.1)–(2.3) has at least three radially symmetric periodic solutions.

In the sequel, we always assume that R, T and μ, β satisfy the conditions in Theorem 2.3, except otherwise stated. Since μ∉σ⁢(L) and μ>0, there exists a constant δ>0 such that

(2.4) | λ j ⁢ k - μ | ≥ δ > 0 , j ∈ ℕ + , k ∈ ℤ .

Noting that μ∈(0,β+-β), we have

(2.5) μ 0 = β + - μ > β .

Moreover, if λj⁢k>β, we have

(2.6) | λ j ⁢ k - μ | = λ j ⁢ k - μ ≥ μ 0 > β , j ∈ ℕ + , k ∈ ℤ .

For each u∈L2⁢(Ω), it can be expanded as a Fourier series u⁢(t,r)=∑j,kαj⁢k⁢ψj⁢k⁢(t,r) with Fourier coefficients αj⁢k. We define the working space

E = { u ∈ L 2 ⁢ ( Ω ) : ∥ u ∥ E 2 = ∑ j , k | λ j ⁢ k - μ | ⁢ | α j ⁢ k | 2 < ∞ , j ∈ ℕ + , k ∈ ℤ } ,

which is a subspace of L2⁢(Ω). Estimate (2.4) implies that ∥⋅∥E is a norm. Furthermore, E is a Hilbert space with the inner product

〈 u , v 〉 0 = ∑ j , k | λ j ⁢ k - μ | ⁢ α j ⁢ k ⁢ β ¯ j ⁢ k for all ⁢ u , v ∈ E ,

where αj⁢k and βj⁢k are the Fourier coefficients of u and v, respectively.

By (2.4), we have

(2.7) ∥ u ∥ L 2 ⁢ ( Ω ) 2 = ∑ j , k | α j ⁢ k | 2 ≤ δ - 1 ⁢ ∑ j , k | λ j ⁢ k - μ | ⁢ | α j ⁢ k | 2 = δ - 1 ⁢ ∥ u ∥ E 2 ,

which implies that E can be embedded into L2⁢(Ω). For u∈E, the combination of Hölder inequality and (2.7) yields that

(2.8) ∥ u ∥ L q ⁢ ( Ω ) ≤ C ⁢ ∥ u ∥ E , 1 ≤ q ≤ 2 ,

for some constant C depending on q.

Now, we consider the energy functional

(2.9) Φ ⁢ ( u ) = 1 2 ⁢ 〈 ( L - μ ) ⁢ u , u 〉 - ∬ Ω F ⁢ ( t , r , u ) ⁢ r n - 1 ⁢ d t ⁢ d r for all ⁢ u ∈ E ,

where F⁢(t,r,u)=∫0uf⁢(t,r,s)⁢ds. Obviously, Φ is a C1 functional on E, and

(2.10) 〈 Φ ′ ⁢ ( u ) , v 〉 = 〈 ( L - μ ) ⁢ u , v 〉 - ∬ Ω f ⁢ ( t , r , u ) ⁢ v ⁢ r n - 1 ⁢ d t ⁢ d r for all ⁢ u , v ∈ E .

Thus u is a weak solution of problem (2.1)–(2.3) if and only if Φ′⁢(u)=0. Since f is a C1 function, we also have

〈 Φ ′′ ⁢ ( u ) ⁢ w , v 〉 = 〈 ( L - μ ) ⁢ w , v 〉 - ∬ Ω ∂ ⁡ f ∂ ⁡ u ⁢ ( t , r , u ) ⁢ v ⁢ w ⁢ r n - 1 ⁢ d t ⁢ d r for all ⁢ u , v , w ∈ E .

In particular,

〈 Φ ′′ ⁢ ( u ) ⁢ v , v 〉 = 〈 ( L - μ ) ⁢ v , v 〉 - ∬ Ω ∂ ⁡ f ∂ ⁡ u ⁢ ( t , r , u ) ⁢ v 2 ⁢ r n - 1 ⁢ d t ⁢ d r for all ⁢ u , v ∈ E .

Thus, the periodic solutions of problem (2.1)–(2.3) are transformed into the critical points of the functional Φ. In what follows, we will prove the existence of multiple critical points of Φ by the saddle point reduction technique developed by Amann [1], Arcoya and Costa [2] and Castro and Lazer [5].

3 The Saddle Point Reduction

Lemma 3.1 ([1, 2, 5]).

Let H be a real Hilbert space with the norm ∥⋅∥H, let Φ∈C1⁢(H,R) and let H1, H2 and H3 be closed subsets of H such that H=H1⊕H2⊕H3. If there exists a constant γ>0 satisfying

〈 Φ ′ ⁢ ( u + w + v 1 ) - Φ ′ ⁢ ( u + w + v 2 ) , v 1 - v 2 〉 ≤ - γ ⁢ ∥ v 1 - v 2 ∥ H 2 for all ⁢ u ∈ H 2 , w ∈ H 3 , v 1 , v 2 ∈ H 1 ,

and

〈 Φ ′ ⁢ ( u + w 1 + v ) - Φ ′ ⁢ ( u + w 2 + v ) , w 1 - w 2 〉 ≥ γ ⁢ ∥ w 1 - w 2 ∥ H 2 for all ⁢ u ∈ H 2 , v ∈ H 1 , w 1 , w 2 ∈ H 3 ,

then the following assertions hold:

There exists a unique continuous mapping h : H 2 → H 1 ⊕ H 3 such that

Φ ⁢ ( u + h ⁢ ( u ) ) = max v ∈ H 1 ⁡ min w ∈ H 3 ⁡ Φ ⁢ ( u + v + w ) = min w ∈ H 3 ⁡ max v ∈ H 1 ⁡ Φ ⁢ ( u + v + w ) .
Define Φ ^ ⁢ ( u ) = Φ ⁢ ( u + h ⁢ ( u ) ) for any u ∈ H 2 . Then Φ ^ ∈ C 1 ⁢ ( H 2 , ℝ ) and

〈 Φ ^ ′ ⁢ ( u ) , v 〉 = 〈 Φ ′ ⁢ ( u + h ⁢ ( u ) ) , v 〉 for all ⁢ u , v ∈ H 2 .
If u ∈ H 2 is a critical point of Φ ^ , then u + h ⁢ ( u ) is a critical point of Φ . On the other hand, if u+v is a critical point of Φ , where u∈H2 and v∈H1⊕H3, then v=h⁢(u), and u is a critical point of Φ^.
Furthermore, if Φ satisfies the Palais–Smale condition (PS)c at the level c∈ℝ, then the functional Φ^ also satisfies the (PS)c condition.

Noting μ,β∉σ⁢(L), we can decompose E into three orthogonal subspaces

E 1 = { u ∈ E : u = ∑ λ j ⁢ k < μ α j ⁢ k ⁢ ψ j ⁢ k ⁢ ( t , r ) } ,

E 2 = { u ∈ E : u = ∑ μ < λ j ⁢ k < β α j ⁢ k ⁢ ψ j ⁢ k ⁢ ( t , r ) } ,

E 3 = { u ∈ E : u = ∑ λ j ⁢ k > β α j ⁢ k ⁢ ψ j ⁢ k ⁢ ( t , r ) } .

Thus, E=E1⊕E2⊕E3. Furthermore, since (μ,β)∩σ⁢(L)≠∅, Lemma 2.2 shows that E2≠∅ and dim⁡(E2)<∞.

For any u∈E1, it can be expanded as u=∑λj⁢k<μαj⁢k⁢ψj⁢k⁢(t,r). Then

(3.1) 〈 ( L - μ ) ⁢ u , u 〉 = - ∑ λ j ⁢ k < μ | λ j ⁢ k - μ | ⁢ | α j ⁢ k | 2 = - ∥ u ∥ E 2 .

Similarly, for any u∈E2⊕E3, we have

(3.2) 〈 ( L - μ ) ⁢ u , u 〉 = ∥ u ∥ E 2 .

Lemma 3.2.

If μ, β satisfy the conditions in Theorem 2.3, then there exist γ1>0 and γ2>0 such that

(3.3) 〈 ( L - μ - β ) ⁢ u , u 〉 ≤ - γ 1 ⁢ ∥ u ∥ E 2 ⁢ for all ⁢ u ∈ E 1 ⊕ E 2 ,

(3.4) 〈 ( L - μ - β ) ⁢ u , u 〉 ≥ γ 2 ⁢ ∥ u ∥ E 2 ⁢ for all ⁢ u ∈ E 3 .

Proof.

For u∈E1⊕E2, it can be expanded as u=∑λj⁢k<βαj⁢k⁢ψj⁢k⁢(t,r). Thus

〈 ( L - μ - β ) ⁢ u , u 〉 = ∑ λ j ⁢ k < β ( λ j ⁢ k - μ - β ) ⁢ | α j ⁢ k | 2

= ∑ λ j ⁢ k < β ( λ j ⁢ k - μ ) ⁢ | α j ⁢ k | 2 - β ⁢ ∑ λ j ⁢ k < β | α j ⁢ k | 2

≤ ∑ λ j ⁢ k < μ ( λ j ⁢ k - μ ) ⁢ | α j ⁢ k | 2 + ∑ μ < λ j ⁢ k < β ( λ j ⁢ k - μ ) ⁢ | α j ⁢ k | 2 - β ⁢ ∑ μ < λ j ⁢ k < β | α j ⁢ k | 2 .

Since μ<λj⁢k<β, by the definition of β- which is present in Section 2, we have |λj⁢k-μ|<β-. Hence

〈 ( L - μ - β ) ⁢ u , u 〉 ≤ - ∑ λ j ⁢ k < μ | λ j ⁢ k - μ | ⁢ | α j ⁢ k | 2 + ∑ μ < λ j ⁢ k < β | λ j ⁢ k - μ | ⁢ | α j ⁢ k | 2 - β β - ⁢ ∑ μ < λ j ⁢ k < β | λ j ⁢ k - μ | ⁢ | α j ⁢ k | 2

= - ∑ λ j ⁢ k < μ | λ j ⁢ k - μ | ⁢ | α j ⁢ k | 2 - ( β β - - 1 ) ⁢ ∑ μ < λ j ⁢ k < β | λ j ⁢ k - μ | ⁢ | α j ⁢ k | 2

≤ - γ 1 ⁢ ∥ u ∥ E 2 ,

where γ1=min⁡{1,ββ--1}>0 because of β-<β.

On the other hand, for u∈E3, we rewrite u=∑λj⁢k>βαj⁢k⁢ψj⁢k⁢(t,r). By (2.6), we obtain

〈 ( L - μ - β ) ⁢ u , u 〉 = ∑ λ j ⁢ k > β | λ j ⁢ k - μ | ⁢ | α j ⁢ k | 2 - β ⁢ ∑ λ j ⁢ k > β | α j ⁢ k | 2

≥ ∑ λ j ⁢ k > β | λ j ⁢ k - μ | ⁢ | α j ⁢ k | 2 - β μ 0 ⁢ ∑ λ j ⁢ k > β | λ j ⁢ k - μ | ⁢ | α j ⁢ k | 2

= ( 1 - β μ 0 ) ⁢ ∑ λ j ⁢ k > β | λ j ⁢ k - μ | ⁢ | α j ⁢ k | 2 .

Set γ2=1-βμ0>0 provided by (2.5). Then

〈 ( L - μ - β ) ⁢ u , u 〉 ≥ γ 2 ⁢ ∥ u ∥ E 2 .

We arrive at the results. ∎

Now, we prove the following lemma which shows that the functional Φ defined in (2.9) satisfies the conditions in Lemma 3.1.

Lemma 3.3.

If the assumptions in Theorem 2.3 hold, then there exists a constant γ>0 such that

〈 Φ ′ ⁢ ( u + v ) - Φ ′ ⁢ ( u + w ) , v - w 〉 ≤ - γ ⁢ ∥ v - w ∥ E 2 for all ⁢ u ∈ E 2 ⊕ E 3 , v , w ∈ E 1 ,

and

〈 Φ ′ ⁢ ( u + v ) - Φ ′ ⁢ ( u + w ) , v - w 〉 ≥ γ ⁢ ∥ v - w ∥ E 2 for all ⁢ u ∈ E 1 ⊕ E 2 , v , w ∈ E 3 .

Proof.

For every u,v,w∈E, we have

(3.5) 〈 Φ ′ ⁢ ( u + v ) - Φ ′ ⁢ ( u + w ) , v - w 〉 = ∫ 0 1 〈 Φ ′′ ⁢ ( u + w + s ⁢ ( v - w ) ) ⁢ ( v - w ) , v - w 〉 ⁢ d s

and

(3.6) 〈 Φ ′′ ⁢ ( u + w + s ⁢ ( v - w ) ) ⁢ ( v - w ) , v - w 〉 = 〈 ( L - μ ) ⁢ ( v - w ) , v - w 〉 - ∬ Ω ( v - w ) 2 ⁢ ∂ ⁡ f ∂ ⁡ u ⁢ r n - 1 ⁢ d t ⁢ d r .

On one hand, for the case v,w∈E1, u∈E2⊕E3, by the assumption ∂⁡f∂⁡u≥0 and (3.1), a direct calculation yields

〈 Φ ′ ⁢ ( u + v ) - Φ ′ ⁢ ( u + w ) , v - w 〉 ≤ - ∥ v - w ∥ E 2 .

On the other hand, for the case v,w∈E3, u∈E1⊕E2, by (3.2) we have

(3.7) 〈 ( L - μ ) ⁢ ( v - w ) , v - w 〉 = ∥ v - w ∥ E 2 .

Moreover, by the assumption 0≤∂⁡f∂⁡u⁢(t,x,u)≤μ0-η and (2.6), we have

(3.8) ∬ Ω ( v - w ) 2 ⁢ ∂ ⁡ f ∂ ⁡ u ⁢ r n - 1 ⁢ d t ⁢ d r ≤ ( μ 0 - η ) ⁢ ∥ v - w ∥ L 2 ⁢ ( Ω ) 2 ≤ ( μ 0 - η ) μ 0 ⁢ ∥ v - w ∥ E 2 = ( 1 - η μ 0 ) ⁢ ∥ v - w ∥ E 2 .

Therefore, from (3.5)–(3.8) we have

〈 Φ ′ ⁢ ( u + v ) - Φ ′ ⁢ ( u + w ) , v - w 〉 ≥ η μ 0 ⁢ ∥ v - w ∥ E 2 .

By choosing γ=min⁡{1,ημ0}, we obtain the desired results. ∎

By Lemma 3.3 and Lemma 3.1, for the functional Φ defined in (2.9) there exists a unique continuous mapping h:E2→E1⊕E3 such that

(3.9) Φ ^ ⁢ ( u ) = Φ ⁢ ( u + h ⁢ ( u ) ) = max v ∈ E 1 ⁡ min w ∈ E 3 ⁡ Φ ⁢ ( u + v + w ) = min w ∈ E 3 ⁡ max v ∈ E 1 ⁡ Φ ⁢ ( u + v + w ) .

Moreover, the problem of finding the critical points of the functional Φ on the infinite-dimensional space E is reduced to the problem of finding the critical points of the reduction functional Φ^ on the finite-dimensional subspace E2. In what follows, we shall apply variational methods (including the mountain pass lemma, see [7]) to obtain critical points of the reduction functional Φ^.

4 Verification of the (PS)c Condition

In order to acquire the critical points of Φ^ by variational methods, with the aid of Lemma 3.1, it suffices to verify that Φ satisfies the (PS)c condition for any c∈ℝ. That means, any sequence {ui}⊂E satisfying Φ⁢(ui)→c and Φ′⁢(ui)→0 as i→∞ contains a convergent subsequence.

Lemma 4.1.

Assume that the assumptions in Theorem 2.3 hold, and that {ui}⊂E satisfies Φ⁢(ui)→c and Φ′⁢(ui)→0 as i→∞. Then there exists a constant C~>0 independent of i such that ∥ui∥E≤C~.

Proof.

We write ui=ui++ui- with ui+∈E3 and ui-∈E1⊕E2, i=1,2,…. Firstly, for ui+∈E3, by (2.10) and Φ′⁢(ui)→0 as i→∞, we have

o ⁢ ( 1 ) ⁢ ∥ u i + ∥ E ≥ 〈 Φ ′ ⁢ ( u i ) , u i + 〉

= 〈 ( L - μ ) ⁢ u i + , u i + 〉 - ∬ Ω f ⁢ ( t , r , u i ) ⁢ u i + ⁢ r n - 1 ⁢ d t ⁢ d r

(4.1) = 〈 ( L - μ - β ) ⁢ u i + , u i + 〉 - ∬ Ω ( f ⁢ ( t , r , u i ) - β ⁢ u i ) ⁢ u i + ⁢ r n - 1 ⁢ d t ⁢ d r .

In virtue of (3.4), we have

(4.2) 〈 ( L - μ - β ) ⁢ u i + , u i + 〉 ≥ γ 2 ⁢ ∥ u i + ∥ E 2 .

In addition, by (1.5), for any ε>0 small enough, there exists a constant C=C⁢(ε)>0 such that

(4.3) | f ⁢ ( t , x , u i ) - β ⁢ u i | < ε ⁢ | u i | + C .

Thus, from (2.7), (2.8) and (4.3) we have

| ∬ Ω ( f ⁢ ( t , r , u i ) - β ⁢ u i ) ⁢ u i + ⁢ r n - 1 ⁢ d t ⁢ d r | ≤ ε ⁢ ∥ u i + ∥ L 2 ⁢ ( Ω ) ⁢ ∥ u i ∥ L 2 ⁢ ( Ω ) + C ⁢ ∥ u i + ∥ L 1 ⁢ ( Ω )

(4.4) ≤ ε 2 ⁢ δ ⁢ ∥ u i + ∥ E 2 + ε 2 ⁢ δ ⁢ ∥ u i ∥ E 2 + C ⁢ ∥ u i + ∥ E

for some constant C independent of i. Therefore, by (4.1), (4.2) and (4.4), we have

(4.5) ( γ 2 - ε 2 ⁢ δ ) ⁢ ∥ u i + ∥ E 2 - ε 2 ⁢ δ ⁢ ∥ u i ∥ E 2 - C ⁢ ∥ u i + ∥ E ≤ 0 .

Secondly, for ui-∈E1⊕E2, we have

o ⁢ ( 1 ) ⁢ ∥ u i - ∥ E ≥ 〈 - Φ ′ ⁢ ( u i ) , u i - 〉 = - 〈 ( L - μ - β ) ⁢ u i - , u i - 〉 + ∬ Ω ( f ⁢ ( t , r , u i ) - β ⁢ u i ) ⁢ u i - ⁢ r n - 1 ⁢ d t ⁢ d r .

A calculation similar to the one in (4.4) yields

(4.6) | ∬ Ω ( f ⁢ ( t , r , u i ) - β ⁢ u i ) ⁢ u i - ⁢ r n - 1 ⁢ d t ⁢ d r | ≤ ε 2 ⁢ δ ⁢ ∥ u i - ∥ E 2 + ε 2 ⁢ δ ⁢ ∥ u i ∥ E 2 + C ⁢ ∥ u i - ∥ E .

Then, in virtue of (3.3) and (4.6), we have

(4.7) ( γ 1 - ε 2 ⁢ δ ) ⁢ ∥ u i - ∥ E 2 - ε 2 ⁢ δ ⁢ ∥ u i ∥ E 2 - C ⁢ ∥ u i - ∥ E ≤ 0 .

Finally, noting that E3 and E1⊕E2 are orthogonal subspaces of E, we have ∥ui∥E2=∥ui+∥E2+∥ui-∥E2. Obviously, ∥ui+∥E+∥ui-∥E≤2⁢∥ui∥E. Set γ0=min⁡{γ1,γ2}. Then the sum of (4.5) and (4.7) yields

(4.8) ( γ 0 - 3 ⁢ ε 2 ⁢ δ ) ⁢ ∥ u i ∥ E 2 - C ⁢ ∥ u i ∥ E ≤ 0 .

Now we choose ε∈(0,2⁢δ⁢γ03). Then (4.8) shows that there exists a constant C~>0 independent of i such that ∥ui∥E≤C~. The proof is completed. ∎

Now, we rewrite E=E1⊕E1⊥ for simplicity, where

E 1 ⊥ = E 2 ⊕ E 3 = { u ∈ E : u = ∑ λ j ⁢ k > μ α j ⁢ k ⁢ ψ j ⁢ k ⁢ ( t , r ) } .

Let E0 be the subspace of those u∈L2⁢(Ω) for which αj⁢k=0 if βj≠τk. That is to say

E 0 = { u ∈ L 2 ⁢ ( Ω ) : u = ∑ j , k α j ⁢ k ⁢ ψ j ⁢ k ⁢ ( t , r ) ⁢ for ⁢ β j = τ k } .

Remark 4.2.

Under the assumptions of Theorem 2.3, Lemma 2.2 shows that if n-3 is not an integer multiple of (4,a), then E0={0}; if n-3 is an integer multiple of (4,a), then E0 is an infinite-dimensional space spanned by the eigenfunctions ψj⁢k and the corresponding eigenvalues accumulate to λ0. Therefore, we have dim⁡(E1⊥∩E0)<∞ for both cases E0={0} and dim⁡(E0)=∞. Moreover, we have dim⁡(E1∩E0)=∞ for dim⁡(E0)=∞.

Proposition 4.3 ([9]).

For all q∈(1,2], the embedding

E ⊖ E 0 ↪ L q ⁢ ( Ω )

is compact.

Since E is a Hilbert space, from Lemma 4.1, without loss of generality, we have

u i ⇀ u in ⁢ E ⁢ as ⁢ i → ∞ ,

where {ui}⊂E satisfies Φ⁢(ui)→c and Φ′⁢(ui)→0 as i→∞. The following lemma shows that we can extract a subsequence of {ui} which converges strongly to some u∈E.

Lemma 4.4.

If the assumptions of Theorem 2.3 hold, then for any c∈R the functional Φ satisfies the (PS)c condition.

Proof.

For any c∈ℝ, suppose that {ui}⊂E satisfies Φ⁢(ui)→c and Φ′⁢(ui)→0 as i→∞. Decompose ui=vi+yi+wi+zi and u=v+y+w+z, where v, y, w and z are the weak limits of {vi}, {yi}, {wi} and {zi}, respectively, and vi,v∈E1⊥⊖E0, yi,y∈E1⊥∩E0, wi,w∈E1⊖E0 and zi,z∈E1∩E0.

(i) For vi,v∈E1⊥⊖E0, we have

∥ v i - v ∥ E 2 = 〈 ( L - μ ) ⁢ ( v i - v ) , v i - v 〉 = 〈 ( L - μ ) ⁢ v i , v i - v 〉 - 〈 ( L - μ ) ⁢ v , v i - v 〉 .

In virtue of vi⇀v in E and (2.7), it is easy to see vi⇀v in L2⁢(Ω). Thus,

〈 ( L - μ ) ⁢ v , v i - v 〉 → 0 as ⁢ i → ∞ .

Consequently,

∥ v i - v ∥ E 2 ≤ 〈 ( L - μ ) ⁢ v i , v i - v 〉 + o ⁢ ( 1 ) .

In what follows, we shall prove 〈(L-μ)⁢vi,vi-v〉→0 as i→∞. By noting that vi,v∈E1⊥⊖E0 and ui=vi+yi+wi+zi, it is easy to see ui-vi∈(E1⊥⊖E0)⊥. Thus, we have

〈 ( L - μ ) ⁢ ( u i - v i ) , v i - v 〉 = 0 .

Consequently, by (2.10), we have

(4.9) 〈 ( L - μ ) ⁢ v i , v i - v 〉 = 〈 ( L - μ ) ⁢ u i , v i - v 〉 = 〈 Φ ′ ⁢ ( u i ) , v i - v 〉 + ∬ Ω f ⁢ ( t , r , u i ) ⁢ ( v i - v ) ⁢ r n - 1 ⁢ d t ⁢ d r .

Since Φ′⁢(ui)→0 as i→∞, we have

(4.10) 〈 Φ ′ ⁢ ( u i ) , v i - v 〉 → 0 as ⁢ i → ∞ .

In virtue of (4.3), a direct calculation yields

| ∬ Ω f ⁢ ( t , r , u i ) ⁢ ( v i - v ) ⁢ r n - 1 ⁢ d t ⁢ d r | ≤ ( β + ε ) ⁢ ∥ u i ∥ L 2 ⁢ ( Ω ) ⁢ ∥ v i - v ∥ L 2 ⁢ ( Ω ) + C ⁢ ∥ v i - v ∥ L 1 ⁢ ( Ω ) .

According to Proposition 4.3 and vi⇀v in E, we obtain that there exists a subsequence of {vi} which converges strongly to v in L2⁢(Ω). For the sake of convenience, we still use {vi} to denote the subsequence. Furthermore, the continuous embedding L2⁢(Ω)↪L1⁢(Ω) shows vi→v in L1⁢(Ω). Therefore,

(4.11) | ∬ Ω f ⁢ ( t , r , u i ) ⁢ ( v i - v ) ⁢ r n - 1 ⁢ d t ⁢ d r | → 0 as ⁢ i → ∞ .

Inserting (4.10) and (4.11) into (4.9), we have

〈 ( L - μ ) ⁢ v i , v i - v 〉 → 0 as ⁢ i → ∞ .

Consequently,

(4.12) ∥ v i - v ∥ E → 0 as ⁢ i → ∞ .

(ii) For yi,y∈E1⊥∩E0, from Remark 4.2 we have dim⁡(E1⊥∩E0)<∞. Thus, the weak convergence of {yi} implies

(4.13) ∥ y i - y ∥ E → 0 as ⁢ i → ∞ .

(iii) For wi,w∈E1⊖E0, a calculation similar to the one for {vi} yields

(4.14) ∥ w i - w ∥ E 2 = - 〈 ( L - μ ) ⁢ ( w i - w ) , w i - w 〉 → 0 as ⁢ i → ∞ .

(iv) If E0={0}, we have ui=vi+wi. From (4.12) and (4.14) we arrive at the conclusion.

If E0≠{0}, it remains to prove that there exists a subsequence of {zi} which converges strongly to z in E.

Since Φ′⁢(ui)→0 and zi⇀z in E, we have

∥ z i - z ∥ E 2 = - 〈 ( L - μ ) ⁢ ( z i - z ) , z i - z 〉

= - 〈 Φ ′ ⁢ ( u i ) , z i - z 〉 - ∬ Ω f ⁢ ( t , r , u i ) ⁢ ( z i - z ) ⁢ r n - 1 ⁢ d t ⁢ d r + 〈 ( L - μ ) ⁢ z , z i - z 〉

(4.15) ≤ - ∬ Ω f ⁢ ( t , r , u i ) ⁢ ( z i - z ) ⁢ r n - 1 ⁢ d t ⁢ d r + o ⁢ ( 1 ) .

Set f⁢(ui)=f⁢(t,r,ui) for convenience. By the definition of the inner product in L2⁢(Ω), we have

∬ Ω f ⁢ ( t , r , u i ) ⁢ ( z i - z ) ⁢ r n - 1 ⁢ d t ⁢ d r = 〈 f ⁢ ( u i ) , z i - z 〉

(4.16) = 〈 f ⁢ ( u i ) - f ⁢ ( u ~ i + z ) , z i - z 〉 + 〈 f ⁢ ( u ~ i + z ) - f ⁢ ( u ) , z i - z 〉 + 〈 f ⁢ ( u ) , z i - z 〉 ,

where u~i=vi+yi+wi.

On one hand, noting that f is nondecreasing in u, we obtain

(4.17) 〈 f ⁢ ( u i ) - f ⁢ ( u ~ i + z ) , z i - z 〉 ≥ 0 .

Therefore, from (4.15)–(4.17) we have

(4.18) ∥ z i - z ∥ E 2 ≤ - 〈 f ⁢ ( u ~ i + z ) - f ⁢ ( u ) , z i - z 〉 - 〈 f ⁢ ( u ) , z i - z 〉 + o ⁢ ( 1 ) .

On the other hand, by (4.3), we have |f⁢(t,x,u)|<(ε+β)⁢|u|+C. Thus f:u↦f⁢(t,r,u) is a continuous mapping from L2⁢(Ω) into itself. By (2.7) and (4.12)–(4.14), we have u~i→u~ in L2⁢(Ω), where u~=x+y+w. Thus,

(4.19) 〈 f ⁢ ( u ~ i + z ) - f ⁢ ( u ) , z i - z 〉 → 0 as ⁢ i → ∞ .

Since zi⇀z in L2⁢(Ω), we have

(4.20) 〈 f ⁢ ( u ) , z i - z 〉 → 0 as ⁢ i → ∞ .

Thus, from (4.18)–(4.20), we have

∥ z i - z ∥ E → 0 as ⁢ i → ∞ .

The proof is completed. ∎

5 Bounds of the Reduction Functional

Assertion (i) of the following lemma focuses on the upper bound of the reduction functional Φ^. We apply it to acquire one critical point on E2. Assertion (ii) shows that if ∥u∥E is sufficiently large, then the value of Φ^ is no more than 0. It will be used to obtain the critical point of mountain pass type later.

Lemma 5.1.

If the assumptions of Theorem 2.3 hold, then we have the following assertions:

There exists a constant M > 0 such that Φ ^ ⁢ ( u ) < M for all u ∈ E 2 .
For u ∈ E 2 , there exists a constant R 0 > 0 such that Φ ^ ⁢ ( u ) ≤ 0 for ∥ u ∥ E ≥ R 0 .

Proof.

For u∈E2, from (3.9) we have

Φ ^ ⁢ ( u ) = min w ∈ E 3 ⁡ max v ∈ E 1 ⁡ Φ ⁢ ( u + v + w ) ≤ max v ∈ E 1 ⁡ Φ ⁢ ( u + v ) .

This estimate shows that, in order to prove assertion (i), it suffices to prove that Φ⁢(u+v) is bounded from above for any u∈E2 and v∈E1.

For any u∈E2 and v∈E1, in virtue of (2.9), we have

(5.1) Φ ⁢ ( u + v ) = 1 2 ⁢ 〈 ( L - μ - β ) ⁢ ( u + v ) , u + v 〉 - ∬ Ω ( F ⁢ ( t , r , u + v ) - β 2 ⁢ ( u + v ) 2 ) ⁢ r n - 1 ⁢ d t ⁢ d r .

From (4.3) we have

(5.2) | F ⁢ ( t , r , u + v ) - β 2 ⁢ ( u + v ) 2 | ≤ ε ⁢ | u + v | 2 + C ⁢ | u + v | .

By (3.3), (5.1) and (5.2), a simple calculation yields

Φ ⁢ ( u + v ) ≤ - γ 1 2 ⁢ ∥ u + v ∥ E 2 + ∬ Ω ( ε ⁢ | u + v | 2 + C ⁢ | u + v | ) ⁢ r n - 1 ⁢ d t ⁢ d r

≤ - γ 1 2 ⁢ ∥ u + v ∥ E 2 + ε ⁢ ∥ u + v ∥ L 2 ⁢ ( Ω ) 2 + C ⁢ ∥ u + v ∥ L 1 ⁢ ( Ω ) .

Thus, in virtue of (2.7), (2.8) and taking ε=δ⁢γ1/4, we have

(5.3) Φ ⁢ ( u + v ) ≤ - γ 1 4 ⁢ ∥ u + v ∥ E 2 + C ⁢ ∥ u + v ∥ E .

Therefore, estimate (5.3) implies that there exists M>0 such that Φ⁢(u+v)≤M for any u∈E2 and v∈E1. Assertion (i) is established.

Noting that ∥u+v∥E2=∥u∥E2+∥v∥E2 and ∥u+v∥E≤∥u∥E+∥v∥E, from (5.3) we have

Φ ⁢ ( u + v ) ≤ - γ 1 4 ⁢ ∥ u ∥ E 2 + C ⁢ ∥ u ∥ E + ( - γ 1 4 ⁢ ∥ v ∥ E 2 + C ⁢ ∥ v ∥ E ) ≤ - γ 1 4 ⁢ ∥ u ∥ E 2 + C ⁢ ∥ u ∥ E + C 0 ,

where C0=maxs≥0⁡{-γ14⁢s2+C⁢s}. Thus, there exists a constant R0>0 such that Φ⁢(u+v)≤0 for ∥u∥E≥R0. We arrive at assertion (ii). ∎

With the help of the following lemma, we can obtain another critical point on some open ball in E2.

Lemma 5.2.

If the assumptions of Theorem 2.3 hold, then for any R~>0 there exists a constant b~ depending on R~ such that Φ^⁢(u)≥b~ for any u∈BR~={u∈E2:∥u∥E<R~}.

Proof.

For u∈E2, from (3.9) we have

Φ ^ ⁢ ( u ) = max v ∈ E 1 ⁡ min w ∈ E 3 ⁡ Φ ⁢ ( u + v + w ) ≥ min w ∈ E 3 ⁡ Φ ⁢ ( u + w ) ,

where

(5.4) Φ ⁢ ( u + w ) = 1 2 ⁢ 〈 ( L - μ - β ) ⁢ ( u + w ) , u + w 〉 - ∬ Ω ( F ⁢ ( t , r , u + w ) - β 2 ⁢ ( u + w ) 2 ) ⁢ r n - 1 ⁢ d t ⁢ d r .

Since E2 and E3 are orthogonal subspaces of E, we have

〈 ( L - μ - β ) ⁢ ( u + w ) , u + w 〉 = 〈 ( L - μ - β ) ⁢ u , u 〉 + 〈 ( L - μ - β ) ⁢ w , w 〉 .

By (2.7) and (3.2), we have

| 〈 ( L - μ - β ) ⁢ u , u 〉 | = | 〈 ( L - μ ) ⁢ u , u 〉 - β ⁢ 〈 u , u 〉 | ≤ ∥ u ∥ E 2 + β ⁢ ∥ u ∥ L 2 ⁢ ( Ω ) 2 ≤ ( 1 + β δ ) ⁢ ∥ u ∥ E 2 = C 1 ⁢ ∥ u ∥ E 2 ,

where C1=1+βδ.

In view of w∈E3 and (3.4), we have

〈 ( L - μ - β ) ⁢ w , w 〉 ≥ γ 2 ⁢ ∥ w ∥ E 2 .

Therefore,

(5.5) 〈 ( L - μ - β ) ⁢ ( u + w ) , u + w 〉 ≥ γ 2 ⁢ ∥ w ∥ E 2 - C 1 ⁢ ∥ u ∥ E 2 .

A calculation similar to the one in (5.2) yields

(5.6) | F ⁢ ( t , r , u + w ) - β 2 ⁢ ( u + w ) 2 | ≤ ε ⁢ | u + w | 2 + C ⁢ | u + w | .

By substituting (5.5) and (5.6) into (5.4) and taking ε=δ⁢γ2/4, a direct calculation yields

Φ ⁢ ( u + w ) ≥ γ 2 2 ⁢ ∥ w ∥ E 2 - C 1 2 ⁢ ∥ u ∥ E 2 - δ ⁢ γ 2 4 ⁢ ∥ u + w ∥ L 2 ⁢ ( Ω ) 2 - C ⁢ ∥ u + w ∥ L 1 ⁢ ( Ω )

≥ γ 2 2 ⁢ ∥ w ∥ E 2 - C 1 2 ⁢ ∥ u ∥ E 2 - γ 2 4 ⁢ ( ∥ u ∥ E 2 + ∥ w ∥ E 2 ) - C ⁢ ( ∥ u ∥ E + ∥ w ∥ E )

≥ - ( C 1 2 + γ 2 4 ) ⁢ ∥ u ∥ E 2 - C ⁢ ∥ u ∥ E + C 2 ,

where C2=mins≥0⁡{γ24⁢s2-C⁢s}.

For any R~>0, set b~=-(C12+γ24)⁢R~2-C⁢R~+C2. Then Φ^⁢(u)≥b~ for ∥u∥E<R~. We complete the proof. ∎

The following lemma, combined with assertion (ii) in Lemma 5.1, helps us to acquire one critical point of mountain pass type on E2 which is different from the previous two.

Lemma 5.3.

If the assumptions of Theorem 2.3 hold, then there exist two constants τ>0 and r¯>0 such that Φ^⁢(u)≥τ for any u∈E2 with ∥u∥E=r¯.

Proof.

For any u∈E2 and w∈E3, we have

(5.7) Φ ⁢ ( u + w ) = 1 2 ⁢ 〈 ( L - μ ) ⁢ ( u + w ) , u + w 〉 - ∬ Ω F ⁢ ( t , r , u + w ) ⁢ r n - 1 ⁢ d t ⁢ d r .

Since E2 and E3 are orthogonal subspaces of E, we have

(5.8) 〈 ( L - μ ) ⁢ ( u + w ) , u + w 〉 = ∥ u ∥ E 2 + ∥ w ∥ E 2 .

Set f⁢(ξ)=f⁢(t,r,ξ) and F⁢(ξ)=F⁢(t,r,ξ) for convenience. It is easy to see that

∫ 0 1 ∫ 0 1 s ⁢ ∂ ⁡ f ∂ ⁡ ξ ⁢ ( u + s ⁢ θ ⁢ w ) ⁢ w 2 ⁢ d θ ⁢ d s = ∫ 0 1 w ⁢ f ⁢ ( u + s ⁢ w ) ⁢ d s - f ⁢ ( u ) ⁢ w

= ∫ 0 u + w f ⁢ ( s ) ⁢ d s - ∫ 0 u f ⁢ ( s ) ⁢ d s - f ⁢ ( u ) ⁢ w

= F ⁢ ( u + w ) - F ⁢ ( u ) - f ⁢ ( u ) ⁢ w .

Thus,

F ⁢ ( u + w ) = ∫ 0 1 ∫ 0 1 s ⁢ ∂ ⁡ f ∂ ⁡ ξ ⁢ ( u + s ⁢ θ ⁢ w ) ⁢ w 2 ⁢ d θ ⁢ d s + f ⁢ ( u ) ⁢ w + F ⁢ ( u ) .

In what follows, we estimate ∬ΩF⁢(u+w)⁢rn-1⁢dt⁢dr by dividing it into three terms:

∬ Ω ( ∫ 0 1 ∫ 0 1 s ⁢ ∂ ⁡ f ∂ ⁡ ξ ⁢ ( u + s ⁢ θ ⁢ w ) ⁢ w 2 ⁢ d θ ⁢ d s ) ⁢ r n - 1 ⁢ d t ⁢ d r , ∬ Ω f ⁢ ( u ) ⁢ w ⁢ r n - 1 ⁢ d t ⁢ d r , ∬ Ω F ⁢ ( u ) ⁢ r n - 1 ⁢ d t ⁢ d r .

(i) Since 0≤∂⁡f∂⁡u≤μ0-η, we have

∫ 0 1 ∫ 0 1 s ⁢ ∂ ⁡ f ∂ ⁡ ξ ⁢ ( u + s ⁢ θ ⁢ w ) ⁢ w 2 ⁢ d θ ⁢ d s ≤ 1 2 ⁢ w 2 ⁢ ( μ 0 - η ) .

Thus,

∬ Ω ( ∫ 0 1 ∫ 0 1 s ⁢ ∂ ⁡ f ∂ ⁡ ξ ⁢ ( u + s ⁢ θ ⁢ w ) ⁢ w 2 ⁢ d θ ⁢ d s ) ⁢ r n - 1 ⁢ d t ⁢ d r ≤ μ 0 - η 2 ⁢ ∥ w ∥ L 2 ⁢ ( Ω ) 2 .

Recalling w∈E3, with the help of (2.6) we obtain

∥ w ∥ L 2 ⁢ ( Ω ) 2 ≤ 1 μ 0 ⁢ ∥ w ∥ E 2 .

Therefore,

(5.9) ∬ Ω ( ∫ 0 1 ∫ 0 1 s ⁢ ∂ ⁡ f ∂ ⁡ ξ ⁢ ( u + s ⁢ θ ⁢ w ) ⁢ w 2 ⁢ d θ ⁢ d s ) ⁢ r n - 1 ⁢ d t ⁢ d r ≤ μ 0 - η 2 ⁢ μ 0 ⁢ ∥ w ∥ E 2 .

(ii) For fixed p>1, assumptions (1.4) and (1.5) imply that, for ε>0 small enough which will be chosen later, there exists a constant C=C⁢(ε)>0 such that

(5.10) | f ⁢ ( u ) | ≤ ε ⁢ | u | + C ⁢ | u | p for all ⁢ ( t , r , u ) ∈ Ω × ℝ .

Thus, a direct calculation yields

| ∬ Ω f ⁢ ( u ) ⁢ w ⁢ r n - 1 ⁢ d t ⁢ d r | ≤ ε ⁢ ∥ u ∥ L 2 ⁢ ( Ω ) ⁢ ∥ w ∥ L 2 ⁢ ( Ω ) + C ⁢ ∥ u ∥ L 2 ⁢ p ⁢ ( Ω ) p ⁢ ∥ w ∥ L 2 ⁢ ( Ω )

≤ ε 2 ⁢ ∥ u ∥ L 2 ⁢ ( Ω ) 2 + C ⁢ ∥ u ∥ L 2 ⁢ p ⁢ ( Ω ) 2 ⁢ p + ε ⁢ ∥ w ∥ L 2 ⁢ ( Ω ) 2 ,

where the last inequality is acquired by the Cauchy inequality with ε. Since dim⁡(E2)<∞, there exists a constant C>0 such that ∥u∥L2⁢p⁢(Ω)2⁢p≤C⁢∥u∥E2⁢p. Thus,

(5.11) | ∬ Ω f ⁢ ( u ) ⁢ w ⁢ r n - 1 ⁢ d t ⁢ d r | ≤ ε 2 ⁢ δ ⁢ ∥ u ∥ E 2 + C ⁢ ∥ u ∥ E 2 ⁢ p + ε μ 0 ⁢ ∥ w ∥ E 2 .

(iii) From (5.10) we have |F⁢(u)|≤ε2⁢|u|2+C⁢|u|p+1. Since dim⁡(E2)<∞, there exists a constant C>0 such that ∥u∥Lp+1⁢(Ω)p+1≤C⁢∥u∥Ep+1. Therefore,

(5.12) | ∬ Ω F ⁢ ( u ) ⁢ r n - 1 ⁢ d t ⁢ d r | ≤ ε 2 ⁢ ∥ u ∥ L 2 ⁢ ( Ω ) 2 + C ⁢ ∥ u ∥ L p + 1 ⁢ ( Ω ) p + 1 ≤ ε 2 ⁢ δ ⁢ ∥ u ∥ E 2 + C ⁢ ∥ u ∥ E p + 1 .

Consequently, the sum of (5.9), (5.11) and (5.12) yields

(5.13) | ∬ Ω F ⁢ ( u + w ) ⁢ r n - 1 ⁢ d t ⁢ d r | ≤ ε δ ⁢ ∥ u ∥ E 2 + C ⁢ ∥ u ∥ E p + 1 + C ⁢ ∥ u ∥ E 2 ⁢ p + μ 0 - η + 2 ⁢ ε 2 ⁢ μ 0 ⁢ ∥ w ∥ E 2 .

Finally, substituting (5.8) and (5.13) into (5.7), we have

Φ ⁢ ( u + w ) ≥ ( 1 2 - ε δ ) ⁢ ∥ u ∥ E 2 - C ⁢ ∥ u ∥ E p + 1 - C ⁢ ∥ u ∥ E 2 ⁢ p + ( 1 2 - μ 0 - η + 2 ⁢ ε 2 ⁢ μ 0 ) ⁢ ∥ w ∥ E 2

(5.14) = ( 1 2 - ε δ ) ⁢ ∥ u ∥ E 2 - C ⁢ ∥ u ∥ E p + 1 - C ⁢ ∥ u ∥ E 2 ⁢ p + ( η - 2 ⁢ ε 2 ⁢ μ 0 ) ⁢ ∥ w ∥ E 2 .

Taking ε=min⁡{δ4,η4} in (5.14), we have

Φ ⁢ ( u + w ) ≥ 1 4 ⁢ ∥ u ∥ E 2 - C ⁢ ∥ u ∥ E p + 1 - C ⁢ ∥ u ∥ E 2 ⁢ p .

Now we consider the function

ϕ ⁢ ( s ) = 1 4 ⁢ s 2 - C ⁢ s p + 1 - C ⁢ s 2 ⁢ p , s ≥ 0 .

Since p>1, it is easy to see that ϕ attains a local minimum at s=0. Therefore, there exist two constants r¯>0 and τ>0 such that ϕ⁢(r¯)≥τ. Recall that Φ^⁢(u)≥minw∈E3⁡Φ⁢(u+w), which leads to Φ^⁢(u)≥τ for u∈E2 with ∥u∥E=r¯. We complete the proof. ∎

6 Proof of Theorem 2.3

In this section, let r¯ and τ be the constants in Lemma 5.3 and let Br¯={u∈E2:∥u∥E<r¯}.

Proof.

Firstly, we assert that there exist a local minimum point and a global maximum point in E2.

On one hand, the assumptions on f in Theorem 2.3 and (1.4) imply that F⁢(t,r,ξ)≥0 for any ξ∈ℝ. For v∈E1, by (3.1), we have

Φ ⁢ ( v ) = 1 2 ⁢ 〈 ( L - μ ) ⁢ v , v 〉 - ∬ Ω F ⁢ ( t , r , v ) ⁢ r n - 1 ⁢ d t ⁢ d r ≤ 0 .

Therefore,

(6.1) Φ ^ ⁢ ( 0 ) = min w ∈ E 3 ⁡ max v ∈ E 1 ⁡ Φ ⁢ ( v + w ) ≤ max v ∈ E 1 ⁡ Φ ⁢ ( v ) ≤ 0 .

With the aid of Lemma 5.2, Lemma 5.3 and Φ^⁢(0)≤0, by taking R~=r¯ and noting 0∈Br¯, the reduction functional Φ^ attains its infimum in Br¯. Let σ1=infu∈Br¯⁡Φ^⁢(u). By Lemma 3.1 and Lemma 4.4, we know that Φ^ satisfies the (PS)σ1 condition. Then there exists u1∈E2 such that Φ^′⁢(u1)=0 and Φ^⁢(u1)=σ1.

On the other hand, by Lemma 5.1, we have Φ^ has upper bound. Thus, set σ2=supu∈E2⁡Φ^⁢(u). By Lemma 3.1 and Lemma 4.4, Φ^ satisfies the (PS)σ2 condition. Then there exists u2∈E2 such that Φ^′⁢(u2)=0 and Φ^⁢(u2)=σ2.

Now we prove that u1 and u2 are two different critical points in E2. By (6.1), Lemma 5.3 and noting that 0∈Br¯, we have

(6.2) inf u ∈ B r ¯ ⁡ Φ ^ ⁢ ( u ) ≤ Φ ^ ⁢ ( 0 ) ≤ max v ∈ E 1 ⁡ Φ ⁢ ( v ) ≤ 0 < τ ≤ inf ∥ u ∥ = r ¯ ⁡ Φ ^ ⁢ ( u ) ≤ sup u ∈ E 2 ⁡ Φ ^ ⁢ ( u ) ,

and thus u1≠u2.

In what follows, we prove that there exists a third critical point which is distinct from u1 and u2. We divide the proof into the following two cases.

Case 1: Assume that Φ^ has another local maximum point which is different from u2. Then there exist at least three critical points of Φ^.

Case 2: Assume that u2 is the unique maximum point of Φ^. By taking u0∈E2 with ∥u0∥E=1, by Lemma 5.1 there exists R0>r¯ such that Φ^⁢(R0⁢u0)≤0. Moreover, one of the following facts holds: either s⁢R0⁢u0≠u2 or -s⁢R0⁢u0≠u2 for all s∈[0,1].

(i) If s⁢R0⁢u0≠u2 for all s∈[0,1], by Lemma 5.3 and (6.2) we have

max ⁡ { Φ ^ ⁢ ( 0 ) , Φ ^ ⁢ ( R 0 ⁢ u 0 ) } ≤ 0 < τ ≤ inf ∥ u ∥ = r ¯ ⁡ Φ ^ ⁢ ( u ) .

Let

c + = inf g ∈ Σ + ⁡ max s ∈ [ 0 , 1 ] ⁡ Φ ^ ⁢ ( g ⁢ ( s ) ) ,

where Σ+={g∈C⁢([0,1],E2):g⁢(0)=0,g⁢(1)=R0⁢u0}. Since Φ^ is a C1 functional and satisfies the (PS)c+ condition, by the mountain pass lemma we obtain that c+ is a critical value of Φ^ and satisfies c+≥τ>0. Therefore, there exists u3+∈E2 such that Φ^′⁢(u3+)=0 and Φ^⁢(u3+)=c+.

Moreover, since s⁢R0⁢u0≠u2 and g0⁢(s)=s⁢R0⁢u0∈Σ+ for s∈[0,1], we obtain

c + ≤ max s ∈ [ 0 , 1 ] ⁡ Φ ^ ⁢ ( g 0 ⁢ ( s ) ) < sup u ∈ E 2 ⁡ Φ ^ ⁢ ( u ) = σ 2 .

Thus,

σ 1 = inf u ∈ B r ¯ ⁡ Φ ^ ⁢ ( u ) ≤ 0 < τ ≤ c + ≤ max s ∈ [ 0 , 1 ] ⁡ Φ ^ ⁢ ( g 0 ⁢ ( s ) ) < sup u ∈ E 2 ⁡ Φ ^ ⁢ ( u ) = σ 2 ,

which implies that u1, u2 and u3+ are three different critical points.

(ii) If -s⁢R0⁢u0≠u2 for all s∈[0,1], similarly we have that

c - = inf g ∈ Σ - ⁡ max s ∈ [ 0 , 1 ] ⁡ Φ ^ ⁢ ( g ⁢ ( s ) )

is a critical value of Φ^, where Σ-={g∈C⁢([0,1],E2):g⁢(0)=0,g⁢(1)=-R0⁢u0}. Therefore, there exists u3-∈E2 such that Φ^′⁢(u3-)=0 and Φ^⁢(u3-)=c-. Furthermore, we have σ1<c-<σ2, which implies that the reduction functional Φ^ has three critical points.

Consequently, according to the above discussion, the reduction functional Φ^ has at least three critical points whenever either s⁢R0⁢u0≠u2 or -s⁢R0⁢u0≠u2 holds. Finally, by Lemma 3.1, it follows that the energy functional Φ has at least three critical points. We complete the proof. ∎

Communicated by Paul Rabinowitz

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11671071

Award Identifier / Grant number: 11322105

Award Identifier / Grant number: 11701077

Award Identifier / Grant number: 11871140

Funding source: Jilin University

Award Identifier / Grant number: 2017TD–18

Funding source: Department of Finance of Jilin Province

Award Identifier / Grant number: 2017C028–1

Funding statement: This work is partially supported by NSFC Grants (nos. 11671071, 11322105, 11701077 and 11871140), the Fundamental Research Funds for the Central Universities at Jilin University (no. 2017TD–18) and the Special Funds of Provincial Industrial Innovation in Jilin Province (no. 2017C028–1).

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Received: 2018-07-02

Accepted: 2018-12-02

Published Online: 2018-12-21

Published in Print: 2019-08-01

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

Articles in the same Issue

https://doi.org/10.1515/ans-2018-2036

Keywords for this article

Existence; Periodic Solutions; Wave Equation

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