Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 111, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE AND MULTIPLICITY OF HOMOCLINIC
SOLUTIONS FOR p(t)-LAPLACIAN SYSTEMS WITH
SUBQUADRATIC POTENTIALS
BIN QIN, PENG CHEN
Abstract. By using the genus properties, we establish some criteria for the
second-order p(t)-Laplacian system
´
d`
|u̇(t)|p(t)−2 u̇(t) − a(t)|u(t)|p(t)−2 u(t) + ∇W (t, u(t)) = 0
dt
to have at least one, and infinitely many homoclinic orbits. where t ∈ R,
u ∈ RN , p(t) ∈ C(R, R) and p(t) > 1, a ∈ C(R, R) and W ∈ C 1 (R × RN , R)
may not be periodic in t.
1. Introduction
Consider the second-order ordinary p(t)-Laplacian system
d
|u̇(t)|p(t)−2 u̇(t) − a(t)|u(t)|p(t)−2 u(t) + ∇W (t, u(t)) = 0,
(1.1)
dt
where p ∈ C(R, R) and p(t) > 1, t ∈ R, u ∈ RN , a : R → R and W : R × RN → R.
As usual, we say that a solution u(t) of (1.1) is homoclinic (to 0) if u(t) → 0 as
t → ±∞. In addition, if u(t) 6≡ 0 then u(t) is called a nontrivial homoclinic solution.
System (1.1) has been studied by Fan, et al. in a series of papers [10, 11, 12, 13].
The p(t)-Laplacian systems can be applied to describe the physical phenomena with
“pointwise different properties” which first arose from the nonlinear elasticity theory
(see [29]). The p(t)-Laplacian operator possesses more complicated nonlinearity
than that of the p-Laplacian, for example, it is not homogeneous, this causes many
troubles, and some classic theories and methods, such as the theory of Sobolev
spaces, are not applicable.
It is well-known that homoclinic orbits play an important role in analyzing the
chaos of dynamical systems. If a system has the transversely intersected homoclinic orbits, then it must be chaotic. Therefore, it is of practical importance and
mathematical significance to consider the existence of homoclinic orbits of (1.1)
emanating from 0.
If p(t) ≡ p is a constant, system (1.1) reduces to the ordinary p-Laplacian system
d
|u̇(t)|p−2 u̇(t) − a(t)|u(t)|p−2 u(t) + ∇W (t, u(t)) = 0.
(1.2)
dt
2000 Mathematics Subject Classification. 34C37, 58E05, 70H05.
Key words and phrases. Homoclinic solutions; p(t)-Laplacian systems; genus.
c
2014
Texas State University - San Marcos.
Submitted December 17, 2013. Published April 16, 2014.
1
2
B. QIN, P. CHEN
EJDE-2014/111
In recent years, the existence and multiplicity of homoclinic orbits for Hamiltonian systems have been investigated in many papers via variational methods and
many results were obtained based on various hypotheses on the potential functions
when p = 2, see, e.g., [2, 3, 4, 14, 15, 20, 21, 24, 25, 27, 28].
In the last decade there has been an increasing interest in the study of ordinary
differential systems driven by the p-Laplacian (or the generalization of Laplacian
[17]). For the existence of solutions for p(t)-Laplacian Dirichlet problems on a
bounded domain we refer to [5, 6, 7, 8, 9, 26]. The study on the existence of solutions for p(t)-Laplacian equations in R is a new topic, which seems not to have been
considered in the literature. We know that in the study of p-Laplacian equations in
R, a main difficulty arises from the lack of compactness. On the other hand, compared with the literature available for W (t, x) being superquadratic as |x| → +∞,
there is less literature available for the case where W (t, x) is subquadratic at infinity. Motivated by papers [2, 27], we will use the genus properties to establish some
existence criteria to guarantee that system (1.1) has infinitely many homoclinic
solutions under more relaxed assumptions on W (t, x).
For our results, we use the following assumptions:
(A1) a ∈ C(R, (0, ∞)) and a(t) → +∞ as |t| → ∞, b(t) = 1/a(t), b(t)
belongs to Lri (t) (R, R), where ri (t) satisfies
αi (t)
p(t)
αi (t)
1
+
= 1, i = 1, 2.
ri (t)
p(t)
(P1) 1 < p− := inf t∈R p(t) ≤ supt∈R p(t) := p+ < ∞;
(W1) W ∈ C 1 (R × RN , R), W (t, 0) = 0 and there exist two bounded functions
ai (t) (i = 1, 2.) such that
|∇W (t, x)| ≤ a1 (t)α1 (t)|x|α1 (t)−1 ,
∀(t, x) ∈ R × RN , |x| ≤ 1,
and for every (t, x) ∈ R × RN with |x| ≥ 1,
|∇W (t, x)| ≤ a2 (t)α2 (t)|x|α2 (t)−1 ,
|W (t, x)| ≤ ca2 (t)(t)|x|α2 (t)
where αi (t) satisfy αi+ < p− , ai (t) ∈ C(R, R+ ) (i = 1, 2), and c is a
constant;
(W2) There exist an open set J ⊂ R and a function γ1 (t) such that
W (t, x) ≥ η|x|γ1 (t) ,
∀(t, x) ∈ J × RN , |x| ≤ 1,
where γ1 (t) satisfy 1 < γ1+ < p− , η > 0 is a constant;
(W3) W (t, −x) = W (t, x) for all (t, x) ∈ R × RN .
Our main results are the following two theorems.
Theorem 1.1. Assume (A1), (P1), (W1), (W2) are satisfied. Then (1.1) possesses
at least one nontrivial homoclinic solution.
Theorem 1.2. Assume (A1), (P1), (W1), (W2), (S3) are satisfied. Then (1.1)
possesses infinitely many nontrivial homoclinic solutions.
The rest of the this article is organized as follows. In Section 2, we introduce
1,p(t)
some notations, preliminary results in space Wa
and establish the corresponding
variational structure. In Section 3, we complete the proofs of Theorems 1.1–1.2. In
Section 4, we give some examples to to illustrate our results.
EJDE-2014/111
EXISTENCE AND MULTIPLICITY
3
2. Preliminaries
In this section, we recall some known results in critical point theory and the
1,p(t)
properties of space Wa
are listed for the convenience of readers. Let Ω be an
open subset of R. Let S = {u|u is a measurable function in Ω}, elements in S that
are equal to each other almost everywhere are considered as one element. Define
Z
N
N
Lp(t)
(Ω,
R
)
=
u
∈
S(Ω,
R
)
:
a(t)|u(t)|p(t) dt < ∞
a
Ω
with the norm
|u|p(t),a = inf λ > 0 :
Z
u
a(t)| |p(t) dt ≤ 1 .
λ
Ω
Define
N
p(t)
Wa1,p(t) (Ω, RN ) = {u ∈ Lp(t)
(Ω, RN )}
a (Ω, R ) : u̇ ∈ L
with the norm
Z
u
u̇
kuk = inf λ > 0 :
| |p(t) + a(t)| |p(t) dt ≤ 1 .
λ
λ
Ω
p(t)
We call the space La a generalized Lebesgue space, it is a special kind of gen1,p(t)
is called a generalized Sobolev space, it
eralized Orlicz spaces. The space Wa
is a special kind of generalized Orlicz-Sobolev spaces. For the general theory of
generalized Orlicz spaces and generalized Orlicz-Sobolev spaces, see [1, 19]. One
1,p(t)
p(t)
in [10, 11, 12, 13].
can find the basic theory of spaces La and Wa
Lemma 2.1 ([11, 12]). Let
Z
ρ(u) =
a(t)|u|p(t) dt,
∀u ∈ Lap(t) ,
Ω
then
(i) |u|p(t),a < 1 (= 1; > 1) if and only if ρ(u) < 1 (= 1; > 1);
−
+
+
−
(ii) |u|p(t),a > 1 implies |u|pp(t),a ≤ ρ(u) ≤ |u|pp(t),a ,
|u|p(t),a < 1 implies |u|pp(t),a ≤ ρ(u) ≤ |u|pp(t),a ;
(iii) |u|p(t),a → 0 if and only if ρ(u) → 0;
|u|p(t),a → ∞ if and only if ρ(u) → ∞.
p(t)
(iv) Let u ∈ La
\ {0}, then kukp(t),a = λ if and only if ρ( λu ) = 1.
Lemma 2.2 ([11, 12]). Let
Z
ϕ(u) = (|u̇|p(t) + a(t)|u|p(t) )dt,
∀u ∈ Wa1,p(t) ,
Ω
(i) kuk < 1 (= 1; > 1) if and only if ϕ(u) < 1 (= 1; > 1);
−
+
(ii) kuk > 1 implies kukp ≤ ϕ(u) ≤ kukp ,
+
−
kuk < 1 implies kukp ≤ ϕ(u) ≤ kukp ;
(iii) kuk → 0 if and only if ϕ(u) → 0;
kuk → ∞ if and only if ϕ(u) → ∞.
R
p(t)
Lemma 2.3 ([11]). Let ρ(u) = Ω a(t)|u|p(t) dt for u, un ∈ La (n = 1, 2, · · · ),
then the following statements are equivalent to each other
(i) limn→∞ |un − u|p(t),a = 0;
(ii) limn→∞ ρ(un − u) = 0;
4
B. QIN, P. CHEN
EJDE-2014/111
(iii) un → u a.e. t ∈ Ω and limn→∞ ρ(un ) = ρ(u).
Lemma 2.4 ([11]). If
1
p(t)
+
1
q(t)
= 1, then
(i) (Lp(t) )∗ = Lq(t) , where (Lp(t) )∗ is the conjugate space of Lp(t) ;
(ii) for all u ∈ Lp(t) and all v ∈ Lq(t) , we have
Z
u(t)v(t)dt ≤ 2|u|p(t) |v|q(t) .
Ω
Lemma 2.5 ([10]).
1
If p(t)
r(t)
Lq(t) (R, R) and w ∈ L
+
1
q(t)
+
1
r(t)
= 1 and for any u ∈ Lp(t) (R, R), v ∈
(R, R), we have
Z
|uvw|dt ≤ 3|u|p(t) |v|q(t) |w|r(t) .
R
Lemma 2.6 ([10]). If |u|q(x) ∈ Ls(x)/q(x) , where q, s ∈ L∞
+ (Ω), q(x) ≤ s(x), then
u ∈ Ls(x) (Ω) and there is a number q ∈ [q − , q + ] such that |u|q(x) |s(x)/q(x) =
(|u|s(x) )q .
p(t)
Lemma 2.7 ([16]). If aα(t)/p(t) |u|α(t) ∈ Lp(t)/α(t) , then u ∈ La (R, R) and
e
|aα(t)/p(t) |u|α(t) |p(t)/α(t) = |u|α
p(t),a , where α, p satisfy the condition (P1) and α(t) <
p(t) for all t ∈ R, α
e ∈ [α− , α+ ] is a constant.
Now, we establish the variational structure of system (1.1). Define
N
p(t)
E = Wa1,p(t) (R, RN ) = {u ∈ Lp(t)
(R, RN )}.
a (R, R )|u̇ ∈ L
Let I : E → R be defined by
Z
Z
1
p(t)
p(t)
I(u) =
(|u̇|
+ a(t)|u| )dt −
W (t, u(t))dt.
R
R p(t)
For convenience, we denote
Z
Z
1
J(u) =
(|u̇|p(t) + a(t)|u|p(t) )dt, F (u) =
W (t, u(t))dt.
R p(t)
R
(2.1)
(2.2)
Lemma 2.8 ([11]).
(i) J ∈ C 1 (E, R), and
Z hJ 0 (u), vi =
|u̇(t)|p(t)−2 (u̇(t), v̇(t)) + a(t)|u(t)|p(t)−2 (u(t), v(t)) dt,
R
for all u, v ∈ E;
(ii) J 0 : E → E ∗ is a mapping of type (S+ ), i.e., if un * u and
lim sup(J 0 (un ), un − u) ≤ 0,
n→∞
then un has a convergent subsequence in E.
If (A1), (W1) or (W2) hold, then I ∈ C 1 (E, R) and one can easily check that
Z
hI 0 (u), vi =
|u̇(t)|p(t)−2 (u̇(t), v̇(t)) + a(t)|u(t)|p(t)−2 (u(t), v(t))
(2.3)
R
− (∇W (t, u(t)), v(t)) dt.
Furthermore, the critical points of I in E are classical solutions of (1.1) with
u(±∞) = 0.
Lemma 2.9 ([16]). For u ∈ E, then u ∈ C(R, RN ), and u(t) → 0, |t| → ∞.
Furthermore, the embedding E ,→ L∞ (R, RN ) is continuous and compact.
EJDE-2014/111
EXISTENCE AND MULTIPLICITY
5
Remark 2.10. By Lemma 2.9, there exists a constant C > 0 such that
kukL∞ ≤ CkukE .
(2.4)
1
Lemma 2.11 ([18]). Let E be a real Banach space and I ∈ C (E, R) satisfy the
(PS)-condition. If I is bounded from below, then c = inf E I is a critical value of I.
To find nontrivial critical points of I, we will use the “genus” properties, so we
recall the following definitions and results (see [18]). Let E be a Banach space,
f ∈ C 1 (E, R) and c ∈ R. We set
Σ = {A ⊂ E − {0} : A is closed in E and symmetric with respect to 0},
Kc = {u ∈ E : f (u) = c, f 0 (u) = 0},
f c = {u ∈ E : f (u) ≤ c}.
Definition 2.12 ([18]). For A ∈ Σ, we say genus of A is n (denoted by γ(A) = n)
if there is an odd map φ ∈ C(A, Rn \ {0}) and n is the smallest integer with this
property.
Lemma 2.13 ([18]). Let f be an even C 1 functional on E and satisfy the (PS)condition. For any n ∈ N, set
Σn = {A ∈ Σ : γ(A) ≥ n},
cn = inf sup f (u).
A∈Σn u∈A
(i) If Σn 6= ∅ and cn ∈ R, then cn is a critical value of f ;
(ii) If there exists r ∈ N such that
cn = cn+1 = · · · = cn+r = c ∈ R,
and c 6= f (0), then γ(Kc ) ≥ r + 1.
3. Proof of main results
Proof of Theorem 1.1. In view of Lemma 2.8 and (W1), I ∈ C 1 (E, R). In what
follows, we first show that I is coercive. By (W1), we have
|W (t, x)| ≤ a1 (t)|x|α1 (t) ,
|x| ≤ 1,
(3.1)
|W (t, x)| ≤ ca2 (t)|x|α2 (t) ,
|x| > 1.
(3.2)
Assume that kuk ≥ 1, by (W1), Lemma 2.2 and Lemma 2.7, we have
Z
Z
1
I(u) =
(|u̇|p(t) + a(t)|u|p(t) )dt −
W (t, u(t))dt
R p(t)
R
Z
Z
−
1
≥ + kukp −
W (t, u(t))dt −
W (t, u(t))dt
p
{t:|u(t)|≤1}
{t:|u(t)|>1}
Z
Z
−
1
≥ + kukp −
a1 (t)|u(t)|α1 (t) dt −
a2 (t)|u(t)|α2 (t) dt
p
{t:|u(t)|≤1}
{t:|u(t)|>1}
Z
−
1
≥ + kukp − C1
bα1 (t)/p(t) aα1 (t)/p(t) |u(t)|α1 (t) dt
p
{t:|u(t)|≤1}
Z
α2 (t)
α2 (t)
− cC2
b p(t) a p(t) |u(t)|α2 (t) dt
{t:|u(t)|>1}
α1 (t)
α2 (t)
−
1
f1
f2
≥ + kukp − 2C1 |b p(t) |Lr1 (t) |u|α
− 2cC2 |b p(t) |Lr2 (t) |u|α
p(t),a
p(t),a
p
α1 (t)
α2 (t)
−
1
≥ + kukp − 2C1 |b p(t) |Lr1 (t) kukαf1 − 2cC2 |b p(t) |Lr2 (t) kukαf2 .
p
(3.3)
6
B. QIN, P. CHEN
EJDE-2014/111
i (t)
Where Ci = supt∈R ai (t), αi (t), ri (t) satisfy ri1(t) + αp(t)
= 1, αei ∈ [αi− , αi+ ] is a
constant, (i = 1, 2). By (W1), αi− < αi+ < p− , this implies that αei < p− . By (A),
we have I(u) → +∞ as kuk → +∞. Consequently, I is bounded from below.
Next, we prove that I satisfies the (PS)-condition. Assume that {uk }k∈N ⊂ E is
a sequence such that {I(uk )}k∈N is bounded and I 0 (uk ) → 0 as k → +∞. Then by
(2.1) and (3.3), there exists a constant A > 0 such that
kuk k ≤ A,
k ∈ N.
(3.4)
So passing to a subsequence if necessary, it can be assumed that uk * u in E.
By (A), bαi (t)/p(t) ∈ Lri (t) (i = 1, 2), for any ε > 0, there exists R > 0 such that
|b(t)αi (t)/p(t) |Lri (t) (Ω2 ) < ε,
(3.5)
where Ω1 = {t ∈ R : |t| ≤ R}, Ω2 = R \ Ω1 , by Lemma 2.9, if uk * u0 , then uk → u
in L∞ , hence, we have
Z
|W (t, uk ) − W (t, u)|dt < ε, k → ∞,
(3.6)
Ω1
Z
|∇W (t, uk ) − ∇W (t, u)|dt < ε, k → ∞.
(3.7)
Ω1
Without loss of generality, suppose that max{kuk k, kuk} ≤ 1, it follows from (A1),
(3.5)-(3.7), Lemma 2.4 and Lemma 2.7 that
|F (uk ) − F (u)|
Z
= (W (t, uk (t)) − W (t, u(t)))dt
Z R
Z
≤
|W (t, uk (t)) − W (t, u(t))|dt +
|W (t, uk (t)) − W (t, u(t))|dt
Ω1
Ω2
Z
≤ ε + [|W (t, uk (t))| + |W (t, u(t))|]dt
R
Z
Z
≤ ε + ε a1 (t)(|uk |α1 (t) + |u|α1 (t) )dt + cε a2 (t)(|uk |α2 (t) + |u|α2 (t) )dt
R
R
α1 (t)/p(t)
≤ ε + 2C1 |b(t)
+ 2C1 |b(t)
α1 (t)/p(t)
+ 2cC2 |b(t)
+ 2cC2 |b(t)
α2 (t)
p(t)
α2 (t)
p(t)
≤ ε + 2C1 ε(|uk |
|Lr1 (t) |a
α1 (t)/p(t)
|Lr1 (t) |a
|Lr2 (t) |a
|Lr2 (t) |a
α
g
1,1
p(t)
La
α1 (t)/p(t)
α2 (t)
p(t)
α2 (t)
p(t)
α
g
1,2
+ |u|
p(t)
La
|uk (t)|
|u(t)|
α1 (t)
α1 (t)
(3.8)
|Lp(t)/alpha1 (t)
|Lp(t)/alpha1 (t)
|uk (t)|α2 (t) |Lp(t)/α2 (t)
|u(t)|α2 (t) |Lp(t)/α2 (t)
α
g
2,1
) + 2cC2 ε|uk |
p(t)
La
+ |u|
α
g
2,2
p(t)
La
)
≤ ε + 4C1 ε + 4cC2 ε,
−
+
−
+
where Ci = supt∈R ai (t)(i = 1, 2), αg
g
g
g
1,1 , α
1,2 ∈ [α1 , α1 ], α
2,1 , α
2,2 ∈ [α2 , α2 ].
0
0
Hence, there exists a constant C such that |F (uk ) − F (u)| < C ε, this implies
that F (uk ) → F (u), k → ∞.
On the other hand, for any v ∈ E with kvk = 1, by (W1), Lemmas 2.5 and 2.7,
we have
|(F 0 (uk ) − F 0 (u), v)|
EJDE-2014/111
EXISTENCE AND MULTIPLICITY
7
Z
≤
|∇W (t, uk (t)) − ∇W (t, u(t)))||v|dt
Ω1
Z
|∇W (t, uk (t)) − ∇W (t, u(t)))||v|dt
Z
≤ εkvkL∞ + (|∇W (t, uk (t))| + |∇W (t, u(t))|)|v|dt
R
Z
≤ Cε + α1 (t)a1 (t)(|uk |α1 (t)−1 + |u|α1 (t)−1 )|v|dt
R
Z
+
α2 (t)a2 (t)(|uk |α2 (t)−1 + |u|α2 (t)−1 )|v|dt
R
Z
Z
α1 (t)−1
α1 (t)−1
≤ Cε + C3 (|uk |
+ |u|
)|v|dt + C4 (|uk |α2 (t)−1 + |u|α2 (t)−1 )|v|dt
R
R
Z
α1 (t)−1
1
α
(t)−1
α1 (t)/p(t)
a p(t) |v|dt
= Cε + C3 b
a p(t) |uk | 1
R
Z
α1 (t)−1
1
+ C3 bα1 (t)/p(t) a p(t) |u|α1 (t)−1 a p(t) |v|dt
ZR
α2 (t)
α2 (t)−1
1
+ C4 b p(t) a p(t) |uk |α2 (t)−1 a p(t) |v|dt
ZR
α2 (t)−1
α2 (t)
1
+ C4 b p(t) a p(t) |u|α2 (t)−1 a p(t) |v|dt
+
Ω2
R
α (t)/p(t)
≤ Cε + 3C3 |b(t)|L1r1 (t)
α (t)/p(t)
+ 3C3 |b(t)|L1r1 (t)
α2 (t)
+ 3C4 |b(t)|Lp(t)
r2 (t) |a
α2 (t)
+ 3C4 |b(t)|Lp(t)
r2 (t) |a
|a
α2 (t)−1
p(t)
p(t)
La
α1 (t)−1
p(t)
α1 (t)−1
p(t)
α2 (t)−1
p(t)
α
g
1,3 −1
≤ Cε + 3C3 ε(|uk |
|a
|uk |α1 (t)−1 |
1
p(t)
L α1 (t)−1
1
|u|α1 (t)−1 |
p(t)
L α1 (t)−1
p(t)
L α2 (t)−1
+ |u|
α
g
1,4 −1
p(t)
La
|a p(t) v|Lp(t)
1
|uk |α2 (t)−1 |
|u|α2 (t)−1 |
|a p(t) v|Lp(t)
|a p(t) v|Lp(t)
1
p(t)
L α2 (t)−1
|a p(t) v|Lp(t)
)|v|Lp(t) + 3C4 ε(|uk |
a
α
g
2,3 −1
p(t)
La
α
g
2,4 −1
+ |u|
p(t)
La
)|v|Lp(t)
a
≤ Cε + 6C3 ε + 6C4 ε.
Where C is defined in (2.4), C3 = supt∈R α1 (t)a1 (t), C4 = supt∈R α2 (t)a2 (t),
−
+
−
+
00
αg
g
g
g
1,3 , α
1,4 ∈ [α1 , α1 ], α
2,3 , α
2,4 ∈ [α2 , α2 ]. Hence, there exists a constant C such
0
0
00
0
0
that |F (uk ) − F (u)| < C ε, this implies that F (uk ) → F (u), k → ∞. This implies that (J 0 (uk ), uk − u) → 0. By Lemma 2.8, J 0 is a mapping type (S+ ), hence,
uk → u. So, I satisfies (PS) condition.
By Lemma 2.13, c = inf E I(u) is a critical value of I, that is there exists a critical
point u∗ ∈ E such that I(u∗ ) = c.
T
1,p(t)
Finally, we show that u∗ 6= 0. Let u0 ∈ (W0
(J) E) \ {0} and ku0 k = 1,
then by (W2) and Lemma 2.2, we have
Z
Z
1
((|su̇|p(t) + a(t)|su|p(t) ))dt − W (t, u(t))dtnonumber
(3.9)
I(su0 ) =
R
R p(t)
Z
−
sp
W (t, su0 (t))dtnonumber
(3.10)
≤ − −
p
J
8
B. QIN, P. CHEN
−
+
sp
≤ − − ηsγ1
p
Z
|u0 (t)|γ1 (t) dt,
EJDE-2014/111
0 < s < 1.
(3.11)
J
Since 1 < γ1+ < p− , it follows from (3.11) that I(su0 ) < 0 for s > 0 small enough.
Hence I(u∗ ) = c < 0, therefore u∗ is nontrivial critical point of I, and so u∗ = u∗ (t)
is a nontrivial homoclinic solution of (1.1). The proof is complete.
Proof of Theorem 1.2. By (W3), I is an even functional. Denote by γ(A) the genus
of A. Set
Σ = {A ⊂ E − {0} : A is closed in E and symmetric with respect to 0},
Σk = {A ∈ Σ : γ(A) ≥ k},
ck = inf sup I(u),
A∈Σk u∈A
k = 1, 2, . . . ,
k = 1, 2, . . . ,
we have
−∞ < c1 ≤ c2 ≤ · · · ≤ ck ≤ ck+1 ≤ . . . .
Now let us prove that ck < 0 for every k.
By (W2), there exists a bounded open set J ⊂ R such that W (t, x) ≥ η|x|γ1 (t) ,
1,p(t)
for all t ∈ J. Since W0
(J) ⊂ E, For any k, we can choose a k-dimensional linear
1,p(t)
subspace Ek ⊂ W0
(J). Since all norms of a finite dimensional normed space
are equivalent, there exists ρk ∈ (0, 1) such that u ∈ Ek with kuk ≤ ρk implies
|u|L∞ ≤ 1. Set
Sρ(k)
= {u ∈ Ek : kuk = ρk },
k
(k)
for any u ∈ Sρk , s ∈ (0, 1), we have
Z
Z
1
[|su̇|p(t) + a(t)|su|p(t) ]dt −
W (t, su(t))dt
I(su) =
J
J p(t)
Z
−
sp p−
γ1+
≤ − ρk − s
|u|γ1 (t) dt
p
J
−
≤
Where dk =
R
J
+
sp p−
ρk − dk sγ1 .
−
p
|u|γ1 (t) dt, since γ1+ < p− , there exist sk ∈ (0, 1), εk > 0 such that
I(sk u) ≤ −εk < 0,
∀u ∈ Sρ(k)
.
k
(k)
We know that γ(Ssk ρk ) = k, so ck ≤ −εk < 0.
By genus theory [22] and Lemma 2.13, each ck is a critical value of I, hence
there is a sequence of solutions {±uk : k = 1, 2, . . . , } of system (1.1) such that
I(±uk ) = ck < 0. By the arbitraries of k, we can conclude that system (1.1) have
infinitely many homoclinic solutions. The proof is complete.
4. An example
In this section, we give an example to illustrate our results. Consider the secondorder ordinary p(t)-Laplacian system
d
|u̇(t)|8+10| sin t| u̇(t) − a(t)|u(t)|8+10| sin t| u(t) + ∇W (t, u(t)) = 0,
(4.1)
dt
EJDE-2014/111
EXISTENCE AND MULTIPLICITY
9
4
where p(t) = 10 + 10| sin t|, a(t) = 1 + t2 , let
W (t, x) =
|x|2| sin t|+2
|x|4| sin t|+4
+
,
2
1+t
1 + t2
then
(4| sin t| + 4)|x|4| sin t|+2 x (2| sin t| + 2)|x|2| sin t| x
+
,
1 + t2
1 + t2
3
|∇W (t, x)| ≤
(2 + 2| sin t|)|x|2| sin t|+1 , ∀(t, x) ∈ R × RN , |x| ≤ 1,
1 + t2
3
|∇W (t, x)| ≤
(4 + 4| sin t|)|x|4| sin t|+3 , ∀(t, x) ∈ R × RN , |x| ≥ 1,
2(1 + t2 )
∇W (t, x) =
Let J = (−2, 2), γ1 (t) = 2| sin t| + 2 and
1
W (t, x) ≥ |x|2| sin t|+2 , ∀(t, x) ∈ J × RN , |x| ≤ 1.
5
These inequalitires show that all conditions of Theorem 1.2 are satisfied, where
α1 (t) = 2 + 2| sin t|, α2 (t) = 4 + 4| sin t|,
3
3
4
a1 (t) =
, a2 (t) =
, c= ,
1 + t2
2(1 + t2 )
3
5
5
r1 (t) = , r2 (t) = .
4
3
By Theorem 1.2, system (1.1) has infinitely many nontrivial homoclinic solutions.
Acknowledgements. The authors would like to thank the referee for his/her valuable comments and suggestions. This work is partially supported by the NSFC
(No:11301297, 11261020) of China, Scientific Research Foundation for talents of
China Three Gorges University (KJ2012B078), Foundation of Hubei Educational
Committee (Q20131308) and Scientific and Technological Innovation Projects of
China Three Gorges University.
References
[1] R. A. Admas; Sobolev Spaces, New York: Academic Press, 1975.
[2] P. Chen, X. H. Tang, Ravi P. Agarwal; Infinitely many homoclinic solutions for nonautonomous p(t)-Laplacian Hamiltonian systems, Computers and Mathematics with Applications 63 (2012) 751-763.
[3] V. Coti Zelati, I. Ekeland, E. Sere; A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann. 288 (1) (1990) 133-160.
[4] V. Coti Zelati, P. H. Rabinowitz; Homoclinic orbits for second second order Hamiltonian
systems possessing superquadratic potentials, J. Amer. Math. Soc. 4(1991) 693-727.
[5] G. Dai; Infinitely many solutions for a Neumann-type differential inclusion problem involving
the p(x)-Laplacian, Nonlinear Anal. 70 (2009) 2297-2305.
[6] G. Dai; Infinitely many solutions for a hemivariational inequality involving the p(x)Laplacian, Nonlinear Anal. 71 (2009) 186-195.
[7] G. Dai; Three solutions for a Neumann-type differential inclusion problem involving the
p(x)-Laplacian, Nonlinear Anal. 70 (2009) 3755-3760.
[8] G. Dai; Infinitely many solutions for a p(x)-Laplacian equation in RN , Nonlinear Anal. 71
(2009) 1133-1139.
[9] G. Dai; Infinitely many solutions for a differential inclusion problem in RN involving the
p(x)-Laplacian, Nonlinear Anal. 71 (2009) 1116-1123.
[10] X. L. Fan, X. Y. Han; Existence and multiplicity of solutions for p(x)-Laplacian equations
in RN , Nonlinear Analysis 59 (2004) 173-188.
10
B. QIN, P. CHEN
EJDE-2014/111
[11] X. L. Fan, D. Zhao; On the space Lp(x) and W m,p(x) (Ω), J. Math. Anal. Appl. 263 (2001)
424-446.
[12] X. L. Fan, Q. H. Zhang; Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003) 1843-1852.
[13] X. L. Fan, Y. Z. Zhao, D. Zhao; Compact embeddings theorems with symmetry of StraussLions type for the space W m,p(x) (Ω), J. Math. Anal. Appl. 255 (2001) 333-348.
[14] P. L. Felmer, E. A. De, B. E. Silva; Homoclinic and periodic orbits for Hamiltonian systems,
Ann. Sc. Norm. Super. Pisa CI. Sci. (4) 26 (2) (1998) 285-301.
[15] M. Izydorek, J. Janczewska; Homoclinic solutions for a class of second order Hamiltonian
systems, J. Differential Equations 219 (2)(2005) 375-389.
[16] Y. H. Ma; Homoclinic orbits for second-order p(t)-Laplacian system, Ph.D. thesis, Lanzhou
University, 2005.
[17] R. Manásevich, J. Mawhin; Periodic solutions for nonlinear systems with p-Laplacian-like
operators, J. Differential Equations, 145 (2) 367-393, 1998.
[18] J. Mawhin, M. Willem; Critical point theory and Hamiltonian systems in Applied Mathematical Sciences, Vol. 74, Springer-Verlag, New York, 1989.
[19] J. Musielak; Orlicz Spaces and Modular Spaces in Lecture Notes in Mathematics, Vol. 1034,
Springer-Verlag, Berlin, 1983.
[20] P. H. Rabinowitz; Periodic and Homoclinic orbits for a periodic Hamiltonian systems, Ann.
Inst. H. Poincaré Anal. Non Linéaire 6 (5) (1989) 331-346.
[21] P. H. Rabinowitz; Homoclinic orbits for a class of Hamiltonian systems, Proc, Roy. Soc.
Edinburgh Sect. A 114 (1-2) (1990) 33-38.
[22] P. H. Rabinowitz, K. Tanaka; Some results on connecting orbits for a class of Hamiltonian
systems, Math. Z. 206 (3) (1991) 473-499.
[23] P. H. Rabinowitz; Minimax methods in critical point theory with applications to differential
equations, CBMS Reg. Conf. Ser. in Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986.
[24] A. Salvatore; Homoclinic orbits for a special class of nonautonomous Hamiltonian systems,
in: Proceedings of the Second World Congress of Nonlinear Analysis, Part 8 (Athens, 1996),
Nonlinear Anal. 30 (8) (1997) 4849-4857.
[25] X. H. Tang, X. Y. Lin; Homoclinic solutions for a class of second-order Hamiltonian systems,
J. Math. Anal. Appl., 354(2)(2009), 539-549.
[26] X. J. Wang, R. Yuan; Existence of periodic solutions for p(t)-Laplacian systems, Nonlinear
Anal. 70 (2009) 866-880.
[27] Z. Zhang, R. Yuan; Homoclinic solutions for some second order non-autonomous systems,
Nonlinear Anal. TMA, 71 (2009) 5790-5798.
[28] L. Zhang, X. H. Tang; Periodic solutions for some nonautonomous p(t)-Laplacian Hamiltonian systems, Applications of Mathematics, 58 (1) (2013) 39-61.
[29] V. V. Zhikov; Averaging of functionals in the calculus of variations and elasticity, Math.
USSR Izv. 29 (4) (1987) 33-66.
Bin Qin
College of Science, China Three Gorges University, Yichang, Hubei 443002, China
E-mail address: 1070751409@qq.com
Peng Chen (corresponding author)
College of Science, China Three Gorges University, Yichang, Hubei 443002, China
E-mail address: pengchen729@sina.com