# Infinitely Many Solutions for a Generalized Periodic Boundary Value Problem without the Evenness Assumption

Authors: Xiaodong Gu; Mingliang Song
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we investigate infinitely many solutions for the generalized periodic boundary value problem−x″−B0tx+B1tx=λ∇xVt,xa.e.t∈0,1,x1=Mx0,x′1=Nx′0 under the potential function Vt,x without the evenness assumption and obtain two new existence results by the multiple critical point theorem. Meanwhile, we give two corollaries for the periodic solutions of second-order Hamiltonian systems and an example that illustrates our results.


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## Body

## 1. Introduction and Main Results

We consider the existence of infinitely many solutions for the following generalized periodic boundary value problem:(1)−x″−B0tx+B1tx=λ∇Vt,xa.e.t∈0,1,x1=Mx0,x′1=Nx′0,where λ>0, B0t,B1t∈L∞0,1,LsRn=Bt=bjkn×n∣bjkt=bkjt,t∈0,1,bjkt∈L∞0,1, M,N∈GLn=A=ajkn×n∣ajk∈RanddetA≠0 with MNT=In, In is the unit matrix of order n, ∇Vt,x denotes the gradient of Vt,x for x∈Rn, and Vt,x satisfies (H0), that is, Vt,x is continuously differentiable in x for a.e. t∈0,1 and measurable in t for every x∈Rn, and there exist ax∈CR+,R+ and bt∈L10,1,R+ such that
(2)Vt,x+∇Vt,x≤axbt,for all x∈Rn and a.e. t∈0,1, where R+=0,+∞.Note that ifM=N=In,B0t≡0, B1t is 1-periodic and Vt,x is 1-periodic in t; then, the solutions of problem (1) are the 1-periodic solutions of second-order Hamiltonian systems.In his pioneer paper [1] of 1978, Rabinowitz studied for the existence of periodic solutions for Hamiltonian systems via the critical point theory. From then on, with the aid of the critical point theory, the existence of infinitely many periodic solutions for Hamiltonian systems has been extensively investigated in some papers (see [2–17]) and the excellent books (see [18–20]).For second-order Hamiltonian systems, under various conditions, the authors in [3–5, 7, 8, 14–17] obtained infinitely many periodic solutions under the evenness assumption of Vt,x. Without the evenness assumption of Vt,x, the authors in [2, 6, 9–13] also obtained infinitely many periodic solutions for first- (or second-) order Hamiltonian systems under the potential function Vt,x∈C2R×Rn,R. In this paper, we are interested in the potential function Vt,x∈C1R×Rn,R, and without the evenness assumption. We study the existence of infinitely many solutions of problem (1) via the multiple critical point theorem established in [21, 22] under Vt,x∈C1R×Rn,R.Now, we use the indexiMB,νMB∈N×N defined in [23] (see Section 2) to state our results.Theorem 1.
Assume thatiMB0=0,νMB0≠0,B1tis positive definite; i.e., there existsb¯0>b_0>0such thatb¯0x2≥B1tx,x≥b_0x2for allx∈Rn. In addition, assume thatVt,xsatisfies(H0) and (H1) (Vt,θ=0) and (H2):
(3)liminf∥x~∥∞→+∞∫01sup∥x∥∞≤∥x~∥∞Vt,xdt∥x~∥∞2<12k02b¯0limsup∥x~∥∞→+∞∫01Vt,x~dt∥x~∥∞2,for all x~∈kerΛ−B0, where Λxt=−x′′t, k0=δ0min1,b_0−1/2, and δ0 is a constant of the compact imbedding Z↪L∞ (or see (30) in Section 2).
Then, for eachλ∈λ1,λ2, problem (1) possesses infinitely many solutions, where
(4)λ1=b¯02limsup∥x~∥∞→+∞∫01Vt,x~dt/∥x~∥∞2and
(5)λ2=14k02liminf∥x~∥∞→+∞∫01sup∥x∥∞≤∥x~∥∞Vt,xdt/∥x~∥∞2.Theorem 2.
The conclusion of Theorem 1still holds if we replace(H2) with (H′2):
(6)liminf∥x~∥∞→0+∫01sup∥x∥∞≤∥x~∥∞Vt,xdt∥x~∥∞2<12k02b¯0limsup∥x~∥∞→0+∫01Vt,x~dt∥x~∥∞2.
We postpone the proofs to the next section and turn to applications to second-order Hamiltonian systems. For systematic researches of second-order Hamiltonian systems, we refer to the excellent books (see [18–20]).As the special case, we consider the periodic solution problem:(7)−x″+B1tx=λ∇Vt,xa.e.t∈0,1,x1−x0=x′1−x′0=0,where λ>0, B1t∈L∞0,1,LsRn, and B1t and Vt,x are 1-periodic. After a simple calculation, we have minσΛ=0, iIn0=0, νIn0≠0, kerΛ=Rn, δ0=1/23, and ∣x∣=∥x∥∞ for all x∈Rn. Therefore, the following corollaries are immediately obtained from Theorems 1 and 2.Corollary 3.
Assume thatB1tis positive definite andVt,xsatisfies (H0), (H1), and (H2,1):
(8)liminf∣x~∣→+∞∫01sup∣x∣≤∣x~∣Vt,xdtx~2<12k02b¯0limsup∣x~∣→+∞∫01Vt,x~dtx~2,for all x~∈Rn, where k0=1/23min1,b_0−1/2.
Then, for eachλ∈λ1,λ2, problem (7) possesses infinitely many 1-periodic solutions, where λ1,λ2 are given by (34).Corollary 4.
The conclusion of Corollary 3still holds if we replace (H2,1) with (H2,2):
(9)liminf∣x~∣→0+∫01sup∣x∣≤∣x~∣Vt,xdtx~2<12k02b¯0limsup∣x~∣→0+∫01Vt,x~dtx~2.Next, an example of problem (7) is given below.Example 5.
LetB1t≡In and
(10)αm=m!m+2!−12m+1!,βm=m!m+2!+12m+1!,for every m∈N\0=N+. Define the continuous function Vt,x:0,1×Rn⟶R as follows:
(11)Vt,x=a0,x·sin2πt+V1x·sinπt,where a0∈Rn\θ and
(12)V1x=4m+1!2m+1!2−m!2π·x−m!m+22·14m+1!2−x−m!m+222+m+1!2−m!2π·arcsin2m+1!x−m!m+2!+m+1!2+m!22−1,ifx∈⋃m∈N+αm,βm,m+1!2−1,ifx∈⋃m∈N+βm,αm+1,0,ifx∈0,54.
Fors≥0, by
(13)V′1s=8m+1!2m+1!2−m!2π·14m+1!2−s−m!m+222,ifs∈⋃m∈N+αm,βm,0,otherwise,it is easy to see that the conditions (H0) and (H1) are satisfied. Noticing that
(14)lim∣x∣→+∞∣a0,x∣x2=0,limm→+∞V1αmαm2=0,limm→+∞V1βmβm2=4,we have
(15)0=liminf∣x~∣→+∞∫01sup∣x∣≤∣x~∣Vt,xdtx~2<12k02b¯0limsup∣x~∣→+∞∫01Vt,x~dtx~2=24π,via a simple computation. This shows that (H2,1) holds. By Corollary 3, there exist infinitely many 1-periodic solutions for problem (7), for each λ∈π/48,+∞. In particular, by 1∈π/48,+∞, we can see that the following second-order Hamiltonian systems in Rn
(16)−x″+x=sin2πta0+V′1∣x∣·∣sinπt∣∣x∣x,x1−x0=x′1−x′0=0also have infinitely many 1-periodic solutions.Remark 6.
In Example5, we discard the evenness assumption of potential function Vt,x, which means that Example 5 does not satisfy the assumptions in [3–5, 7, 8, 14–17]. Noticing that the potential function Vt,x∈C2R×Rn,R in Example 5, we can see that Example 5 does also not satisfy the assumptions in [2, 6, 9–13]. Therefore, our result is also new even in the case of periodic solutions for second-order Hamiltonian systems.


## 2. Variational Setting and Proof of the Main Result

In this section, we first recall the multiple critical point theorem due to [21, 22] and some conclusions of index theory due to [23, 24], respectively.Lemma 7 ([21], Theorem 2.1; [22], Theorem 2.5).
LetZbe a reflexive real Banach space,Φ:Z→Rbe a (strongly) continuous, coercive, sequentially weakly lower semicontinuous and Gâteaux differentiable functional, andΨ:Z⟶Rbe a sequentially weakly upper semicontinuous and Gâteaux differentiable functional. For allr>infZΦ, let
(17)φr=infx∈Φ−1−∞,rsupy∈Φ−1−∞,rΨy−Ψxr−Φx,γ=liminfr→+∞φr,δ=liminfr→infZΦ+φr.Then,(a)
For everyr>infZΦ and every λ∈0,1/φr, the restriction of the functional Φ−λΨ to Φ−1−∞,r admits a global minimum, which is a critical point (local minimum) of Φ−λΨ in Z
(b)
Ifγ<+∞, then, for each λ∈0,1/γ, the following alternative holds: either
(b1)
Φ−λΨ possesses a global minimum or
(b2)
there is a sequencexn of critical points (local minima) of Φ−λΨ such that limn→+∞Φxn=+∞
(c)
Ifδ<+∞, then, for each λ∈0,1/δ, the following alternative holds: either
(c1)
there is a global minimum ofΦ which is a local minimum of Φ−λΨ or
(c2)
there is a sequence of pairwise distinct critical points (local minima) ofΦ−λΨ, with limn→+∞Φxn=infZΦ which weakly converges to a global minimum of ΦIndex theory in [23] deals with a classification of L∞0,1,LsRn associated with the following system:
(18)−x″−Btx=0,(19)x1=Mx0,x′1=Nx′0,where Bt∈L∞0,1,LsRn, M,N∈GLn, and MNT=In.LetX=L20,1,Rn. Set Λxt=−x″t with DΛ=x∈H20,1,Rn:xsatisfies19. By Section 2.4 in [23], we see that Λ is self-adjoint and σΛ=σdΛ⊂0,+∞. In particular, if M=N=In, then σΛ=σdΛ=2k2π2∣k∈Z, and if M=N=−In, then σΛ=σdΛ=2k−12π2∣k∈Z. By Corollary 1.21 in [25], we know that Z=DΛ1/2=x∈H10,1,Rn:x1=Mx0 is a Hilbert space with the norm xZ=∫01x′t2+xt2dt1/2 for each x∈Z, and the embeddings Z↪L2=X, Z↪Lpp≥1, and Z↪L∞ are compact.For anyBt∈L∞0,1,LsRn, we define a bilinear form as follows:
(20)qBx,y=∫01x′,y′−Btx,ydt,∀x,y∈Z.Proposition 8 ([24], Proposition 7.2.1).
For anyBt∈L∞0,1,LsRn, the spaceZhas aqB-orthogonal decomposition:
(21)Z=Z+B⊕Z0B⊕Z−B,such that qB is positive definite, null, and negative definite on Z+B, Z0B, and Z−B, respectively. Moreover, Z0B and Z−B are finite-dimensional.Definition 9 ([23], Definition 2.4.1; [24], Definition 7.1.3).
For anyBt∈L∞0,1,LsRn, we define
(22)νMB=dimkerΛ−B,iMB=∑λ<0νMB+λIn.
We callνMB and iMB the nullity and index of B with respect to the bilinear form qB·,·, respectively.Proposition 10 ([24], Proposition 7.2.2).
For anyBt∈L∞0,1,LsRn, we haveνMB=dimZ0BandiMB=dimZ−B.Proposition 11 ([23], Proposition 2.4.2 (1); [24], Corollary 7.2.2 (i)).
For anyB∈L∞0,1,LsRn, we have thatZ0Bis the solution subspace of systems (18) and (19), andνMB∈0,1,2,⋯,2n.Remark 12 ([23], Example 2.4.3; [24], Remark 7.1.3).
Leta1≤a2≤⋯≤an be the eigenvalues of a constant n×n symmetric matrix B. For ζ∈R\0 with ξ0=arccos2/ζ−1+ζ, we have
(23)iζInB=∑k=1n#j∈N:2jπ+ξ02<ak+∑k=1n#j∈N:2π−ξ0+2jπ2<ak,(24)νζInB=∑k=1n#j∈N:2jπ+ξ02=ak+∑k=1n#j∈N:2π−ξ0+2jπ2=ak,where S# denotes the number of elements in set S. In particular, formulae (23) and (24) when ζ=1 were given first by Mawhin and Willem in [19].Next, we establish the variational setting for problem (1).It is known that the operatorΛ−B0 is also self-adjoint and σΛ−B0=σdΛ−B0 is bounded from below. Noticing that iMB0=0 and νMB0≠0, by Definition 9 and Proposition 10, we know that the operator Λ−B0 has a sequence of eigenvalues:
(25)0=minσΛ−B0=λ0=λ1=⋯=λn0<λn0+1≤λn0+2≤⋯≤λn0+n≤⋯⟶+∞,and the system of eigenfunctions en:n∈N corresponding to λn:n∈N forming an orthogonal basis in L2=X. Hence, we can define another inner product:
(26)x,y=qB0x,y+∫01xt,ytdt,∀x,y∈Z,with the corresponding norm x=qB0x,x+xL221/2,∀x∈Z. Clearly, · is equivalent to ·Z. Put
(27)xB1=qB0x,x+∫01B1txt,xtdt1/2,∀x∈Z.SinceB1t is positive definite, there exist b¯0>b_0>0 such that
(28)∫01b¯0x2dt≥∫01B1tx,xdt∫01b_0x2dt,for all x∈Z. So, we have
(29)min1,b_0x2≤xB12≤max1,b¯0x2,for all x∈Z.Noticing the compactness of the embeddingZ↪L∞, from (29), we know that there is an embedded constant δ0>0 such that
(30)x≤x∞≤δ0x≤k0xB1,for all x∈Z, where k0=δ0min1,b_0−1/2 and x∞ is the norm of L∞0,1,Rn.Now, we define(31)Iλx=xB122−λ∫01Vt,xdt,∀x∈Z.From the assumption (H0) and Theorem 1.2 in [19], it is easy to verify that Iλ∈C1Z,R is weakly lower semicontinuous on Z and I′λ is weakly continuous with
(32)I′λxy=qB0x,y+∫01B1tx,ydt−λ∫01∇Vt,x,ydt,for all x,y∈Z. If I′λx=0, we easily find that the critical points of I correspond to the solutions of problem (1) and omit the details.Finally, we give the proofs of Theorems1 and 2.For convenience, put(33)Γ≔liminfx~∞→+∞∫01supx∞≤x~∞Vt,xdt∥x~∥∞2,Y=limsupx~∞→+∞∫01Vt,x~dtx~∞2,(34)λ1≔b¯02Y,λ2≔14k02Γ.SinceVt,θ=0, we have Γ≥0. So, if Γ=0, we put λ2=+∞, and if Y=+∞, we put λ1=0.Proof of Theorem 1.
Set that(35)Φx=xB122,Ψx=∫01Vt,xdt,∀x∈Z.
Obviously,Φ is (strongly) continuous, coercive, and Gâteaux differentiable, Ψ is sequentially weakly upper semicontinuous and Gâteaux differentiable, and infZΦ=0. On the other hand, the critical points of Φ−λΨ in Z are the solutions of problem (1).
Take thatλ∈λ1,λ2. Pick a sequence x~n⊂kerΛ−B0 such that limn→+∞x~n∞=+∞ and limn→+∞∫01supx∞≤x~n∞Vt,xdt/x~n∞2=Γ. Set rn=x~n∞2/2k02. By (30), we have xt∞≤x~n∞ for any x∈Z with Φx=xB12/2<rn. From definition of φ· and (3) of H2, we obtain
(36)φrn=infxB12<2rnsupyB12<2rn∫01Vt,ydt−∫01Vt,xdtrn−xB12/2≤supyB12<2rn∫01Vt,ydtrn≤∫01supx∞≤x~n∞Vt,xdtrn=2k02∫01supx∞≤x~n∞Vt,xdtx~n∞2,which shows that
(37)γ=liminfn→+∞φrn≤2k02Γ<+∞.
Next, we prove thatΦ−λΨ in Z is unbounded from below.
Again, pick a sequencey~n⊂kerΛ−B0 such that limn→+∞y~n∞=+∞ and
(38)limn→+∞∫01Vt,y~ndty~n∞2=Y.
Noticing thaty~n⊂kerΛ−B0, by Proposition 8, we have
(39)Φy~n−λΨy~n=12qB0y~n,y~n+12∫01B1ty~n,y~ndt−λ∫01Vt,y~ndt=12∫01B1ty~n,y~ndt−λ∫01Vt,y~ndt.
IfY<+∞, put ε∈0,Y−b¯0/2λ. By (34), we know that there exists Nε>0 such that
(40)∫01Vt,y~ndt>Y−εy~n∞2,∀n>Nε.
This implies that(41)Φy~n−λΨy~n≤b¯02∫01y~n2dt−λ∫01Vt,y~ndt≤y~n∞2b¯02−λY−ε.
Choosing small enoughε, from (34) and λ>λ1, we have
(42)limn→+∞Φy~n−λΨy~n=−∞.
IfY=+∞, put M>b¯0/2λ. From (38), there exists NM>0 such that
(43)∫01Vt,y~ndt>My~n∞2,∀n>NM.
This shows that(44)Φy~n−λΨy~n≤y~n∞2b¯02−λM⟶−∞asn⟶+∞.
Noting thatλ1,λ2⊂0,1/γ, from (b) of Lemma 7, we know that Φ−λΨ admits a sequence xn of critical points with limn→+∞Φxn=+∞. The proof is complete.Furthermore, from (c) of Lemma7, we can prove Theorem 2 similar to Theorem 1. Here, we omit it.



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*Source: 8406719-2020-04-03.xml*