# Nonconstant Periodic Solutions of Discretep-Laplacian System via Clark Duality and Computations of the Critical Groups

Authors: Bo Zheng
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study the existence of periodic solutions to a discretep-Laplacian system. By using the Clark duality method and computing the critical groups, we find a simple condition that is sufficient to ensure the existence of nonconstant periodic solutions to the system.


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## Body

## 1. Introduction

Letℕ, ℤ, and ℝ denote the respective sets of natural numbers, integers, and real numbers. For m,n∈ℤ with m≤n, write ℤ(m,n)=ℤ∩[m,n]={m,m+1,…,n}. Let T∈ℕ. We are concerned with the existence of nonconstant T-periodic solutions to the following discrete p-Laplacian system:
(1)Δ(ϕp(Δx(n-1)))+∇F(n,x(n))=0,n∈ℤ(1,T),
where Δ is the forward difference operator defined by Δx(n)=x(n+1)-x(n) and ϕp is the p-Laplace operator defined by ϕp(x)=|x|p-2x (p>1). Consider F∈C2(ℤ×ℝN,ℝ), and ∇F(n,x) denotes the gradient of F with respect to x∈ℝN. We assume that F is T-periodic in the first variable n; that is, F(n+T,·)=F(n,·).Whenp=2, (1) reduces to the second order discrete T-periodic Hamiltonian system:
(2)Δ2x(n-1)+∇F(n,x(n))=0,n∈ℤ.
In 2003, Guo and Yu [1] introduced the critical point theory (see, e.g., [2]) to the study of the existence of T-periodic solutions of (2). By using Rabinowitz's saddle point theorem, they proved existence of kT-periodic solutions, k∈ℕ, when either ∇F is bounded and F is coercive with respect to x or F satisfies a subquadratic Ambrosetti-Rabinowitz condition and a related coercivity condition. In the same year, they also proved existence of at least two nontrivial T-periodic solutions of (2) when ∇F satisfies a superlinear condition near x=0 and F satisfies a superquadratic Ambrosetti-Rabinowitz condition [3]. The growth condition of ∇F was later removed in Zhou et al. [4] by using the linking theorem. For additional studies on the existence and multiplicity of solutions to (2) subject to various boundary value conditions through the use of critical point theory, we refer the reader to [4–13].There have been tremendous efforts devoted to the study of thep-Laplacian system (1) and the systems involving the p-Laplace operator in recent years [14–20]. Many interesting results have been proved on the existence and multiplicity of solutions to (1) subject to the Dirichlet boundary condition x(0)=x(T+1)=0 by using the critical point theory [16, 17, 19, 21]. However, we have seen a very limited success in the application of this theory to the study of the existence of periodic solutions for (1). For the general case p>1, He and Chen [22] obtained a result on the existence of periodic solutions for (1) with a convex F by using Clark duality and the perturbation technique. For the case p≥2, Luo and Zhang [23] proved the existence of nonconstant periodic solutions for
(3)Δ(ϕp(Δx(n-1)))-a(n)|x(n)|p-2x(n)+∇F(n,x(n))=0,n∈ℤ,
by making use of the linking theorem.Our major goal in this paper is to prove the following theorem which gives a simple sufficient condition for the existence of nonconstantT-periodic solutions to the p-Laplacian system (1).Theorem 1.
Let1<p<2. Assume that F(n,x) is strictly convex in x for every n∈ℤ(1,T) and there exist α>0 and δ>0 such that, for all x∈ℝN and n∈ℤ(1,T), one has
(4)δ|x|qq≤F(n,x)≤α|x|qq,
where p-1+q-1=1 with
(5)2δp-1≥pqp-1αp/q.
Then (1) has at least one nonconstant T-periodic solution.Our assumptions differ from those in [22, 23] considerably and can be verified easily. For instance, let n∈ℤ(1,T), and
(6)F(n,x)=(2-sinn)(x12+x22+⋯+xN2)9/29.
Then F(n,x) is strictly convex in x for every n∈ℤ(1,T), and (4) and (5) are satisfied with p=9/8, q=9, δ=1, and α=3. For this function F, Theorem 1 confirms the existence of at least one nonconstant T-periodic solution of (1).In the remainder of this section, we outline our approach based on the Clark duality and computation of the critical groups. LetH be a real Hilbert space and f∈C1(H,ℝ). In Morse theory, the local behavior of f near an isolated critical point x0 at the level c is described by the critical groups:
(7)Cq(f,x0):=Hq(fc∩U,fc∩U∖{x0}),
where fc={x∈H:f(x)≤c}, U is a neighbourhood of x0 containing no other critical points, and H denotes singular homology. The critical groups distinguish between different types of critical points and are extremely useful for obtaining the existence and multiplicity of solutions for variational problems [24].NonzeroT-periodic solutions of (1) are the nontrivial critical points of the variational functional
(8)φ(x)=∑n=1T‍1p|Δx(n)|p-F(n,x(n))
defined on the finite dimensional space
(9)ΩT={x={x(n)}∣x(n)∈ℝN,x(n+T)=x(n),n∈ℤ};
see [9, 23] for details. However, it is difficult to compute the critical groups for the case p≠2 because there are few results on the nonlinear eigenvalue problem:
(10)Δ(ϕp(Δx(n-1)))+λϕp(x(n))=0,n∈ℤ(1,T),x(0)=x(T),x(1)=x(T+1).
As a result, there are no known eigenspaces to work with. As usual, if (10) has nonzero solutions, then we say that λ is an eigenvalue of the discrete p-Laplacian with periodic boundary condition. To overcome this difficulty, we introduce
(11)u1(n)=x(n),u2(n)=ϕp(Δx(n-1))
to transform (1) into the following equivalent first order nonautonomous system:
(12)Δu1(n)=ϕq(u2(n+1)),Δu2(n)=-∇F(n,u1(n)).
Denote
(13)u(n)=(u1(n)u2(n)),J=(0IN-IN0),Lu(n)=(u1(n)u2(n+1)).
System (12) can be rewritten in the compact form
(14)JΔu(n)+∇H(n,Lu(n))=0,
where H(n,u)=H1(n,u1)+H2(n,u2) with
(15)H1(n,u1)=F(n,u1),H2(n,u2)=|u2|qq.
And nonzero solutions of (14) correspond to nontrivial critical points of
(16)I(u)=12∑n=1T‍(JΔLu(n-1),u(n))+∑n=1T‍H(n,Lu(n))
defined on
(17)E={u={u(n)=(u1(n)u2(n))∈ℝ2N}∣xxxxu(n+T)=u(n),ui(n)∈ℝN,i=1,2,n∈ℤ(u1(n)u2(n))}.
For u,v∈E, E can be equipped with the inner product
(18)〈u,v〉=∑n=1T‍(u(n),v(n)),
by which the norm ∥·∥ can be induced by
(19)∥u∥:=∥u∥2=(∑n=1T‍|u(n)|2)1/2,
where |·| denotes the Euclidean norm in ℝ2N and (·,·) denotes the usual inner product in ℝ2N. It is easy to know that E is a finite dimensional Hilbert space which can be identified with ℝ2TN. The variational functional I can be rewritten as
(20)I(u)=12〈A1u,u〉+Q1(u),
where
(21)〈A1u,u〉=∑n=1T‍(JΔLu(n-1),u(n)),Q1(u)=∑n=1T‍H(n,Lu(n));
see [11] for details. Very fortunately, the linear eigenvalue problem
(22)JΔu(n)+λLu(n)=0,n∈ℤ(1,T),u(0)=u(T),u(1)=u(T+1)
can be worked out with eigenvalues
(23)λ=λk=2sinkπT,k∈ℤ(-T+1,T-1).
So, 0 lies in the spectrum of A1 which brings another difficulty in computing the critical groups of I at infinity (e.g., to compute the critical groups of I at infinity, the variational functional I may be required to satisfy the angle condition proposed in [25]). To conquer this difficulty, motivated by [2, 9, 22], we introduce a dual action functional J in the form
(24)J(v)=12〈Av,v〉+Q(v)
and 0 is not in the spectrum of A. Furthermore, nontrivial critical points of J correspond to nonconstant T-periodic solutions of (1). To show that the dual action functional J has at least one nontrivial critical point, firstly, we show that J satisfies the condition (C) which guarantees that the critical groups C*(J,∞) make sense. Then we compute the critical groups C*(J,∞). And finally, we show that 0 is a local minimum of J and hence the critical groups at infinity of J are different from the critical groups at zero of J which is sufficient for the existence of at least one nontrivial critical point of J and hence the existence of at least one nonconstant T-periodic solution of (1).


## 2. The Dual Action Functional and Related Lemmas

In this section, we present several technical lemmas to facilitate our proof of Theorem1 in Section 3. In order to decompose the space E appropriately, we consider the eigenvalue problem
(25)JΔu(n)+λLu(n)=0,n∈ℤ(1,T),u(0)=u(T),u(1)=u(T+1)
with λ∈ℝ. Apparently, λ=0 is an eigenvalue of (25) with the eigenfunction
(26)η0(n)=(a1,a2,…,a2N)τ,ai∈ℝ,i=1,2,…,2N,xxxxxxxxxxxxxxxxxxxxxxxxxxxn=1,2,…,T.
Through a simple calculation, we see that (25) is equivalent to
(27)Δu1(n)=λu2(n+1),u1(n+T)=u1(n),Δu2(n)=-λu1(n),u2(n+T)=u2(n).
If λ≠0, then (27) is equivalent to
(28)Δ2u1(n-1)+λ2u1(n)=0,u1(n+T)=u1(n),Δ2u2(n-1)+λ2u2(n)=0,u2(n+T)=u2(n).
It has been proved that (28) has a nontrivial solution if and only if λ2=λk2=4sin2(kπ/T) with k∈ℤ(1,T-1) [1, 3]. So in this case (25) has a nontrivial solution if and only if λ=λk=2sin(kπ/T) with k∈ℤ(-T+1,T-1)∖{0}. The multiplicity of λ0=0 is 2N and the multiplicities of λk≠0 are of the same number N. So, on the eigenvalue problem (25), the following results hold.Proposition 2.
For the eigenvalue problem (25), the eigenvalues are
(29)λ=λk=2sinkπT,k∈ℤ(-T+1,T-1)
which can be arranged as
(30)λ-r<λ-r+1<⋯<λ-1<0=λ0<λ1<λ2<⋯<λr
with r=(T-1)/2 if T is odd, and r=T/2 if T is even.To make an explicit decomposition of the Hilbert spaceE, we also need to compute the eigenfunctions of (25) corresponding to each λk,k≠0.For each fixedk∈ℤ(-r,-1)∪ℤ(1,r), a solution of (28) can be written as
(31)u1(n)=c1cos⁡(kwn)+c2sin(kwn),u2(n)=d1cos⁡(kwn)+d2sin(kwn),
where w=2π/T and c1,c2,d1,d2 are constant vectors in ℝN. By using the relation between u1 and u2, that is, (27) with λ=λk, we obtain
(32)c1sin(kw2)+c2cos⁡(kw2)=d1,-c1cos⁡(kw2)+c2sin(kw2)=d2.
Let ej, j=1,2,…,N, denote the canonical basis of ℝN. If we choose c1=ej and c2=0, then d1=sin(kw/2)ej, and d2=-cos⁡(kw/2)ej; if we choose c1=0 and c2=ej, then d1=cos⁡(kw/2)ej, and d2=sin(kw/2)ej. Therefore, the eigenfunctions of (25) corresponding to each λk(k≠0) can be given as
(33)ηk,j(1)(n)=(cos⁡⁡(kwn)ej-sin⁡(kw(n-12))ej),n=1,2,…,T,ηk,j(2)(n)=(sin⁡(kwn)ejcos⁡⁡(kw(n-12))ej),n=1,2,…,T.
Let
(34)W0={u={u(n)}∣u(n)=c1e1+c2e2+⋯xxxxx+c2Ne2N,ci∈ℝ,i=1,2,…,2N};W+={∑j=1N‍u={u(n)}∣u(n)=∑j=1N‍∑k=1r‍[αk,j(1)ηk,j(1)(n)+αk,j(2)ηk,j(2)(n)],xxxxαk,j(1),αk,j(2)∈ℝ∑j=1N‍};W-={∑j=1N‍u={u(n)}∣u(n)=∑j=1N‍∑k=-r-1‍[βk,j(1)ηk,j(1)(n)+βk,j(2)ηk,j(2)(n)],xxxxβk,j(1),βk,j(2)∈ℝ∑j=1N‍}.
Note that if T is even and k=T/2, then
(35)ηk,j(1)(n)=ηk,j(1)(T-n),ηk,j(2)(n)=ηk,j(2)(T-n)
which shorten the dimension of the eigensubspace corresponding to λT/2 to N. Hence
(36)dim⁡W0=2N,dim⁡W+=dim⁡W-=(T-1)N,
and E has a eigensubspace decomposition as
(37)E=W0⊕W-⊕W+.
Thus, for any u∈E, u can be expressed in the form
(38)u=u~+u¯,
where u~=(u~1,u~2)τ∈W-⊕W+ and u¯=(u¯1,u¯2)τ∈W0. Obviously, ui=u~i+u¯i,i=1,2.Denote(39)μmin⁡=min⁡{λk2=4sin2(kπT),k∈ℤ(1,T-1)}=4sin2πT.μmax⁡=max⁡⁡{λk2=4sin2(kπT),k∈ℤ(1,T-1)}={4,Tiseven;4sin2(T-1)π2T,Tisodd.
Then we have the following Wirtinger type inequalities:
(40)μmin⁡∥u~i∥2≤∥Δu~i∥2≤μmax⁡∥u~i∥2,i=1,2.
On the other hand, we can also define the norm ∥·∥p on ℝ2NT as follows:
(41)∥u∥p=(∑n=1T‍|u(n)|p)1/p
with p>1. Since ∥u∥p and ∥u∥ are equivalent, there exist constants c1,c2 such that c2≥c1>0 and
(42)c1∥u∥≤∥u∥p≤c2∥u∥.Now, we introduce the Clark duality [2, 26]. The Legendre transform L*(n,·) of L(n,·) with respect to the second variable can be defined by
(43)L*(n,y)=sup⁡x∈ℝM{(x,y)-L(n,x)},
where (·,·) denotes the usual inner product in ℝM with M a given positive integer.From [2, Theorem 2.2 and Proposition 2.4], we have the following lemma.Lemma 3.
Let, for everyn∈ℤ(1,T), L(n,x) be continuously differentiable and strictly convex in x∈ℝM, and
(44)L(n,x)|x|⟶+∞as|x|⟶∞.
Then for every n∈ℤ(1,T), L*(n,y) is continuously differentiable in y∈ℝM and
(45)L*(n,y)=(x,y)-L(n,x)⟺y=∇L(n,x)⟺x=∇L*(n,y).Remark 4.
If, forx=(x1,x2)τ, x1,x2∈ℝM, L(n,x) can be split into two parts L(n,x)=L1(n,x1)+L2(n,x2), then by (45), we have L*(n,y)=L1*(n,y1)+L2*(n,y2), y=(y1,y2)τ, and y1,y2∈ℝM.From [2, Proposition 2.2], we have the following lemma.Lemma 5.
Let convex lower semicontinuous functionL(n,x) be such that, for some ξ>0, t>1, β≥0, and γ≥0, one has
(46)-β≤L(n,x)≤ξt-1|x|t+γ
whenever x∈ℝM. Then, if
(47)y∈∂L(n,x)={w∈ℝM:L(n,z)≥L(n,x)xxxxxxxxxxxx+(w,z-x)forallz∈ℝM},
one has
(48)|y|≤{sξs/t[|x|+β+γ]+1}t-1,
where s-1+t-1=1.Remark 6.
From assumption (4) and p>1,
(49)F(n,u1)|u1|⟶+∞as|u1|⟶∞.
And by Lemma 3, F*(n,v1) is continuously differentiable in v1 for every n∈ℤ(1,T).Furthermore, on theF*(n,v1) we have the following.Lemma 7.
For allv1∈ℝN and every n∈ℤ(1,T), one has
(50)α-p/q|v1|pp≤F*(n,v1)≤δ-p/q|v1|pp,(51)|∇F*(n,v1)|≤(qδ|v1|+1)p-1.Proof.
By Lemma3, F*(n,v1)=(u1,v1)-F(n,u1)⇔v1=∇F(n,u1). Let F1:ℝN→ℝ be defined by
(52)F1(u1)=c|u1|qq,
where c>0. Then
(53)F1*(v1)=sup⁡u1∈ℝN{(v1,u1)-c|u1|qq}=c-p/q|v1|pp.
Hence (4), (53), and the fact that F1*≥F2* if F1≤F2 give (50).
To show that (51) holds, we can apply Lemma 5 to F*(n,v1) with β=γ=0, ξ=δ-p/q, p=t and x=v1, and y=∇F*(n,v1). Then (48) implies that (51) holds.By Lemmas3 and 7, H1* and H2* are well defined and
(54)H1*(n,v1)=F*(n,v1),H2*(n,v1)=|v1|pp.The variational functional of (14) defined on E is
(55)I(u)=12∑n=1T‍(JΔLu(n-1),u(n))+∑n=1T‍H(n,Lu(n));
see [11] for details. If
(56)u(n)=Jv(n)orv(n)=-Ju(n),
then, by (45) and (54), we obtain
(57)I(u)=12∑n=1T‍(JΔLu(n-1),u(n))+∑n=1T‍H(n,Lu(n))=12∑n=1T‍[(Δu2(n),u1(n))-(Δu1(n-1),u2(n))]+∑n=1T‍[H1(n,u1(n))+H2(u2(n+1))]=∑n=1T‍[12(Δu2(n),u1(n))-12(Δu1(n-1),u2(n))xxxxx+(u1(n),Δv1(n))+(u2(n+1),Δv2(n))12]+∑n=1T‍[-(u1(n),Δv1(n))+H1(n,u1(n))xxxxxx-(u2(n+1),Δv2(n))+H2(u2(n+1))]=12∑n=1T‍[(Δv1(n),v2(n))-(Δv2(n-1),v1(n))]-∑n=1T‍[F*(n,Δv1(n))+|Δv2(n)|pp]=-12∑n=1T‍(LJΔv(n-1),v(n))-∑n=1T‍[F*(n,Δv1(n))+|Δv2(n)|pp],
where
(58)F*(n,Δv1(n))=(u1(n),Δv1(n))-F(n,u1(n))⟺Δv1(n)=∇F(n,u1(n))⟺u1(n)=∇F*(n,Δv1(n)).
Now, we introduce a dual action functional
(59)J(v)=∑n=1T‍12(LJΔv(n-1),v(n))+F*(n,Δv1(n))+|Δv2(n)|pp,
where v=(v1,v2)τ∈E. Note that J(v)=J(v~+v¯)=J(v~) for v=v~+v¯∈E with v~∈Y:=W+⊕W- and v¯∈W0; hence it is sufficient to consider the functional J on the subspace Y of E. By Remark 6, the functional J is continuously differentiable on Y. And, for any h=(h1,h2)∈Y, we have
(60)〈J′(v),h〉=lim⁡s→0J(v+sh)-J(v)s=12∑n=1T‍[(LJΔh(n-1),v(n))+(LJΔv(n-1),h(n))]+∑n=1T‍(∇F*(n,Δv1(n)),Δh1(n))+∑n=1T‍(ϕp(Δv2(n)),Δh2(n))=∑n=1T‍(-v2(n)+∇F*(n,Δv1(n)),Δh1(n))+(v1(n+1)+ϕp(Δv2(n)),Δh2(n))=∑n=1T‍(Δ(v2(n-1)-∇F*(n-1,Δv1(n-1))),h1(n))-(Δ(v1(n)+ϕp(Δv2(n-1))),h2(n)).
If v is a critical point of J on the subspace Y, that is, 〈J′(v),h〉=0 for any h∈Y, then
(61)v2(n)=∇F*(n,Δv1(n))-c1,-v1(n+1)=ϕp(Δv2(n))-c2.
That is,
(62)LJv(n)+c=∇H*(n,Δv(n)),
where c=(c1,c2)∈ℝ2N. Setting u(n)=(u1(n),u2(n))τ=Jv(n)+c, we get Δu(n)=JΔv(n), and by relation (45) and (62), Δv(n)=∇H(n,Lu(n)). Therefore, JΔu(n)+∇H(n,Lu(n))=0. And u(n+T)=u(n) due to v(n+T)=v(T). Hence, we have the following.Remark 8.
Ifv∈Y is a critical point of the dual action functional J, then there exists a constant c∈ℝ2N such that u=(u1,u2)τ with u1,u2∈ℝN,
(63)u(n)=Jv(n)+c
is a solution of (14), and u1 is a solution of (1). If v is a nontrivial critical point, then u is a nonconstant T-periodic solution of (14), and u1 is a nonconstant T-periodic solution of (1).


## 3. Proof of the Main Result

As our proof of Theorem1 is mainly based on the computation of the critical groups in Morse theory, we recall several basic concepts about critical groups [2, 24]. Let H be a real Hilbert space, and f∈C1(H,ℝ). Denote
(64)fc={x∈H:f(x)≤c},𝒦c={x∈H:f′(x)=0,f(x)=c}
for c∈ℝ.Definition 9.
The functionalf satisfies the Palais-Smale (PS) condition if any sequence {xm}⊂H such that {f(xm)} is bounded and f′(xm)→0 as m→∞ has a convergent subsequence.In [27], Cerami introduced a weak version of the (PS) condition as follows.Definition 10.
The functionalf satisfies the Cerami condition (the (C) condition for short) if any sequence {xm}⊂H such that {f(xm)} is bounded and (1+∥xm∥)∥f′(xm)∥→0 as m→∞ has a convergent subsequence.Iff satisfies the (PS) condition or the (C) condition, then f also satisfies the following deformation condition which is essential in Morse theory [28, 29].Definition 11 (deformation condition).
The functionalf satisfies the (Dc) condition at the level c∈ℝ if, for any ϵ¯>0 and any neighborhood 𝒩 of 𝒦c, there are ϵ>0 and a continuous deformation η:[0,1]×H→H such that(1)
η(0,x)=x for all x∈H;(2)
η(t,x)=x for all x∉f-1([c-ϵ¯,c+ϵ¯]);(3)
f(η(t,x)) is nonincreasing in t for any x∈H;(4)
η(1,fc+ϵ∖𝒩)⊂fc-ϵ.f satisfies the (D) condition if f satisfies the (Dc) condition for all c∈ℝ.Letx0 be an isolated critical point of f with f(x0)=c∈ℝ, and let U be a neighborhood of x0; the group
(65)Cq(f,x0):=Hq(fc∩U,fc∩U∖{x0}),q∈ℤ
is called the qth critical group of f at x0, where Hq(A,B) denotes the qth singular relative homology group of the pair (A,B) over a field 𝔉, which is defined to be quotient Hq(A,B)=Zq(A,B)/Bq(A,B), where Zq(A,B) is the qth singular relative closed chain group and Bq(A,B) is the qth singular relative boundary chain group [30].Bartsch and Li [25] defined the qth critical group of f at infinity as
(66)Cq(f,∞):=Hq(H,fa),q∈ℤ,
provided that f(𝒦) is bounded from below by a∈ℝ with 𝒦={x∈E:f′(x)=0} and f satisfies the (Dc) condition for all c≤a.Assume♯𝒦<∞ and f satisfies the (D) condition. The Morse-type numbers of the pair (H,fa) are defined by
(67)Mq=Mq(H,fa)=∑x∈𝒦‍dim⁡Cq(f,x),
and the Betti numbers of the pair (H,fa) are
(68)βq≔dim⁡Cq(f,∞);
see [2, 24]. Furthermore, the following relations hold:
(69)∑j=0q‍(-1)q-jMj≥∑j=0q‍(-1)q-jβj,q∈ℤ,∑q=0∞‍Mq=∑q=0∞‍βq.
Thus, if Cq(f,∞)≇0, that is, βq≠0 for some q∈ℤ, then there must exist a critical point x of f with Cq(f,x)≇0. Furthermore, the following results hold.Proposition 12 (see [24]).
LetH be a real Hilbert space and f∈C2(H,ℝ). Assume that ♯𝒦<∞ and that f satisfies the (D) condition. If there exists some q∈ℤ such that(i)
Cq(f,∞)≇0, thenfmust have a critical pointxwithCq(f,x)≇0,(ii)
Cq(f,0)≇Cq(f,∞), thenfmust have a nontrivial critical point.We will use the following result to compute the critical groups ofJ at infinity.Proposition 13 (see [24]).
Let the functionalf:H→ℝ be of the form
(70)f(x)=12〈Ax,x〉+Q(x),
where A:H→H is a self-adjoint linear operator such that 0 is not in the spectrum of A, V± are invariant subspaces corresponding to the positive/negative of spectrum of A, respectively, A±:=A|V± has a bounded inverse on V±, and Q∈C1(H,ℝ) has a compact differential Q′(x) with
(71)lim⁡∥x∥→∞∥Q′(x)∥∥x∥=0.
Assume that m=dim⁡V- is finite and f satisfies the deformation condition. Then
(72)Cq(f,∞)=δq,m𝔉,q∈ℤ.For the proof of Theorem1, in what follows we may assume that J has only finitely many critical points. Firstly, we show that J satisfies the condition (C) which guarantees that the critical groups C*(J,∞) of J at infinity make sense. Then, via computations of critical groups of J at infinity and at zero, we complete the proof of Theorem 1.Lemma 14.
Under the conditions of Theorem1, the functional J defined by (59) satisfies (C) condition.Proof.
Let{v(l)}⊂Y be a Cerami sequence of J. Since dim⁡Y is finite, we only need to show that {v(l)} is bounded.
If{v(l)} is unbounded, up to a subsequence, still denoted by {v(l)}, we may assume that, for some c∈ℝ,
(73)J(v(l))⟶c,∥v(l)∥⟶∞,∥J′(v(l))∥∥v(l)∥⟶0.
In particular,
(74)lim⁡l→∞(J(v(l))-12〈J′(v(l)),v(l)〉)=c.
However, by (50) and (51), we have
(75)J(v(l))-12〈J′(v(l)),v(l)〉=∑n=1T‍F*(n,Δv1(l)(n))-12(∇F*(n,Δv1(l)(n)),Δv1(l)(n))+(1p-12)|Δv2(l)(n)|p≥∑n=1T‍[α-p/qp-12(qδ)p-1]|Δv1(l)(n)|p+(1p-12)|Δv2(l)(n)|p.
By relation (11) and u(l)(n)=Jv(l)(n), one has
(76)v1(l)(n)=-ϕp(Δ(x(n-1))),v2(l)(n)=x(n),
which implies that v1(l)(n)=-ϕp(Δ(v2(l)(n-1))), and hence
(77)∥v2(l)∥⟶∞as∥v(l)∥⟶∞.
At the same time, note that
(78)∑n=1T‍|Δv2(l)(n)|p=∥Δv2(l)∥pp≥c1p∥Δv2(l)∥p≥c1pμmin⁡p/2∥v2(l)∥p,
so,
(79)∑n=1T‍1q|Δv2(l)(n)|p⟶∞as∥v(l)∥⟶∞.
And assumption (5) implies that
(80)α-p/qp-12(qδ)p-1≥0,
so
(81)lim⁡l→∞(J(v(l))-12〈J′(v(l)),v(l)〉)=+∞
holds which gives a contradiction to (74). This completes the proof.Lemma 15.
Under the conditions of Theorem1, Cq(J,∞)≅δq,l𝔉 with l=(T-1)N, q∈ℤ.Proof.
Recall the dual action functionalJ:Y→ℝ being of the form
(82)J(v)=∑n=1T‍12(LJΔv(n-1),v(n))+F*(n,Δv1(n))+|Δv2(n)|pp.
For any u~,v∈Y, if we define a bilinear function as a(u~,v)=∑n=1T(LJΔu~(n-1),v(n)), then by (40) one has
(83)|a(u~,v)|≤(∑n=1T‍|LJΔu~(n-1)|2)1/2(∑n=1T‍|v(n)|2)1/2=(∑n=1T‍|Δu~(n)|2)1/2(∑n=1T‍|v(n)|2)1/2≤μmax⁡∥u~∥∥v∥.
By Riese representation theorem [31], we can define the unique continuous self-adjoint linear operator A on Y by 〈Av,v〉=∑n=1T‍(LJΔv(n-1),v(n)). If λ is in the spectrum of A, then the equation LJΔv(n-1)=λv(n) yields nontrivial solutions in E which turns out to be the same as (28) with u being replaced by v and the same invariant subspaces W+ and W- which can be given by eigensubspace. It is obvious that 0 is not in the spectrum of A from the definition of the subspaces of W+ and W-. Hence, by Proposition 13, we only need to prove that
(84)∥Q′(v)∥=o(∥v∥)as∥v∥⟶∞,
where
(85)Q(v)=∑n=1T‍F*(n,Δv1(n))+1p|Δv2(n)|p.
Note that
(86)〈Q′(v),h〉=∑n=1T‍(∇F*(n,Δv1(n)),Δh1(n))+(|Δv2(n)|p-2Δv2(n),Δh2(n))
with h(n)=(h1(n),h2(n))τ∈Y. And
(87)lim⁡∥v∥→∞〈Q′(v),v〉∥v∥2=lim⁡∥v∥→∞∑n=1T‍(∇F*(n,Δv1(n)),Δv1(n))∥v∥2xxxxxxxxxx+(|Δv2(n)|p-2Δv2(n),Δv2(n))∥v∥2=lim⁡∥v∥→∞∑n=1T‍(∇F*(n,Δv1(n)),Δv1(n))∥Δv1∥2∥Δv1∥2∥v∥2xxxxxxxxxx+(|Δv2(n)|p-2Δv2(n),Δv2(n))∥Δv2∥2∥Δv2∥2∥v∥2.
By (51), 1<p<2, and the equivalence of ∥v∥ and ∥Δv∥, we see that (84) holds. Hence Cq(J,∞)=δq,(T-1)N𝔉 by Proposition 13 and (36).Proof of Theorem1.
First, we prove that0 is a local minimum of J and, hence,
(88)Cq(J,0)≅δq,0𝔉.
Firstly, by (40), for all v∈Y, we have
(89)∑n=1T‍(LJΔv(n-1),v(n))≥-(∑n=1T‍|LJΔv(n-1)|2)1/2(∑n=1T‍|v(n)|2)1/2=-(∑n=1T‍|Δv(n)|2)1/2(∑n=1T‍|v(n)|2)1/2≥-μmax⁡1/2∑n=1T‍|v(n)|2≥-2∥v∥2.
Hence, by (40), (42), and (50), for v∈Y, we have
(90)J(v)=∑n=1T‍12(LJΔv(n-1),v(n))+F*(n,Δv1(n))+|Δv2(n)|pp≥-∑n=1T‍|v(n)|2+α-p/q∑n=1T‍|Δv1(n)|pp+∑n=1T‍|Δv2(n)|pp≥-∥v∥2+α-p/qc1pp∥Δv1∥p+c1pp∥Δv2∥p≥-∥v∥2+α-p/qc1pμmin⁡p/2p∥v1∥p+c1pμmin⁡p/2p∥v2∥p=-∥v1∥2-∥v2∥2+α-p/qc1pμmin⁡p/2p∥v1∥p+c1pμmin⁡p/2p∥v2∥p=∥v1∥p(α-p/qc1pμmin⁡p/2p-∥v1∥2-p)+∥v2∥p(c1pμmin⁡p/2p-∥v2∥2-p).
Take
(91)δ=min⁡{(α-p/qc1pμmin⁡p/22p)1/(2-p),(c1pμmin⁡p/22p)1/(2-p),1}.
Then, for ∥v∥≤δ, one has
(92)J(v)≥δ(∥v1∥p+∥v2∥p)≥δ(∥v1∥2+∥v2∥2)=δ∥v∥2≥0=J(0).
Hence 0 is a local minimum of J, and (88) must hold. By Lemma 15, (88), and Proposition 12, J must have at least one nontrivial critical point. The proof is completed.



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*Source: 567471-2014-02-12.xml*