# Existence and Multiplicity of Homoclinic Orbits for Second-Order Hamiltonian Systems with Superquadratic Potential

Authors: Ying Lv; Chun-Lei Tang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We investigate the existence and multiplicity of homoclinic orbits for second-order Hamiltonian systems with local superquadratic potential by using the Mountain Pass Theorem and the Fountain Theorem, respectively.


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## Body

## 1. Introduction and Main Result

Consider the second-order nonautonomous Hamiltonian systems(1)u¨(t)-L(t)u(t)+∇W(t,u(t))=0,
where L∈C(R,RN2) is a symmetric matrix valued function, W∈C1(R×RN,R), and ∇W(t,x)=(∂W/∂x)(t,x). We say that a nonzero solution u of problem (1) is homoclinic (to 0) if u(t)→0 and u˙(t)→0 as |t|→∞.The existence of homoclinic orbits for Hamiltonian systems and their importance in the study of the behavior of dynamical systems have been already recognized by Poincaré [1]. Only during the last two decades such problem has been studied by using critical point theory.IfL(t) and W(t,x) are independent of t or periodic in t, many authors have studied the existence of homoclinic orbits for Hamiltonian systems, see, for instance, [2–9], and a more general case is considered in recent papers [10, 11]. In this case, the existence of homoclinic orbits is obtained by going to the limit of periodic solutions of approximating problems. In recent years, concentration compactness principle has also been widely used to deal with the perturbations of periodic or autonomous problems, for example, [12, 13].IfL(t) and W(t,x) are neither autonomous nor periodic, the problem is quite different from the ones just described, because of the lack of compactness of the Sobolev embedding. Rabinowitz and Tanaka [14] study without any periodicity assumption and obtain the existence of homoclinic orbits of problem (1) by using a variant of the Mountain Pass Theorem without the Palais-Smale condition under the following condition.(L)  L∈C(R,RN2) is a symmetric and positively definite matrix for all t∈R, and there exists a continuous function l:R→R such that l(t)>0 for all t∈R and
(2)(L(t)x,x)≥l(t)|x|2,l(t)→∞as|t|→∞.
Assuming coercivity assumption (L), Omana and Willem [15] obtain an improvement on the latter result by employing a new compact embedding theorem; in fact, they show that the (PS) condition is satisfied and obtain the existence and multiplicity of homoclinic orbits of problem (1) by using the usual Mountain Pass Theorem. After [14] and [15], many results [16–22] are obtained for the case where L(t) is neither constant nor periodic in t.Korman and Lazer [23] remove the technical coercivity in case that L(t) and W(t,x) are even in t and L(t) is positively definite for all t∈R, by approximating homoclinic orbits from solutions of boundary value problems, which is complemented by [24].Most of the papers mentioned previously tackle the superquadratic case (see [2–10, 14–16, 18–21, 23, 24]) and the subquadratic case (see [17–19, 22, 25]). The following Ambrosetti-Rabinowitz condition is widely used in almost all papers tackling the superquadratic case.(AR) There exists a constantμ>2 such that, for every t∈R and x∈Rn∖{0},
(3)0<μW(t,x)≤(∇W(t,x),x).
Many recent papers have complemented the (AR) condition, for example, [6, 16, 20–22, 24].There are also many papers that tackle the multiplicity of homoclinic orbits, for example [20, 21, 23–25]. In particular, based on the variant Fountain Theorem of [26], Yang and Han [19] consider the multiplicity of homoclinic orbits for problem (1).Theorem A (see [19, Theorem  1.2]).
Suppose thatL(t) satisfies (L) and (L′). For some a>0 and r->0, one of the following is true: (i)
L∈C1(R,RN2) and |L′(t)|≤a|L(t)|, for all |t|≥r-, or(ii)
L∈C2(R,RN2) and L′′(t)≤aL(t), for all |t|≥r-,where L′(t)=(d/dt)L(t) and L′′(t)=(d2/dt2)L(t) and W(t,x) satisfies the following.(W1)
W(t,0)≡0 for all t∈R.(W2)
(∇W(t,x),x)≥0, for all (t,x)∈R×RN.(W3)
There existd1>0,μ>1 such that
(4)|∇W(t,x)|≤d1(1+|x|μ),
for all (t,x)∈R×RN.(W4)
lim|x|→0(∇W(t,x)/|x|)=0 uniformly for t∈R.(W5)
There existν>2 and c>0 such that
(5)liminf|x|→∞(∇W(t,x),x)|x|ν≥c>0
uniformly for t∈R.(W6)
s-1(∇W(t,sx),x) is an increasing function of s∈(0,1], for all (t,x)∈R×RN.(W7)
W(t,-x)=W(t,x), for all (t,x)∈R×RN.
Then system (1) has infinitely many homoclinic solutions uk satisfying
(6)12∫R|u˙k(t)|2+(L(t)uk(t),uk(t))dt-∫RW(t,uk(t))dt→+∞
as k→∞.In the present paper, based on the Fountain Theorem, we can prove the same result under more generic conditions, which generalizes Theorem A. Our first result can be stated as follows.Theorem 1.
Assume thatL satisfies (L) and (L′) and W satisfies (W1), (W4), (W7), (W8), and (W9). (W8)
For anyT>0,
(7)lim|x|→∞W(t,x)|x|2=+∞,
uniformly in t∈[-T,T].(W9)
there existsθ≥1, such that
(8)θF(t,x)≥F(t,sx)
for all (t,x)∈R×RN and s∈[0,1], where F(t,x)=(∇W(t,x),x)-2W(t,x).
Then problem (1) has infinitely many homoclinic orbits {uk} satisfying
(9)12∫R|u˙k(t)|2+(L(t)uk(t),uk(t))dt-∫RW(t,uk(t))dt→+∞
ask→∞.Remark 2.
Theorem1 generalizes and improves Theorem A. Firstly, in Theorem 1 we remove the positiveness condition (W2) and the growth condition (W3), which are indispensable in Theorem A. Now we compare conditions (W5) and (W8), (W6), and (W9). Our condition (W8) is a local superquadratic condition and is really weaker than condition (W5). Under condition (W6), for all s∈[0,1] we have
(10)F(t,x)-F(t,sx)=(∇W(t,x),x)-(∇W(t,sx),sx)-2[W(t,x)-W(t,sx)]=2[∫01(∇W(t,x),τx)dτ-∫0s(s-1∇W(t,sx),τx)dτ]-2∫s1(∇W(t,τx),x)dτ=2∫s1(∇W(t,x)-τ-1∇W(t,τx),τx)dτ+2∫0s(∇W(t,x)-s-1∇W(t,sx),τx)dτ≥0
for all (t,x)∈R×RN, which means that (W9) holds in the case that θ=1. We consider the multiplicity of homoclinic orbits for problem (1) by using the Fountain Theorem in [27] which is simpler than the variant Fountain Theorem [26].
Moreover, under all conditions of Theorem1 except (W7) we obtain an existence result.Theorem 3.
Assume thatL satisfies (L) and (L′) and W satisfies (W1), (W4), (W8) and (W9). Then problem (1) possesses a nontrivial homoclinic orbit.Remark 4.
In Theorem3, we consider the existence of homoclinic orbits for problem (1) under a class of local superquadratic conditions without the (AR) condition and any periodicity assumptions on both L and W. There are functions L and W which satisfy Theorem 3, but do not satisfy the corresponding results in [2–10, 14–16, 18–21, 23, 24]. For example,
(11)L(t)=(t2+1)IN,W(t,x)=2a(t)|x|2ln(1+|x|2),
where IN is the unit matrix of order N, a(t)=1/(1+t2).


## 2. Preliminary Results

In order to establish our results via critical point theory, we firstly describe some properties of the spaceE on which the variational functional associated with problem (1) is defined. Let
(12)E={u∈H1(R,RN):∫R[|u˙(t)|2u∈=H1(R,RN):∫R‍+|u˙(t)|2(L(t)u(t),u(t))]∫Rdt<+∞}.
Then the space E is a Hilbert space with the inner product
(13)(u,v)=∫R[(u˙(t),v˙(t))+(L(t)u(t),v(t))]dt
and the corresponding norm
(14)∥u∥2=(u,u).
Note that E⊂H1(R,RN)⊂Lp(R,RN) for all p∈[2,+∞] with the embedding being continuous. In particular, for p=2 and p=+∞, there exist constants C2 and C∞ such that
(15)∥u∥L2≤C2∥u∥,∀u∈E,(16)∥u∥L∞≤C∞∥u∥,∀u∈E.
Here Lp(R,RN)  (2≤p≤+∞) and H1(R,RN) denote the Banach spaces of functions on R with values in RN under the norms
(17)∥u∥Lp=(∫R|u(t)|pdt)1/p,(18)∥u∥H1=(∥u∥L22+∥u˙∥L22)1/2,
respectively. L∞(R,RN) is the Banach space of essentially bounded functions from R into RN equipped with the norm
(19)∥u∥∞=esssup{|q(t)|:t∈R}.Lemma 5 (see [18]).
Suppose that assumption (L) holds. Then the embedding of E into Lp(R,RN) is compact for all p∈[2,+∞].Denote byA the self-adjoint extension of the operator -(d2/dt2)+L(t) with the domain D(A)⊂L2≡L2(R,RN).Lemma 6 (see [18]).
If L satisfies(L) and (L′), then D(A) is continuously embedded in H2(R,RN), and, consequently, one has
(20)u(t)→0,u˙(t)→0
as |t|→∞, for all u∈D(A).Lemma 7.
Suppose that assumptions (W1), (W4) and (W9) hold. ThenW(t,x)≥0 for all t∈R and x∈RN.Proof.
Givent∈R and x∈RN, let
(21)f(s):=W(t,sx)s2
for s>0; then
(22)f′(s)=(∇W(t,sx),sx)-2W(t,sx)s3.
By (W1) and (W9), we have
(23)(∇W(t,x),x)-2W(t,x)≥0
for all t∈R and x∈RN. Hence,
(24)f′(s)≥0
for all s>0, which shows that f(s) is nondecreasing in (0,+∞). It is clear that
(25)lims→0f(s)=lims→0W(t,sx)s2=lims→0(∇W(t,sx),x)2s.
On the other hand, by (W4) one has
(26)lims→0|(∇W(t,sx),x)2s|≤lims→0|∇W(t,sx)|2|sx||x|2→0.
Therefore
(27)lims→0f(s)=0.
Now we get f(s)≥0 for all s>0, which implies that
(28)W(t,x)=f(1)≥0
for all t∈R and x∈RN.Lemma 8.
Assume that assumptions (L) and (W4) hold and un⇀u (weakly) in E. Then ∇W(t,un)→∇W(t,u) in L2(R,RN).Proof.
Assume thatun⇀u in E. Then there exists a constant d>0 such that
(29)supn∈N∥un∥L∞≤d,∥u∥L∞≤d.
By (W4), for every ε>0, there exists δ>0 such that
(30)|∇W(t,x)|<ε|x|
for all t∈R and x∈RN with |x|<δ. Now we claim that given d>0, for any δ>0, there exists T0>0 such that
(31)|u(t)|<δ
for all |t|>T0 and all u∈E with ∥u∥≤d. If not, there exists δ0>0, for all n∈N, and there exists un∈E with ∥un∥≤d and tn>n such that
(32)|un(tn)|≥δ0.
On the other hand, by Lemma 5(33)un→u
as n→∞ in L∞. In view of (32) and (33), we have
(34)|u(tn)|≥|un(tn)|-∥un-u∥L∞≥δ0-∥un-u∥L∞≥12δ0
when n is large enough, which is a contradiction to the fact that
(35)lim|t|→∞|u(t)|=0.
Hence, (31) holds. It follows from (29), (30), and (31) that
(36)|∇W(t,un(t))|≤ε|un(t)|,|∇W(t,u(t))|≤ε|u(t)|
for all n∈N and |t|>T0. By Lemma 5, un→u in L2(R,RN), and un(t)→u(t) for almost every t∈R and passing to a subsequence if necessary:
(37)∑n=1∞‍∥un-u∥L2<∞,
which implies
(38)vk(t):=∑n=1k‍|un(t)-u(t)|∈L2(R,R)
for k∈N+ and
(39)∥vk1-vk2∥L2≤∑n=k1k2‍∥un-u∥L2
for k2>k1∈N+. Since {un} is a Cauchy sequence in L2(R,RN), so by (39) we know that {vk} is also a Cauchy sequence in L2(R,R), which together with (38) and the completeness of L2(R,R) shows that
(40)ν(t)≔∑n=1∞‍|un(t)-u(t)|
is well defined and
(41)ν∈L2(R,R).
In consequence,
(42)|∇W(t,un(t))-∇W(t,u(t))|≤ε(|un(t)|+|u(t)|)≤ε(|un(t)-u(t)|+2|u(t)|)≤ε(ν(t)+2|u(t)|)
for all n∈N and |t|>T0. Consequently,
(43)∫R|∇W(t,un(t))-∇W(t,u(t))|2dt=∫|t|≤T0|∇W(t,un(t))-∇W(t,u(t))|2dt+∫|t|>T0|∇W(t,un(t))-∇W(t,u(t))|2dt≤∫|t|≤T0(|∇W(t,un(t))|+|∇W(t,u(t))|)2dt+∫|t|>T0ε2(|ν(t)|2+4|ν(t)||u(t)|+4|u(t)|2)dt≤∫|t|≤T0(max|x|≤d|∇W(t,x)|+max|x|≤d|∇W(t,x)|)2dt+ε2(∥ν∥L22+4∥ν∥L2∥u∥L2+4∥u∥L22)<∞
for all n∈N. Then using Lebesgue’s convergence theorem, the lemma is proved.In our paper we will also use the following lemma which is a special case of Lemma  1.1 in [28], due to Arioli and Szulkin [29].Lemma 9 (see [28, 29]).
Let{un} be a bounded sequence in Ls(R,RN), 1≤s<∞ such that {u˙n} is bounded in Lq(R,RN), 1≤q<∞. If, in addition, there exists l>0 such that
(44)supy∈R∫y-ly+l|un(t)|sdt→0
as n→∞, then
(45)un→0
in Lp(R,RN) for all p∈(s,∞).Now we introduce some notations and some necessary definitions which will be used later. LetB be a real Banach space, I∈C1(B,R), which means that I is continuously Frechet-differentiable functional defined on B. Recall that I∈C1(B,R) is said to satisfy the (PS) condition if any sequence {un}n∈N⊂B, for which {I(un)} is bounded and I′(un)→0 as n→+∞ possesses a convergent subsequence in B.Moreover, letBr be the open ball in B with the radius r and centered at 0, and ∂Br denotes its boundary; we obtain the existence of homoclinic orbits of problem (1) by the use of the following well-known Mountain Pass Theorem [30].Lemma 10 (see [30]).
LetB be a real Banach space and let I∈C1(B,R) satisfying the (PS) condition. Suppose that I(0)=0 and that(A1)
there are constantsρ,α>0 such that I|∂Bρ≥α,(A2)
there is ane∈B∖B-ρ such that I(e)<0.Then I possesses a critical value c≥α. Moreover c can be characterized as
(46)c=inff∈Γmaxt∈[0,1]I(g(t)),
where
(47)Γ={f∈C([0,1],B):f(0)=0,f(1)=e}.
As shown in [31], a deformation lemma can be proved with the (C)c condition replacing the usual (PS) condition, and it turns out that Lemma 10 holds true under the (C)c condition.In order to prove the multiplicity of homoclinic orbits, we will use the Fountain Theorem. SinceE is a Hilbert space, then there exists a basis {vn}⊂X such that X=⊕j≥1Xj¯, where Xj=span{vj}. Letting Yk=⊕j=1kXj,Zk=⊕j≥kXj¯, now we show the following Fountain Theorem.Lemma 11 (see [27]).
IfI∈C1(X,R) satisfies the (C)c condition, I(-u)=I(u), and for every k∈N, there exists ρk>rk>0 such that(i)
bk:=infu∈Zk,∥u∥=rkI(u)→+∞, as k→+∞;(ii)
ak:=maxu∈Yk,∥u∥=ρkI(u)≤0.Then I has a sequence of critical points {uk} such that I(uk)→+∞ as k→∞.In the proof of Theorem1, the following lemma will also be used. A similar result with respect to elliptic problem has been proved in [27].Lemma 12.
Suppose that2≤p≤+∞; then one has
(48)βk(p):=supu∈Zk,∥u∥=1∥u∥Lp→0
as k→∞.Proof.
It is clear that0<βk+1(p)≤βk(p), so there exists β(p) such that
(49)βk(p)→β(p)≥0
as k→∞ for every 2≤p≤+∞. By the definition of βk(p), there exists uk(p)∈Zk with ∥uk(p)∥=1 such that
(50)∥uk(p)∥Lp>βk(p)2
for every 2≤p≤+∞ and k∈N+. Since {uk(p)}k∈N is bounded, then there exists u(p)∈E such that
(51)uk(p)⇀u(p)
as k→∞. Now since {vj} is a basis of E, it follows that for all j∈N(52)0=(uk(p),vj)∀k>j→(u(p),vj)
as k→∞, which shows that u(p)=0. By Lemma 5 we have
(53)uk(p)→0
in Lp for all 2≤p≤+∞, which together with (49) and (50) implies that β(p)=0.


## 3. Proof of Theorems

Define the functionalI:E→R by
(54)I(u)=∫R[12|u˙(t)|2+12(L(t)u(t),u(t))-W(t,u(t))]dt=12∥u∥2-∫RW(t,u(t))dt.Lemma 13.
Under the conditions (L), (L′), and (W4), I∈C1(E,R), and for all u,v∈E one has
(55)〈I′(u),v〉=∫R[(u˙(t),v˙(t))+(L(t)u(t),v(t))aaa-(∇W(t,u(t)),v(t))]dt.
Moreover, any critical point of I on E is a solution of problem (1) with u(±∞)=0 and u˙(±∞)=0.Proof.
We firstly show thatI is well defined. It follows from (30) that for any ε>0, there exists δ>0 such that
(56)0≤W(t,x)≤12ε|x|2
for all t∈R and x∈RN with |x|<δ. Letting u∈E, then u∈C0(R,RN), the space of continuous function u on R, such that u(t)→0 as |t|→∞. Therefore there exists T1>0 such that
(57)|u(t)|<δ
for all |t|>T1. Hence, one has
(58)∫RW(t,u(t))dt≤∫-T1T1W(t,u(t))dt+12ε∫|t|>T1|u(t)|2dt≤∫-T1T1W(t,u(t))dt+12ε∥u∥L22<∞,
so I is well defined.
Next we prove thatI∈C1(E,R). Rewrite I as follows
(59)I=I1-I2,
where
(60)I1=∫R[12|u˙(t)|2+12(L(t)u(t),u(t))]dt,I2=∫RW(t,u(t))dt.
It is easy to check that I1∈C1(E,R) and
(61)〈I1′(u),v〉=∫R[(u˙(t),v˙(t))+(L(t)u(t),v(t))]dt.
It remains to show that I2∈C1(E,R). By the mean value theorem, for any u,v∈E and h∈[0,1] we have
(62)W(t,u(t)+hv(t))-W(t,u(t))=(∇W(t,u(t)+hθ(t)v(t)),v(t)),
where θ(t)∈(0,1). For any u,v∈E, there exists T2>0 such that
(63)|u(t)|+|v(t)|<δ
for all |t|>T2, so that
(64)|u(t)+hθ(t)v(t)|<δ
for all |t|>T2, which together with (15) and (30) implies
(65)∫Rmaxh∈[0,1]|(∇W(t,u(t)+hθ(t)v(t)),v(t))|dt=∫|t|≤T2maxh∈[0,1]|(∇W(t,u(t)+hθ(t)v(t)),v(t))|dt+∫|t|>T2maxh∈[0,1]|(∇W(t,u(t)+hθ(t)v(t)),v(t))|dt≤∫|t|≤T2max|x|≤∥u∥L∞+∥v∥L∞|∇W(t,x)||v(t)|dt+∫|t|>T2ε[|u(t)||v(t)||v(t)|2]dt≤∫|t|≤T2max|x|≤∥u∥L∞+∥v∥L∞|∇W(t,x)||v(t)|dt+ε(∥u∥L2∥v∥L2+∥v∥L22)<∞.
Then by Lebesgue’s convergence theorem, we have(66)〈I2′(u),v〉=limh→0+I2(u+hv)-I2(u)h=limh→0+∫R[W(t,u(t)+hv(t))-W(t,u(t))]dt=limh→0+∫R(∇W(t,u(t)+hθ(t)v(t)),v(t))dt=∫R(∇W(t,u(t)),v(t))dt.
Now we show that I2′ is continuous. Supposing that un→u in E, by an easy computation, one has
(67)sup∥v∥=1|〈I2′(un)-I2′(u),v〉|=sup∥v∥=1|∫R(∇W(t,un(t))-∇W(t,u(t)),v(t))dt|≤sup∥v∥=1∥∇W(t,un(t))-∇W(t,u(t))∥L2∥v∥L2≤C2∥∇W(t,un(t))-∇W(t,u(t))∥L2.
Hence by Lemma 8, we obtain
(68)〈I2′(un)-I2′(u),v〉→0
as n→∞ uniformly with respect to v, which implies the continuity of I2′. Now we have proved
(69)I∈C1(E,R).
Finally, we show that any critical pointu of I is a solution of problem (1) satisfying u(±∞)=0 and u˙(±∞)=0. If u∈E is a critical point of I, a standard argument shows that u∈C2(R,RN) satisfies (1). By Lemma 6, we only need to show that u is an element of D(A). It follows from (30) and (57) that
(70)|∇W(t,u(t))|<ε|u(t)|
for all |t|>T1. Hence, one has
(71)∫R|∇W(t,u(t))|2dt≤∫-T1T1|∇W(t,u(t))|2dt+ε2∫|t|>T1|u(t)|2dt≤∫-T1T1|∇W(t,u(t))|2dt+ε2∥u∥L22<∞,
so ∇W(t,u)∈L2(R,RN), which together with (1) implies that Au∈L2(R,RN). This means; u∈D(A), and the proof is completed.Lemma 14.
Under conditions (L), (W4), and (W9), I satisfies the (C)c condition.Proof.
I satisfies the (C)c condition; that is, for every {un}⊂E, {un} has a convergent subsequence if I(un) is bounded and (1+∥un∥)∥I′(un)∥→0 as n→∞. Assume that {un}⊂E is a sequence such that
(72)I(un)→c
is bounded and
(73)(1+∥un∥)∥I′(un)∥→0
as n→∞. Hence, we have
(74)limn→∞∫R[12(∇W(t,un(t)),un(t))-W(t,un(t))]dt=limn→∞(I(un)-12〈I′(un),un〉)=c.
Firstly, we show that{un} is bounded; if not, up to a subsequence we have
(75)∥un∥→∞
as n→∞. Letting vn=un/∥un∥, then {vn} is bounded in E. By Lemma 5, we have
(76)vn⇀vinE,vn→vinL2(R,RN)
as n→∞. We claim the following.
Claim1. consider
(77)limn→∞supy∈R∫y-1y+1|vn(t)|2dt=0.
Otherwise, for some σ>0, up to a subsequence we have
(78)supy∈R∫y-1y+1|vn(t)|2dt≥σ>0.
We can choose {yn}⊂R such that
(79)∫yn-1yn+1|vn(t)|2dt≥σ2.
In view of vn→v in L2(R,RN) and (79), we have
(80)∥v∥L22+σ4≥∫R|v(t)|2dt+∫R|vn(t)-v(t)|2dt≥∫yn-1yn+1|v(t)|2dt+∫yn-1yn+1|vn(t)-v(t)|2dt≥∫yn-1yn+1|vn(t)|2dt≥σ2
when n is large enough. By (80), there exists ε0>0, such that the set Θ={t∈R:|v(t)|≥ε0} has a positive Lebesgue measure. Moreover similar to (57), there exists T3>0 such that |v(t)|<ε0 for all |t|>T3, which implies that Θ⊂[-T3,T3]. For all t∈Θ, one has |un(t)|→∞ as n→∞, which together with (W8) shows
(81)W(t,un(t))∥un(t)∥2=W(t,un(t))|un(t)|2|vn(t)|2→+∞
as n→∞ uniformly for all t∈Θ. Hence by Lemma 6 and the fact that W(t,x)≥0 for all t∈R and x∈RN, we have
(82)12-c+o(1)∥un∥2=(1/2)∥un∥2-I(un)∥un∥2=∫RW(t,un(t))∥un∥2dt=∫RW(t,un(t))|un(t)|2|vn(t)|2dt≥∫ΘW(t,un(t))|un(t)|2|vn(t)|2dt→∞
as n→∞, which is a contradiction. Therefore we have proved Claim 1. Since ∥vn∥ is bounded, by Lemma 9, we have
(83)vn→0
in Lq(R,RN) for all q>2. Next, we will derive a contradiction. For any given r>0, ∥rvn∥=r. Similar to (31), for δ>0 defined in (56), there exists T4>0 such that
(84)|rvn(t)|<δ
for all |t|>T4 and all n∈N, which together with (56) shows that
(85)0≤W(t,rvn(t))≤ε2r2|vn(t)|2
for all n∈N and |t|>T4. In view of (83), vn→0 in L3(R,RN), which implies that
(86)∫|t|≤T4|vn(t)|2dt=(∫|t|≤T41dt)1/3(∫|t|≤T4|vn(t)|3dt)2/3→0
as n→∞. We can derive from (W4) that
(87)max|x|≤rC∞|∇W(t,x)||x|
is bounded for all |t|≤T4. Combining (86) and (87), we have
(88)∫|t|≤T4max|x|≤rC∞|∇W(t,x)||x||vn(t)|2dt<ε
when n is large enough. It follows from (85) and (88) that
(89)0≤∫RW(t,rvn(t))dt=∫|t|>T4W(t,rvn(t))dt+∫|t|≤T4W(t,rvn(t))dt=∫|t|>T4ε2r2|vn(t)|2dt+∫|t|≤T4∫01(∇W(t,srvn(t)),rvn(t))dsdt=∫|t|>T4ε2r2|vn(t)|2dt+∫|t|≤T4∫01|∇W(t,srvn(t))||srvn(t)||rvn(t)|2sdsdt≤∫Rε2r2|vn(t)|2dt+∫|t|≤T4r22max|x|≤rC∞|∇W(t,x)||x||vn(t)|2dt≤ε2r2C22+r22ε
when n is large enough, which implies that
(90)limn→∞∫RW(t,rvn(t))dt=0
for any given r>0. Choose a sequence {rn}⊂[0,1], such that
(91)I(rnun)=maxr∈[0,1]I(run).
Given M>0, since n is large enough, we have 2M∥un∥-1∈[0,1]; using (90) with r=2M, we obtain
(92)I(rnun)≥I(2M∥un∥un)=I(2Mvn)=2M-∫RW(t,2Mvn)dt≥M
for n large enough, which together with the arbitrary of M implies that
(93)I(rnun)→+∞
as n→∞. In view of (91) and the fact that rn∈(0,1), we have
(94)∫R[|rnu˙n|2+(L(t)rnun,rnun)-(∇W(t,rnun),rnun)]dt=〈I′(rnun),rnun〉=rnddt|r=rnI(run)=0.
By (W9), we get
(95)∫RF(t,un)dt≥1θ∫R‍F(t,rnun)dt
for all t∈R and n∈N. It follows from (93) and (94) that
(96)∫R[12(∇W(t,un),un)-W(t,un)]dt≥1θ∫R[12(∇W(t,rnun),rnun)-W(t,rnun)]dt=1θ∫R[12(|rnu˙n|2+(L(t)rnun,rnun))-W(t,rnun)12]dt=1θI(rnun)→+∞
as n→∞, which contradicts (74). Therefore we have proved that {un} is bounded.
By Lemma5 and the fact that {un} is bounded in E, there exist u∈E, d>0, and a subsequence of {un} again denoted by {un} such that
(97)supn∈N∥un∥L∞≤d∥u∥≤d,(98)un⇀uinE,un→uinL2(R,RN)
as n→∞. Arguing as in Lemma 8, we can also define
(99)ν(t):=∑n=1∞‍|un(t)-u(t)|
and ν∈L2(R,R). It is obvious that
(100)〈I′(un)-I′(u),u〉→0
as n→∞. By (L), (97), and Lemma 8 one has
(101)|∫R(∇W(t,un(t))-∇W(t,u(t)),un(t))dt|≤∥∇W(t,un(t))-∇W(t,u(t))∥L2∥un(t)∥L2≤C2∥∇W(t,un(t))-∇W(t,u(t))∥L2∥un∥≤C2d∥∇W(t,un(t))-∇W(t,u(t))∥L2→0
as n→∞, which implies that
(102)〈I′(un)-I′(u),un〉→0
as n→∞. Summing up (100) and (102), we have
(103)〈I′(un)-I′(u),un-u〉→0
as n→∞. On the other hand, by Lemma 7 and (97) we get
(104)|∫R‍(∇W(t,un(t))-∇W(t,u(t)),un(t)-u(t))dt|≤∥(∇W(t,un(t))-∇W(t,u(t))∥L2×∥(un(t)-u(t))∥L2≤C2∥∇W(t,un(t))-∇W(t,u(t))∥L2∥un-u∥≤2C2d∥∇W(t,un(t))-∇W(t,u(t))∥L2→0
as n→∞. An easy computation shows that
(105)〈I′(un)-I′(u),un-u〉=∥un-u∥2-∫R(∇W(t,un(t)-∇W(t,u(t)),un(t)-u(t)))dt.
Consequently, ∥un-u∥→0 as n→∞.Proof of Theorem1.
By Lemma13 and Lemma 14, I∈C1(E,R) satisfies the (C)c condition and I(u)=I(-u); hence to prove Theorem 1 we should just show that I has the geometric properties (i) and (ii) of Lemma 11.
(i) By Lemma12(106)βk(p)=supu∈Zk,∥u∥=1∥u∥Lp→0
as k→∞ for p∈[2,+∞]. We choose rk=min{1/βk(∞),1/βk(4)}; then rk→∞ as k→∞, and for every u∈Zk with ∥u∥=rk, we have
(107)∥u∥L∞≤βk(∞)rk<1,∥u∥L4≤βk(4)rk<1.
Similar to (31), there exists T5>0 such that
(108)|u(t)|<δ
for all |t|>T5 and all u∈Zk with ∥u∥=rk, where δ is defined in (56). Consequently, by (56), for any ε>0(109)0≤W(t,u(t))≤12ε|u(t)|2
for all |t|>T5 and all u∈Zk with ∥u∥=rk. Hence, we have for all u∈Zk with ∥u∥=rk(110)I(u)=12∥u∥2-∫|t|>T5W(t,u(t))dt-∫|t|≤T5W(t,u(t))dt≥12∥u∥2-∫|t|>T512ε|u(t)|2dt-∫|t|≤T5max|x|≤1W(t,x)dt≥12∥u∥2-12ε(βk(2))2∥u∥2-∫|t|≤T5max|x|≤1W(t,x)dt≥14∥u∥2-∫|t|≤T5max|x|≤1W(t,x)dt
when ε is small enough. Therefore, one has
(111)bk=infu∈Zk,∥u∥=rkI(u)≥infu∈Zk,∥u∥=rk(14∥u∥2-∫|t|≤T5max|x|≤1W(t,x)dt)=14rk2-∫|t|≤T5max|x|≤1W(t,x)dt→∞
as k→∞.
(ii) Firstly, we claim that there exists a constantε>0 such that
(112)meas{t∈R:|u(t)|≥ε∥u∥}≥ε
for all u∈Yk∖{0}. Otherwise, for every n∈N, there exists un∈Yk∖{0} such that
(113)meas{t∈R:|un(t)|≥1n∥un∥}<1n.
Without loss of generality, we suppose that ∥un∥=1; then there is
(114)meas{t∈R:|un(t)|≥1n}<1n.
In view of the compactness of the unit sphere of Yk, there exists a subsequence which is still denoted by {un} such that un converges to some u0∈Yk as n→∞. It is clear that ∥u0∥=1. Since all the norms in Yk are equivalent, we have un→u0 in L2 as n→∞; that is,
(115)∫R|un(t)-u0(t)|2dt→0
as n→∞. Thus there exist constants τ1 and τ2 such that
(116)meas{t∈R:|u0(t)|≥τ1}≥τ2.
If not, we have
(117)meas{t∈R:|u0(t)|≥1n}=0
for all n∈N, which implies that
(118)0≤∫R|u0(t)|4dt≤∥u0∥L∞2∥u0∥L22<1n2C22∥u0∥2=1n2C22→0
as n→∞. Hence u0=0, which contradicts ∥u0∥=1. Therefore (116) holds. Let Ω0={t∈R:|u0(t)|≥τ1}, Ωn={t∈R:|un(t)|<1/n}, and Ωnc=R∖Ωn. It follows from (114) and (116) that
(119)meas(Ωn∖Ω0)=meas(Ω0∖(Ωnc∩Ω0))≥meas(Ω0)-meas(Ωnc∩Ω0)=τ2-1n.
We can choose n large enough, such that
(120)τ1-1n≥12τ1,τ2-1n≥12τ2.
Then we have
(121)|un(t)-u0(t)|2≥||un(t)|-|u0(t)||2≥(τ1-1n)2≥14τ12
for all t∈Ωn∩Ω0. In consequence,
(122)∫R|un(t)-u0(t)|2dt≥∫Ωn∩Ω0|un(t)-u0(t)|2dt≥14τ12·meas(Ωn∩Ω0)≥14τ12(τ2-1n)≥18τ12τ2>0
for all n being large enough, which is a contradiction to (115). Hence (112) holds.
Similar to (31), for any ε>0 there exists T6>0 such that
(123)|v(t)|<ε
for all |t|>T6 and all v∈Yk with ∥v∥≤1. Consequently, for all u∈Yk,
(124)|u(t)|∥u∥<ε
for all |t|>T6, which implies that Ωu:={t∈R:|u(t)|≥ε∥u∥}⊂[-T6,T6] for all u∈Yk. By (W8), there exists G>0 such that
(125)W(t,x)≥1ε3|x|2
for all x∈RN with |x|≥G and |t|<T6. Hence we have
(126)W(t,u(t))≥1ε3|u(t)|2≥1ε∥u∥2
for all t∈Ωu and u∈Yk with ∥u∥≥G/ε, which together with Lemma 7 implies that
(127)I(u)∥u∥2=12-∫RW(t,u(t))∥u∥2dt≤12-∫ΩuW(t,u(t))∥u∥2dt=12-1ε·meas(Ωu)≤12-1<0
for all u∈Yk and ∥u∥≥G/ε. So we can choose ρk>max{G/ε,rk}; then
(128)ak=maxu∈Yk,∥u∥=ρkI(u)≤0.
Hence by Lemma11, we obtain that problem (1) has infinitely many homoclinic solutions {uk} satisfying
(129)12∫R|u˙k(t)|2+(L(t)uk(t),uk(t))dt-∫RW(t,uk(t))dt→+∞
as k→∞.Proof of Theorem3.
We divide the proof of Theorem3 into the following three steps.
Step1. It is clear that I(0)=0, and we have proved that I∈C1(E,R) satisfies the (C)c condition in Lemma 13 and Lemma 14.
Step2. Letting ρ=δ/C∞ and ∥u∥=ρ, we have ∥u∥L∞≤δ, where C∞>0 is defined in (16) and δ is defined in (56). Hence, by (56), for any ε>0(130)0≤W(t,u(t))≤12ε|u(t)|2
for all t∈R and u∈E with ∥u∥=ρ. In consequence, combining this with (54), we obtain
(131)I(u)=12∥u∥2-∫R‍W(t,u)dt≥12∥u∥2-ε2∥u∥L22≥(12-ε2C22)∥u∥2
for all t∈R and u∈E with ∥u∥=ρ. Setting ε=1/2C22, the inequality (131) implies that
(132)I|∂Bρ≥ρ24:=α>0.
Step3. It remains to prove that there exists an e∈E such that ∥e∥>ρ and I(e)≤0, where ρ is defined in Step 2. By (W8), for any T,M>0, there exists G>0 such that
(133)W(t,x)|x|2≥M
for all |x|>G and t∈[-T,T]. Letting u0(t)=e-t2e1, where e1=(1,0,…,0), hence |u0(t)|>e-4 for all t∈[1,2]. It is clear that when r>Ge4,
(134)|ru0(t)|>G
for all t∈[1,2], which together with (133) shows that when r>Ge4(135)W(t,ru0(t))|ru0(t)|2≥M
for all t∈[1,2]. Combining (135), Lemma 7, and the fact that ∫12‍|u0(t)|2dt>0, we have
(136)I(ru0)r2=12∥u0∥2-∫R‍W(t,ru0(t))|ru0(t)|2|u0(t)|2dt=12∥u0∥2-∫12‍W(t,ru0(t))|ru0(t)|2|u0(t)|2dt-∫R∖[1,2]‍W(t,ru0(t))|ru0(t)|2|u0(t)|2dt≤12∥u0∥2-∫12‍W(t,ru0(t))|ru0(t)|2|u0(t)|2dt≤12∥u0∥2-M∫12‍|u0(t)|2dt<0,
when r and M are both large enough.
By Lemma10, I possesses a critical value c≥α>0 given by
(137)c=inff∈Γmaxt∈[0,1]I(f(t)),
where
(138)Γ={f∈C([0,1],B):f(0)=0,f(1)=e}.
Hence there is a u∈E such that
(139)I(u)=c,I′(u)=0.
Therefore u is a nontrivial homoclinic orbit of problem (1). Theorem 3 is proved now.



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*Source: 328630-2013-02-20.xml*