# Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems

Authors: Wenping Qin; Jian Zhang; Fukun Zhao
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the following nonperiodic Hamiltonian systemż=𝒥Hz(t,z), where H∈C1(ℝ×ℝ2N,ℝ) is the form H(t,z)=(1/2)B(t)z·z+R(t,z). We introduce a new assumption on B(t) and prove that the corresponding Hamiltonian operator has only point spectrum. Moreover, by applying a generalized linking theorem for strongly indefinite functionals, we establish the existence of homoclinic orbits for asymptotically quadratic nonlinearity as well as the existence of infinitely many homoclinic orbits for superquadratic nonlinearity.


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## Body

## 1. Introduction and Main Results

In this paper, we are interested in the existence of homoclinic orbits of the Hamiltonian system(HS)ż=JHz(t,z),
where z=(p,q)∈ℝN×ℝN=ℝ2N,𝒥=(0IN-IN0), and H∈C1(ℝ×ℝ2N,ℝ) is the form
(1.1)H(t,z)=12B(t)z⋅z+R(t,z)
with B(t)∈C(ℝ,ℝ4N2) being a 2N×2N symmetric matrix valued function, and R∈C1(ℝ×ℝ2N,ℝ). Here, by a homoclinic orbit of (HS), we mean a solution of the equation satisfying z(t)≢0 and z(t)→0 as |t|→∞.Establishing the existence of homoclinic orbits for system like (HS) is one of the most important problems in the theory of Hamiltonian systems. In very recent years, many authors devoted to the existences of homoclinic orbits for Hamiltonian systems via critical point theory. For example, see [1–5] for the second-order systems and [6–18] for the first-order systems. Coti-Zelati et al. first considered the system (HS) in [6], and they obtained a homoclinic orbit for strictly convex Hamiltonian system. The existence of infinitely many homoclinic orbits was established in [15]. Subsequently, Hofer and Wysocki removed the convexity assumptions in [13]. Later, suppose that R(t,z) and B(t) depend periodically on t, the existence of homoclinic orbit for (HS) was considered in [7, 8, 12, 16, 18].Without assumption of periodicity, the problem is quite different in nature. In [10], Ding and Li first obtained one homoclinic orbit for the nonperiodic system (HS), see also [9, 11] and the references therein for recent works on this direction.Motivated by [9, 11], in this paper, we introduce a new nonperiodic assumption on B(t) as the following. (B0)
B(t)∈C(ℝ,ℝ2N×2N), there exists r0>0 such that, for any h>0,
(1.2)|{t∈R:|t-t1|≤r0,J0B(t)<h}|⟶0,as|t1|⟶∞,
where 𝒥0 will be be defined in Section 2; we regard a real function U(x) as a symmetric matrix U(x)I2N×2N, and, for two given matrix valued functions L1(t) and L2(t), we say that L1(t)≤L2(t) if and only if
(1.3)max⁡ξ∈R2N,|ξ|=1(L1(t)-L2(t))ξ⋅ξ≤0,
and L1(t)>L2(t) if and only if L1(t)≤L2(t) does not hold. Obviously, (B0) holds if B(t) satisfies.(B1)
For anyb>0, the set Λb:={t∈ℝ:𝒥0B(t)<b} is nonempty and has finite Lebesgue measure.One of the main ingredients of our work is two steps, the first is to show that the spectrum of Hamiltonian operator(1.4)A:=-(Jddt+B)
consists of a sequence of eigenvalues with finite multiplicity which is unbounded from above and below (see Lemma 2.2), and hence the energy functional corresponding to (HS) is strongly indefinite; the second is to show that the working space E has compact embedding property (see Lemma 2.3).First, we handle the asymptotically quadratic case. In what follows,R̂(t,z):=(1/2)Rz(t,z)z-R(t,z). Suppose that  (R1)
R∈C1(ℝ×ℝ2N,[0,∞)) and Rz(t,z)=o(|z|) as |z|→0 uniformly in t,(R2)
there exists a bounded functionR∞∈C(ℝ,ℝ) such that |Rz(t,z)-R∞(t)z|/|z|→0 as |z|→∞ uniformly in t, and b:=inft∈ℝR∞(t)>inf[(0,∞)∩σ(A)], where σ(A) denotes the spectrum of A,(R3)
R̂(t,z)>0 if z≠0, R̂(t,z)→∞ as |z|→∞.We recall that a solutionz1 of (HS) is called a least energy solution if it possesses minimum energy among all solutions, that is,
(1.5)Φ(z1)=θ:=inf⁡{Φ(z)∣zisanonzerosolutionof(HS)},
where Φ is the energy functional
(1.6)Φ(z):=-∫R(12Jż⋅z+H(t,z))dt.Letk be the number of eigenvalues of the operator A lying in (0,b), our main results are the following.Theorem 1.1.
Suppose (B0), (R1)–(R3) be satisfied. Then, (HS) has a least energy solution. Moreover, if R(t,z) is even in z, then (HS) has at least k pairs of solutions.Next, we consider the superquadratic case. Assume(R4)
R(t,z)·|z|-2→∞ as |z|→∞ uniformly in t,(R5)
R̂(t,z)>0 if z≠0, and there are μ>1 and r,c1>0, such that, for |z|≥r, |Rz(t,z)|μ≤c1R̂(t,z)|z|μ.Theorem 1.2.
Suppose (B0), (R1), and (R4)-(R5) be satisfied. Then, (HS) has a least energy solution. Moreover, if R(t,z) is even in u. Then, (HS) has infinitely many solutions.Remark 1.3.
The assumptions (R1), (R4)-(R5) imply that R̂(t,z)→∞ uniformly in t as |z|→∞.


## 2. Variational Framework and Linking Structure

In order to establish a variational setting for the system (HS), in this section, we first study the spectrum of the Hamiltonian operator.Observe that, since we have assumed (B0) about B(t), A=-(𝒥(d/dt)+B(t)) is a self-dajoint operator on L2(ℝ,ℝ2N) with 𝒟(A)⊂H1(ℝ,ℝ2N). Let |·|q denote the usual Lq- norm, (·,·)2 denote the usual L2 inner product, c,ci,d, or di stand for different positive constants. Let σ(A),σd(A), and σe(A) be the spectrum of A, the discrete spectrum of A, and the essential spectrum of A, respectively. Observe that 𝒟(A) is a Hilbert space with the graph inner product
(2.1)(z1,z2)A:=(Az1,Az2)2+(z1,z2)2,
and the induced norm ∥z∥A:=(z,z)A1/2.Set𝒥0:=(0ININ0) and A0:=𝒥(d/dt)+𝒥0 (self-adjoint with 𝒟(A0)=H1(ℝ,ℝ2N)); thus, A02=-d2/dt2+1. Let |A0| denote the absolute value of A0, and we have
(2.2)||A0|z|22=|A0z|22=(A0z,A0z)2=(A02z,z)=((-d2dt2+1)z,z)2=|∇z|22+|z|22,
which implies that
(2.3)‖z‖H1=||A0|z|2,
for all z∈𝒟(A02)=H2(ℝ,ℝ2N), hence for all z∈H1(ℝ,ℝ2N) because of the density of H2 in H1.Lemma 2.1.
For anyz∈𝒟(A)⊂H1(ℝ,ℝ2N), there exists d>0 such that
(2.4)‖z‖H1=‖A0|z|2≤d‖z∥A.Proof.
LetA1 be the restriction of A0 to 𝒟(A), A1 is a linear operator from 𝒟(A) to L2. We claim that A1 is closed. Indeed, let zn→∥·∥Az and A1zn→|·|2w. Then, z∈𝒟(A), and since A0 is closed, A1zn=A0zn→|·|2A0z=A1z, hence the claim holds. Now the Closed Graph Theorem implies that A1:𝒟(A)→L2 is a bounded linear operator, so |A0z|2=|A1z|2≤d∥z∥A for all z∈𝒟(A). This together with (2.3) implies (2.4).Lemma 2.2.
Suppose(B0) holds. Then σ(A)=σd(A).Proof.
Similar to the idea of [9, 19], for any h>0, if (B0) holds, set
(2.5)(J0B(t)-h)+:={J0B(t)-h,ifJ0B(t)-h≥0,0,ifJ0B(t)-h<0,
and (𝒥0B(t)-h)-=(𝒥0B(t)-h)-(𝒥0B(t)-h)+. Thus, A=A2+𝒥0(𝒥0B(t)-h)-, where
(2.6)A2=-Jddt+J0(J0B(t)-h)++hJ0.
Since 𝒥02=I and 𝒥0𝒥=-𝒥𝒥0, we have, for z∈𝒟(A),
(2.7)(A2z,A2z)2=|(-Jddt+J0(J0B(t)-h)+)z+hJ0z|22=|(-Jddt+J0(J0B(t)-h)+)z|22+h2|z|22+(-Jż,hJ0z)2+(hJ0z,-Jż)2+((J0(J0B(t)-h)+)z,hJ0z)2+((hJ0z,J0(J0B(t)-h)+)z)2=|(-Jddt+J0(J0B(t)-h)+)z|22+h2|z|22+2h(((J0B(t)-h)+)z,z)2≥h2|z|22.
Thus, σ(A2)⊂ℝ∖(-h,h).
We claim thatσe(A)∩(h,h)=∅. Assume by contradiction that there is ν∈σe(A) with |ν|<h. Let {zn}⊂𝒟(A) with |zn|2=1, zn⇀0 in L2 and |(A-ν)zn|2→0. Moreover, by (B0), one can check that the multiplication operator z→𝒥0(𝒥0B(t)-h)-z is compact. In fact, let {zn} be bounded in 𝒟(A), without loss of generality, we may assume zn⇀0 in 𝒟(A). Next we show that |𝒥0(𝒥0B(t)-h)-zn|2→0 in L2. For every R>0, define BR(0)={t∈ℝ:|t|<R} and BRc=ℝ∖BR(0). Let {ti} be a sequence of points in BRc satisfying BRc⊂∪i=1∞B(ti,r0) and such that each point t is contained in at most 2 such balls B(ti,r0). Let B:={t∈BRc:𝒥0B(t)<h}, choose s∈(1,3) and s′=s/(s-1). We get
(2.8)∫BRc|J0(J0B(t)-h)-zn|2≤∑i=1∞∫B(ti,r0)∩B|J0(J0B(t)-h)-zn|2≤∑i=1∞(∫B(ti,r0)∩B|zn|2s)1/s(∫B(ti,r0)∩B|J0(J0B(t)-h)-|2s′)1/s′≤CR2s′∑i=1∞|B(ti,r0)∩B|1/s′(∫B(ti,r0)∩B|zn|2s)1/s≤2CR2s′ɛR‖zn‖A2,
where ɛR=supi|B(ti,r0)∩B|1/s′,CR=supi∥𝒥0(𝒥0B(t)-h)-∥M (for real matrix valued function Q(x)=(qjk(x))2N×2N, ∥Q(x)∥M=max1≤j≤2N∑k=12N|qjk(x)|).
Here, we used (2.4) of Lemma 2.1 and that H1 embeds into L2s. Assumption (B0) implies that ɛR→0 as R→∞; thus,
(2.9)∫BRc|J0(J0B(t)-h)-zn|2⟶0.
On the other hand,
(2.10)∫BR(0)|J0(J0B(t)-h)-zn|2≤(∫BR(0)|zn|2s)1/s(∫BR(0)|J0(J0B(t)-h)-|2s′)1/s′⟶0,
since H1↪Lloc2 is compact. Thus, |𝒥0(𝒥0B(t)-h)-zn|2→0, we have
(2.11)o(1)=|(A-ν)zn|2=|A2zn-νzn+J0(J0B(t)-h)-zn|2≥|A2zn|2-|ν|-o(1)≥h-|ν|-o(1),
which implies that 0<h-|ν|≤0, a contradiction. So σe(A)∩(-h,h)=∅. Since h>0 is arbitrary, it follows that σ(A)=σd(A).From Lemma2.2, we know that the operator A has a sequence of eigenvalues
(2.12)⋯λ-k≤⋯≤λ-1≤0<λ1≤⋯≤λk⋯
with λ±k→±∞ as k→∞, and corresponding eigenfunctions {e±k}k∈ℕ form an orthogonal basis in L2. Observe that we have an orthogonal decomposition
(2.13)L2=L-⊕L0⊕L+,z=z-+z0+z+,
such that A is negative definite on L- and positive definite on L+ and L0=kerA. Let P0:L2→L0 be the projection. Set E:=𝒟(|A|1/2) be the domain of the self-adjoint operator |A|1/2 which is a Hilbert space equipped with the inner product
(2.14)(z,w)=(|A|1/2z,|A|1/2w)2+(P0z,P0w)2
and norm ∥z∥=(z,z)1/2. Let E±:=span{e±k}¯k∈ℕ,E0=kerA. Then, E=E-⊕E0⊕E+ is an orthogonal decomposition of E.Similar to [19], on 𝒟(A), we introduce an inner product
(2.15)〈z,w〉A=(Az,Aw)2+(P0z,P0w)2=(|A|z,|A|w)2+(P0z,w)2
whose induced norm will be denoted by |z|A. Since 0 is at most an isolated eigenvalue of finite multiplicity, it is clear that |·|A and ∥·∥A are equivalent on 𝒟(A): d1|z|A≤∥u∥A≤d2|z|A, for all z∈𝒟(A). Define
(2.16)Â:=|A|+P0.
Then, 𝒟(A)=𝒟(Â). Noting that P0|A|=|A|P0=0, for z,w∈𝒟(Â), we have (2.17)(Âz,Âw)2=(|A|z,|A|w)2+(P0z,|A|w)2+(|A|z,P0w)2+(P0z,P0w)2=(|A|z,|A|w)2+(P0z,P0w)2=〈z,w〉A,
hence
(2.18)|z|A=|Âz|2,∀z∈D(A).
Observe that, for all z∈𝒟(A) and w∈𝒟(|A|1/2),
(2.19)(Â1/2z,Â1/2w)2=(Âz,w)2=((|A|+P0)z,w)2=(|A|z,w)2+(P0z,w)2=(|A|1/2z,|A|1/2w)2+(P0z,P0w)2=(z,w).
Consequently, since 𝒟(A)=𝒟(Â) is a core of Â1/2, we have
(2.20)(z,w)=(Â1/2z,Â1/2w)2∀z,w∈D(|A|1/2),
which implies in particular that
(2.21)‖z‖=|Â1/2z|2∀z∈E.
By complex interpolation theory, we have H1/2=[L2,H1]1/2 (see Theorem 2.4.1 [20]). Since 𝒟(|A0|0)=L2 and ∥z∥H1=∥A0|z|2, one has
(2.22)H1/2=[D(|A0|0),D(|A0|)]1/2
with equivalent norms. It then follows from Theorem 1.18.10 of [20] that
(2.23)H1/2=[D(|A0|0)D(|A0|0),D(|A0|)]1/2=D(|A0|1/2),
hence ∥z∥H1/2 and ||A0|1/2z|2 are equivalent norms on H1/2,
(2.24)d3‖z‖H1/2≤||A0|1/2z|2≤d4‖z‖H1/2∀z∈H1/2(R,R2N).Lemma 2.3.
E embeds continuously into H1/2(ℝ,ℝ2N). Moreover, E embeds compactly into Lp(ℝ,ℝ2N) for all p∈[2,∞).Proof.
By (2.4) and (2.18), there exists d5>0 such that
(2.25)||A0|z|2≤d5|Âz|2=|(d5Â)z|2∀z∈D(A).
Thus, (|A0|z,z)2≤(d5Âz,z)2 for all u∈𝒟(A) (see Proposition III 8.11 of [21]). This implies
(2.26)||A0|1/2z|22=(|A0|z,z)2≤(d5Âz,z)2=d5|Â1/2z|22
for all z∈𝒟(A) (see Proposition III  8.12 of [21]). Since 𝒟(A) is core of Â1/2, we obtain that ||A0|1/2z|22≤d5|Â1/2z|22 for all z∈E. This combination with (2.21) shows that
(2.27)||A0|1/2z|22≤d5‖z‖2∀z∈E,
which, together with (2.24), implies that
(2.28)‖z‖H1/2≤d6‖z‖∀z∈E.
This proves that the embedding E↪H1/2(ℝ,ℝ2N) is continuous. In order to show that the embedding E↪Lp(ℝ,ℝ2N) is compact for all p∈[2,∞), it suffices to prove that E↪L2 is compact. Set Lk:=span{e-k,…,e-1,e1,…,ek}. Let Pk:E→Lk denote the orthogonal projector. Consider a weakly converging sequence zn⇀z in E. Denote wn=zn-z and K:=supn∥wn∥2. Given ɛ>0, we choose k∈ℕ so that K/νk<ɛ/2, where νk:=|λ-k|+λk. Since Pkwn→0 as n→∞, then there exists n0∈ℕ such that ∥Pkwn∥2<ɛ/2 for all n≥n0. Let {E(τ)}τ∈ℝ be the spectral family of A. It follows from
(2.29)‖wn‖2≥‖(I-Pk)wn‖2=(|A|1/2(I-Pk)wn,|A|1/2(I-Pk)wn)2=∫-∞λ-k|τ|d|E(τ)(I-Pk)wn|22+∫λk∞τd|E(τ)(I-Pk)wn|22≥(|λ-k|+λk)|(I-Pk)wn|22
that
(2.30)|(I-Pk)wn|22≤‖wn‖2|λ-k|+λk=‖wn‖2νk<ɛ2.
Then,
(2.31)|wn|22=|Pkwn|22+|(I-Pk)wn|22<ɛ2+ɛ2=ɛ
for all n≥n0. This proves that zn→z in L2.SetΨ(z)=∫ℝR(t,z(t))dt, by assumptions and Lemma 2.3, Ψ(z)∈C1(E,ℝ) and Ψ′(z)w=∫ℝRz(t,z(t))w(t)dt, for all z,w∈E. Now, let us consider the function
(2.32)Φ(z)=12(‖z+‖2-‖z-‖2)-Ψ(z)
for z=z++z0+z-∈E, then Φ∈C1(E,ℝ). Moreover, for Ψ∈C0∞(ℝ), (2.33)Φ′(z)Ψ=∫R(-Jż(t)+B(t)z(t)-Rz(t,z(t)),Ψ(t))dt.
It follows that critical points of Φ(z) are solutions of (HS). Moreover, if z is a solution of (HS), by Lemma 2.3, Rz(t,z)∈Ls(ℝ,ℝ2N) for any s∈[2,∞). Thus, Rz(t,z)∈L2. A standard argument shows that z is also a homoclinic orbit of (HS) (see [12]). So if z≠0 is a solution of (HS), then z is a homoclinic orbit of (HS).Now, we discuss the linking structure ofΦ.Lemma 2.4.
Suppose (R1), (R2) are satisfied. Then, there is a ρ>0 such that κ:=infΦ(∂Bρ∩E+)>0.Proof.
Observe that, givenɛ>0, there is Cɛ>0 such that
(2.34)|Rz(t,z)|≤ɛ|z|+Cɛ|z|p-1,(2.35)|R(t,z)|≤ɛ|z|2+Cɛ|z|p,
for all (t,z), where p∈[2,∞). For z∈E+, by Lemma 2.3 and (2.35), we have
(2.36)Φ(z)=12‖z‖2-∫RR(t,z)≥12‖z‖2-C(ɛ‖z‖2+Cɛ‖z‖p).
Choosing an appropriate ɛ, we see that the desired conclusion holds for some ρ>0.For the asymptotically quadratic case, letb:=infR∞(t), and we arrange all the eigenvalues (counted with multiplicity) of A in (0,b) by 0<λ1≤λ2≤⋯≤λk<b and let ej denote the corresponding eigenfunctions Aej=λjej for j=1,2,…,k. Set Y0:=span{e1,e2,…,ek}. Note that(2.37)λ1|z|22≤‖z‖2≤λk|z|22∀z∈Y0.
For any subspace W of Y0, set EW=E-⊕E0⊕W.Lemma 2.5.
Let (R1)–(R3) be satisfied and ρ>0 given by Lemma 2.4. Then, for any subspace W of Y0, supΦ(EW)<∞, and there is a RW>0 such that supΦ(EW∖B¯RW)<infΦ(Bρ∩E+).Proof.
It is sufficient to prove thatΦ(z)→-∞ in EW as ∥z∥→∞. If not, then there are M>0 and {zj}⊂EW with ∥zj∥→∞ such that Φ(zj)≥-M for all j. Denote vj:=zj/∥zj∥, passing to a subsequence if necessary, vj⇀v,vj-⇀v-,vj0→v0, and vj+→v+. Now, we have
(2.38)12(‖vj+‖2-‖vj-‖2)-∫RR(t,zj)‖zj‖2=Φ(zj)‖zj‖2≥-M‖zj‖2,
since R(t,z)≥0. If v+=0 and v0=0, it follows from (2.38) that
(2.39)12‖vj-‖2+∫RR(t,zj)‖zj‖2≤12‖vj+‖2+M‖zj‖2⟶0
as j→∞. Thus, we have ∥vj-∥→0 and ∫ℝ(R(t,zj)/∥zj∥2)→0, this is a contradiction with ∥vj∥=1. Therefore, there are three possibilities: (i) v+≠0 and v0=0, (ii) v+=0 and v0≠0, (iii) v+≠0 and v0≠0.
If (i) holds, by (2.37), one has
(2.40)‖v+‖2-‖v-‖2-∫RR∞(t)v2≤‖v+‖2-‖v-‖2-b|v|22≤-(b-λk)|v+|22-‖v-‖2-b|v-|22-b|v0|22-b|v-|22<0.
If (ii) or (iii) holds, similar to (i), one has(2.41)‖v+‖2-‖v-‖2-∫RR∞(t)v2<0.
Then, there exists a>0 such that
(2.42)‖v+‖2-‖v-‖2-∫-aaR∞(t)v2<0.
Letting R̃(t,z):=R(t,z)-(1/2)R∞(t)z2, then |R̃(t,z)|≤cz2 for some c>0 and R̃(t,z)/z2→0 as |z|→∞ uniformly in t. Hence, by Lebesgues dominated convergence theorem, we have
(2.43)lim⁡j→∞∫-aaR̃(t,zj)‖zj‖2=lim⁡j→∞∫-aaR̃(t,zj)|zj|2|vj|2=0.
Thus, (2.38)–(2.43) imply that
(2.44)0≤lim⁡j→∞(12(‖vj+‖2-‖vj-‖2)-∫-aaR(t,zj)‖zj‖2)≤12(‖v+‖2-‖v-‖2-∫-aaR∞(x)v2)<0.
Now, the desired conclusion is obtained from this contradiction.For the superquadratic case, we defineYn:=span{e1,…,en} and En=E-⊕E0⊕Yn, n∈ℕ. Then, Yn is a finite dimensional subspace of E+ and(2.45)λ1|z|22≤‖z‖2≤λn|z|22∀z∈Yn.
By (R4), there exists R>0 and α>0 such that(2.46)R(t,z)≥α|z|2≥λn|z|2∀|z|≥R.Lemma 2.6.
Let (R1), (R4)–(R5) be satisfied and ρ>0 given by Lemma 2.4. Then, there is a sequence {Rn} with Rn>ρ such that supΦ(En)<∞ and supΦ(En∖B¯n)<infΦ(Bρ∩E+), where B¯n={z∈En:∥z∥≤Rn}.Proof.
Similar to proof of Lemma2.5. If (i) holds, by (2.45) and (2.46), it is easy to prove that
(2.47)‖v+‖2-‖v-‖2-α∫R|v|2≤λn‖v+‖2-‖v-‖2-α|v|22=-(α-λn)|v+|22-‖v-‖2-α|v-|22-α|v0|22<0.
If (ii) or (iii) holds, similar to (i), one has
(2.48)‖v+‖2-‖v-‖2-α∫R|v|2<0.
Hence, there exists a>0 such that
(2.49)‖v+‖2-‖v-‖2-α∫-aa|v|2<0.
Note that
(2.50)Φ(zj)‖zj‖2≤12(‖vj+‖2-‖vj-‖2)-α∫-aaR(t,zj)‖zj‖2=12(‖vj+‖2-‖vj-‖2-α∫-aa|vj|2)-∫-aaR(t,zj)-(α/2)|zj|2‖zj‖2≤12(‖vj+‖2-‖vj-‖2-α∫-aa|vj|2)+∫(-a,a)∩{t∈R:|zj|≤R}(α/2)|zj|2-R(t,zj)‖zj‖2≤12(‖vj+‖2-‖vj-‖2-α∫-aa|vj|2)+2aCR‖zj‖2,
where CR=sup{R(t,z):t∈(-a,a),|z|≤R}. Thus, (2.38) and (2.49) imply that
(2.51)0≤lim⁡j→∞(12(‖vj+‖2-‖vj-‖2)-∫-aaR(t,zj)‖zj‖2)≤12(‖v+‖2-‖v-‖2-α∫Ω|v|2)<0,
a contradiction. This proves the lemma.As a consequence, we have the folllowing.Lemma 2.7.
Under the assumptions of Theorem1.1 (Theorem 1.2), then letting e∈Y0(Yn) with ∥e∥=1, there is R1>ρ>0, such that Φ|∂Q≤κ, where κ>0 be given by Lemma 2.4, Q:={u=u-+u0+se:u-+u0∈E-⊕E0,s≥0,∥u∥≤R1}.


## 3. The(C)c Condition

In this section, we discuss the properties of the(C)c sequences. Recall that a sequence {zn}⊂E is said to be a (C)c sequence if Φ(zn)→c and (1+∥zn∥)Φ′(zn)→0, Φ is said to satisfy the (C)c condition if any (C)c sequence has a convergent subsequence.Lemma 3.1.
Under the assumptions of Theorem1.1 or Theorem 1.2, then any (C)c-sequence is bounded.Proof.
Let{zj}⊂E be such that
(3.1)Φ(zj)⟶c,(1+‖zj‖)Φ′(zj)⟶0.
Then, there is constant C>0 such that we have
(3.2)Φ(zj)-12Φ′(zj)zj=∫RR̂(t,zj)≤C.
Suppose to the contrary that {zj} is unbounded. Setting vj:=zj/∥zj∥, then ∥vj∥=1,vj=vj++vj0+vj-∈E+⊕E0⊕E-. After passing a subsequence, we have vj⇀v,vj0→v0, and γ=limj→∞∥vj++vj-∥2 exists. Moreover, |vj|s≤Cs for all s∈[2,∞). For γ, we have only the following two cases: γ=0 or γ>0.
First, we considerγ=0, then ∥vj0∥=|vj0|2→1=|v0|2. Denote z̃j=zj++zj-,ṽj=vj++vj-. Note that by (R3), for any ℓ>0, there exists a h>0 such that
(3.3)l≤R̂(t,z),∀|z|≥h.
For δ>0, set Ωδ:={t∈ℝ:|v0(t)|≥2δ} and Ωjδ:={t∈ℝ:|ṽj(t)|≥δ}. Since v0∈C1(ℝ) and |v0|2=1,|Ωδ|>0 for all δ small. Moreover, we have
(3.4)|Ωjδ|≤1δ2∫Ωjδ|ṽj|2≤cδ2‖ṽj‖2=cδ2‖vj++vj+‖⟶0asj⟶∞.
Hence, |Ωδ∖Ωjδ|→|Ωδ| as j→∞. Therefore, there exists N>0 such that |vj(t)|≥δ/2 for all t∈Ωδ∖Ωjδ with j≥N; thus, |zj(t)|≥(δ/2)∥zj∥≥h for j≥N. From this and (3.3), we get
(3.5)∫RR̂(t,zj)≥∫Ωδ∖Ωjδl=l|Ωδ∖Ωjδ|,
contradicting (3.2) since ℓ is arbitrary.
Hence,γ>0, observe that
(3.6)Φ′(zj)(zj+-zj-)=‖zj‖2(‖ṽj‖2-∫RRz(t,zj)(vj+-vj-)|vj||zj|).
Hence,
(3.7)∫RRz(t,zj)(vj+-vj-)|vj||zj|⟶γ2.
Set
(3.8)g(r):=inf⁡{R̂(t,z)∣t∈R,z∈R2Nwith|z|≥r}
for r≥0. By (R3), g(r)>0 for all r>0 and g(r)→∞ as r→∞. For 0≤a<b, let
(3.9)Ωj(a,b):={t∈R∣a≤|zj(t)|<b}Cab:=inf⁡{R̂(t,z)|z|2∣t∈R,z∈R2Nwitha≤|z(x)|<b}.
By (3.2), it is easy to prove that
(3.10)|Ωj(b,∞)|≤Cg(b)⟶0
as b→∞ uniformly in t, and, for any fixed 0<a<b,
(3.11)∫Ωj(a,b)|vj|2=1‖zj‖2∫Ωj(a,b)|zj|2≤CCab‖zj‖2⟶0
as j→∞, and
(3.12)∫Ωj(b,∞)|vj|s≤(∫Ωj(b,∞)|vj|2s)1/2|Ωj(b,∞)|1/2⟶0
for any s∈[2,∞), as b→∞ uniformly in j. Let 0<ɛ<γ2/3. By (R1), there is a aɛ>0 such that
(3.13)|Rz(t,z)|<ɛ|z|,∀|z|≤aɛ.
Consequently,
(3.14)∫Ωj(0,aɛ)Rz(t,zj)(vj+-vj-)|vj||zj|≤∫Ωj(0,aɛ)ɛ|vj+-vj-||vj|≤ɛC2|vj|22<ɛ
for all j.
By (R2), there is some c>0 such that
(3.15)|Rz(t,z)|<c|z|,∀(t,z).
By (3.11) and Hölder inequality, setting μ=2ν/(ν-1)>2 and ν′=μ/2=ν/(ν-1), we can take large bɛ such that
(3.16)∫Ωj(bɛ,∞)Rz(t,zj)(vj+-vj-)|vj||zj|≤(∫Ωj(bɛ,∞)|Rz(t,zj)|ν|zj|ν)1/ν(∫Ωj(bɛ,∞)(|vj+-vj-||vj|)ν′)1/ν′≤c|Ωj(bɛ,∞)|1/ν(∫Ωj(bɛ,∞)|vj+-vj-|μ)1/μ(∫Ωj(bɛ,∞)|vj|μ)1/μ≤ɛ
for all j. By (3.11), there is j0 such that
(3.17)∫Ωj(aɛ,bɛ)Rz(t,zj)(vj+-vj-)|vj||zj|≤c∫Ωj(aɛ,bɛ)|vj+-vj-||vj|≤c|vj|2(∫Ωj(aɛ,bɛ)|vj|2)1/2≤ɛ
for all j≥j0. By (3.14)–(3.17), we have
(3.18)lim⁡j→∞sup⁡∫RRz(t,zj)(vj+-vj-)|vj||zj|≤3ɛ<γ2
which contradicts (3.7).Next, we deal with the superquadratic case. Note that2μ′=2μ/(μ-1)<∞, by (3.12), we can take large bɛ>r such that
(3.19)∫Ωj(bɛ,∞)Rz(t,zj)(vj+-vj-)|vj||zj|≤(∫Ωj(bɛ,∞)(|Rz(t,zj)||zj|)μ)1/μ(∫Ωj(bɛ,∞)(|vj+-vj-||vj|)μ′)1/μ′≤(∫Rc1R̂(t,zj))1/μ(∫R|vj+-vj-|2μ′)1/2μ′(∫Ωj(bɛ,∞)|vj|2μ′)1/2μ′<ɛ
for all j. Clearly, (3.17) remains true. Now, the combination of (3.14), (3.17) and (3.19) implies again a contradiction with (3.7).Let{zn} be an arbitrary (C)c sequence. By Lemma 3.1, it is bounded; hence, we may assume without loss of generality that zn⇀z in E, zn→z in Lq for q≥2. Plainly, z is a critical point of Φ.Lemma 3.2.
Under the assumptions of Theorem1.1 or Theorem 1.2, then Φ satisfies (C)c condition for all c>0.Proof.
Let{zn} be an arbitrary (C)c sequence. By (2.35), Lemmas 2.3 and 3.1, it follows from
(3.20)∫R(Rz(t,zm)-Rz(t,zn))(zm+-zn+)≤(∫R(Rz(t,zm)-Rz(t,zn))2)1/2(∫R|zm+-zn+|)1/2≤c(∫R|zm-zn|2)1/2⟶0o(1)=(Φ′(zm)-Φ′(zn),zm+-zn+)=‖zm+-zn+‖2+∫R(Rz(t,zm)-Rz(t,zn))(zm+-zn+)=‖zm+-zn+‖2+o(1)
that z∥zm+-zn+∥2=o(1) as m,n→∞. Similarly, we can prove that z∥zm--zn-∥2=o(1) as m,n→∞. Since dimE0<∞, thus zn0→z0. So {zn} is a Cauchy sequence in E. Now, the conclusion follows from the completeness of E.


## 4. Proof of Theorems

In this section, we give the proof for our Theorems1.1 and 1.2. Let E be a Banach space with direct sum E=X⊕Y and corresponding projections PX,PY onto X,Y. Let 𝒮⊂X* be a dense subset, for each s∈𝒮, there is a seminorm on E defined by
(4.1)ps:E⟶R,ps(z):|s(x)|+‖y‖forz=x+y∈E.
We denote by 𝒯𝒮 the topology induced by seminorm family {ps}, w* denotes the weak*-topology on E*. Now, some notations are needed. For a functional Φ∈C1(E,ℝ), we write Φa={z∈E∣Φ(z)≥a}, Φb={z∈E∣Φ(z)≤b}, and Φab=Φa∩Φb. Recall that Φ is said to be weakly sequentially lower semicontinuous, if for any zj⇀z in E, one has Φ(z)≤liminfj→∞Φ(zj), and Φ′ is said to be weakly sequentially continuous if limj→∞Φ′(zj)w=Φ′(z)w for each w∈E.Suppose(Φ0)
for anyc∈ℝ, superlevel Φc is 𝒯𝒮-closed, and Φ′:(Φc,𝒯𝒮)→(E*,w*) is continuous;(Φ1)
for anyc>0, there exists ξ>0 such that ∥z∥<ξ∥PYz∥ for all z∈Φc;(Φ2)
there existsρ>0 such that κ:=infΦ(Sρ∩Y)>0, where Sρ:={z∈E:∥z∥=ρ};(Φ3)
there exists a finite-dimensional subspaceY0⊂Y and R>ρ such that we have for E0:=X⊕Y0 and B0:={z∈E0:∥z∥≤R} that c¯:=supΦ(E0)<∞ and supΦ(E0∖B0)<infΦ(Bρ∩Y);(Φ4)
there exists an increasing sequence of finite-dimensional subspacesYn⊂Y and a sequence of numbers Rn>ρ such that we have for En:=X⊕Yn and Bn:={z∈En:∥z∥≤Rn} that c¯:=supΦ(En)<∞ and supΦ(En∖Bn)<infΦ(Bρ∩Y);(Φ5)
Φ satisfies the (C)c condition for c>0.Now, we state three critical point theorems which will be used later (see [22]).Theorem 4.1.
LetΦ0–Φ2 be satisfied, and suppose there are R>ρ>0 and e∈Y with ∥e∥=1 such that supΦ(∂Q)≤κ where Q:={z=x+te:x∈X,t≥0,∥z∥<R}. If Φ satisfies the (C)c condition for all c≤c¯, then Φ has a critical point u with κ≤Φ(z)≤c¯.Theorem 4.2.
AssumeΦ is even and (Φ0), (Φ2)–(Φ3) be satisfied. Then, Φ has at least m:=dimY0 pairs of critical points with critical values less or equal to c¯ provided Φ satisfies the (C)ccondition for all c∈[κ,c¯].Theorem 4.3.
AssumeΦ is even with Φ(0)=0, and (Φ0)–(Φ2), (Φ4)–(Φ5) be satisfied. Then, Φ has an unbounded sequence of critical values.Lemma 4.4.
Under the assumptions of Theorem1.1 or Theorem 1.2, then, Ψ is nonnegative, weakly sequentially lower semicontinuous, and Ψ′ is weakly sequentially continuous. Moreover, Ψ′ is compact.In virtue of a standard way of [23], by Lemma 2.3, it is easy to prove this lemma, so we omit it here.Lemma 4.5.
Φ satisfies (Φ1).Proof.
For anyc>0 and u∈Φc, using the fact that R(t,z)≥0, one has
(4.2)0<c≤12(‖z+‖-‖z-‖)-∫RR(t,z)≤12‖z+‖.
This yields
(4.3)‖z+‖>‖z-‖,‖z+‖2≥2c.
Thus it suffices to show that there exists C>0 such that ∥z0∥≤C∥z+∥. If it is not true, then there is a sequence {zj}⊂Φc such that
(4.4)‖zj‖2≥‖zj0‖2≥j‖zj+‖2.
By (4.3), ∥zj0∥→∞ as j→∞, hence ∥zj∥→∞ as j→∞. Set vj=zj/∥zj∥, then ∥vj∥2=1 and ∥vj+∥2≤1/j→0, hence ∥vj-∥→0 as j→∞. Therefore, we can assume there is a subsequence denoted still by {vj}, vj⇀v, thus vj→v0 and ∥v0∥=1. Recall that R̃(t,z)=R(t,z)-(1/2)R∞(t)z2, then R̃(t,z)/|z|2→0 as |z|→∞ uniformly in t. Therefore, since |zj(t)|→∞ for v0(t)≠0, we have
(4.5)∫RR̃(t,zj)‖zj‖2≤∫RR̃(t,zj)|vj-v|2|zj|2+∫RR̃(t,zj)|v|2|zj|2≤∫v0(t)≠0R̃(t,zj)|v|2|zj|2+c|vj-v|22⟶0.
By (4.4),
(4.6)‖zj-+zj0‖2≥(j-1)‖zj+‖2≥(j-1)(2c+‖zj-‖2+2∫RR(t,zj))
or
(4.7)‖zj0‖2≥(j-1)2c+(j-2)‖zj-‖2+2(j-1)∫RR(t,zj),
which implies
(4.8)12(j-1)≥∫RR(t,zj)‖zj‖2=12∫RR∞(t)|vj|2+∫RR̃(t,zj)|vj|2|zj|2≥12b|vj|22+o(1),
consequently, v0=0, a contradiction.
Next, we handle the superquadratic case. Similar to the above proof, passing to a subsequence if necessary, we havevj→v0 and ∥v0∥=1. So there exists r>0 such that
(4.9)∫-rr|v0|2>0.
By (R4), we know that there exist m>0 and Cm such that R(t,z)≥Cm|z|2 for |z|≥m, and we have
(4.10)c‖zj‖2≤Φ(zj)‖zj‖2≤12(‖vj+‖2-‖vj-‖2)-Cm∫-rrR(t,zj)‖zj‖2=12(‖vj+‖2-‖vj-‖2-Cm∫-rr|vj|2)-∫-rrR(t,zj)-(Cm/2)|zj|2‖zj‖2≤12(‖vj+‖2-‖vj-‖2-Cm∫-rr|vj|2)+∫(-r,r)∩{t∈R:|zj|≤m}(Cm/2)|zj|2-R(t,zj)‖zj‖2≤12(‖vj+‖2-‖vj-‖2-Cm∫-rr|vj|2)+2rCm‖zj‖2.
Then,
(4.11)0≤lim⁡j→∞(12(‖vj+‖2-‖vj-‖2)-∫-rrR(t,zj)‖zj‖2)≤-Cm∫-rr|v0|2<0,
a contradiction.Proof of Theorem1.1.
Existence of a Least Energy Solution
WithX=E-⊕E0 and Y=E+, the condition (Φ0) holds by Lemma 4.4 and (Φ1) holds by Lemma 4.5. Lemma 2.4 implies (Φ2). Lemma 2.7 shows that Φ possesses the linking structure of Theorem 4.1, and Lemma 3.2 implies Φ satisfies (Φ5). Therefore, Φ has at least one critical point z with Φ(z)≥κ>0. Let 𝒦={z∈E∖{0}:Φ′(z)=0} be the set of nontrivial points of Φ. Then, 𝒦≠∅. We claim that θ=inf{Φ(z)∣z∈𝒦} is achieved. Let {zj}⊂𝒦 be a minimizing sequence for θ, then {zj} is bounded by Lemma 3.1. By Lemma 3.2, there exists a renamed subsequence such that zj→z, and
(4.12)θ=lim⁡j→∞Φ(zj)=Φ(z)
from which it follows that θ is achieved.Multiplicity
Φ is even provided R(t,z) is even in z. Lemma 2.5 says that Φ satisfies (Φ3) with dimY0=k. Therefore, Φ has at least k pairs of nontrivial critical points by Theorem 4.2.Proof of Theorem1.2.
Existence of a Least Energy Solution
Repeating the above proof, we know thatθ is achieved by some nonzero critical point.Existence of Infinitely Many Solutions
Φ is even provided R(t,z) is even in z. Lemma 2.6 says that Φ satisfies (Φ4). Therefore, Φ has an unbounded critical values by Theorem 4.3, and hence the equation (HS) has infinitely many solutions.



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*Source: 769232-2012-04-04.xml*