# Ground State Solutions for the Periodic Discrete Nonlinear Schrödinger Equations with Superlinear Nonlinearities

Authors: Ali Mai; Zhan Zhou
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider the periodic discrete nonlinear Schrödinger equations with the temporal frequency belonging to a spectral gap. By using the generalized Nehari manifold approach developed by Szulkin and Weth, we prove the existence of ground state solutions of the equations. We obtain infinitely many geometrically distinct solutions of the equations when specially the nonlinearity is odd. The classical
Ambrosetti-Rabinowitz superlinear condition is improved.


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## Body

## 1. Introduction

The following discrete nonlinear Schrödinger equation (DLNS):(1)iψ˙n=-Δψn+εnψn-σχnfn(ψn),n∈ℤ,
where σ=±1 and
(2)Δψn=ψn+1+ψn-1-2ψn
is the discrete Laplacian operator, appears in many physical problems, like polarons, energy transfer in biological materials, nonlinear optics, and so forth (see [1]). The parameter σ characterizes the focusing properties of the equation: if σ=1, the equation is self-focusing, while σ=-1 corresponds to the defocusing equation. The given sequences {εn} and {χn} are assumed to be T-periodic in n, that is, εn+T=εn and χn+T=χn. Moreover, {χn} is a positive sequence. Here, T is a positive integer. We assume that fn(0)=0 and the nonlinearity fn(u) is gauge invariant, that is,
(3)fn(eiθu)=eiθfn(u),θ∈ℝ.
We are interested in the existence of solitons of (1), that is, solutions which are spatially localized time-periodic and decay to zero at infinity. Thus, ψn has the form
(4)ψn=une-iωt,lim|n|→∞ψn=0,
where {un} is a real-valued sequence and ω∈ℝ is the temporal frequency. Then, (1) becomes
(5)-Δun+εnun-ωun=σχnfn(un),n∈ℤ,(6)lim|n|→∞un=0
holds. Naturally, if we look for solitons of (1), we just need to get the solutions of (5) satisfying (6).Actually, we consider a more general equation:(7)Lun-ωun=σχnfn(un),n∈ℤ,
with the same boundary condition (6). Here, L is a second-order difference operator
(8)Lun=anun+1+an-1un-1+bnun,
where {an} and {bn} are real-valued T-periodic sequences. When an≡-1 and bn=2+εn, we obtain (5).We consider (7) as a nonlinear equation in the space l2 of two-sided infinite sequences. Note that every element of l2 automatically satisfies (6).As it is well known, the operatorL is a bounded and self-adjoint operator in l2. The spectrum σ(L) is a union of a finite number of closed intervals, and the complement ℝ∖σ(L) consists of a finite number of open intervals called spectral gaps. Two of them are semi-infinite (see [2]). If T=1, then finite gaps do not exist. However, in general, finite gaps exist, and the most interesting case in (7) is when the frequency ω belongs to a finite spectral gap. Let us fix any spectral gap and denote it by (α,β).DNLS equation is one of the most important inherently discrete models. DNLS equation plays a crucial role in the modeling of a great variety of phenomena, ranging from solid state and condensed matter physics to biology (see [1, 3–6] and references therein). In the past decade, solitons of the periodic DNLS have become a hot topic. The existence of solitons for the periodic DNLS equations with superlinear nonlinearity [7–10] and with saturable nonlinearity [11–13] has been studied, respectively. If ω is below or above the spectrum of the difference operator -Δ+εn, solitons were shown by using the Nehari manifold approach and a discrete version of the concentration compactness principle in [14]. If ω is a lower edge of a finite spectral gap, the existence of solitons was obtained by using variant generalized weak linking theorem in [10]. If ω lies in a finite spectral gap, the existence of solitons was proved by using periodic approximations in combination with the linking theorem in [8] and the generalized Nehari manifold approach in [9], respectively. The results were extended by Chen and Ma in [7]. In this paper, we employ the generalized Nehari manifold approach instead of periodic approximation technique to obtain the existence of a kind of special solitons of (7), which called ground state solutions, that is, nontrivial solutions with least possible energy in l2. We should emphasize that the results are obtained under more general super nonlinearity than the classical Ambrosetti-Rabinowitz superlinear condition [8, 9, 15].This paper is organized as follows. In Section2, we first establish the variational framework associated with (7) and transfer the problem on the existence of solutions in l2 of (7) into that on the existence of critical points of the corresponding functional. We then present the main results of this paper and compare them with existing ones. Section 3 is devoted to the proofs of the main results.


## 2. Preliminaries and Main Results

The following are the basic hypotheses to establish the main results of this paper:(V1)
ω∈(α,β),(f1)
fn∈C(ℝ,ℝ) and fn+T(u)=fn(u), and there exist a>0 and p∈(2,∞) such that
(9)|fn(u)|≤a(1+|u|p-1)∀n∈ℤ,u∈ℝ,(f2)
fn(u)=o(|u|)asu→0,(f3)
lim|u|→∞Fn(u)/u2=∞, where Fn(u) is the primitive function of fn(u), that is,
(10)Fn(u)=∫0ufn(s)ds,(f4)
u↦fn(u)/|u| is strictly increasing on (-∞,0) and (0,∞).To state our results, we introduce some notations. Let(11)A=L-ω,E=l2(ℤ).Consider the functionalJ defined on E by
(12)J(u)=12(Au,u)E-σ∑n∈ℤ‍χnFn(un),
where (·,·)E is the inner product in E and ∥·∥E is the corresponding norm in E. The hypotheses on fn(u) imply that the functional J∈C1(E,ℝ) and (7) is easily recognized as the corresponding Euler-Lagrange equation for J. Thus, to find nontrivial solutions of (7), we need only to look for nonzero critical points of J in E.For the derivative ofJ, we have the following formula:
(13)(J′(u),v)=(Au,v)E-σ∑n∈ℤ‍χnfn(un)vn,∀v∈E.By(V1), we have σ(A)⊂ℝ∖(α-ω,β-ω). So, E=E+⊕E- corresponds to the spectral decomposition of A with respect to the positive and negative parts of the spectrum, and
(14)(Au,u)E≥(β-ω)∥u∥E2,u∈E+,(Au,u)E≤(α-ω)∥u∥E2,u∈E-.
For any u,v∈E, letting u=u++u- with u±∈E± and v=v++v- with v±∈E±, we can define an equivalent inner product (·,·) and the corresponding norm ∥·∥ on E by
(15)(u,v)=(Au+,v+)E-(Au-,v-)E,∥u∥=(u,u)1/2,
respectively. So, J can be rewritten as
(16)J(u)=12∥u+∥2-12∥u-∥2-σ∑n∈ℤ‍χnFn(un).We define foru∈E∖E-, the subspace
(17)E(u):=ℝu+E-=ℝu+⊕E-,
and the convex subset
(18)E^(u):=ℝ+u+E-=ℝ+u+⊕E-,
of E, where, as usual, ℝ+=[0,∞). Let
(19)ℳ={u∈E∖E-:J′(u)u=0,J′(u)v=0∀v∈E-},(20)c=infu∈ℳJ(u).In this paper, we also consider the multiplicity of solutions of (7).For eachℓ∈ℤ, let
(21)ℓ*u=(un+ℓT)n∈ℤ,∀u=(un)n∈ℤ,
which defines a ℤ-action on E. By the periodicity of the coefficients, we know that both J and J′ are ℤ-invariants. Therefore, if u∈E is a critical point of J, so is ℓ*u. Two critical points u1,u2∈E of J are said to be geometrically distinct if u1≠ℓ*u2 for all ℓ∈ℤ.Now, we are ready to state the main results.Theorem 1.
Suppose that conditions(V1),(f1)–(f4) are satisfied. Then, one has the following conclusions.(1)
If eitherσ=1 and β≠∞ or σ=-1 and α≠-∞, then (7) has at least a nontrivial ground state solution.(2)
If eitherσ=1 and β=∞ or σ=-1 and α=-∞, then (7) has no nontrivial solution.Theorem 2.
Suppose that conditions(V1),(f1)–(f4) are satisfied and fn is odd in u. If either σ=1 and β≠∞ or σ=-1 and α≠-∞, then (7) has infinitely many pairs of geometrically distinct solutions.In what follows, we always assume thatσ=1. The other case can be reduced to σ=1 by switching L to -L and ω to -ω.Remark 3.
In [8], the author considered (7) with fn defined by
(22)fn(u)=|u|2u,
which obviously satisfies (f1)–(f4); the author also discussed the case where f satisfies the Ambrosetti-Rabinowitz condition; that is, there exists μ>2 such that
(23)0<μFn(u)≤fn(u)u,u≠0.
Clearly, (23) implies that Fn(u)≥c|u|μ>0 for |u|≥1. So, it is a stronger condition than (f3).Remark 4.
In [9], the author assumed that fn satisfies the following condition: there exists θ∈(0,1) such that
(24)0<u-1fn(u)≤θfn′(u),u≠0.
Obviously, (24) implies (23) with μ=1+(1/θ), so it is a stronger condition than the Ambrosetti-Rabinowitz condition. In our paper, the nonlinearities satisfy more general superlinear assumptions instead of (24) which also implies (f4). However, we do not assume that fn is differentiable and satisfies (24), ℳ is not a C1 manifold of E, and the minimizers on ℳ may not be critical points of J. Hence, the method of [9] does not apply any more. Nevertheless, ℳ is still a topological manifold, naturally homeomorphic to the unit sphere in E+ (see in detail in Section 3). We use the generalized Nehari manifold approach developed by Szulkin and Weth which is based on reducing the strongly indefinite variational problem to a definite one and prove that the minimizers of J on ℳ are indeed critical points of J.Remark 5.
In [7], it is shown that (7) has at least a nontrivial solution u∈l2 if f satisfies (V1), (f2), (f3), and the following conditions:(B1)
Fn(u)≥0 for any u∈ℝ and Hn(u):=(1/2)fn(u)u-Fn(u)>0 if u≠0,(B2)
Hn(u)→∞ as |u|→∞, and there exist r0>0 and γ>1 such that |fn(u)|γ/|u|γ≤c0Hn(u) if |u|≥r0, where c0 is a positive constant,In our paper, we use (9) and (f4) instead of (B1) and (B2).


## 3. Proofs of Main Results

We assume that(V1) and (f1)–(f4) are satisfied from now on.Lemma 6.
F
n
(
u
)
>
0 and (1/2)fn(u)u>Fn(u) for all u≠0.Proof.
By(f2) and (f4), it is easy to get that
(25)Fn(u)>0∀u≠0.
Set Hn(u)=(1/2)fn(u)u-Fn(u). It follows from (f4) that
(26)Hn(u)=u2fn(u)-∫0ufn(s)dsHn(u)>u2fn(u)-fn(u)u∫0usds=0.
So, (1/2)fn(u)u>Fn(u) for all u≠0.To continue the discussion, we need the following proposition.Proposition 7 (see [16, 17]).
Letu,s,v∈ℝ be numbers with s≥-1 and w:=su+v≠0. Then,
(27)fn(u)[s(s2+1)u+(1+s)v]+Fn(u)-Fn(u+w)<0.Lemma 8.
Ifu∈ℳ, then
(28)J(u+w)<J(u)foreveryw∈U:={su+v:s≥-1,v∈E-},w≠0.
Hence, u is the unique global maximum of J|E^(u).Proof.
We rewriteJ by
(29)J(u)=12(Au+,u+)E+12(Au-,u-)E-σ∑n∈ℤ‍χnFn(un).
Sinceu∈ℳ, we have
(30)0=(J′(u),2s+s22u+(1+s)v)0=2s+s22(Au+,u+)E+2s+s22(Au-,u-)E0=+(1+s)(Au-,v)E0=-∑n∈ℤ‍χnfn(un)(2s+s22un+(1+s)vn).
Together with Proposition7, we know that
(31)J(u+w)-J(u)=12{(A(1+s)u+,(1+s)u+)E-(Au+,u+)E}+12{(A((1+s)u-+v),(1+s)u-+v)E-(Au-,u-)E}+∑n∈ℤ‍χnFn(un)-∑n∈ℤ‍χnFn(un+wn)=2s+s22(Au+,u+)E+2s+s22(Au-,u-)E+12(Av,v)E+(1+s)(Au-,v)E+∑n∈ℤ‍χnFn(un)-∑n∈ℤ‍χnFn(un+wn)=12(Av,v)E+∑n∈ℤ‍χn{fn(un)[s(s2+1)un+(1+s)vn]=12(Av,v)E+∑n∈ℤ‍χn+Fn(un)-Fn(un+wn)(s2+1)}<0.
The proof is complete.Lemma 9.
(a) There existsα>0 such that c:=infℳJ(u)≥infSαJ(u)>0, where Sα:={u∈E+:∥u∥=α}.
(b)∥u+∥≥max{∥u-∥,2c} for every u∈ℳ.Proof.
(a) By(f1) and (f2), it is easy to show that for any ε>0, there exists cε>0 such that
(32)|fn(u)|≤ε|u|+cε|u|p-1,|Fn(u)|≤ε|u|2+cε|u|p.∥·∥ is equivalent to the E norm on E+ and E⊂lq for 2≤q≤∞ with ∥u∥lq≤∥u∥E. Hence, for any ε∈(0,1/2) and u∈E+, we have
(33)J(u)≥12∥u∥2-ε∥u∥2-cεχ-∥u∥p,
which implies infSαJ(u)>0 for some α>0 (small enough), where χ-=max{χn}.
The first inequality is a consequence of Lemma8 since for every u∈ℳ, there is s>0 such that su+∈E^(u)∩Sα.
(b) Foru∈ℳ, by (25), we have
(34)c≤12∥u+∥2-12∥u-∥2-∑n∈ℤ‍χnFn(un)c≤12(∥u+∥2-∥u-∥2).
Hence, ∥u+∥≥max{∥u-∥,2c}.Lemma 10.
Let𝒲⊂E+∖{0} be a compact subset. Then, there exists R>0 such that J≤0 on E(u)∖BR(0) for every u∈𝒲, where BR(0) denotes the open ball with radius R and center 0.Proof.
Suppose by contradiction that there existu(k)∈𝒲 and w(k)∈E(u(k)), k∈ℕ, such that J(w(k))>0 for all k and ∥w(k)∥→∞ as k→∞. Without loss of generality, we may assume that ∥u(k)∥=1 for k∈ℤ. Then, there exists a subsequence, still denoted by the same notation, such that u(k)→u∈E+. Set v(k)=w(k)/∥w(k)∥=s(k)u(k)+v(k)-. Then,
(35)0<J(w(k))∥w(k)∥2=12((s(k))2-∥v(k)-∥2)-∑n∈ℤ‍χnFn(wn(k))(wn(k))2(vn(k))2.
By (25), we have
(36)∥v(k)-∥2≤(s(k))2=1-∥v(k)-∥2.
Consequently, we know that ∥v(k)-∥≤1/2 and 1/2≤s(k)≤1. Passing to a subsequence if necessary, we assume that s(k)→s∈[1/2,1], v(k)⇀v, v(k)-⇀v*-∈E-, and vn(k)→vn for every n. Hence, v=su+v*-≠0 and v*-=v-. It follows that for n0∈ℤ with vn0≠0, |wn0(k)|=∥w(k)∥·|vn0(k)|→∞, as k→∞. Then, by (f3), we have
(37)∑n∈ℤ‍χnFn(wn(k))(wn(k))2(vn(k))2⟶∞,
which contradicts with (35).Lemma 11.
For eachu∈E+∖{0}, the set ℳ∩E^(u) consists of precisely one point which is the unique global maximum of J|E^(u).Proof.
By Lemma8, it suffices to show that ℳ∩E^(u)≠∅. Since E^(u)=E^(u+/∥u+∥), we may assume that u∈S+. By Lemma 10, there exists R>0 such that J≤0 on E(u)∖BR(0) provided that R is large enough. By Lemma 9 (a), J(tu)>0 for small t>0. Moreover, J≤0 on E^(u)∖BR(0). Hence, 0<supE^(u)J<∞.
Letv(k)⇀v in E^(u). Then, vn(k)→vn as k→∞ for all n after passing to a subsequence if necessary. Hence, Fn(vn(k))→Fn(vn). Let φ(v)=∑n∈ℤχnFn(vn). Then,
(38)φ(v)=∑n∈ℤ‍limk→∞χnFn(vn(k))φ(v)≤liminfk→∞∑n∈ℤ‍χnFn(vn(k))φ(v)=liminfk→∞φ(v(k));
that is, φ is a weakly lower semicontinuous. From the weak lower semi-continuity of the norm, it is easy to see that J is weakly upper semicontinuous on E^(u). Therefore, J(u0)=supE^(u)J for some u0∈E^(u)∖{0}. By the proof of Lemma 10, u0 is a critical point of J|E^(u). It follows that (J′(u0),u0)=(J′(u0),z)=0 for all z∈E and hence u0∈ℳ. To summarize, u0∈ℳ∩E^(u).According to Lemma11, for each u∈E+∖{0}, we may define the mapping m^:E+∖{0}→ℳ, u↦m^(u), where m^(u) is the unique point of ℳ∩E^(u).Lemma 12.
J is coercive on ℳ; that is, J(u)→∞ as ∥u∥→∞, u∈ℳ.Proof.
Suppose, by contradiction, that there exists a sequence{u(k)}⊂ℳ such that ∥u(k)∥→∞ and J(u(k))≤d for some d∈[c,∞). Let v(k)=u(k)/∥u(k)∥. Then, there exists a subsequence, still denoted by the same notation, such that v(k)⇀v and vn(k)→vn for every n as k→∞.
First, we know that there existδ>0 and nk∈ℤ such that
(39)|vnk(k)+|≥δ.
Indeed, if not, then v(k)+→0 in l∞ as k→∞. By Lemma 9(b), 1/2≤∥v(k)+∥2≤1, which means that ∥v(k)+∥l2 is bounded. For q>2,
(40)∥v(k)+∥lqq≤∥v(k)+∥l∞q-2∥v(k)+∥l22.
Then, v(k)+→0 in all lq,q>2. By (32), for any s∈ℝ,
(41)∑n∈ℤ‍χnFn(svn(k)+)≤εs2χ-∥v(k)+∥l22+cεspχ-∥v(k)+∥lpq,
which implies that ∑n∈ℤχnFn(svn(k)+)→0 as k→∞.
Sincesv(k)+∈E^(u(k)) for s≥0, Lemma 8 implies that
(42)d≥J(u(k))≥J(sv(k)+)d≥J(u(k))=s22∥v(k)+∥2-∑n∈ℤ‍χnFn(svn(k)+)d≥J(u(k))≥s24-∑n∈ℤ‍χnFn(svn(k)+)⟶s24,
as k→∞. This is a contradiction if s>4d.
Due to the periodicity of coefficients, bothJ and ℳ are invariant under T-translation. Making such shifts, we can assume that 1≤nk≤T-1 in (39). Moreover, passing to a subsequence if needed, we can assume that nk=n0 is independent of k. Next, we may extract a subsequence, still denoted by {v(k)}, such that vn(k)+→vn+ for all n∈ℤ. In particular, for n=n0, inequality (39) shows that |vn0+|≥δ and hence v+≠0.
Since|un(k)|→∞ as k→∞, it follows again from (f3) and Fatou’s lemma that
(43)0≤J(u(k))∥u(k)∥2=12(∥v(k)+∥2-∥v(k)-∥2)-∑n∈ℤ‍χnFn(un(k))(un(k))2×(vn(k))V⟶-∞ask⟶∞,
a contradiction again. The proof is finished.Lemma 13.
(a) The mappingm^:E+∖{0}→ℳ is continuous.
(b) The mappingm=m^|S+:S+→ℳ is a homeomorphism between S+ and ℳ, and the inverse of m is given by m-1(u)=u+/∥u+∥, where S+:={u∈E+:∥u∥=1}.
(c) The mappingm-1:ℳ↦S+ is the Lipschitz continuous.Proof.
(a) Let(u(k))⊂E+∖{0} be a sequence with u(k)→u. Since m^(w)=m^(w+/∥w+∥), without loss of generality, we may assume that ∥u(k)∥=1 for all k. Then, m^(u(k))=∥m^(u(k))+∥u(k)+m^(u(k))-. By Lemma 10, there exists R>0 such that
(44)J(m^(u(k)))=supE(u(k))J≤supBR(0)JJ(m^(u(k)))≤supu∈BR(0)∥u+∥2=R2foreveryk.
It follows from Lemma 12 that m^(u(k)) is bounded. Passing to a subsequence if needed, we may assume that
(45)t(k):=∥m^(u(k))+∥⟶t,m^(u(k))-⇀u*-inEask→∞,
where t≥2c>0 by Lemma 9(b). Moreover, by Lemma 11,
(46)J(m^(u(k)))≥J(t(k)u(k)+m^(u)-)⟶J(tu+m^(u)-)J(m^(u(k)))=J(m^(u)).
Therefore, using the weak lower semicontinuity of the norm and φ (defined in Lemma 11), we get
(47)J(m^(u))≤limk⟶∞J(m^(u(k)))J(m^(u))=limk→∞(∑n∈ℤ‍12(t(k))2-12∥m^(u(k))-∥2J(m^(u))≤limk⟶∞J-∑n∈ℤ‍χnFn(m^(un(k))))J(m^(u))≤12t2-12∥u*-∥2-∑n∈ℤ‍χnFn(tun+u*,n-)J(m^(u))=J(tu+u*-)≤J(m^(u)),
which implies that all inequalities above must be equalities and m^(u(k))-→u*-. By Lemma 11, u*-=m^(u)- and hence m^(u(k))→m^(u).
(b) This is an immediate consequence of (a).
(c) Foru,v∈ℳ, by (b), we have
(48)∥m-1(u)-m-1(v)∥=∥u+∥u+∥-v+∥v+∥∥∥m-1(u)-m-1(v)∥=∥u+-v+∥u+∥+(∥v+∥-∥u+∥)v+∥u+∥∥v+∥∥∥m-1(u)-m-1(v)∥≤2∥u+∥∥(u-v)+∥≤2c∥u-v∥.We will consider the functionalΨ^:E+∖{0}→ℝ and Ψ:S+→ℝ defined by
(49)Ψ^:=J(m^(w)),Ψ:=Ψ^|S+.Lemma 14.
(a)Ψ^∈C1(E+∖{0},ℝ) and
(50)Ψ^′(w)z=∥m^(w)+∥∥w∥J′(m^(w))z∀w,z∈E+,w≠0.
(b)Ψ∈C1(S+,ℝ) and
(51)Ψ′(w)z=∥m^(w)+∥J′(m(w))z∀z∈TwS+Ψ′(w)z={v∈E+:(w,v)=0}.
(c){wn} is a Palais-Smale sequence for Ψ if and only if {m(wn)} is a Palais-Smale sequence for J.
(d)w∈S+ is a critical point of Ψ if and only if m(w)∈ℳ is a nontrivial critical point of J. Moreover, the corresponding values of   Ψ and J coincide and infS+Ψ=infℳJ=c.Proof.
(a) We putu=m^(w)∈ℳ, so we have u=(∥u+∥/∥w∥)w+u-. Let z∈E+. Choose δ>0 such that wt:=w+tz∈E+∖{0} for |t|<δ and put ut=m^(wt)∈ℳ. We may write ut=stwt+ut- with st>0. From the proof of Lemma 13, the function t↦st is continuous. Then, s0=∥u+∥/∥w∥. By Lemma 11 and the mean value theorem, we have
(52)Ψ^(wt)-Ψ^(w)=J(ut)-J(u)Ψ^(wt)-Ψ^(w)=J(stwt+ut-)-J(s0w+u-)Ψ^(wt)-Ψ^(w)≤J(stwt+ut-)-J(stw+ut-)Ψ^(wt)-Ψ^(w)=J′(st[w+ηt(wt-w)]+ut-)sttz
with some ηt∈(0,1). Similarly,
(53)Ψ^(wt)-Ψ^(w)=J(stwt+ut-)-J(s0w+u-)Ψ^(wt)-Ψ^(w)≥J(s0wt+u-)-J(s0w+u-)Ψ^(wt)-Ψ^(w)=J′(s0[w+τt(wt-w)]+u-)s0tz,
with some τt∈(0,1). Combining these inequalities and the continuity of function t↦st, we have
(54)limt⟶0Ψ^(wt)-Ψ^(w)t=s0J′(u)z=∥m^(w)+∥∥w∥J′(m^(w))z.
Hence, the Gâteaux derivative of Ψ^ is bounded linear in z and continuous in w. It follows that Ψ^ is of class C1 (see [15]).
(b) It follows from (a) by noting thatm(w)=m^(w) since w∈S+.
(c) Let{wn} be a Palais-Smale sequence for Ψ, and let un=m(wn)∈ℳ. Since for every n∈ℤ, we have an orthogonal splitting E=TwnS+⊕E(wn); using (b), we have
(55)∥Ψ′(wn)∥=supz∈TwnS+∥z∥=1Ψ′(wn)z∥Ψ′(wn)∥=∥m(wn)+∥supz∈TwnS+∥z∥=1J′(m(wn))z∥Ψ′(wn)∥=∥un+∥supz∈TwnS+∥z∥=1J′(un)z,
because J′(un)v=0 for all v∈E(wn) and E(wn) is orthogonal to TwnS+. Using (b) again, we have
(56)∥Ψ′(wn)∥≤∥un+∥∥J′(un)∥=∥un+∥supz∈TwnS+,v∈E(wn)z+v≠0J′(un)(z+v)∥z+v∥≤∥un+∥supz∈TwnS+∖{0}J′(un)(z)∥z∥=∥Ψ′(wn)∥.
Therefore,
(57)∥Ψ′(wn)∥=∥un+∥∥J′(un)∥.
According to Lemma 9(b) and Lemma 12, 2c≤∥un+∥≤supn∥un+∥<∞. Hence, {wn} is a Palais-Smale sequence for Ψ if and only if {un} is a Palais-Smale sequence for J.
(d) By (57), Ψ′(w)=0 if and only if J′(m(w))=0. The other part is clear.Proof of Theorem1.
(1) We know thatc>0 by Lemma 9(a). If u0∈ℳ satisfies J(u0)=c, then m-1(u0)∈S+ is a minimizer of Ψ and therefore a critical point of Ψ and also a critical point of J by Lemma 14. We shall show that there exists a minimizer u∈ℳ of J|ℳ. Let {w(k)}⊂S+ be a minimizing sequence for Ψ. By Ekeland’s variational principle, we may assume that Ψ(w(k))→c and Ψ′(w(k))→0 as k→∞. Then, J(u(k))→c and J′(u(k))→0 as k→∞ by Lemma 14(c), where u(k):=m(w(k))∈ℳ. By Lemma 12, {u(k)} is bounded, and hence {u(k)} has a weakly convergent subsequence.
First, we show that there existδ>0 and nk∈ℤ such that
(58)|unk(k)|≥δ.
Indeed, if not, then u(k)→0 in l∞ as k→∞. From the simple fact that for q>2,
(59)∥u(k)∥lqq≤∥u(k)∥l∞q-2∥u(k)∥l22,
we have u(k)→0 in all lq,q>2. By (32), we know that
(60)∑n∈ℤ‍χnfn(un(k))un(k)+≤εχ-∑n∈ℤ‍|un(k)|·|un(k)+|∑n∈ℤ‍χnfn(un(k))un(k)+≤+cεχ-∑n∈ℤ‍|un(k)|p-1·|un(k)+|∑n∈ℤ‍χnfn(un(k))un(k)+≤εχ-∥u(k)∥l2·∥u(k)+∥l2∑n∈ℤ‍χnfn(un(k))un(k)+≤+cεχ-∥u(k)∥lpp-1·∥u(k)+∥lp∑n∈ℤ‍χnfn(un(k))un(k)+≤εχ-∥u(k)∥l2·∥u(k)+∥∑n∈ℤ‍χnfn(un(k))un(k)+≤+cεχ-∥u(k)∥lpp-1·∥u(k)+∥,
which implies that ∑n∈ℤχnfn(un(k))un(k)+=o(∥u(k)+∥) as k→∞. Therefore,
(61)o(∥u(k)+∥)=(J′(u(k)),u(k)+)o(∥u(k)+∥)=∥u(k)+∥2-∑n∈ℤ‍χnfn(un(k))un(k)+o(∥u(k)+∥)=∥u(k)+∥2-o(∥u(k)+∥).
Then, ∥u(k)+∥2→0 as k→∞, contrary to Lemma 9(b).
From the periodicity of the coefficients, we know thatJ and J′ are both invariant under T-translation. Making such shifts, we can assume that 1≤nk≤T-1 in (58). Moreover, passing to a subsequence, we can assume that nk=n0 is independent of k.
Next, we may extract a subsequence, still denoted by{u(k)}, such that u(k)⇀u and un(k)→un for all n∈ℤ. Particularly, for n=n0, inequality (58) shows that |un0|≥δ, so u≠0. Moreover, we have
(62)(J′(u),v)=limk→∞(J′(u(k)),v)=0,∀v∈E;
that is, u is a nontrivial critical point of J.
Finally, we show thatJ(u)=c. By Lemma 6 and Fatou’s lemma, we have
(63)c=limk→∞(J(u(k))-12J′(u(k))u(k))c=limk→∞∑n∈ℤ‍χn(12fn(un(k))un(k)-Fn(un(k)))c≥∑n∈ℤ‍χn(12fn(un)un-Fn(un))c=J(u)-12J′(u)u=J(u)≥c.
Hence, J(u)=c. That is, u is a nontrivial ground state solution of (7).
(2) Ifβ=∞, by way of contradiction, we assume that (7) has a nontrivial solution u∈E. Then, u is a nonzero critical point of J in E. Thus, J′(u)=0. But by Lemma 6,
(64)(J′(u),u)=((L-ω)u,u)-∑n∈ℤ‍χnfn(un)un<0.
This is a contradiction, so the conclusion holds.
This completes the proof of Theorem1.Now, we are ready to prove Theorem2. From now on, we always assume that fn is odd in u. We need some notations. For a≥b≥c, denote
(65)Ja={u∈ℳ:J(u)≤a},Jb:={u∈ℳ:J(u)≥b},Jba=Ja∩Jb,Ψa={w∈S+:Ψ(w)≤a},Ψb:={w∈S+:Ψ(w)≥b},Ψba=Ψa∩Ψb,K={w∈S+:Ψ′(w)=0},Ka={w∈K:Ψ(w)=a},ν(a)=sup{∥u∥:u∈Ja}.It is easy to see thatν(a)<∞ for every a by Lemma 12.Proof of Theorem2.
It is easy to see that mappingsm,m-1 are equivariant with respect to the ℤ-action by Lemma 13; hence, the orbits 𝒪(u)⊂ℳ consisting of critical points of J are in 1-1 correspondence with the orbits 𝒪(w)⊂S+ consisting of critical points of Ψ by Lemma 14(d). Next, we may choose a subset ℱ⊂K such that ℱ=-ℱ and ℱ consists of a unique representative of ℤ-orbits. So, we only need to prove that the set ℱ is infinite. By contradiction, we assume that
(66)ℱisafiniteset.
Let
(67)Γj={A⊂S+:A=-A,Aisclosedandγ(A)≥j},
where γ denotes genus and j∈ℕ. We consider the sequence of the Lusternik-Schnirelmann values of Ψ defined by
(68)ck=inf{d∈ℝ:γ(Ψd)≥k,k∈ℕ}.
Now, we claim that(69)Kck≠∅,ck<ck+1.
Firstly, we show that(70)κ=inf{∥v-w∥:v,w∈K,v≠w}>0.
In fact, there exist v(k),w(k)∈ℱ, and gk,lk∈ℤ such that v(k)*gk≠w(k)*lk for all k and
(71)∥v(k)*gk-w(k)*lk∥⟶κask⟶∞.
Let mk=gk-lk. Passing to a subsequence, v(k)=v∈ℱ, w(k)=w∈ℱ, and either mk=m∈ℤ for all k or |mk|→∞. In the first case, 0<∥v(k)*gk-w(k)*lk∥=∥v-w*m∥=κ for all k. In the second case, w*mk⇀0 and therefore κ=limk→∞∥v-w*mk∥≥∥v∥=1. By (70), γ(Kck)=0 or 1.
Next, we consider a pseudogradient vector field ofΨ[18]; that is, there exists a Lipschitz continuous map V: S+∖K→TwS+ and for all w∈S+∖K,
(72)∥V(w)∥<2∥Ψ′(w)∥,〈V(w),Ψ′(w)〉>12∥Ψ′(w)∥2.
Let η:𝒟→S+∖K be the corresponding Ψ-decreasing flow defined by
(73)ddtη(t,w)=-V(η(t,w)),η(0,w)=w,
where 𝒟={(t,w):w∈S+∖K,T-(w)<t<T+(w)}⊂ℝ×(S+∖K), and T-(w)<0, T+(w)>0 are the maximal existence times of the trajectory t→η(t,w) in negative and positive direction. By the continuity property of the genus, there exists δ>0 such that γ(U-)=γ(Kck), where U=Nδ(Kck):={w∈S+:dist(w,Kck)<δ} and δ<κ/2. Following the deformation argument (Lemma A.3), we choose ε=ε(δ)>0 such that
(74)limt→T+(w)Ψ(η(t,w))<ck-εforw∈Ψck+ε∖U.
Then, for every w∈Ψck+ε∖U, there exists t∈[0,T+(w)) such that Ψ(η(t,w))<ck-ε. Hence, we may define the entrance time map
(75)r:w∈Ψck+ε∖U⟶[0,∞),r(w)=inf{t∈[0,T+(w)):Ψ(η(t,w))≤ck-ε},
which satisfies r(w)<T+(w) for every w∈Ψck+ε∖U. Since ck-ε is not a critical value of Ψ by (74), it is easy to see that r is a continuous and even map. It follows that the map
(76)g:Ψck+ε∖U⟶Ψck-ε,g(w)=η(r(w),w)
is odd and continuous. Then, γ(Ψck+ε∖U)≤γ(Ψck-ε)≤k-1, and consequently,
(77)γ(Ψck+ε)≤γ(U-)+k-1=γ(Kck)+k-1.
So, γ(Kck)≥1. Therefore, Kck≠∅. Moreover, the definition of ck and of ck+1 implies that γ(Kck)≥1 if ck<ck+1 and γ(Kck)>1 if ck=ck+1. Since γ(ℱ)=γ(Kck)≤1, ck<ck+1. Therefore, there is an infinite sequence {±wk} of pairs of geometrically distinct critical points of Ψ with Ψ(wk)=ck, which contradicts with (66). Therefore, the set ℱ is infinite.
This completes the proof of Theorem2.



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*Source: 317139-2013-04-15.xml*