for bounded selfadjoint operator equations and homoclinic orbits of Hamiltonian systems

A nonzero solution

for bounded selfadjoint operator equations and homoclinic orbits of Hamiltonian systems

Mingliang Song

and Runzhen Li

1Mathematics and Information Technology School, Jiangsu Second Normal University, Nanjing, 210013, P. R. China.

2School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097, P. R. China

Received 20 November 2020, appeared 10 September 2021 Communicated by Petru Jebelean

Abstract. We obtain an existence theorem of nonzero solution for a class of bounded selfadjoint operator equations. The main result contains as a special case the existence result of a nontrivial homoclinic orbit of a class of Hamiltonian systems by Coti Zelati, Ekeland and Séré. We also investigate the existence of nontrivial homoclinic orbit of indefinite second order systems as another application of the theorem.

Keywords:bounded selfadjoint operator equations, nonzero solution, homoclinic orbit, Hamiltonian systems, indefinite second order systems.

2020 Mathematics Subject Classification: 34A34, 34C37, 37N05, 58E05.

1 Introduction

In recent years several authors studied the existence of homoclinic orbits for first or second order Hamiltonian systems via variational methods and critical point theory, see for instance [2,4–6,9,12–16]. In particular, with the aid of a bounded self-adjoint linear operator and the dual action principle, Coti Zelati, Ekeland and Séré [4] obtained some existence theorems of nonzero homoclinic orbit for first order Hamiltonian systems

(x⁰ = J Ax+JH⁰(t,x), x(±_∞) =0,

via the Ambrosetti–Rabinowitz mountain-pass theorem and concentration compactness prin- ciple. Inspired by the ideas of [4], we consider the more generalized operator equation

Lu−G⁰(t,u) =0, (1.1)

where L : L^β(_R,R^N) →W^1,β(_R,R^N)^TL^γ(_R,R^N)is a bounded linear operator for allγ ≥ β and for some β∈(1, 2)andR

R((Lu)(t),v(t))dt= R

R((Lv)(t),u(t))dtfor all u,v∈ L^β(_R,R^N),

BCorresponding author. Email: mlsong2004@163.com

G : R×_R^N → _R and G⁰(t,u) denotes the gradient of G with respect to u. u = u(t) ∈ L^β(_R,R^N)is called a solution of (1.1) if(Lu)(t)−G⁰(t,u(t)) =0 a.e. t ∈_R.

We need the following assumptions:

(L₁) For any bounded {u_n} ⊂ L^β(_R,R^N)and R > 0, there exists a subsequence{u_nj}such thatLu_nj →winC([−R,R],R^N).

(L₂) There existsv₀ ∈ L^β(_R,R^N)such thatR+_∞

−_∞(Lv₀,v₀)dt>0.

(L₃) (Lu(·+T))(t) = (Lu)(t+T)for allt ∈_{R, where}T >0 is a constant.

(L₄) |(Lu)(t)| ≤ c₀R+_∞

−_∞ e^−l|t−τ||u(τ)|dτ for all u ∈ L^β(_R,R^N), where c₀,l > 0 are two con- stants.

(G₁) G(t,·)andG⁰(t,·)are continuous for a.e. t ∈ _R,G(·,u)andG⁰(·,u)are measurable for allu∈_R^N_,G(t,·)is convex for allt∈_RandG^∗0(t,·)exists for a.e. t∈_R.

(G₂) G(t+T,u) =G(t,u)for allt∈_R.

(G3) c₁|u|^β ≤G(t,u)≤c₂|u|^β_{, where}c₂≥c₁ >0 are two constants.

(G₄) 0≤ ¹

β(G⁰(t,u),u)≤ G(t,u).

(G₅) |G⁰(t,u)| ≤c₃|u|^β−1, wherec₃ >0 is a constant.

Now we state our main result as follows.

Theorem 1.1. Assume L and G satisfy(L₁)–(L₄)and(G₁)–(G₅). Then(1.1)has a nonzero solution.

Remark 1.1. Although the equation (1.1) also appeared in the proof of Theorem 4.2 in [4], the bounded linear operator L there equal (2.2) which comes from first order Hamiltonian systems. In this paper,L discussed in (1.1) contains not only (2.2) but also (2.4) coming from indefinite second order Hamiltonian systems. In addition, introducing the condition (L₁) makes the proof of conclusion clearer and simpler.

The rest of this paper is organized as follows. In Section 2, we firstly establish a preliminary lemma, and then, we give two application examples for homoclinic orbit of Hamiltonian systems. In Section 3, we give the proof of our main result.

2 Preliminaries and examples

To complete the proof of Theorem1.1, we need a lemma.

Lemma 2.1. Let ¹_α + ¹

β =1.

(1) If u∈ L^β(_R,R^N)and b> a>0, then _Z

|t|≥b

_{Z a}

−ae^−l|t−τ||u(τ)|dτ α

dt ¹_α

≤2(αl)^−2αe^−l(b−a) _{Z a}

−a

|u(τ)|^βdτ ¹_β

.

(2) If w,u∈ L^β(_R,R^N)and b≥a >r≥0, then Z

|t|≥b

|u(t)|

Z

a≥|τ|≥re^−l|t−τ||w(τ)|dτdt

≤2(αl)^−2αe^−l(b−a) _Z

|t|≥b

|u(t)|^βdt ¹

β _Z

a≥|τ|≥r

|w(τ)|^βdτ ¹

β

. (3) If w,u∈ L^β(_R,R^N)and b>a >r>0, then

Z

a≤|t|≤b

|u(t)|

Z

|τ|≤ae^−l|t−τ||w(τ)|dτdt

≤2(αl)^−2αkuk_Lβ

"

e^−l(a−r)kwk_Lβ+ _Z

r≤|τ|≤a

|w(τ)|^βdτ ¹

β

# .

Proof. Foru∈ L^β(_R,R^N)andb>a>0, by some simple calculations, we have _Z

|t|≥b

_{Z a}

−ae^−l|t−τ||u(τ)|dτ α

dt ¹_α

≤

_{Z +∞}

b

+

Z _−b

−_∞

_{Z a}

−ae^{−αl|t−τ|}dτdt

¹_{αZ a}

−a

|u(τ)|^βdτ ¹

β

=2^1α(αl)^−α2 1−e^−2αal1_α

e^−l(b−a) _{Z a}

−a

|u(τ)|^βdτ ¹_β

≤2(αl)^−2αe^−l(b−a) _{Z a}

−a

|u(τ)|^βdτ ¹

β

,

which implies that(1)holds. The same arguments also prove that(2)holds.

By(2), we have Z

a≤|t|≤b

|u(t)|

Z

|τ|≤ae^−l|t−τ||w(τ)|dτdt

=

Z

a≤|t|≤b

|u(t)|

_Z

|τ|≤r

+

Z

r≤|τ|≤a

e^−l|t−τ||w(τ)|dτdt

≤2(αl)^−2αkuk_Lβ

"

e^−l(a−r)kwk_Lβ+ Z

r≤|τ|≤a

|w(τ)|^βdτ ¹_β#

. This shows that(₃)_holds.

Next, we return to applications to homoclinic orbit of Hamiltonian systems. For systematic researches of homoclinic orbit of Hamiltonian systems, we refer to the excellent papers [2,4–

6,9,12–16] and references therein.

As the first example we consider

(x⁰ = J Ax+JH⁰(t,x),

x(±_∞) =0, (2.1)

where J = ₋⁰_I ^In

n 0

is the standard symplectic matrix in R^2N, A is a 2N×2N symmetric matrix and all the eigenvalues of J Ahave non-zero real part,H(t,x)_satisfies

(H₁) H∈ C(_R×_R^2N,R),H⁰ ∈C(_R×_R^2N,R^2N)andH(t,·)is strictly convex;

(H2) H(t+T,x) =H(t,x)_{for some}T>_0;

(H₃) k₁|x|^α ≤H(t,x)≤k₂|x|^α for someα>2 and 0<k₁≤ k₂; (H₄) H(t,x)≤ ¹

α(H⁰(t,x),x).

As in [4], defineG(t,u) =sup_x∈R2N{(u,x)−H(t,x)}andGsatisfies(G₁)–(G₅). DefineL: L^β(_R,R^2N)→W^1,β(_R,R^N)^TL^α(_R,R^2N)byz= Lusatisfies

−Jz⁰−Az =u,z(±_∞) =0 Then

z(t) =

Z _t

−_∞e^E(t−τ)P_sJu(τ)dτ−

Z _+∞

t e^E(t−τ)P_uJu(τ)dτ, (2.2) where E = J A,R^2N = Eu⊕Es and Ps and Pu are the projections ontoEs and Eurespectively satisfying |e^tEP_sξ| ≤ ke^−bt|ξ| for t ≥ 0 and |e^tEP_uξ| ≤ ke^bt|ξ| for t ≤ 0,ξ ∈ _R^2N and some b,k>0. So

|(Lu)(t)| ≤

Z _t

−_∞ke^−b(t−τ)|u(τ)|dτ+

Z +_∞

t ke^b(t−τ)|u(τ)|dτ

= k Z +_∞

−_∞ e^−b|t−τ||u(τ)|dτ,

which implies that (L₄) holds. From Lemma 2.1 of [4], we know that L : L^β(_R,R^2N) → W^1,β(_R,R^2N)^TL^γ(_R,R^2N)is a bounded linear operator forγ≥ β,β∈(1, 2)and

Z

R((Lu)(t),v(t))dt=

Z

R((Lv)(t),u(t))dt for allu,v∈ L^β(_R,R^2N).

Byz⁰(t) = Ju(t) +Ez(t)for allt∈_{R, we have}

|z(t₁)−z(t₂)|=

Z _t2

t1

(Ju(t) +Ez(t))dt

≤ |t₂−t₁|^1αkuk_Lβ+M₀|t₂−t₁|kzk_∞

where M₀ >0, which implies that (L₁)holds. Note that the proof of (b) of Lemma 4.1 in [4], we see that there existsv0∈ L^β(_R,R^2N)such that(L2)holds. The validity of(L3)is obvious.

Moreover, G^∗(t,x) = H(t,x) and a solution u ∈ L^β(_R,R^2N)\ {0} of Lu−G⁰(t,u) = 0 corresponds to a nonzero solutionx= Luof

(−Jx⁰−Ax= H⁰(t,x), x(±_∞) =0.

Therefore, we have the following corollary.

Corollary 2.2([4, Theorem 4.2]). Assume H satisfies(_H1)–(_H4). Then(_2.1)has a nonzero solution, i.e., the Hamiltonian system

−Jx⁰−Ax= H⁰(t,x) has at least one nontrivial homoclinic orbit.

Remark 2.1. The above corollary was essentially [4, Theorem 4.2] by Coti Zelati, Ekeland and Séré using the Ekeland variational principle and concentration compactness principle, and the equation (1.1) also appeared in the proof the theorem already.

As a second example we consider

(Dx⁰⁰−Bx=V⁰(t,x),

x(±_∞) =0, (2.3)

where D,BareN×Nsymmetric matrix,(±σ(D))^T(0,+_∞)6= _∅,Dis invertible,D⁻¹B=Q² with Q being a N×N matrix and all the eigenvalues of Q have positive real part, V : R× R^N → _R and V⁰(t,x) denotes the gradient of V with respect to x. The system was called indefinite second order system in [3].

Let

(Dx⁰⁰−Bx=u, x(±_∞) =0.

Then

(x⁰⁰−D⁻¹Bx=x⁰⁰−Q²x=D⁻¹u, x(±_∞) =0

and

(e^tQ(x⁰−Qx)⁰ =e^tQD⁻¹u, x(±_∞) =0.

Assumex⁰(−_∞) =0 (and this will be verified later). Then x⁰−Qx=e^−tQ

Z _t

−_∞e^τQD⁻¹u(τ)dτ and

(e^−tQx)⁰ =e^−2tQ Z _t

−_∞e^τQD⁻¹u(τ)dτ.

So, we have

x =−e^tQ Z +_∞

t e^−2sQ _{Z s}

−_∞e^τQD⁻¹u(τ)dτ

ds

=−^Q

−1

2 e^tQ Z _t

−_∞e^−2tQe^τQD⁻¹u(τ)dτ− ^Q

−1

2 e^tQ Z _+∞

t e^−τQD⁻¹u(τ)dτ

=−^Q

−1

2

Z +_∞

−_∞ e^−|t−τ|QD⁻¹u(τ)dτ.

For u∈ L^β(_R,R^N), set

x= Lu= −^Q

−1

2

Z _+∞

−_∞ e^−|t−τ|QD⁻¹u(τ)dτ. (2.4) We claim that

x= Lu∈W^1,β(_R,R^N)^\L^γ(_R,R^N)

forγ ≥ β,β ∈ (1, 2)andu ∈ L^β(_R,R^N). In fact, from all the eigenvalues of Qhave positive real part, we know that there existλ₀>0 andc₄>0 such that|e^−|t|Qξ| ≤c₄e^−λ0|t||ξ|fort ∈_R andξ ∈_R^N. ByR+_∞

−_∞ e^−η|t|dt= ²

η, we have

e^−λ0|t| ∈ L^η(_R,R) and ke^−λ0|t|k^η_Lη = ²

λ₀η ∀η≥1.

Using the convolution inequality, we have _{Z +∞}

−_∞

|Lu|^rdt ¹_r

≤ ^c4kQ⁻¹k · kD⁻¹k 2

_{Z +∞}

−_∞

_{Z +∞}

−_∞ e^{−λ0|t−τ|}|u(τ)|dτ r

dt ¹_r

≤ ^c4kQ⁻¹k · kD⁻¹k

2 ke^−λ0|t|k_L^p· kuk_Lβ (2.5) for ¹_r = ¹_p + ¹

β −1 andr,p ≥ 1, which shows that Lu ∈ L^r(_R,R^N) ∀r ∈ [β,+_∞]. Similarly, from the equation

x⁰ = Qx+

Z _t

−_∞e^−(t−τ)QD⁻¹u(τ)dτ, (2.6) it is easy to see thatLu∈W^1,β(_R,R^N). Moreover, by (2.5), we can also see thatL: L^β(_R,R^N)→ W^1,β(_R,R^N) ^TL^γ(_R,R^N) is a bounded linear operator for γ ≥ β. This implies x(±_∞) = 0 andx⁰(−_∞) =0 via the above equation.

Letx =Luandy= Lv. Then Z

R((Lu)(t),v(t))dt=

Z +_∞

−_∞ (x,Dy⁰⁰−By))dt

=

Z _+∞

−_∞ (Dx⁰⁰−Bx),y)dt

=

Z

R(u(t),(Lv)(t))dt for allu,v∈ L^β(_R,R^N), which implies thatL: L^β →L^α is self-adjoint.

By (2.6) for allt₁,t2 ∈_{R, we have}

|x(t₁)−x(t₂)|=

Z _t2

t1

Qx+

Z _t

−_∞e^−(t−τ)QD⁻¹u(τ)dτ

dt

≤ kQk · kxk_∞· |t₂−t₁|+c₄(λ₀α)^−α1kD⁻¹k · kuk_Lβ· |t₂−t₁|, which implies that(L₁)holds.

Since(±σ(D))^T(0,+_∞)6=_∅, we know that there existλ₁<0 andξ₀∈_R^N\{0}such that

|ξ0|=1 andDξ0= λ₁ξ0. Let

x₀(t) =















ξ0sinkt, t ∈[0, 2mπ],

ξ₀[ ^k

π²(t−2mπ−π)³+ ^k

π(t−2mπ−π)²], t ∈[2mπ, 2mπ+π],

0, t ≥2mπ+π,

−x₀(−t)_, t <_0,

wherek,m∈N\{0}. Then

(Dx₀⁰⁰−Bx₀ =v₀, x₀(±_∞) =0

and

Z _+∞

−_∞

(Lv₀,v₀)dt=2 Z _+∞

0

(Dx⁰⁰₀(t)−Bx₀(t),x₀(t))dt

=2

_{Z 2mπ}

0

+

Z _2mπ+π 2mπ

(Dx⁰⁰₀(t)−Bx₀(t),x₀(t)dt

≥2

Z _2mπ

0

+

Z _2mπ+π

2mπ

−λ₁|x⁰₀(t)|²− kBk · |x0(t)|²dt

=−2λ₁

k²mπ+ ² 15k²π

−2kBk ·

mπ+^k

2π³ 105

>−2λ₁mk²−2πkBk ·(m+k²)

>0 provided m= k² andk² > ^2π₋^kBk

λ₁ . This shows that there exists v₀ ∈ L^β(_R,R^N)such that(L₂) holds. The validity of(L3)and(L₄)are obvious.

Further, assumeV satisfies (H₁)–(H₄) with H(t,x)replaced withV(t,x)and 2N replaced with N. Define V^∗(t,u) = sup_x∈RN{(u,x)−V(t,x)}. Then V^∗(t,u) satisfies(G₁)–(G₅)with G(t,u)replaced withV^∗(t,u). By the Legendre reciprocity formula

V^∗0(t,u) =x ⇔u=V⁰(t,x)_, we see that (2.3) is equivalent to

Lu−V^∗0(t,u) =0, u∈L^β(_R,R^N). (2.7) Therefore, we have the following result from Theorem1.1.

Corollary 2.3. Assume V satisfies (H₁)–(H₄)with H(t,x) replaced with V(t,x)and 2N replaced with N. Then(2.3)has a nonzero solution.

3 Proof of Theorem 1.1

In this section we prove Theorem1.1. The method comes from [4] with some modifications.

Proof of Theorem1.1. We define the functional I on L^β(_R,R^N)_by I(u) =

Z

RG(t,u)dt− ¹ 2

Z

R(Lu,u)dt (3.1)

for all u∈L^β(_R,R^N). From(G₃), we have 0≤

Z

RG(t,u)dt≤c₂ Z

R|u|^βdt<+_∞.

Noticing that Lu ∈ L^α(_R,R^N) and L is a bounded linear operator, then R

R(Lu,u)dt is well defined. SinceG(t,·)andG⁰(t,·)are continuous for a.e. t ∈_{R, from}(G5), we know that the functional I is a C¹ functional. Moreover, a solution of (1.1) correspond to a critical point of the functional I.

Next, we take five steps to prove the existence of the critical point of the functionalI. Step 1. There exists a sequence{u_n} ⊂ L^β(_R,R^N)_{such that} I(u_n)→c>_{0 and} I⁰(u_n)→_0.

By(L₂)and(G₃), forv₀ ∈L^β(_R,R^N),β∈ (1, 2)ands>0 we have I(sv₀) =

Z

RG(t,sv₀)dt− ^s

2

2 Z

R(Lv₀,v₀)dt

≤c₂s^β Z

R|v₀|^βdt− ^s

2

2 Z

R(Lv₀,v₀)dt

→ −_∞ ass →+_∞,

which shows there iss₀ >0 such thatI(s₀v₀)<0. Setu₀=s₀v₀ and define c= inf

γ∈_Γ sup

u∈γ([0,1])

I(u), whereΓ={γ∈C([0, 1],L^β(_R,R^N))|γ(0) =0,γ(1) =u₀}.

By(G3), we have

I(u)≥ c₁ Z

R|u|^βdt− ¹ 2

Z

R(Lu,u)dt

≥ c₁kuk^β

L^β− ^M 2 kuk²_Lβ,

where M > 0 and kLuk_L^α ≤ Mkuk_Lβ. Since β ∈ (1, 2), there existsr ∈ (0,ku0k_Lβ)such that c₁r^β− ^M₂r² > 0. So sup_{u∈γ([0,1])}I(u) ≥ c₁r^β− ^M₂r² > 0 and c> 0. By [7, Theorem V.1.6], the result follows.

Step 2. We prove that the sequence{un} ⊂ L^β(_R,R^N)is bounded and there existδ2 >δ₁ >0 such thatku_nk_Lβ ∈[δ₁,δ₂].

Clearly,

hI⁰(u_n),u_ni=

Z

R(G⁰(t,u_n),u_n)dt−

Z

R(Lu_n,u_n)dt.

Using(G₃)and(G₄), we have I(un) + ¹

2kI⁰(un)k_L^α· kunk_Lβ ≥ I(un)− ¹

2hI⁰(un),uni

=

Z

RG(t,u_n)dt− ¹ 2

Z

R(G⁰(t,u_n)_,u_n)dt

≥(1− ^β 2)

Z

RG(t,u_n)dt

≥(1− ^β

2)c₁kunk^β

L^β.

Sincec₁ >_{0, 1} < β< _2, I(u_n)→c> _{0 and}kI⁰(u_n)k_Lα →0, we deduce that{u_n}_{is bounded} inL^β(_R,R^N).

Again, from (3.1) and(G₃), we have I(u_n)≤c₂

Z

R|u_n|^βdt− ¹ 2

Z

R(Lu_n,u_n)dt

≤c₂ku_nk^β

L^β+ ^M 2 ku_nk²

L^β. If there is a subsequence{u_nk}such thatku_nkk_Lβ →0, then

I(u_nk)≤c₂ku_nkk^β

L^β+ ^M

2 ku_nkk²_Lβ →0⇒c≤0,

which contradicts c>0.

Setρ_n(t) = ^|un(t)|β

kunk^β

Lβ

. ThenR+_∞

−_∞ ρ_n(t)dt = 1. By [4, page 145, Lemma] (also see [10,11]), we have three possibilities:

(i) vanishing

sup

y∈_R Z _y+R

y−R ρ_n(t)dt→0 as n→_∞ ∀R>0;

(ii) concentration

∃y_n∈_R:∀ε>₀∃R>_{0 :}

Z _yn+R

yn−R

ρ_n(t)dt≥₁−ε ∀n;

(iii) dichotomy

∃yn∈_R,∃λ∈ (0, 1),∃R¹_n,R²_n∈ _Rsuch that (a) R¹_n,R²_n→+_∞, ^R_R¹ⁿ₂

n →0;

(b) Ryn+R¹_n

yn−R¹_n ρ_n(t)dt→λ as n→_∞;

(c) ∀ε>0 ∃R>0 such thatRyn+R

yn−R ρ_n(t)dt≥λ−ε∀n;

(d) Ryn+R²_n

yn−R²_n ρ_n(t)dt→λ as n→_∞.

Step 3. Vanishing cannot occur.

Otherwise, there exists a nonnegative sequenceεn→0 such that Z _s+1

s−1

|u_n(t)|^βdt≤ε_nku_nk^β

L^β ∀s∈_R.

By(_L4)_{, we have}

|(Lun)(t)| ≤c0

Z _+∞

−_∞ e^−l|t−τ||un(τ)|dτ

=c₀ Z +_∞

t e^−l|t−τ||u_n(τ)|dτ+c₀ Z _t

−_∞e^−l|t−τ||u_n(τ)|dτ

≤c₀e^lt

+_∞ k

∑

_{Z t+k+1}

t+k e^−αlτdτ

¹_{αZ t+k+1}

t+k

|un(τ)|^βdτ ¹_β

+c₀e^−lt

+_∞ k

∑

_{Z t−k}

t−k−1e^αlτdτ

¹_{αZ t−k}

t−k−1

|u_n(τ)|^βdτ ¹_β

≤2c₀ε

1 β

nkunk_Lβ

1−e^−αl αl

_α¹

· ¹

1−e^−l →0 asn→_∞uniformly for t∈R, which implies thatkLunk_∞ →0.

FromL : L^β(_R,R^N) →W^1,β(_R,R^N)^TL^γ(_R,R^N)is a bounded linear operator forγ ≥ β, we obtain kLu_nk_Lβ ≤ c₅ku_nk_Lβ, wherec₅ >0. Since

kLunk^α_Lα =

Z

R|Lun|^αdt≤ kLunk^α_∞^−β

Z

R|Lun|^βdt≤c₅kunk_LβkLunk^α_∞^−β,

we have kLu_nk_Lα → 0. By (G₃) and the convexity of G(t,·), G(t, 0) ≡ 0 and G(t,u_n) ≤ (G⁰(t,un),un). So

Z

R|u_n|^βdt≤ ¹ c₁

Z

R(G⁰(t,u_n)_,u_n)dt≤ ¹

c₁kG⁰(t,u_n)k_Lα· ku_nk_Lβ →_0,

sinceG⁰(t,un) =Lun+I⁰(un)→0 in L^α(_R,R^N). This is a contradiction tokunk_Lβ ≥δ₁>0.

Step 4. Concentration implies the existence of a nontrivial solution of (1.1).

If concentration occurs, we set

w_n(t) =u_n(t+y_n), v_n(t) = ^wn(t) kwnk_Lβ

. ThenR

R|vn(t)|^βdt=1 and for everyε₁ >0 there existsR>0 such that 1−ε₁≤

Z _R

−R

|v_n(t)|^βdt≤1. (3.2)

We claim there iszand a subsequence denoted also by itself such that

Lvn→z inL^α(_R,R^N). (3.3)

In fact it suffices to show that for everyε>0 there exist zε ∈ L^α(_R,R^N)and subsequencevn_j

such that

kLv_nj −z_εk_Lα ≤ε.

Let v⁽_n¹⁾(t) = v_n(t)χ_[−R,R](t) and v⁽_n²⁾(t) = v_n(t)−v⁽_n¹⁾(t). By (L₁), for every t₀ > 0 there exist {v⁽_n¹_j⁾} and u⁽_ε¹⁾ ∈ C([−t₀,t₀],R^N) such that Lv⁽_n¹_j⁾ → u⁽_ε¹⁾ in C([−t₀,t₀],R^N). Define u_ε(t) =u⁽_ε¹⁾(t)_fort ∈[−t₀,t₀]_andu_ε(t) =0 otherwise. Then

kLv_nj −u_εk_Lα ≤ kLv⁽_n²_j⁾k_Lα+kLv⁽_n¹_j⁾−u_εk_Lα ≤ Mε

1 β

1 +kLv⁽_n¹_j⁾−u_εk_Lα, and

_{Z +∞}

−_∞

|Lv⁽_n¹_j⁾−u_ε|^αdt ¹

α

≤ _Z

|t|≥t0

|Lv⁽_n¹_j⁾|^αdt+2t₀kLv⁽_n¹_j⁾−uεk^α_C[−t

0,t₀]

_α¹

≤c_{0 Z}

|t|≥t₀

_{Z R}

−Re^−l|t−τ||v⁽_n¹_j⁾(τ)|dτ α

dt ¹_α

+ (2t₀)^1αkLv⁽_n¹_j⁾−u_εk_C[−t0,t0]

≤2c₀(αl)^−2αe^−l(t0−R) Z _R

−R

|v⁽_n¹_j⁾(τ)|^βdτ ¹_β

+ (2t₀)^1αkLv⁽_n¹_j⁾−uεk_C[−t0,t0]

≤2c₀(αl)^−2αe^−l(t0−R)+ (2t₀)^1αkLv⁽_n¹_j⁾−u_εk_C[−t0,t0] via(₁)_{of Lemma2.1 and}R

R|v_n(t)|^βdt=_{1, where}t₀ >R.

For anyε>0, there isε₁ >0 such that Mε

1 β

1 ≤ ₃^ε, and there existsR= R(ε₁)>0 such that (3.2) is satisfied. For the aboveR>0, there existst₀> Rsuch that

2c₀(αl)^−2αe^−l(t0−R) ≤ ^ε 3.

Then we can choose subsequence v_nj such that

(2t₀)^1αkLv⁽_n¹_j⁾−uεk_C[−t0,t0] ≤ ^ε 3 via(L₁). It follows that

kLv_nj −u_εk_Lα ≤ ^ε 3+ ^ε

3 + ^ε 3 =_ε.

From (3.3) and the boundedness of kw_nk_Lβ, there existsz ∈ L^α(_R,R^N)such that Lw_n → z in L^α(_R,R^N). We assume ^y_Tⁿ ∈ Z. It follows that I(w_n) = I(u_n) _{and that} I⁰(w_n)(t) = I⁰(un)(t+yn), andI(wn)→c,I⁰(wn)→0 inL^α(_R,R^N). Then

z_n(t) =G⁰(t,w_n) =I⁰(w_n)(t) + (Lw_n)(t)→z in L^α(_R,R^N).

We havew_n =G^∗0(t,z_n)→G^∗0(t,z) =won L^β(_R,R^N). Taking limit on both sides of G⁰(t,wn)−Lwn = I⁰(wn),

we have G⁰(t,w)−Lw=0, i.e.,u=wis a nontrivial solution of (1.1).

Step 5. Dichotomy also leads to a nontrivial solution of (1.1).

If dichotomy occurs, we set

wn(t) =un(t+yn), w⁽_n¹⁾(t) =w_n(t)χ_[−R1

n,R¹_n](t), w⁽_n²⁾(t) =w_n(t)(1−χ_[−R2

n,R²_n](t)), w⁽_n³⁾(t) =w_n(t)−w⁽_n¹⁾(t)−w⁽_n²⁾(t)_,

v⁽_n¹⁾(t) = ^w

(1) n (t) kw⁽_n¹⁾k_Lβ

. By (b) of the dichotomy, we have

Z _+∞

−_∞

|w⁽_n¹⁾(t)|^β kw_nk^β

L^β

dt=

Z _R¹_n

−R¹_n

|w_n(t)|^β kw_nk^β

L^β

dt→λ>0.

Fromδ₂ ≥ kw_nk_Lβ = ku_nk_Lβ ≥δ₁, we can see that there existsδ₃ >0 such thatkw⁽_n¹⁾k_Lβ > δ₃. By Step 4 and(L₁),w⁽_n¹⁾(t)→zin L^β(_R,R^N)andkzk_Lβ ≥δ₃. We will show that I⁰(w⁽_n¹⁾)→0, and hence I⁰(z) = 0, that is, u = z is a nontrivial solution of (1.1). In fact, for any u ∈ L^β(_R,R^N), as the splitting ofwn,u=u⁽¹⁾+u⁽²⁾+u⁽³⁾, and

hI⁰(w⁽_n¹⁾),ui=

Z +_∞

−_∞ (G⁰(t,w⁽_n¹⁾),u⁽¹⁾)dt−

Z +_∞

−_∞ (Lw⁽_n¹⁾,u)dt

=hI⁰(w_n),u⁽¹⁾i −

Z +_∞

−_∞ (Lw_n⁽¹⁾,u⁽²⁾+u⁽³⁾)dt+

Z +_∞

−_∞ (L(w⁽_n²⁾+w⁽_n³⁾),u⁽¹⁾)dt.

In the following we assumekuk_Lβ ≤1 and the limits will be taken asn→+∞. From (b) and (d) of the dichotomy, we have

kw⁽_n³⁾k^β

L^β =

Z +_∞

−_∞ |w⁽_n³⁾|^βdt=

Z

|t|≤R²_n

|wn|^βdt−

Z

|t|≤R¹_n

|w⁽_n¹⁾|^βdt→0,