This paper concerns solutions for the Hamiltonian system: z˙=𝒥Hz(t,z).
Here H(t,z)=(1/2)z·Lz+W(t,z), L
is a 2N×2N
symmetric matrix, and W∈C1(ℝ×ℝ2N,ℝ). We consider the case that 0∈σc(−(𝒥(d/dt)+L)) and W
satisfies some superquadratic condition different from the type of Ambrosetti-Rabinowitz. We study this
problem by virtue of some weak linking theorem recently developed and prove the
existence of homoclinic orbits.

1. Introduction and the Main Results

In this paper, we consider the existence of homoclinic orbits for the following Hamiltonian system:
ż=𝒥Hz(t,z),
where H(t,z)=(1/2)z·Lz+W(t,z), L is a 2N×2N symmetric matrix-valued function, and W∈C1(ℝ×ℝ2N,ℝ) is superquadratic both around 0 and at infinity in z∈ℝ2N.

A solution of (1.1) is called to be homoclinic to 0 if z(t)≢0 and z(t)→0 as |t|→∞.

In recent years, the existence and multiplicity of homoclinic orbits for Hamiltonian systems have been investigated in many papers via variational methods. See, for example, [1–7] for the second-order systems and [8–12] for the first-order systems. We note that in most of the papers on the first order system (1.1) it was assumed that

L is constant such that sp(𝒥L)∩iℝ=∅, where sp(𝒥L) denotes the set of all eigenvalues of 𝒥L.

Thus, if we let σ(A) denote the spectrum of A, (◊) means that L is independent of t and there is α>0 such that (-α,α)∩σ(A)=∅. Consequently, the operator A:=-(𝒥(d/dt)+L):W1,p(ℝ,ℝ2N)→Lp(ℝ,ℝ2N) is a homeomorphism for all p>1. This is important for the variational arguments. Later in [13], Ding considered the case that L depends periodically on t. He made assumptions on L such that 0 lies in a gap of σ(A). If additionally W(t,z) is periodic in t and satisfies some superquadratic or asymptotically quadratic conditions in z at infinity, then infinitely many homoclinic orbits were obtained.

If 0∈σc(A), then the problem is quite different in nature since the operator A cannot lead the behavior at 0 of the equation. Ding and Willem considered this case in [14]. They assumed that

L(t)∈C(ℝ,ℝ4N2) is 1-periodic. There exists α>0 such that (0,α)∩σ(A)=∅.

Under (A0), 0 may belong to continuous spectrum of A. The authors managed to construct an appropriate Banach space, on which some embedding results necessary for variational arguments were obtained. Using a generalized linking theorem developed by Kryszewski and Szulkin in [15], they got one homoclinic orbit of (1.1). Later, Ding and Girardi obtained infinitely many homoclinic orbits in [16] under the conditions of [14] with an additional evenness assumption on W. Note that in both papers W satisfies a condition of the type of Ambrosetti-Rabinowitz (see [17]), that is,

∃μ>2suchthat0<μW(t,z)≤Wz(t,z)z,t∈ℝ,z∈ℝ2N∖{0}.
The (A-R) condition is essential to prove the Palais-Smale condition since the variational functional Φ is strongly indefinite and 0∈σc(-𝒥(d/dt)+L). The argument of Palais-Smale condition is rather technical and not standard without the (A-R) condition. In this paper, we consider the existence of solutions of (1.1) under (A0) without the (A-R) condition on W.

We observed that just recently some abstract linking theorems were developed by Bartsch and Ding in [18]. These theorems are impactful to study the existence and multiplicity of solutions for the strongly indefinite problem. Many new results have been obtained by these theorems based on the use of (C)c sequence. See [19–21] for applications of these ideas. Note that in [19–21] 0 either is not a spectral point or is at most an isolated eigenvalue of finite multiplicity. Thus (C)c condition was checked by virtue of some very technical analysis. However, if 0∈σc(A), then we can find a sequence {zn}⊂H1 with |zn|L2=1 and |Azn|L2→0. Thus the operator A cannot lead the behavior at 0 of the equation. Consequently, besides (C)c condition, it seems also hard to check the following condition necessary for the linking theorems in [19–21]:

for any c>0, there exists ζ>0 such that ∥z∥<ζ∥PYz∥ for all z∈Φc.

Our work benefits from [14] and some weak linking theorem recently developed by Schechter and Zou in [22]. This theorem permits us first to study a sequence of approximating problems Φλ for λ∈[1,2] (the initial problem corresponds to λ=1) for which a bounded Palais-Smale sequence of Φλ is given for almost each λ∈[1,2]. Then by monotonicity, we find a sequence of {λn} and {wn} such that λn→1, Φλn′(wn)=0, and Φλn(wn)≤d. Since the sequence {wn} consists of critical points of Φλn, then its boundedness can be checked. Consequently one solution of (1.1) is obtained. The idea of first studying approximating problems for which the existence of a bounded Palais-Smale sequence is given freely and then proving that the sequence of approximated critical points is bounded was originally introduced in [23]. See also [24].

We make the following assumptions.

W(t,z)∈C1(ℝ×ℝ2N,ℝ) is 1-periodic in t. W(t,0)=0 for all t∈ℝ. There exist constants c1>0 and μ>2 such that Wz(t,z)z≥c1|z|μ for (t,z)∈ℝ×ℝ2N.

there exist c2, r>0 such that |Wz(t,z)|≤c2|z|μ-1 for t∈ℝ and |z|≤r.

there exist c3, R≥r and p≥μ such that |Wz(t,z)|≤c3|z|p-1 for t∈ℝ and |z|≥R.

there exists b0>2 such that lim infz→0(Wz(t,z)z/W(t,z))≥b0 uniformly for t∈ℝ;

W̃(t,z):=(1/2)Wz(t,z)z-W(t,z)>0 for allt∈ℝ, z∈ℝ2N∖{0}. There exist constants b∞>0 and β>p(p-2)/(p-1) such that lim inf|z|→∞W̃(t,z)/|z|β≥b∞ uniformly for t∈ℝ.

Theorem 1.1.

Let (A0), (A1)–(A5) be satisfied, then (1.1) has at least one homoclinic orbit.

Remark 1.2.

We can easily check that the (A-R) condition implies (A4) and (A5). But the converse proposition is not true. See the following example:
W(t,z)=|z|μ+(μ-2)|z|μ-ϵsin2(|z|ϵϵ),
where 2<μ<∞, 0<ϵ<min{μ-2,μ/(μ-1)} (see [25] or [26] for details).

If Wz(t,z)=a|z|μ-2z+Rz(t,z), a>0, μ∈(2,∞) with R satisfying

R∈C1(ℝ×ℝ2N,ℝ) is 1-periodic in t and
Rz(t,z)=o(|z|μ-1)as|z|→0,Rz(t,z)=o(|z|μ-1)as|z|→∞,
uniformly in t∈ℝ, then

0<Rz(t,z)z≤(a(μ-2)/2)|z|μ for allt∈ℝ, z∈ℝ2N∖{0}.

Theorem 1.3.

Let (A0),(B1), and (B2) be satisfied, then (1.1) has at least one homoclinic orbit.

This paper is organized as follows. In Section 2 we will construct some appropriate variational space and obtain some embedding results necessary for our variational arguments. In Section 3 we will recall a weak linking theorem, by which we will give the proof of Theorems 1.1 and 1.3 in Section 4.

2. Some Embedding Results

In what follows, by |·|p we denote the usual Lp-norm and by (·)2 the usual L2-inner product. A standard Floquet reduction argument shows that σ(A)=σc(A) (see [14]).

Let {E(λ);λ∈ℝ} be the spectral family of A. A possesses the polar decomposition A=U|A| with U=I-E(0)-E(-0). By (A0), 0 is at most a continuous spectrum of A. L2 has an orthogonal decomposition

L2=L2-⊕L2+,
where L2±:={u∈L2;Uu=±u}.

Let 𝒟(|A|1/2) denote the domain of |A|1/2 and let E be the space of the completion of 𝒟(|A|1/2) under the norm

∥u∥E:=||A|1/2u|2.E becomes a Hilbert space under the inner product

(u,v)E:=(|A|1/2u,|A|1/2v)2.E possesses an orthogonal decomposition

E=E-⊕E+,
where E±⊇L2±∩𝒟(|A|1/2).

Under (A0), it is easy to check

E+=L2+∩𝒟(|A|1/2),∥·∥E~∥·∥H1/2onE+.
Therefore, E+ can be embedded continuously into Lp(ℝ,ℝ2N) for any p≥2 and compactly into Llocp(ℝ,ℝ2N) for any p∈[1,∞).

For any ϵ>0, set Lϵ2-:=E(-ϵ)L2 and Eϵ-:=Lϵ2-∩𝒟(|A|1/2)=Lϵ2-∩E-. Then on Eϵ-, we also have ∥·∥E~∥·∥H1/2 and the same embedding conclusion as that of E+.

Let L̃ϵ2-:=L2-∩(clL2(⋃λ<-ϵE(λ)L2))⊥ where clL2(·) stands for the closure of · in L2.

For μ>2, let Eϵ,μ- be the completion of L̃ϵ2- under the norm
∥u∥μ:=(||A|1/2u|22+|u|μ2)1/2,
and let Eμ- denote the completion of 𝒟(A)∩L2- with respect to the norm ∥·∥μ. Then Eϵ- is a closed subspace of Eμ-, and Eμ- possesses the following decomposition:

Eμ-=Eϵ-⊕Eϵ,μ-.
Moreover, Eϵ- is orthogonal to Eϵ,μ- with respect to (·)E.

Let Eμ be the completion of 𝒟(A) under the norm ∥·∥μ. The following result holds true.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B14">14</xref>]).

Under (A0), Eμ has the direct sum decomposition
Eμ=Eμ-⊕E+,
and Eμ is embedded continuously in Lν for any ν∈[μ,∞) and compactly in Llocν for any ν∈[2,∞).

3. A Weak Linking Theorem

In this section we state some weak linking theorem due to [22] which was first built in a Hilbert space. This theorem is still true in a reflexive Banach space (cf. Willem and Zou [25]).

Let E be a reflexive Banach space with norm ∥·∥ and possess a direct sum decomposition E=N⊕M, where N⊂E is a closed and separable subspace. Since N is separable, we can define a new norm |z|w satisfying |z|w≤∥z∥, for all z∈N such that the topology induced by this norm is equivalent to the weak topology of N on bounded subsets of N. For z=v+w∈E with v∈N and w∈M, we define |z|w2=|v|w2+∥w∥2, then |z|w≤∥z∥, for all z∈E. In particular, if zn=vn+wn is |·|w-bounded and zn→z under the norm |·|w in E, then vn⇀v weakly in N, wn→w strongly in M, and zn⇀v+w weakly in E. Let Q⊂N be a ∥·∥-bounded open convex subset and let p0∈Q be a fixed point. Let F be a |·|w-continuous map from E onto N satisfying the following.

F|Q=id; F maps bounded sets to bounded sets.

there exists a fixed finite-dimensional subspace E0 of E such that F(u-v)-(F(u)-F(v))⊂E0, for all u,v∈E.

F maps finite-dimensional subspaces of E into finite-dimensional subspaces of E.

Set A:=∂Q, B:=F-1(p0), where ∂Q denotes the ∥·∥-boundary of Q. For Φ∈C1(E,ℝ), we introduce the class Γ of mappings h:[0,1]×Q̅→E with the following properties.

h:[0,1]×Q̅→E is |·|w-continuous.

for any (s0,u0)∈[0,1]×Q̅, there is a |·|w-neighborhood U(s0,u0) such that {u-h(t,u):(t,u)∈U(s0,u0)∩([0,1]×Q̅)}⊂Efin, where Efin denotes some finite-dimensional subspaces of E.

h(0,u)=u,Φ(h(s,u))≤Φ(u),for all u∈Q̅.

The following is a variant weak linking theorem in [22].

Theorem 3.1.

Let the family of C1-functionals (Φλ) have the form
Φλ(u):=I(u)-λJ(u),∀λ∈[1,2].
Assume that the following conditions hold.

J(u)≥0,for all u∈E;Φ1:=Φ.

I(u)→∞ or J(u)→∞ as ∥u∥→∞.

Φλ is |·|w-upper semicontinuous; Φλ′ is weakly sequentially continuous on E. Moreover, Φλ maps bounded sets into bounded sets.

supAΦλ<infBΦλ,for all λ∈[1,2].

Then for almost all λ∈[1,2], there exists a sequence {un} such that
supn∥un∥<∞,Φλ′(un)→0,Φλ(un)→Cλ,
where Cλ:=infh∈Γsupu∈Q̅Φλ(h(1,u))∈[infBΦλ,supQ̅Φλ].Remark 3.2.

Consider Eμ=Eμ-⊕E+ defined as in Section 2. Obviously, Eμ is reflexive. For z0∈E+ with ∥z0∥μ=1,E+=ℝz0⊕E1+. Let N=Eμ-⊕ℝz0 and M=E1+, then Eμ=N⊕M. It is easy to see that N is a closed and separable subspace of Eμ. For any u∈Eμ,u can be written as u=u-+sz0+w+ with u-∈Eμ- and w+∈E1+. For R>0, let Q:={u:=u-+sz0,s>0,u-∈Eμ-,∥u∥μ<R}, then p0:=s0z0∈Q for 0<s0<R. Define F:Eμ→N as Fu:=u-+∥sz0+w+∥μz0. Then it is easy to check that F,Q,and p0 satisfy (i), (ii), and (iii). If we let A:=∂Q and B:=F-1(s0z0)={u:=sz0+w+,s≥0,w+∈E1+,∥sz0+w+∥μ=s0}, then A links B (see Lemmas 4.2 and 4.3 in Section 4).

4. The Proof of the Main Results

Consider the functional

Φ(z):=12∥z+∥E2-12∥z-∥E2-∫ℝW(t,z),
for z=z++z-∈Eμ. Then by assumptions (A1)–(A3) and Lemma 2.1, Φ∈C1(Eμ,ℝ). A standard argument shows that any critical point of Φ is a homoclinic orbit of (1.1) (cf. [14]).

Set

Φλ(z):=12∥z+∥E2-λ(12∥z-∥E2+∫ℝW(t,z))=I(z)-λJ(z),λ∈[1,2].
Then Φ1=Φ and J(z)≥0. By (A2) and (A3),

|Wz(t,z)|≤C(|z|μ-1+|z|p-1),
where, as below,C stands for some generic positive constant.

Together with (A1), one has

c1μ|z|μ≤W(t,z)≤C(|z|μ+|z|p).
Thus I(z)→∞ or J(z)→∞ if ∥z∥μ2=∥z+∥E2+∥z-∥E2+|z|μ2→∞.

Lemma 4.1.

Φλ is |·|w-upper semicontinuous and Φλ′ is weakly sequentially continuous.

Proof.

For any c∈ℝ, assume that zn∈{z∈Eμ,Φλ(z)≥c} with zn⇀z. Let zn=zn++zn- with zn+∈E+ and zn-∈Eμ-. Then zn+→z+ in Eμ and hence sup∥zn+∥<∞. Since Φλ(zn)≥c and W(t,zn)≥0, we have sup∥zn-∥<∞.

By (4.4),

C|zn|μμ≤∫ℝW(t,zn)dt≤1λ(12∥zn+∥E2-λ2∥zn-∥E2-c)<∞.
Then sup∥zn∥μ<∞. By Lemma 2.1, zn⇀z in Eμ,zn→z in Llocμ, and zn(t)→z(t) a.e. for t∈ℝ. By Fatou's Lemma, Φλ(z)≥c. Therefore, Φλ is |·|w-upper semicontinuous.

Let zn⇀z in Eμ, then zn→z in Llocp,2≤p<∞. By (4.3), Wz(t,zn)→Wz(t,z) in Llocμ/(μ-1) and ∫ℝWz(t,zn)v→∫ℝWz(t,z)v for any v∈Eμ. Therefore, Φλ′(zn)→Φλ′(z).

Lemma 4.2.

There exist b>0, r>0 such that Φλ(z)≥b>0, for all z∈E+ with ∥z∥μ=r, for all λ∈[1,2].

Proof.

By (4.4) and Lemma 2.1,
∫ℝW(t,z)≤C(|z|μμ+|z|pp)≤C(∥z∥μμ+∥z∥μp).
The conclusion is obvious.

Lemma 4.3.

There exists R>r>0 such that Φλ|∂M=0 and supMΦλ<d<∞ for all λ∈[1,2], where M:={z=x+sz0,x∈Eμ-,∥z∥μ≤R,s>0} and z0∈E+, ∥z0∥μ=1.

Proof.

For z=x+sz0, by (4.4),
Φλ(z)≤s22∥z0∥E2-12∥x∥E2-C∫ℝ|x+sz0|μ.

Since Eμ is continuously embedded in Lt for μ≤t<∞, there exists a continuous projection from Eμ-⊕ℝz0 in Lμ to ℝz0. Thus, |sz0|μ≤C|x+sz0|μ for some C>0 and then

Φλ(z)≤Cs2-C∥x∥E2-Csμ,
and thus the lemma follows easily.

Combining Lemmas 4.1–4.3 and Theorem 3.1, we get the following.

Lemma 4.4.

Under (A0) and (A1)–(A3), for almost every λ∈[1,2], there exist {zn}⊆Eμ such that
sup∥zn∥μ<∞,Φλ′(zn)→0,Φλ(zn)→Cλ∈[b,d].

We need the following lemma which is a special case of a more general result due to Lions [27, 28].

Lemma 4.5.

Let a>0 and 2≤p<∞. If {zn}⊂H1 is bounded and if
sups∈ℝ∫B(s,a)|un|p→0,n→∞,
where B(s,a):=(s-a,s+a), then un→0 in Lt for 2<t<∞.

Lemma 4.6.

Under (A0)–(A3), let λ∈[1,2] be fixed. For the sequence {zn} in Lemma 4.4, there exist {kn}⊂ℤ such that, up to a subsequence, un(t):=zn(t+kn) satisfies un⇀uλ≠0, Φλ′(uλ)=0 and Φλ(uλ)≤d.

Proof.

Write zn=zn++zn- with zn+∈E+ and zn-∈Eμ-. Since sup∥zn∥μ<∞, sup∥zn+∥E<∞, let A+ denote the part of A in 𝒟(A)∩L2+=H1∩L2+:=H+1. Then by (A0),
A+=∫α∞λdE(λ).
Obviously, A+:H+1⊂L2+→L2+ has a bounded inverse A+-1. Since
|A+z|22=∫α∞λ|dE(λ)z|22≥α|z|22,|ż|2=|Az+Lz|2≤|A+z|2+|Lz|2forz∈H+1,
then we have
∥z∥H1≤C|A+z|2forz∈H+1.
Set vn=A+-1zn+∈H+1, then
∥vn∥H1≤C|zn+|2≤C∥zn+∥E<∞.
We claim that vn is nonvanishing, that is, there exist M>0, a>0, and yn∈ℝ such that
lim infn→∞∫B(yn,a)|vn|2dt≥M.
Indeed, if not, by (4.14), {vn} is bounded in H1. Lemma 4.5 shows that vn→0 in Lt for 2<t<∞. By (4.3),
|∫ℝWz(t,zn)vn|≤C∫ℝ(|zn|μ-1+|zn|p-1)|vn|≤C(|zn|μμ-1|vn|μ+|zn|pp-1|vn|p)→0.
Hence
(zn+,vn)E=Φλ′(zn)vn+λ∫ℝWz(t,zn)vn→0.
Thus,
|zn+|22=(zn+,A+vn)L2=(zn+,vn)E→0.
Therefore, for any 2≤t<∞,
|zn+|t≤|zn+|21/t|zn+|2(t-1)1-1/t≤C∥zn+∥E1-1/t|zn+|21/t→0.
Thus we obtain
∥zn+∥E2=Φλ′(zn)zn++λ∫ℝWz(t,zn)zn+→0,
and then
Φλ(zn)≤12∥zn+∥E2→0,
a contradiction. Choose kn∈ℤ such that |kn-yn|=min{|s-yn|,s∈ℤ} and let un:=kn*zn=zn(t+kn):=un++un-. In view of the invariance of E+ under the action *, un+=kn*zn+∈E+. Since A commutes with *, then A+-1 also does. Therefore v̅n:=kn*vn=A+-1un+. By (4.15),|v̅n|L2(B(0,a+1))2≥M2.
Clearly,
∥un∥μ=∥zn∥μ<∞.
Thus, up to a subsequence, we assume that
un⇀uλinEμ,un→uλinLloctfort≥2.
We now establish that uλ≠0. If not, un+⇀0 in L2, and then
(v̅n,z)E=(un+,z)L2→0for all z∈H1/2,
which implies that
v̅n⇀0inH1/2,v̅n→0inLloctfort≥2,
contradicting (4.22). By Lemma 4.1, Φλ′ is weakly continuous, hence we have
Φλ′(uλ)=limn→∞Φλ′(un)=0.
By Fatou's Lemma, we obtain
Φλ(uλ)=Φλ(uλ)-12Φλ′(uλ)uλ≤limn→∞(Φλ(un)-12Φλ′(un)un)=limn→∞Φλ(un)=limn→∞Φλ(zn)≤d.

As a straightforward consequence of Lemmas 4.4 and 4.6, we have the following.

Lemma 4.7.

Under (A0)–(A3), there exist {λn}⊂[1,2], {wn}⊂Eμ∖{0} such that λn→1,Φλn′(wn)=0, and Φλn(wn)≤d.

Lemma 4.8.

{wn} is bounded in Eμ.

Proof.

Our argument is motivated by [26]. Write wn=wn++wn- with wn+∈E+ and wn-∈Eμ-.Since Φλn′(wn)wn=0, by (A1), then
∥wn+∥E2-λn∥wn-∥E2=λn∫ℝWz(t,wn)wn≥C|wn|μμ.
Hence,
|wn|μμ≤C∥wn+∥E2,∥wn-∥E≤C∥wn+∥E,|wn-|μ≤C∥wn+∥E+C∥wn+∥E2/μ.
In the following, we show that ∥wn+∥E is bounded. Choose ϵ0>0 small enough such that b0-ϵ0>2. By (A4), there exists 0<r0≤1 such that
Wz(t,z)z≥(b0-ϵ0)W(t,z)
for allt∈ℝ and |z|≤r0. By (A3) and (A5), for allt∈ℝ and |z|≥r0, we can choose C,C′>0 such that
|Wz(t,z)|≤C|z|p-1,Wz(t,z)z-2W(t,z)≥C′|z|β.
Since Φλn(wn)≤d and Φλn′(wn)=0, then we have
(12-1b0-ϵ0)(∥wn+∥E2-λn∥wn-∥E2)+λn∫ℝ(1b0-ϵ0Wz(t,wn)wn-W(t,wn))≤d.
Thus, by (4.31), (4.32), and (A5), we obtain
∥wn+∥E2-λn∥wn-∥E2≤C(∫|wn|<r0+∫|wn|≥r0)(W(t,wn)-1b0-ϵ0Wz(t,wn)wn)dt+C≤C∫|wn|≥r0(W(t,wn)-1b0-ϵ0Wz(t,wn)wn)dt+C≤C∫|wn|≥r0Wz(t,wn)wndt+C≤C∫|wn|≥r0|wn|pdt+C.
By (4.33) and (A5),
C≥Φλn(wn)-12Φλn′(wn)wn=∫ℝ(12Wz(t,wn)wn-W(t,wn))dt≥C∫|wn|≥r0|wn|βdt.
Choose ν>p sufficiently large such that (νp(p-2))/(ν(p-1)-p)<β. Let t:=ν(p-β)/(ν-β)p, then by (A5), 0<t<1/(p-1) for ν being large enough. By Hölder's inequality and Lemma 2.1, we have
∫|wn|≥r0|wn|pdt≤(∫|wn|≥r0|wn|β)(1-t)p/β(∫ℝ|wn|ν)tp/ν≤C|wn|νtp≤C(∥wn+∥E+∥wn-∥E+|wn|μ)tp≤C(∥wn+∥E+∥wn+∥E2/μ)tp≤C(∥wn+∥Etp+∥wn+∥E2tp/μ).
By (4.29), (4.35), and (4.37),
∫ℝ|wn|μ≤C(∥wn+∥Etp+∥wn+∥E2tp/μ+1),
and then
|wn|μ≤C(∥wn+∥Etp/μ+∥wn+∥E2tp/μ2+1).
Using (4.3), (4.37), and (4.39), from Φλn′(wn)wn+=0, we obtain
∥wn+∥E2≤C∫ℝ(|wn|μ-1|wn+|+|wn|p-1|wn+|)dt≤C(|wn|μμ-1+(∫ℝ|wn|p)(p-1)/p)∥wn+∥E≤C(|wn|μμ-1+(∫{t∣∣wn(t)∣≤r0}|wn|p+∫{t∣∣wn(t)∣≥r0}|wn|p)(p-1)/p)∥wn+∥E≤C(|wn|μμ-1+(∫ℝ|wn|μ)(p-1)/p+(∫{t∣∣wn(t)∣≥r0}|wn|p)(p-1)/p)∥wn+∥E≤C(|wn|μμ-1+|wn|μμ(p-1)/p+∥wn+∥Et(p-1)+∥wn+∥E2t(p-1)/μ)∥wn+∥E≤C(∥wn+∥Etp(μ-1)/μ+∥wn+∥E2tp(μ-1)/μ2+∥wn+∥Et(p-1)+∥wn+∥E2t(p-1)/μ+1)∥wn+∥E,
which implies sup∥wn+∥E<∞ since t(p-1)<1.

Proof of Theorem <xref ref-type="statement" rid="thm1">1.1</xref>.

Since {wn} is bounded, wn⇀w in Eμ and wn→w in Lloct for 2≤t<∞. We show that w≠0.

∥wn+∥E2=λn∫ℝWz(t,wn)wn+≤C(∥wn+∥Eμ+∥wn+∥E2(μ-1)/μ+1+∥wn+∥Ep+∥wn+∥E2(p-1)/μ+1),
which implies that there exists C0>0 such that ∥wn+∥E≥C0.

If {wn+} is vanishing, then

∥wn+∥E2=λn∫ℝWz(t,wn)wn+→0,
a contradiction. Hence {wn+} is nonvanishing.

Just along the proof of Lemma 4.6, we can see that there exist M>0 and a>0 such that

∫B(0,a+1)|w̅n+|dt≥M2,
where w̅n+:=wn+(t+yn).

Set w̅n-:=wn-(t+yn) and w̅n=w̅n++w̅n-. Then sup∥w̅n∥μ<∞ and then w̅n⇀w̅, w̅n+⇀w̅+, and w̅n-⇀w̅-. By Lemma 2.1, w̅n+→w̅+ in Lloc2, and hence

∫B(0,a+1)|w̅+|2≥M2>0.
It follows that w̅≠0.

Since Φλn′(w̅n)=0, using Lebesgue's theorem, then we obtain

-Φ′(w̅)ϕ=Φλn′(w̅n)ϕ-Φλn′(w̅)ϕ+Φλn′(w̅)ϕ-Φ′(w̅)ϕ=〈w̅n+-w̅+,ϕ〉E-λn〈w̅n--w̅-,ϕ〉E-λn∫ℝ(Wz(t,w̅n)-Wz(t,w̅))ϕ+(1-λn)〈w̅,ϕ〉E+(1-λn)∫ℝWz(t,w̅)ϕ→0,
for any ϕ∈C0∞, that is, Φ′(w̅)=0.

Proof of Theorem <xref ref-type="statement" rid="thm2">1.3</xref>.

It is easy to check that Wz(t,z)=a|z|μ-2z+Rz(t,z) satisfies all the assumptions of Theorem 1.1 with b0=β=μ.

Acknowledgments

The authors would like to thank the reviewers for the valuable suggestions. This work was supported by the Natural Science Foundation of China.

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