Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 138, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
SUBHARMONIC SOLUTIONS FOR NON-AUTONOMOUS
SECOND-ORDER SUBLINEAR HAMILTONIAN SYSTEMS WITH
p-LAPLACIAN
ZHIYONG WANG
Abstract. In this article, we study the existence of subharmonic solutions
to the non-autonomous second-order sublinear Hamiltonian systems with pLaplacian. Introducing some kinds of control functions, infinitely many subharmonic solutions are obtained by using the minimax methods in critical
point theory. We point out that our results are new even in the case p = 2.
1. Introduction and main results
Consider the second-order system
d
(|u̇(t)|p−2 u̇(t)) + ∇F (t, u(t)) = 0 a.e. t ∈ R.
(1.1)
dt
where p > 1, F : R × RN → R is T -periodic (T > 0) in its first variable for all
x ∈ RN , and satisfies the assumption
(A1) F (t, x) is measurable in t for every x ∈ RN and continuously differentiable
in x for a.e. t ∈ [0, T ], and there exist a ∈ C(R+ , R+ ), b ∈ L1 (0, T ; R+ )
such that
|F (t, x)| + |∇F (t, x)| ≤ a(|x|)b(t)
N
for all x ∈ R and a.e. t ∈ [0, T ].
A solution is called subharmonic solution if it is kT -periodic solution for some
positive integer k (see for example [10]).
Recently, considerable attention has been paid to subharmonic solutions of second-order Hamiltonian systems with p-Laplacian; see [2, 7, 11, 18, 20, 21, 23]. When
p = 2, Equation (1.1) reduces to the second-order non-autonomous Hamiltonian
system
ü(t) + ∇F (t, u(t)) = 0 a.e. t ∈ R.
(1.2)
Using the variational methods, many existence results are obtained under suitable
conditions, we refer the reader to [1, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 22]
and the reference therein. In particular, in [6], Tang and Wu have proved the
following result.
2000 Mathematics Subject Classification. 34B15, 34K13, 58E30.
Key words and phrases. Control functions; subharmonic solutions; p-Laplacian systems;
minimax methods.
c
2011
Texas State University - San Marcos.
Submitted August 1, 2011. Published October 20, 2011.
1
2
Z. WANG
EJDE-2011/138
Theorem 1.1 ([6]). Suppose that F satisfies assumption (A1) and the following
conditions:
(S1) There exist f, g ∈ L1 (0, T ; R+ ) and α ∈ [0, 1) such that
|∇F (t, x)| ≤ f (t)|x|α + g(t)
for all x ∈ RN and a.e. t ∈ [0, T ];
(S2) There exists γ ∈ L1 (0, T ; R) such that
F (t, x) ≥ γ(t)
for all x ∈ R and a.e. t ∈ [0, T ];
(S3) There exists a subset E of [0, T ] with meas(E) > 0 such that
1
F (t, x) → +∞ as |x| → +∞
|x|2α
for a.e. t ∈ E.
Then problem (1.2) has a kT -periodic solution uk for every positive integer k, and
max0≤t≤kT |u(t)| → +∞ as k → +∞.
N
Subsequently, Pasca and Tang in [7] dealt with the second order differential inclusions systems with p-Laplacian. They generalized Theorem 1.1 in a more general
sense. Note that in [6, 7], it is usually assumed that (S1) holds, for p-Laplacian
systems, α ∈ [0, p − 1). This means that nonlinearity ∇F (t, x) is sublinear.
Recently, the author and Zhang [22], introduced a control function h(t), consider
the case in which nonlinearity ∇F (t, x) is only weak sublinear: It is assumed that
there exists a positive function h ∈ C(R+ , R+ ) satisfied the following restrictions
(i) h(s) ≤ h(t) for all s ≤ t, s, t ∈ R+ ;
(ii) h(s + t) ≤ C ∗ (h(s) + h(t)) for all s, t ∈ R+ ;
(iii) 0 < h(t) ≤ K1 tα + K2 for all t ∈ R+ ;
(iv) h(t) → +∞ as t → +∞.
Here C ∗ , K1 , K2 are positive constants, α ∈ [0, 1), and h(t) need just to satisfy
conditions (i)-(iii) if α = 0. Moreover, conditions
|∇F (t, x)| ≤ f (t)h(|x|) + g(t),
Z T
F (t, x)dx → ±∞ as |x| → +∞
1
h2 (|x|) 0
are posed. Under these assumptions, they show that second-order system
ü(t) = ∇F (t, u(t)) a.e. t ∈ [0, T ],
u(0) − u(T ) = u̇(0) − u̇(T ) = 0
(1.3)
has a T -periodic solution. In addition, if the nonlinearity ∇F (t, x) grows slightly
slower than |x|p−1 at infinity, such as
∇F (t, x) =
t|x|p−1
,
ln(e + |x|2 )
solutions are saddle points of problem
d
(|u̇(t)|p−2 u̇(t)) = ∇F (t, u(t)) a.e. t ∈ [0, T ],
dt
u(0) − u(T ) = u̇(0) − u̇(T ) = 0,
which have been obtained in [23] by minimax methods.
(1.4)
(1.5)
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SUBLINEAR HAMILTONIAN SYSTEMS
3
In the present article, we will focus on the subharmonic solutions for (1.1) by
replacing in assumptions (S1) and (S3) the term |x| with more general control
functions h(|x|). Here, we emphasize that our results are still new when p = 2.
We will establish our main results:
Theorem 1.2. Suppose that F satisfies assumption (A1) and the following conditions:
(H1) There exist constants C ∗ > 0, K1 > 0, K2 > 0, α ∈ [0, p − 1) and a positive
function h ∈ C(R+ , R+ ) with the properties (i)–(iv). Moreover, there exist
f, g ∈ L1 (0, T ; R+ ) such that
|∇F (t, x)| ≤ f (t)h(|x|) + g(t)
for all x ∈ R and a.e. t ∈ [0, T ];
(H2) There exists γ ∈ L1 (0, T ; R) such that
N
F (t, x) ≥ γ(t)
for all x ∈ RN and a.e. t ∈ [0, T ];
(H3) There exist a positive function h ∈ C(R+ , R+ ) which satisfies the conditions
(i)–(iv), and a subset E of [0, T ] with meas(E) > 0 such that
1
hq (|x|)
F (t, x) → +∞
for a.e. t ∈ E, here q :=
as |x| → +∞
p
p−1 .
1,p
Then (1.1) has kT -periodic solution uk ∈ WkT
for every positive integer k such
that kuk k∞ → +∞ as k → +∞, where
1,p
WkT
:= u : [0, kT ] → RN | u is absolutely continuous,
u(0) = u(kT ), u̇ ∈ Lp (0, kT ; RN )
is a Banach space with the norm
Z
Z kT
p
kuk :=
|u(t)| dt +
0
and kuk k∞ := max0≤t≤kT |u(t)| for u ∈
kT
|u̇(t)|p dt
1/p
0
1,p
WkT
.
Remark 1.3. Theorem 1.2 generalizes Theorem 1.1. In fact, when p = 2, Theorem
1.1 is a special case of Theorem 1.2 with control function h(t) = tα , α ∈ [0, p − 1),
t ∈ R+ . Furthermore, there are functions F (t, x) satisfying Theorem 1.2 and not
satisfying Theorem 1.1 and earlier results in the references [1, 2, 3, 4, 5, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], even for p = 2.
Example 1.4. Consider the function
3
F (t, x) = sin[(1 + |x|2 )1/2 ln1/2 (e + |x|2 )] + | sin ωt| ln 2 (e + |x|2 )
for all x ∈ RN and t ∈ R, where ω = 2π/T . It is apparent that
|∇F (t, x)| ≤ ln1/2 (e + |x|2 ) + 10,
which implies that F (t, x) is not bounded. Moreover, one has
1
F (t, x) → 0
|x|2α
as |x| → +∞
4
Z. WANG
EJDE-2011/138
for any α ∈ (0, 1) and t ∈ R. Hence, this example can not be solved by Theorem
1.1 and the results in [1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23].
On the other hand, put h(t) = ln1/2 (e+t2 ), it is not difficult to see that the properties (i)-(iv) of h(t) are all satisfied. By simple computation, (H1)-(H3) remains
true. Therefore, F (t, x) satisfies all the conditions of Theorem 1.2, then problem
1,p
(1.1) with p = 2 has kT -periodic solution uk ∈ WkT
for every positive integer k
such that kuk k∞ → +∞ as k → +∞.
Theorem 1.5. Suppose that F satisfies assumption (A1), (H2) and the following
conditions:
(H4) There exists constant C ∗ > 0 and a positive function h∗ ∈ C(R+ , R+ ) with
the properties:
(i*) h∗ (s) ≤ h∗ (t) for all s ≤ t, s, t ∈ R+ ;
(ii*) h∗ (s + t) ≤ C ∗ (h∗ (s) + h∗ (t)) for all s, t ∈ R+ ;
Rt
(ii*) th∗ (t) − pH ∗ (t) → −∞ as t → +∞, where H ∗ (t) = 0 h∗ (s)ds;
(iv*) H ∗ (t)/tp → 0 as t → +∞.
Moreover, there exist f, g ∈ L1 (0, T ; R+ ) such that
|∇F (t, x)| ≤ f (t)h∗ (|x|) + g(t)
for all x ∈ RN and a.e. t ∈ [0, T ];
(H5) There exist a positive function h∗ ∈ C(R+ , R+ ) which satisfies the conditions (i*)–(iv*), and a subset E of [0, T ] with meas(E) > 0 such that
1
F (t, x) > 0
H ∗ (|x|)
as |x| → +∞
for a.e. t ∈ E.
1,p
Then (1.1) has kT -periodic solution uk ∈ WkT
for every positive integer k such
that kuk k∞ → +∞ as k → +∞.
Remark 1.6. (1) In contrast to the result in Theorem 1.2, if ∇F (t, x) grows faster
|x|p−1
at infinity, with the rate like ln(e+|x|
2 ) , from the proof we see that, the approach
of Theorem 1.2 can not be repeated unless f (t) satisfies certain restrictions, and
α has a wider range, say, α ∈ [0, p − 1]. Meanwhile Theorem 1.5 needs only f (t)
belonging to L1 (0, T ; R+ ).
(2) Comparing with [23], we emphasize that Theorem 1.5 can not only treat the
case like (1.4), but also cases like (S1)-(S3). Details for this assertion can be found
in Example 1.7 below, also in the example in Section 4. Furthermore, our methods
here are simpler and more direct than those in [23].
(3) We must point out that assumption (H4) leads to H ∗ (t) → +∞ as t → +∞
(for details see Lemma 2.2), then (H5) is stronger than (H3) (or (S3)) with α = 0.
Therefore, Theorem 1.5 is a new result, and do not cover Theorem 1.1.
(4) There are functions F (t, x) satisfying Theorem 1.5 and not satisfying Theorem 1.1, Theorem 1.2, or the assumptions in [1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23].
Example 1.7. Consider the function
F (t, x) = | sin ωt||f (t)|
|x|p
,
ln(e + |x|2 )
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SUBLINEAR HAMILTONIAN SYSTEMS
5
where f (t) ∈ L1 (R, R+ ). Clearly, for all x ∈ RN and t ∈ R, one has
|∇F (t, x)| ≤ (2 + p)|f (t)|
|x|p−1
,
ln(e + |x|2 )
which implies that (H1) does not hold for any α ∈ [0, p−1). Moreover, as mentioned
before f (t) only belongs to L1 (R, R+ ) and no other further requirements on f (t)
are posed, then the approach of Theorem 1.2 can not be applied. This is the key
feature that Theorem 1.5 is different from Theorem 1.2. Thus, this example can
not be solved by earlier results even if p = 2.
R t sp−1
tp−1
∗
On the other hand, take h∗ (t) = ln(e+t
ds, s, t ∈ R+ , then
2 ) , H (t) =
0 ln(e+s2 )
we can find that conditions (H2), (H4) and (H5) are all satisfied, by Theorem 1.5,
1,p
problem (1.1) has kT -periodic solution uk ∈ WkT
for every positive integer k such
that kuk k∞ → +∞ as k → +∞.
Remark 1.8. Without loss of generality, we may assume that functions b in assumption (A1), f, g in (H1), (H4) and γ in (H2) are T -periodic. Then assumptions
(A1), (H1), (H2), (H4) hold for all t ∈ R by the T -periodicity of F (t, x) in the first
variable.
The remainder of this article is organized as follows. In Section 2 we give some
notations and the estimates of control functions h∗ (t) and H ∗ (t). Section 3 are
devoted to the proofs of main theorems. Finally, we will give a new example to
illustrate our results in Section 4.
2. Preliminaries
Let k be a positive integer. For convenience, in the following we will denote
R kT
1,p
1
various positive constants as Ci , i = 0, 1, 2, · · · . For u ∈ WkT
, let ū := kT
u(t)dt
0
and ũ(t) := u(t) − ū, then one has: Sobolev’s inequality
Z kT
kũk∞ ≤ C0
|u̇(t)|p dt
0
and Wirtinger’s inequality
Z kT
0
|ũ(t)|p dt ≤ C0
Z
kT
|u̇(t)|p dt.
0
1,p
It follows from assumption (A1) that functional ϕk on WkT
give by
Z kT
Z kT
1
ϕk (u) =
|u̇(t)|p dt −
F (t, u(t))dt
p 0
0
1,p
is continuously differentiable on WkT
(see [10]). Moreover, one has
Z kT
Z kT
(ϕ0k (u), v) =
(|u̇(t)|p−2 u̇(t), v̇(t))dt −
(∇F (t, u(t)), v(t))dt
0
1,p
WkT
.
0
for all u, v ∈
It is well known that the solutions to problem (1.1) correspond
to the critical points of the functional ϕk .
To proof of our main theorems, we need the following auxiliary results.
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Z. WANG
EJDE-2011/138
Lemma 2.1 ([4]). Suppose that F satisfies assumption (A1), and E is a measurable
subset of [0, T ]. Assume that
F (t, x) → +∞
as |x| → ∞
for a.e. t ∈ E. Then for every δ > 0, there exists subset Eδ of E with meas(E\Eδ ) <
δ such that
F (t, x) → +∞ as |x| → +∞
uniformly for all t ∈ Eδ .
Lemma 2.2. Suppose that there exists a positive function h∗ which satisfies the
conditions (i*), (iii*), (iv*), then we have the following estimates:
(a) 0 < h∗ (t) ≤ tp−1 + C1 for any > 0, t ∈ R+ ,
(b) h∗ q (t)/H ∗ (t) → 0 as t → +∞,
(c) H ∗ (t) → +∞ as t → +∞.
Proof. It follows from (iv*) that, for any > 0, there exists M1 > 0 such that
H ∗ (t) ≤ εtp
∀t ≥ M1 .
Observe that by (iii*), there exists M2 > 0 such that
th∗ (t) − pH ∗ (t) ≤ 0
∀t ≥ M2 ,
which implies that
pH ∗ (t)
≤ ptp−1
t
where M := max{M1 , M2 }. Hence we obtain
h∗ (t) ≤
∀t ≥ M,
h∗ (t) ≤ ptp−1 + h∗ (M )
for all t > 0 by (i∗ ) of (H4). Obviously, h∗ (t) satisfies (a) due to the definition of
h∗ (t) and the above inequality.
p
Next, we turn to (b). Recalling property (iv*) and the fact q = p−1
, we obtain
h∗ q (t)
h∗ q (t)
p q
=
· H ∗ q−1 (t) ≤
· H ∗ q−1 (t)
q
∗
∗
H (t)
H (t)
t
H ∗ (t) 1/(p−1)
H ∗ q−1 (t)
= pq
→ 0 as t → +∞.
= pq ·
q
t
tp
Therefore, estimate (b) holds.
Finally, we show that (c) is also true. By (iii*), one arrives at, for every L > 0,
there exists M3 > 0 such that
0<
th∗ (t) − pH ∗ (t) ≤ −L ∀t ≥ M3 .
So, one has
θth∗ (θt) − pH ∗ (θt) ≤ −L
for all |θt| ≥ M3 . Then we have
d H ∗ (θt) θt · h∗ (θt) − pH ∗ (θt)
L
d L =
≤ − p+1 =
.
dθ
θp
θp+1
θ
dθ pθp
Let θ > 1, integrating both sides of the above inequality from 1 to θ, we obtain
H ∗ (θt)
L
L
L
1 − H ∗ (t) ≤ p − =
.
θp
pθ
p
p θp − 1
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SUBLINEAR HAMILTONIAN SYSTEMS
7
Let θ → +∞ in the above inequality, and by (iv*), one has
H ∗ (t) ≥
L
p
for all t ≥ M3 . By the arbitrariness of L, we have H ∗ (t) → +∞ as t → +∞. This
completes the proof.
3. Proof of main results
In this section, firstly, we discuss the (PS) condition.
Lemma 3.1. Assume that F satisfies assumptions (A1), (H1)–(H3). Then ϕk
satisfies the (PS) condition, that is, {un } has a convergent subsequence whenever
it satisfies ϕ0k (un ) → 0 as n → +∞ and {ϕk (un )} is bounded.
Proof. It follows from (H1) and Sobolev’s inequality that
Z kT
(∇F (t, un (t)), ũn (t))dt
0
Z
kT
kT
Z
≤
f (t)h(|ūn + ũn (t)|)|ũn (t)|dt +
g(t)|ũn (t)|dt
0
Z
0
kT
Z
∗
≤
kT
f (t)[C (h(|ūn |) + h(|ũn (t)|))]|ũn (t)|dt + kũn k∞
g(t)dt
0
≤ C ∗ [h(|ūn |) + h(|ũn (t)|)]kũn k∞
Z
kT
0
kT
Z
f (t)dt + kũn k∞
0
g(t)dt
0
Z kT
Z kT
q i
1
p
∗ p q
kũ
k
+
2pC
C
h
(|ū
|)
f
(t)dt
+
kũ
k
g(t)dt
n ∞
n
n ∞
0
2pC ∗ C0p
0
0
Z kT
∗
+ C h(kũn k∞ )kũn k∞
f (t)dt
(3.1)
≤ C∗
h
1
≤
2p
Z
0
kT
p
q
|u̇n (t)| dt + C2 h (|ūn |) + C
∗
[K1 kũn kα
∞
Z
+ K2 ]kũn k∞
0
kT
f (t)dt
0
Z
kT
+ kũn k∞
g(t)dt
0
≤
1
2p
kT
Z
|u̇n (t)|p dt + C2 hq (|ūn |) + C3
0
+ C4
Z
kT
|u̇n (t)|p dt
(α+1)/p
0
Z
kT
|u̇n (t)|p dt
1/p
.
0
Hence, we see that
kũn k∞ ≥ |(ϕ0k (un ), ũn )|
Z
Z
kT
=
|u̇n (t)|p dt −
kT
(∇F (t, un (t)), ũn (t))dt
(3.2)
0
0
Z
Z kT
(α+1)/p
1 kT
|u̇n (t)|p dt − C2 hq (|ūn |) − C3
|u̇n (t)|p dt
2p 0
0
1/p
Z kT
− C4
|u̇n (t)|p dt
≥ 1−
0
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Z. WANG
EJDE-2011/138
for large n. It follows from Wirtinger’s inequality that
!1/p
Z kT
1/p
p
kũn k∞ ≤ (C0 + 1)
|u̇n (t)| dt
0
for all n, thus we obtain
C5 hq (|ūn |) ≥
kT
Z
|u̇n (t)|p dt − C6
(3.3)
0
for all large n, which implies that
Z
kũn k∞ ≤ C0
kT
|u̇n (t)|p dt
1/p
0
≤ [C0 (C5 hq (|ūn |) + C6 )]1/p
≤ [C7 (|ūn |qα + 1)]1/p .
Then one has
|un (t)| ≥ |ūn | − |ũn (t)| ≥ |ūn | − kũn (t)k∞ ≥ |ūn | − [C7 (|ūn |qα + 1)]1/p
(3.4)
for all large n and every t ∈ [0, kT ].
We claim that {|ūn |} is bounded. Arguing indirectly, if {|ūn |} is unbounded, we
may assume that, going to a subsequence if necessary,
|ūn | → +∞
as n → +∞,
(3.5)
which, together with (3.4), implies
1
|ūn |.
2
Then for all large n and every t ∈ [0, kT ], we have
|un (t)| ≥
h(|ūn |) ≤ h(2|un (t)|) ≤ 2C ∗ h(|un (t)|).
(3.6)
(3.7)
1
2
Set δ = meas(E). In virtue of (H3) and Lemma 2.1, there exists a subset Eδ of
E with meas(E\Eδ ) < δ such that
1
F (t, x) → +∞ as |x| → +∞
(3.8)
q
h (|x|)
uniformly for all t ∈ Eδ , which implies
meas(Eδ ) = meas(E) − meas(E\Eδ ) > δ > 0,
and for every β > 0, there exists M ≥ 1 such that
1
F (t, x) ≥ β
q
h (|x|)
(3.9)
(3.10)
for all |x| ≥ M and all t ∈ Eδ . By (3.5) and (3.6), one has |un (t)| ≥ M for all large
n and every t ∈ [0, kT ]. It follows from (3.3), (3.7), (3.9), (3.10) that
Z kT
Z
1 kT
p
F (t, un (t))dt
|u̇n (t)| dt −
ϕk (un ) =
p 0
0
Z
Z
1
≤ (C5 hq (|ūn |) + C6 ) −
γ(t)dt −
βhq (|un (t)|)dt
p
[0,kT ]\Eδ
Eδ
Z
q
1
q
h(|ūn |) dt
≤ C8 h (|ūn |) + C9 − β
∗
Eδ 2C
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SUBLINEAR HAMILTONIAN SYSTEMS
≤ C8 hq (|ūn |) + C9 − β
9
1
hq (|ūn |)δ
(2C ∗ )q
for all large n. So, we obtain
1
lim sup
hq (|ū
n→+∞
n |)
ϕk (un ) ≤ C8 − β
1
δ.
(2C ∗ )q
By the arbitrariness of β > 0, one has
lim sup
n→+∞
1
ϕk (un ) = −∞,
hq (|ūn |)
which contradicts the boundedness of ϕk (un ). Hence {|ūn |} is bounded. Furthermore, by (3.2) and (3.3), we know {un } is bounded. Arguing then as in [11,
Proposition 4.1], we conclude that (PS) condition is satisfied.
Lemma 3.2. Assume that F satisfies assumption (A1), (H2), (H4), (H5). Then
ϕk satisfies the (PS) condition.
Proof. It follows from (3.1) and Lemma 2.2 that
Z
kT
(∇F (t, un (t)), ũn (t))dt
0
Z kT
Z kT
q i
1
p
∗ p ∗q
≤C
kũn k∞ + 2pC C0 h (|ūn |)
f (t)dt
+ kũn k∞
g(t)dt
2pC ∗ C0p
0
0
Z kT
+ C ∗ h∗ (kũn k∞ )kũn k∞
f (t)dt
∗
h
0
≤
1
2p
Z
kT
p−1
|u̇n (t)|p dt + C2 h∗ q (|ūn |) + C ∗ [kũn k∞
+ C1 ]kũn k∞
Z
0
kT
f (t)dt
0
kT
Z
+ kũn k∞
g(t)dt
(3.11)
0
≤
1
+ C10
2p
Z
kT
|u̇n (t)|p dt + C2 h∗ q (|ūn |) + C11
0
Z
kT
|u̇n (t)|p dt
1/p
,
0
which implies
kũn k∞ ≥ |(ϕ0k (un ), ũn )|
≥ 1−
1
− C10
2p
Z
kT
|u̇n (t)|p dt − C2 h∗ q (|ūn |) − C11
0
Z
kT
|u̇n (t)|p dt
1/p
0
for large n. Thus, by (3.2), one has
C12 h∗ q (|ūn |) ≥
Z
kT
|u̇n (t)|p dt − C13
(3.12)
0
for all large n and small enough, which implies
kũn k∞ ≤ [C0 (C12 h∗ q (|ūn |) + C13 )]1/p ≤ [C14 (|ūn |p + 1)]1/p
by Lemma 2.2. Consequently, we get
|un (t)| ≥ |ūn | − [C14 (|ūn |p + 1)]1/p
for all large n, every t ∈ [0, kT ] and small enough.
(3.13)
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Z. WANG
EJDE-2011/138
Assume {|ūn |} is unbounded, by (3.13), for small enough, we obtain
|un (t)| ≥
1
|ūn |.
2
(3.14)
Combine (3.14) with (H4 ), one has
H ∗ (|ūn |) ≤ 2C ∗ H ∗ (|un (t)|).
With the same arguments of (3.8)-(3.10), by (H5), we know
1
H ∗ (|x|)
F (t, x) ≥ C15
(3.15)
for all |x| ≥ M and all t ∈ Eδ . We see that, jointly with (3.12), (3.15) and Lemma
2.2, for all large n,
Z kT
Z
1 kT
p
|u̇n (t)| dt −
F (t, un (t))dt
ϕk (un ) =
p 0
0
Z
Z
1
≤ (C12 h∗ q (|ūn |) + C13 ) −
γ(t)dt −
C15 H ∗ (|un (t)|)dt
p
[0,kT ]\Eδ
Eδ
Z
1
∗
∗q
H (|ūn |)dt
≤ C16 h (|ūn |) + C17 − C15
∗
Eδ 2C
≤ C16 h∗ q (|ūn |) + C17 − C18 δH ∗ (|ūn |),
which implies
1
ϕk (un )
H ∗ (|ūn |)
C17
h∗ q (|ūn |)
+ ∗
− C18 δ ≤ −C18 δ,
≤ lim sup C16 ∗
H (|ūn |)
H (|ūn |)
n→+∞
0 = lim sup
n→+∞
a contradiction. Hence {|ūn |} is bounded, moreover, we can get {un } is bounded.
So (PS) condition is satisfied, which completes the proof.
Now, we are ready to proof our main results.
Proof of Theorem 1.2. It follows from Lemma 3.1 that ϕk satisfies the (PS) condition. In order to use the saddle point theorem, we only need to verify the following
conditions
1,p
(I1) ϕk (u) → +∞ as kuk → +∞ in W̃kT
,
(I2) ϕk (x + ek (t)) → −∞ as |x| → +∞ in RN ,
1,p
1,p
1,p
where W̃kT
:= {u ∈ WkT
: ū = 0}, ek (t) = k cos(k −1 ωt)x0 ∈ W̃kT
, x0 ∈ RN ,
2π
|x0 | = 1 and ω = T . Next, we show that ϕk satisfies (I1) and (I2).
For all x ∈ RN , it follows from (3.10) that
EJDE-2011/138
SUBLINEAR HAMILTONIAN SYSTEMS
1
p
Z
1
≤
p
Z
ϕk (x + ek (t)) =
kT
|ėk (t)|p dt −
kT
Z
0
11
F (t, x + k cos(k −1 ωt)x0 )dt
0
kT
|ω(sin k
−1
Z
p
ωt)x0 | dt −
γ(t)dt
0
[0,kT ]\Eδ
Z
−β
(3.16)
−1
q
h (|x + k cos(k ωt)x0 |)dt
Z
≤ C19 k −
γ(t)dt − βhq (M ) meas(Eδ )
Eδ
[0,kT ]\Eδ
for all |x| ≥ M + k. By the arbitrariness of β, one has
ϕk (x + ek (t)) → −∞ as |x| → +∞ in RN .
Thus (I2) is satisfied.
1,p
For all u ∈ W̃kT
, it follows from Sobolev’s inequality that
Z kT
Z
kT
[F (t, u(t)) − F (t, 0)]dt = (∇F (t, su(t)), u(t))dsdt
0
0
Z
kT
1
Z
≤
0
Z
Z
0
kT
0
kT
Z
α
f (t)[K1 |u(t)| + K2 ]|u(t)|dt +
0
K1 kukα+1
∞
kT
Z
f (t)dt + K2 |uk∞
Z
|u̇(t)|p dt
(α+1)/p
+ C21
kT
f (t)dt + kuk∞
g(t)dt
0
kT
Z
g(t)|u(t)|dt
0
kT
Z
0
≤ C20
1
g(t)|u(t)|dsdt
0
≤
≤
kT
Z
f (t)h(|su(t)|)|u(t)|dsdt +
0
Z
kT
|u̇(t)|p dt
1/p
.
0
0
Hence, we have
1
p
Z
1
≥
p
Z
ϕk (u) =
kT
|u̇(t)|p dt −
Z
kT
Z
[F (t, u(t)) − F (t, 0)]dt −
0
0
|u̇(t)| dt − C20
0
− C21
F (t, 0)dt
0
kT
p
kT
kT
Z
|u̇(t)|p dt
(α+1)/p
0
Z
kT
|u̇(t)|p dt
1/p
− C22 ,
0
1,p
then we can conclude that ϕk (u) → +∞ as kuk → +∞ in W̃kT
. Plainly, condition
(I1) holds.
1,p
By (I1), (I2) and the saddle point theorem, there exists a critical point uk ∈ WkT
for ϕk such that
−∞ < inf
ϕk ≤ ϕk (uk ) ≤ sup ϕk .
1,p
W̃kT
RN +ek
For fixed x ∈ R , set
N
Ak := {t ∈ [0, kT ]||x + k(cos k −1 ωt)x0 | ≤ M }.
Using the same argument of [6], we have
meas(Ak ) ≤
1
kδ.
2
(3.17)
12
Z. WANG
EJDE-2011/138
Let Ek := ∪k−1
j=0 (jT + Eδ ), then it follows from (3.17) that
1
kδ
2
for large k. Taking into account (3.16), we have
Z
k −1 ϕk (x + ek (t)) ≤ C19 − k −1
γ(t)dt − k −1 βhq (M ) meas(Eδ )
meas(Ek \Ak ) ≥
[0,kT ]\(Eδ \Ak )
Z
≤ C19 +
0
T
1
|γ(t)|dt − δhq (M )β
2
for every x ∈ R and all large k. Hence one has
Z
lim sup sup k −1 ϕk (x + ek ) ≤ C19 +
N
k→+∞ x∈RN
0
T
1
|γ(t)|dt − δhq (M )β.
2
Observe the arbitrariness of β, we obtain
lim sup sup k −1 ϕk (x + ek ) = −∞,
k→+∞ x∈RN
which implies
lim sup k −1 ϕk (uk ) = −∞.
(3.18)
k→+∞
Finally, we prove that kuk k∞ → +∞ as k → +∞. If not, going to a subsequence
if necessary, we may assume that
kuk k∞ ≤ C23
for all k ∈ N. Hence we have
Z kT
Z
−1
−1
−1
k ϕk (uk ) ≥ −k
F (t, uk (t))dt ≥ −k
max a(s)
0≤s≤C23
0
Z
= − max a(s)
0≤s≤C23
kT
b(t)dt
0
T
b(t)dt.
0
It follows that
lim inf k −1 ϕk (uk ) > −∞,
k→+∞
which contradicts (3.18). This completes the proof.
Proof of Theorem 1.5. By Lemma 3.2, ϕk satisfies the (PS) condition. By the
argument of Theorem 1.2, we only need to check that ϕk satisfies (I1) and (I2). In
fact from (3.15), for all x ∈ RN it follows that
Z kT
Z
1 kT
p
|ėk (t)| dt −
F (t, x + k cos(k −1 ωt)x0 )dt
ϕk (x + ek (t)) =
p 0
0
Z
Z
γ(t)dt − C15
H ∗ (|x + k cos(k −1 ωt)x0 |)dt
≤ C19 k −
[0,kT ]\Eδ
Eδ
for all |x| ≥ M + k. Using the fact H ∗ (t) → +∞ as t → +∞ of Lemma 2.2 and
(3.9), one has
ϕk (x + ek (t)) → −∞ as |x| → +∞ in RN .
Thus (I2) is safisfied.
EJDE-2011/138
SUBLINEAR HAMILTONIAN SYSTEMS
13
1,p
For all u ∈ W̃kT
, we have
kT
Z
[F (t, u(t)) − F (t, 0)]dt
0
kT
Z
1
Z
f (t)h∗ (|su(t)|)|u(t)|dsdt + kuk∞
≤
0
0
kT
Z
f (t)[|u(t)|p−1 + C1 ]|u(t)|dt + kuk∞
≤
0
Z
kT
g(t)dt
0
Z kT
g(t)dt
0
Z
≤ C24
kT
|u̇(t)|p dt + C25
0
Z
kT
|u̇(t)|p dt
1/p
,
0
which implies
ϕk (u) ≥
1
− C24
p
Z
kT
|u̇(t)|p dt − C25
0
Z
kT
|u̇(t)|p dt
1/p
− C22 .
0
1,p
Then for any small enough, we have ϕk (u) → +∞ as kuk → +∞ in W̃kT
. So
condition (I1) holds, and the proof hereby is complete.
4. Example
In this section, we give a new example to illustrate our results. Consider function
F (t, x) = | sin ωt||x|1+α ,
where 0 < α < p − 1 and ω = 2π/T . We claim both Theorem 1.2 and Theorem 1.5
can handle this case.
Indeed, choose h(t) = tα , α ∈ (0, p − 1), clearly, all conditions of Theorem 1.2
1,p
are satisfied. So that (1.1) has a kT -periodic solution uk ∈ WkT
for every positive
integer k, and kuk k∞ → +∞ as k → +∞.
Rt
Let h∗ (t) = tα , H ∗ (t) = 0 sα ds and C ∗ is a suitable positive constant. We know
that
(i*)
(ii*)
(iii*)
(iv*)
h∗ (s) ≤ h∗ (t) for all s ≤ t, s, t ∈ R+ ;
h∗ (s + t) = (s + t)α ≤ C ∗ (h∗ (s) + h∗ (t)) for all s, t ∈ R+ ;
p
th∗ (t) − pH ∗ (t) = 1 − 1+α
t1+α → −∞ as t → +∞;
∗
p
H (t)/t → 0 as t → +∞.
Moreover, we can check that (H2) and (H5) are satisfied. Therefore, by Theorem
1,p
1.5, problem (1.1) has kT -periodic solution uk ∈ WkT
for every positive integer k,
and kuk k∞ → +∞ as k → +∞.
Acknowledgments. The author wants to thank Professor Xiangsheng Xu and
the members of Department of Mathematics and Statistics at Mississippi State
University for their warm hospitality and kindness. This Project is Supported by
National Natural Science Foundation of China (Grant No.11026213, 10871096),
Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant
No.10KJB110006) and Foundation of Nanjing University of Information Science &
Technology (Grant No.20080280).
14
Z. WANG
EJDE-2011/138
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Zhiyong Wang
Department of Mathematics, Nanjing University of Information Science & Technology,
Nanjing 210044, Jiangsu, China
E-mail address: mathswzhy1979@gmail.com