Infinitely many fast homoclinic solutions for some second-order nonautonomous systems
We investigate the existence of infinitely many fast homoclinic solutions for a class of second-order nonautonomous systems. Our main tools are based on the variant fountain theorem. A criterion guaranteeing that the second-order system has infinitely many fast homoclinic solutions is obtained. Rece...
Gespeichert in:
| Datum: | 2014 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2014
|
| Schriftenreihe: | Український математичний журнал |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/165985 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Infinitely many fast homoclinic solutions for some second-order nonautonomous systems / Liu Yang, Liping Luo, Zhenguo Luo // Український математичний журнал. — 2014. — Т. 66, № 3. — С. 404–414. — Бібліогр.: 16 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-165985 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1659852020-02-18T01:27:51Z Infinitely many fast homoclinic solutions for some second-order nonautonomous systems Yang, Liu Luo, Liping Luo, Zhenguo Статті We investigate the existence of infinitely many fast homoclinic solutions for a class of second-order nonautonomous systems. Our main tools are based on the variant fountain theorem. A criterion guaranteeing that the second-order system has infinitely many fast homoclinic solutions is obtained. Recent results from the literature are generalized and significantly improved. Досліджєно існування нескінченної кількості швидких гомоклінічних розв'язків для класу неавтономних систем другого порядку. Наш основний метод базується на модифікації теореми про фонтан. Отримано критерій, що гарантує наявність нескінченної кількості швидких гомоклінічних розв'язків системи другого порядку. Узагальнено та значно покращено нещодавно опубліковані результати. 2014 Article Infinitely many fast homoclinic solutions for some second-order nonautonomous systems / Liu Yang, Liping Luo, Zhenguo Luo // Український математичний журнал. — 2014. — Т. 66, № 3. — С. 404–414. — Бібліогр.: 16 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165985 517.9 en Український математичний журнал Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Статті Статті |
| spellingShingle |
Статті Статті Yang, Liu Luo, Liping Luo, Zhenguo Infinitely many fast homoclinic solutions for some second-order nonautonomous systems Український математичний журнал |
| description |
We investigate the existence of infinitely many fast homoclinic solutions for a class of second-order nonautonomous systems. Our main tools are based on the variant fountain theorem. A criterion guaranteeing that the second-order system has infinitely many fast homoclinic solutions is obtained. Recent results from the literature are generalized and significantly improved. |
| format |
Article |
| author |
Yang, Liu Luo, Liping Luo, Zhenguo |
| author_facet |
Yang, Liu Luo, Liping Luo, Zhenguo |
| author_sort |
Yang, Liu |
| title |
Infinitely many fast homoclinic solutions for some second-order nonautonomous systems |
| title_short |
Infinitely many fast homoclinic solutions for some second-order nonautonomous systems |
| title_full |
Infinitely many fast homoclinic solutions for some second-order nonautonomous systems |
| title_fullStr |
Infinitely many fast homoclinic solutions for some second-order nonautonomous systems |
| title_full_unstemmed |
Infinitely many fast homoclinic solutions for some second-order nonautonomous systems |
| title_sort |
infinitely many fast homoclinic solutions for some second-order nonautonomous systems |
| publisher |
Інститут математики НАН України |
| publishDate |
2014 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/165985 |
| citation_txt |
Infinitely many fast homoclinic solutions for some second-order nonautonomous systems / Liu Yang, Liping Luo, Zhenguo Luo // Український математичний журнал. — 2014. — Т. 66, № 3. — С. 404–414. — Бібліогр.: 16 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT yangliu infinitelymanyfasthomoclinicsolutionsforsomesecondordernonautonomoussystems AT luoliping infinitelymanyfasthomoclinicsolutionsforsomesecondordernonautonomoussystems AT luozhenguo infinitelymanyfasthomoclinicsolutionsforsomesecondordernonautonomoussystems |
| first_indexed |
2025-07-14T20:27:29Z |
| last_indexed |
2025-07-14T20:27:29Z |
| _version_ |
1837655498066755584 |
| fulltext |
UDC 517.9
Liu Yang, Liping Luo, Zhenguo Luo (Hengyang Normal Univ., China)
INFINITELY MANY FAST HOMOCLINIC SOLUTIONS
FOR SOME SECOND-ORDER NONAUTONOMOUS SYSTEMS*
НЕСКIНЧЕННА КIЛЬКIСТЬ ШВИДКИХ ГОМОКЛIНIЧНИХ РОЗВ’ЯЗКIВ
НЕАВТОНОМНИХ СИСТЕМ ДРУГОГО ПОРЯДКУ
We investigate the existence of infinitely many fast homoclinic solutions for a class of second-order nonautonomous
systems. Our main tools are based on the variant fountain theorem. A criterion guaranteeing that the second-order system
have infinitely many fast homoclinic solutions is obtained. Recent results from the literature are generalized and significantly
improved.
Дослiджено iснування нескiнченної кiлькостi швидких гомоклiнiчних розв’язкiв для класу неавтономних систем
другого порядку. Наш основний метод базується на модифiкацiї теореми про фонтан. Отримано критерiй, що
гарантує наявнiсть нескiнченної кiлькостi швидких гомоклiнiчних розв’язкiв системи другого порядку. Узагальнено
та значно покращено нещодавно опублiкованi результати.
1. Introduction. In this article, we are concerned with the existence of infinitely many fast homo-
clinic solutions for the following second-order nonautonomous systems:
ü(t) + cu̇− L(t)u(t) +Wu(t, u(t)) = 0 ∀t ∈ R, (FHS)
where u ∈ Rn, c ≥ 0 is a constant, W (t, u) ∈ C1(R,Rn), and L(t) ∈ C(R,Rn×n) is a symmetric
matrix valued function. A nontrivial solution u of (FHS) is said to be homoclinic to zero if u ∈
∈ C2(R,Rn), u(t)→ 0 and u̇(t)→ 0 as |t| → ∞.
When c = 0, (FHS) is just the following second-order Hamiltonian system:
ü(t)− L(t)u(t) +Wu(t, u(t)) = 0. (HS)
In the last ten years, the existence and multiplicity of homoclinic solutions of (HS) have been
intensively studied by many mathematicians (see [1 – 14] and the references therein). Compared with
the case that W (t, u) is superquadratic growth as |u| → ∞, there is less literature for the case that
W (t, u) is subquadratic growth as |u| → ∞ (see [12 – 14]). In [13], Zhang and Yuan established the
following theorem.
Theorem 1.1 [13]. Assume that L and W satisfy the following conditions:
(H1) L(t) ∈ C(R, Rn×n) is a symmetric and positive definite matrix for all t ∈ R and there is
a continuous function α : R → R such that α(t) > 0 for all t ∈ R and (L(t)u, u) ≥ α(t)|u|2 and
α(t)→∞ as |t| → +∞;
(H2) W (t, u) = a(t)|u|γ where a(t) : R → R+ is a positive continuous function such that
a(t) ∈ L2(R,R) ∩ L
2
2−γ (R,R) and 1 < γ < 2 is a constant.
Then (HS) possesses a nontrivial homoclinic solution.
* This work was supported by the Natural Science Foundation of Hunan Province (12JJ9001), Hunan Provincial Science
and Technology Department of Science and Technology Project (2012SK3117) and Construct program of the key discipline
in Hunan Province.
c© LIU YANG, LIPING LUO, ZHENGUO LUO, 2014
404 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
INFINITELY MANY FAST HOMOCLINIC SOLUTIONS . . . 405
There are many mathematicians introduced the concept of fast heteroclinic solutions for the
second-order ordinary differential equation u′′ + cu′ + f(u) = 0, (see [15]). When c 6= 0 in (FHS),
as far as we know, there is few research about the existence of homoclinic solutions for (FHS). In
[15], Zhang and Yuan introduced the concept of fast homoclinic solutions for (FHS) and established
some criteria to guarantee the existence of fast homoclinic solutions for the first time. In order to
state the concept of the fast homoclinic solutions conveniently, we first introduce some properties of
the weighted Sobolev space Ec. For c ≥ 0, we define the weighted Sobolev space Ec as follows:
Ec =
u ∈ H1(R,Rn) :
∫
R
ect
[
|u′|2 + (L(t)u(t), u(t))
]
dt < +∞
.
If L satisfies (H1), Ec is a Hilbert space with the inner product
(x, y) =
∫
R
ect
[
(x′(t), y′(t)) + (L(t)x(t), y(t))
]
dt
and the corresponding norm ‖x‖2Ec = (x, x). Here, we denote by Lp(ect), 2 ≤ p < +∞, the Banach
space of functions on R with values in Rn under the norm
‖u‖p :=
∫
R
ect|u(t)|pdt
1/p
.
Here, we still use the notation ‖ · ‖p to denote the norm of Lp(ect). Hence, there exists a constant
β = min{α(t), t ∈ R} > 0 such that
β‖u‖22 ≤ ‖u‖2Ec ∀u ∈ Ec. (1.1)
Definition 1.1. For c > 0, a homoclinic solution u of (FHS) is called one fast homoclinic
solution if u ∈ Ec.
Theorem 1.2 [15]. Assume that L and W satisfy (H1) and the following condition:
(H2) W (t, u) = a(t)|u|γ where a(t) : R → R is a continuous function such that a(t1) > 0 for
some t1 ∈ R and a(t) ∈ L
2
2−γ (ect) and 1 < γ < 2 is a constant.
Then (FHS) has at least one nontrivial fast homoclinic solution.
Motivated by the above facts, in this paper, we will use the following conditions to generalize
and improve Theorem 1.2. To the best of our knowledge, there is no paper studying the existence of
infinitely many fast homoclinic solutions for (FHS).
(H2
′
) a(t)|u|γ ≤ Wu(t, u)u, |Wu(t, u)| ≤ b(t)|u|γ−1 + c(t)|u|δ−1 where a(t), b(t), c(t) : R→
→ R+ are positive continuous functions such that a(t), b(t) ∈ L
2
2−γ (ect), c(t) ∈ L
2
2−δ (ect) and
1 < γ < 2, 1 < δ < 2 are constants, W (t, 0) = 0, W (t, u) = W (t,−u).
We can see that if b(t) = a(t), c(t) = 0, then W (t, u) =
a(t)
γ
|u|γ . Therefore, the condition of
(H2) is a special case of the condition of (H2
′
). Here is our main result.
Theorem 1.3. Suppose that the conditions of (H1) and (H2
′
) hold. Then (FHS) possesses
infinitely many fast homoclinic solutions.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
406 LIU YANG, LIPING LUO, ZHENGUO LUO
The organization of this paper is as follows. In Section 2, we shall give some lemmas and some
preliminary results. In Section 3, main result are verified.
2. Preliminaries. In this section, we will present some lemmas that will be used in the proof of
our main result.
Lemma 2.1 [15]. Suppose that L satisfies (H1). Then the embedding of Ec in L2(ect) is com-
pact.
Lemma 2.2. Suppose that (H1), (H2
′
) hold. If uk ⇀ u in Ec, then Wu(t, uk) → Wu(t, u) in
L2(ect).
Proof. Assume that uk ⇀ u in Ec. By (H2
′
) we have
|Wu(t, uk)−Wu(t, u)| ≤ b(t)
[
|uk|γ−1 + |u|γ−1
]
+ c(t)
[
|uk|δ−1 + |u|δ−1
]
, (2.1)
which yields that
|Wu(t, uk)−Wu(t, u)|2 ≤ 4b2(t)
[
|uk|2γ−2 + |u|2γ−2
]
+ 4c2(t)
[
|uk|2δ−2 + |u|2δ−2
]
. (2.2)
Multiplying ect and integrating on R, by (1.1) and Hölder inequality, we get∫
R
ect|Wu(t, uk)−Wu(t, u)|2dt ≤
≤ 4
∫
R
ectb2(t)[|uk(t)|2γ−2 + |u(t)|2γ−2]dt+ 4
∫
R
ectc2(t)[|uk(t)|2δ−2 + |u(t)|2δ−2]dt ≤
≤ 4‖b‖2 2
2−γ
(‖uk‖2γ−2
2 + ‖u‖2γ−2
2 ) + 4‖c‖2 2
2−δ
(‖uk‖2δ−2
2 + ‖u‖2δ−2
2 ) ≤
≤ 4β1−γ‖b‖2 2
2−γ
(‖uk‖2γ−2
Ec
+ ‖u‖2γ−2
Ec
) + 4β1−δ‖c‖2 2
2−δ
(‖uk‖2δ−2
Ec
+ ‖u‖2δ−2
Ec
). (2.3)
Moreover, since uk ⇀ u in Ec, there exists a constant M > 0 such that, by Banach – Steinhaus
theorem,
‖uk‖Ec ≤M, ‖u‖Ec ≤M.
Therefore, we can obtain∫
R
ect|Wu(t, uk)−Wu(t, u)|2dt ≤ 8β1−γ‖b‖2 2
2−γ
M2γ−2 + 8β1−δ‖c‖2 2
2−δ
M2δ−2.
Since, by Lemma 2.1, uk → u in L2(ect), which yields that ectuk(t) → ectu(t) for almost every
t ∈ R, i.e., uk(t) → u(t) for almost every t ∈ R since ect > 0 for every t ∈ R. Then, by the using
the Lebesgue convergence theorem.
Lemma 2.2 is proved.
Define the functional
I(u) =
1
2
∫
R
ect[|u̇|2 + (L(t)u(t), u(t))]dt−
∫
R
ectW (t, u(t))dt =
1
2
‖u‖2Ec −B(u), (2.4)
where B(u) =
∫
R
ectW (t, u(t))dt.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
INFINITELY MANY FAST HOMOCLINIC SOLUTIONS . . . 407
Lemma 2.3. Under the conditions of Theorem 1.3, we get
I ′(u)v =
∫
R
ect[(u̇, v̇) + (L(t)u(t), v(t))]dt−
∫
R
ect(Wu(t, u(t)), v(t))dt =
=
∫
R
ect[(u̇, v̇) + (L(t)u(t), v(t))]dt−B′(u)v (2.5)
for any u, v ∈ Ec, which yields that
I ′(u)u = ‖u‖2Ec −
∫
R
ect(Wu(t, u(t)), u(t))dt. (2.6)
Moreover, I ∈ C1(Ec,R), B′ : Ec → E∗c is compact, and any critical point of I on Ec is a
classical solution of (FHS) satisfying u ∈ C2(R,Rn), u(t)→ 0 and u̇(t)→ 0 as |t| → ∞.
Proof. We firstly show that I : Ec → R. Since W (t, 0) = 0, by (H2′), we have
0 ≤
∫
R
ect
1∫
0
a(t)|u|γhγ−1dh
dt ≤ ∫
R
ectW (t, u(t))dt =
=
∫
R
ect
1∫
0
Wu(t, hu)udh
dt ≤ ∫
R
ect
1∫
0
|Wu(t, hu)||u|dh
dt ≤
≤
∫
R
ect
b(t)
γ
|u(t)|γdt+
∫
R
ect
c(t)
δ
|u(t)|δdt ≤ 1
γ
‖b‖ 2
2−γ
‖u‖γ2 +
1
δ
‖c‖ 2
2−δ
‖u‖δ2 ≤
≤ 1
γ
‖b‖ 2
2−γ
β−γ‖u‖γEc +
1
δ
‖c‖ 2
2−δ
β−δ‖u‖δEc . (2.7)
Next we prove that I ∈ C1(Ec,R). Rewrite I as follows:
I = A(u)−B(u), (2.8)
where
A(u) =
1
2
∫
R
ect
[
|u̇|2 + (L(t)u(t), u(t))
]
dt.
It is easy to check that A ∈ C1(Ec,R) and A′(u)v =
∫
R
ect [(u̇, v̇) + (L(t)u(t), v(t))] dt. Therefore,
it is sufficient to show that this is the case for B. In the process we will see that
B′(u)v =
∫
R
ect(Wu(t, u(t)), v(t))dt. (2.9)
For any given u ∈ Ec, let us define J(u) : Ec → R as follows:
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
408 LIU YANG, LIPING LUO, ZHENGUO LUO
J(u)v =
∫
R
ect(Wu(t, u(t)), v(t))dt, v ∈ Ec. (2.10)
It is obvious that J(u) is linear. Now we show that J(u) is bounded. Indeed, for any given
u ∈ Ec, we have
|J(u)v| =
∫
R
ect(Wu(t, u(t)), v(t))dt ≤
≤
∫
R
ectb(t)|u(t)|γ−1||v(t)|dt+
∫
R
ectc(t)|u(t)|δ−1||v(t)|dt ≤
≤
∫
R
ectb2(t)|u(t)|2γ−2dt
1/2∫
R
ect|v(t)|2dt
1/2
+
+
∫
R
ectc2(t)|u(t)|2δ−2dt
1/2∫
R
ect|v(t)|2dt
1/2
≤
≤
∫
R
ectb2(t)|u(t)|2γ−2
1/2
‖v‖2 +
∫
R
ectc2(t)|u(t)|2δ−2
1/2
‖v‖2 ≤
≤ ‖b‖ 2
2−γ
‖u‖γ−1
2 ‖v‖2 + ‖c‖ 2
2−δ
‖u‖δ−1
2 ‖v‖2 ≤
≤ β−γ‖b‖ 2
2−γ
‖u‖γ−1
Ec
‖v‖Ec + β−δ‖c‖ 2
2−δ
‖u‖δ−1
Ec
‖v‖Ec .
Moreover, for u, v ∈ Ec, by the mean value theorem, we obtain∫
R
ectW (t, u(t) + v(t))dt−
∫
R
ectW (t, u(t))dt =
∫
R
ect(Wu(t, u(t) + h(t)v(t)), v(t))dt,
where h(t) ∈ (0, 1). Therefore, by Lemma 2.2, we get∫
R
ect(Wu(t, u(t) + h(t)v(t)), v(t))dt−
∫
R
ect(Wu(t, u(t)), v(t))dt =
=
∫
R
ect (Wu(t, u(t) + h(t)v(t))−Wu(t, u(t)), v(t)) dt→ 0
as v → 0. Suppose that u→ u0 in Ec and note that
B′(u)v −B′(u0)v =
∫
R
ect(Wu(t, u(t))−Wu(t, u0(t)), v(t))dt. (2.11)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
INFINITELY MANY FAST HOMOCLINIC SOLUTIONS . . . 409
By Lemma 2.2 and the Hölder inequality, we obtain that
B′(u)v −B′(u0)v → 0 as u→ u0, (2.12)
which implies the continuity of B′ and we show that I ∈ C1(Ec,R). Let uk ⇀ u in Ec, we have
‖B′(u
k
)−B′(u)‖E∗c = sup
‖v‖=1
‖(B′(u
k
)−B′(u))v‖ =
= sup
‖v‖=1
∣∣∣∣∣∣
∫
R
ect〈Wu(t, uk)−Wu(t, u), v(t)〉dt
∣∣∣∣∣∣ ≤
≤ sup
‖v‖=1
∫
R
ect|Wu(t, uk)−Wu(t, u)|2dt
1/2
‖v‖2 ≤
≤ C2
∫
R
ect|Wu(t, uk)−Wu(t, u)|2dt
1/2
→ 0
as k → ∞. Consequently, B′ is weakly continuous. Therefore, B′ is compact by the weakly conti-
nuity of B′ since E is a Hilbert space. Proofs of the other conclusions can be found in Lemma 3.1
of [15], so we omit them here.
In order to prove our main results, we recall the variant fountain theorem. Let E be a Ba-
nach space with the norm ‖ · ‖ and E =
⊕k
j=0Xj with dimXj < ∞ for any j ∈ N. Set
Yk =
⊕k
j=0Xj , Zk =
⊕∞
j=kXj . Consider the following C1-functional Iλ : E → R defined by
Iλ(u) = A(u)− λB(u), λ ∈ [1, 2]. (2.13)
Theorem 2.1 [16]. Suppose that the functional Iλ(u) defined above satisfies:
(C1) Iλ maps bounded sets to bounded sets uniformly for λ ∈ [1, 2]. Furthermore, Iλ(−u) =
= Iλ(u) for all (λ, u) ∈ [1, 2]× E.
(C2) B(u) ≥ 0;B(u)→∞ as ‖u‖ → ∞ on any finite dimensional subspace of E.
(C3) There exist ρk > rk > 0 such that
ak(λ) := inf
u∈Zk,‖u‖=ρk
Iλ(u) ≥ 0 > bk(λ) := max
u∈Yk,‖u‖=rk
Iλ(u)
for all λ ∈ [1, 2] and dk(λ) := infu∈Zk‖u‖≤ρk Iλ(u) → 0 as k → ∞ uniformly for λ ∈ [1, 2]. Then
there exist λn → 1, uλn ∈ Yn such that I ′λn |Yn(u(λn)) = 0, Iλn(u(λn)) → ck ∈ [dk(2), bk(1)] as
n → ∞. In particular, if {u(λn)}} has a convergent subsequence for every k, then I1 has infinitely
many nontrivial critical points {un} ⊂ E \ {0} satisfying I1(uk)→ 0− as k →∞.
3. Main results. Proof of Theorem 1.3. In order to apply Theorem 2.1 to prove Theorem 1.3,
we define the functionals A,B and Iλ on our working space Ec by
A(u) =
1
2
‖u‖2Ec , B(u) =
∫
R
ectW (t, u)dt, (3.1)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
410 LIU YANG, LIPING LUO, ZHENGUO LUO
Iλ(u) = A(u)− λB(u) (3.2)
for all u ∈ Ec and λ ∈ [1, 2]. From Lemma 2.3, we know that Iλ ∈ C1(Ec,R) for all λ ∈ [1, 2].
We choose a completely orthonormal basis {ej} of Ec and define Xj := Rej . Then Zk, Yk can be
defined as that in Section 2.
Step 1. In the condition of Theorem 1.3, we have B(u) ≥ 0. Moreover, B(u)→∞ as ‖u‖ → ∞
on any finite dimensional subspace of Ec.
Obviously, B(u) ≥ 0 follows by the definition of the functional B and (H2′). For any finite
dimensional subspace F ⊂ Ec, there exists ε1 > 0 such that
meas
{
t ∈ R : ecta(t)|u(t)|γ ≥ ε1‖u‖γEc
}
≥ ε1 ∀u ∈ F \ {0}, (3.3)
where meas denotes that Lebesgue measure in Rn. Otherwise, for any positive integer n, there exists
un ∈ F \ {0} such that
meas
{
t ∈ R : ecta(t)|u(t)|γ ≥ 1
n
‖u‖γEc
}
<
1
n
. (3.4)
Set vn(t) :=
un(t)
‖un‖Ec
∈ F \ {0}, then ‖vn‖Ec = 1 for all n ∈ N and
meas
{
t ∈ R : ecta(t)|vn(t)|γ ≥ 1
n
}
<
1
n
.
Since dimF < ∞, it follows from the compactness of the unit sphere of F that there exists a
subsequence, say {vn}, such that vn converges to some v0 in F . Hence, we have ‖v0‖Ec = 1. By
the equivalence of the norms on the finite dimensional space F , we have vn → v0 in L2(ect). By the
Hölder inequality, one has
∫
R
ecta(t)|vn − v0|γdt ≤ ‖a‖ 2
2−γ
∫
R
ect|vn − v0|2dt
γ/2
→ 0 as n→∞. (3.5)
Thus there exist ξ1, ξ2 > 0 such that
meas
{
t ∈ R : ecta(t)|v0(t)|γ ≥ ξ1
}
≥ ξ2. (3.6)
In fact, if not, we have
meas
{
t ∈ R : ecta(t)|v0(t)|γ ≥ 1
n
}
≥ 0 (3.7)
for all positive integer n, which implies that
0 ≤
∫
R
e2cta(t)|v0(t)|γ+2dt <
1
n
‖v0‖22 ≤
1
nβ2
‖v0‖2Ec =
1
nβ2
→ 0 (3.8)
as n→∞. Hence v0 = 0 which contradicts that ‖v0‖Ec = 1. Therefore, (3.6) holds. Now let
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
INFINITELY MANY FAST HOMOCLINIC SOLUTIONS . . . 411
Ω0 =
{
t ∈ R : ecta(t)|v0(t)|γ ≥ ξ1
}
, Ωn =
{
t ∈ R : ecta(t)|vn(t)|γ < 1
n
}
and Ωc
n = R \ Ωn. Then we get
meas(Ωn ∩ Ω0) ≥ meas(Ω0)−meas(Ωc
n ∩ Ω0) ≥ ξ2 −
1
n
(3.9)
for all positive integer n. Let n be large enough such that ξ2 −
1
n
≥ 1
2
ξ2 and
1
2γ−1
ξ1 −
1
n
≥ 1
2γ
ξ1.
Then we have ∫
R
ecta(t)|vn − v0|γdt ≥
∫
Ωn∩Ω0
ecta(t)|vn − v0|γdt ≥
≥ 1
2γ−1
∫
Ωn∩Ω0
ecta(t)|v0|γdt−
∫
Ωn∩Ω0
ecta(t)|vn|γdt ≥
≥
(
1
2γ−1
ξ1 −
1
n
)
meas(Ωn ∩ Ω0) ≥ ξ1ξ2
2γ+1
> 0
for all large n, which is a contradiction to (3.5). Therefore, (3.3) holds. For the ε1 given in (3.1), let
Ωu =
{
t ∈ R : ecta(t)|u(t)|γ ≥ ε1‖u‖γEc
}
∀u ∈ F \ {0}. (3.10)
Then by (3.1),
meas(Ωu) ≥ ε1 ∀u ∈ F \ {0}.
Combining (H2′) and (3.10), for any u ∈ F \ {0}, we obtain
B(u) =
∫
R
ect[W (t, u)−W (t, 0)]dt =
∫
R
ect
1∫
0
Wu(t, hu)udh
dt ≥
≥
∫
R
ect
1∫
0
a(t)|u|γhγ−1dh
dt ≥ 1
γ
∫
R
ecta(t)|u(t)|γdt ≥
≥ 1
γ
∫
Ωu
ecta(t)|u(t)|γdt ≥ 1
γ
ε1‖u‖γEcmeas(Ωu) ≥
≥ 1
γ
ε2
1‖u‖
γ
Ec
.
This implies B(u)→∞ as ‖u‖Ec →∞ on any finite dimensional subspace of E.
Step 2. Under the assumptions of Theorem 1.3, then there exists a sequence ρk → 0+ as k →∞
such that
ak(λ) := inf
u∈Zk,‖u‖Ec=ρk
Iλ(u) ≥ 0,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
412 LIU YANG, LIPING LUO, ZHENGUO LUO
and
dk(λ) := inf
u∈Zk,‖u‖Ec≤ρk
Iλ(u)→ 0 as k →∞ uniformly for λ ∈ [1, 2].
Set βk := supu∈Zk,‖u‖Ec=1 ‖u‖2. Then βk → 0 as k →∞ since Ec is compactly embedded into
L2(ect). By (H2′), we have
Iλ(u) =
1
2
‖u‖2Ec − λ
∫
R
ectW (t, u)dt ≥
≥ 1
2
‖u‖2Ec − 2
∫
R
ectW (t, u)dt ≥ 1
2
‖u‖2Ec −
2
γ
‖b‖ 2
2−γ
‖u‖γ2 −
2
δ
‖c‖ 2
2−δ
‖u‖δ2 ≥
≥ 1
2
‖u‖2Ec −
2
γ
βγk‖b‖ 2
2−γ
‖u‖γEc −
2
δ
βδk‖c‖ 2
2−δ
‖u‖δEc .
Let
ρk =
(
16βγk
γ
‖b‖ 2
2−γ
) 1
2−γ
+
(
16βδk
δ
‖c‖ 2
2−δ
) 1
2−δ
.
Obviously, ρk → 0 as k →∞. Combining this with the above inequality, straightforward computation
shows that
ak(λ) ≥ 1
4
ρ2
k > 0. (3.11)
Furthermore, for any u ∈ Zk with ‖u‖Ec ≤ ρk, we get
Iλ(u) ≥ −2
γ
βγk‖b‖ 2
2−γ
‖u‖γEc −
2
δ
βδk‖c‖ 2
2−δ
‖u‖δEc .
Therefore,
0 ≥ dk(λ) ≥ −2
γ
βγk‖b‖ 2
2−γ
‖u‖γEc −
2
δ
βδk‖c‖ 2
2−δ
‖u‖δEc . (3.12)
Since βk, ρk → 0 as k →∞, we obtain
dk(λ) := inf
u∈Zk,‖u‖Ec≤ρk
Iλ(u)→ 0 as k →∞ uniformly for λ ∈ [1, 2].
Step 3. Under the assumptions of Theorem 1.3, for the sequence {ρk}k∈N obtained in Step 2,
there exist 0 < rk < ρk for all k ∈ N such that
bk(λ) := max
u∈Yk,‖u‖Ec=rk
Iλ(u) < 0 for all λ ∈ [1, 2].
For any u ∈ Yk (a finite dimensional subspace of Ec) and λ ∈ [1, 2], we have
Iλ(u) =
1
2
‖u‖2Ec − λ
∫
R
ectW (t, u)dt ≤
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
INFINITELY MANY FAST HOMOCLINIC SOLUTIONS . . . 413
≤ 1
2
‖u‖2Ec −
∫
R
ectW (t, u)dt ≤ 1
2
‖u‖2Ec −
1
γ
∫
Ωu
ecta(t)|u(t)|γdt ≤
≤ 1
2
‖u‖2Ec −
1
γ
ε1‖u‖γEc meas (Ωu) ≤ 1
2
‖u‖2Ec −
1
γ
ε2
1‖u‖
γ
Ec
,
where Ωu is defined in (3.10). Choosing 0 < rk < min
ρk,
(
ε2
1
γ
) 1
2−γ
. Direct computation
shows that
bk(λ) ≤ −
r2
k
2
< 0 ∀k ∈ N.
Step 4. Evidently, the condition (C1) in Theorem 2.1 holds. By Step 1, 2, 3, Conditions (C2),
(C3) in Theorem 2.1 are also satisfied. Therefore, by Theorem 2.1, there exist λn → 1, u(λn) ∈ Yn
such that
I ′λn |Yn(u(λn)) = 0, Iλn(u(λn))→ ck ∈ [dk(2), bk(1)]
as n → ∞. For the sake of notational simplicity, in what follows we always set un = uλn for all
n ∈ N. Now we show that {un} is bounded in Ec. Indeed, we have
1
2
‖un‖2Ec ≤ Iλn(un) + λn
∫
R
[
1
γ
b(t)|un(t)|γ +
1
δ
c(t)|un(t)|δ
]
dt ≤
≤M +
2
γ
‖b‖ 2
2−γ
‖un‖γ2 +
2
δ
‖c‖ 2
2−δ
‖un‖δ2 ≤
≤M +
2
γ
β−γ‖b‖ 2
2−γ
‖un‖γEc +
2
δ
β−δ‖c‖ 2
2−δ
‖un‖δEc ∀n ∈ N
for some M > 0. Since 1 < γ < 2, 1 < δ < 2, it yields {un} is bounded in Ec. Finally, we show that
{un} possesses a strong convergent subsequence in Ec. In fact, in view of the boundness of {un},
without loss of generality, we may assume
un ⇀ u0 (3.13)
as n → ∞ for some u0 ∈ Ec. By virtue of the Riesz representation theorem, I ′λn |Yn : Yn → Y ∗n and
I ′ : Ec → E∗c can be viewed as I ′λn |Yn : Yn → Yn and I ′ : Ec → Ec respectively, where Y ∗n is the
dual space of Yn. Note that
0 = I ′λn |Yn(un) = un − λnPnB′(un) ∀n ∈ N, (3.14)
where Pn is the orthogonal projection for all n ∈ N. That is,
un = λnPnB
′(un) ∀n ∈ N. (3.15)
Due to the compactness of B′ and (3.13), the right-hand side of (3.15) converges strongly in Ec
and hence un → u0 in Ec. Now, we know that I = I1 has infinitely many nontrivial critical points.
Therefore, (FHS) possesses infinitely many nontrivial fast homoclinic solutions.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
414 LIU YANG, LIPING LUO, ZHENGUO LUO
1. Rabinowitz P. H. Homoclinic orbits for a class of Hamiltonian systems // Proc. Roy. Soc. Edinburgh A. – 1990. –
114, № 1-2. – P. 33 – 38.
2. Coti Zelati V., Rabinowitz P. H. Homoclinic orbits for second order Hamiltonian systems possessing superquadratic
potenials // J. Amer. Math. Soc. – 1991. – 4. – P. 693 – 727.
3. Ding Y., Girardi M. Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing
sign // Dynam. Systems Appl. – 1993. – 2. – P. 131 – 145.
4. Fei G. The existence of homoclinic orbits for Hamiltonian systems with the potential changing sign // Chinese Ann.
Math. Ser. B. – 1996. – 17. – P. 403 – 410.
5. Izydorek M., Janczewska J. Homoclinic solutions for a class of second order Hamiltonian systems // J. Different.
Equat. – 2005. – 219, № 2. – P. 375 – 389.
6. Ding Y. Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems // Nonlinear
Anal. – 1995. – 25. – P. 1095 – 1113.
7. Korman P., Lazer A. C. Homoclinic orbits for a class of symmetric Hamiltonian systems // Electron. J. Different.
Equat. – 1994. – 1994. – P. 1 – 10.
8. Ou Z., Tang C. Existence of homoclinic solutions for the second order Hamiltonian systems // J. Math. Anal. and
Appl. – 2004. – 291. – P. 203 – 213.
9. Zou W. Infinitely many homoclinic orbits for the second order Hamiltonian systems // Appl. Math. Lett. – 2003. –
16. – P. 1283 – 1287.
10. Omana W., Willem M. Homoclinic orbits for a class of Hamiltonian systems // Different. Integral Equat. – 1992. – 5,
№ 5. – P. 1115 – 1120.
11. Yuan R., Zhang Z. Infinitely many homoclinic orbits for the second order Hamiltonian systems with super-quadratic
potentials // Nonlinear Anal.: Real World Appl. – 2009. – 10. – P. 1417 – 1423.
12. Yang L., Chen H., Sun J. Infinitely many homoclinic solutions for some second order Hamiltonian systems // Nonlinear
Anal.: Real World Appl. – 2011. – 74. – P. 6459 – 6468.
13. Zhang Z., Yuan R. Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian
systems // Nonlinear Anal. – 2009. – 71. – P. 4125 – 4130.
14. Zhang Z., Yuan R. Homoclinic solutions of some second order non-autonomous systems // Nonlinear Anal. – 2009.
– 71. – P. 5790 – 5798.
15. Zhang Z., Yuan R. Fast homoclinic solutions for some second order non-autonomous systems // J. Math. Anal. and
Appl. – 2011. – 376. – P. 51 – 63.
16. Zou W. Variant fountain theorems and their applications // Manuscr. math. – 2001. – 104. – P. 343 – 358.
Received 04.04.12,
after revision — 18.10.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
|