R E S E A R C H
Open Access
Existence of nonconstant periodic
solutions for a class of second-order systems
with
p
(
t
)
-Laplacian
Yukun An
1*, Yuanfang Ru
2,3and Fanglei Wang
3*Correspondence: [email protected] 1College of Science, Nanjing
University of Aeronautics and Astronautics, Nanjing, 210016, P.R. China
Full list of author information is available at the end of the article
Abstract
In this paper, we investigate a class of second-orderp(t)-Laplacian systems with local ‘superquadratic’ potential. By using the generalized mountain pass theorem, we obtain an existence result for nonconstant periodic solutions.
Keywords: p(t)-Laplacian; generalized Sobolev space; generalized mountain pass theorem; periodic solution
1 Introduction
This paper is concerned with the existence of periodic solutions for the following p(t)-Laplacian system:
⎧ ⎨ ⎩
(|u(t)|p(t)–u(t))+∇F(t,u(t)) = , t∈[,T], u() –u(T) =u() –u(T) = ,
()
whereT> ,u∈RN.F(t,u) andp(t) satisfy the following conditions:
(F) F: [,T]×RN →R is measurable andT-periodic int for eachu∈RN and
con-tinuously differentiable in u for a.e. t∈[,T], and there exist a∈C(R+,R+) and b∈L([,T],R+)such that
F(t,u)≤a|u|b(t), ∇F(t,u)≤a|u|b(t)
for allu∈RN and a.e.t∈[,T].
(P) p(t)∈C([,T],R+),p(t) =p(t+T)and
<p–:=minp(t)≤p+:=maxp(t) < +∞.
The p(t)-Laplacian system can be applied to describe the physical phenomena with ‘pointwise different properties’ which first arose from the nonlinear elasticity theory (see []).
Ifp(t) =pis a constant, system () reduces to thep-Laplacian system
⎧ ⎨ ⎩
(|u(t)|p–u(t))+∇F(t,u(t)) = , t∈[,T],
u() –u(T) =u() –u(T) = . ()
Especially, whenp= , system () or () becomes the well-known second-order Hamilto-nian system
⎧ ⎨ ⎩
u(t) +∇F(t,u(t)) = , t∈[,T],
u() –u(T) =u() –u(T) = . () In , Rabinowitz [] published his pioneer paper on the existence of periodic solu-tions for problem () under the following Ambrosetti-Rabinowitz superquadratic condi-tion:
(AR) There existμ> andL∗> such that
<μF(t,u)≤∇F(t,u),u
for all|u| ≥L∗and a.e.t∈[,T].
From then on, many researchers have tried to replace the Ambrosetti-Rabinowitz (short-enedAR) condition by other superquadratic conditions. Some new superquadratic condi-tions under which there exist periodic solucondi-tions for problem () have been discovered in literature, see, for example, the references [–]. In [], the authors obtained the following existence theorem for () under the ‘local superquadratic conditions’.
Theorem A([], Theorem .) Suppose that F(t,u)satisfies(F)and the following
condi-tions:
(V) F(t,u)≥for allt∈[,T]andu∈RN;
(V) There are constantsm> andα≤ m
T such thatF(t,u)≤α for allu∈RN,|u|<m
and a.e.t∈[,T].
(V) There are constantsμ> ,≤γ < ,G> and the functiond(t)∈L([,T],R+)such
that
μF(t,u)≤∇F(t,u),u+d(t)|u|γ
for allu∈RN,|u| ≥Gand a.e.t∈[,T].
(V) There exist a constantM> and a subsetEof[,T]withmeas(E) > such that
(a) lim inf|u|→∞F|(t,u)
u| > uniformly for a.e.t∈E; (b) d(t)≤Mfor a.e.t∈E.
Then problem()has at least one nonconstant T -periodic solution.
Theorem B([], Theorem .) Suppose that F(t,u)satisfies(F), (V)and the following conditions:
(H) lim inf|u|→F|(ut|,up)= uniformly for a.e.t∈[,T];
(H) There are constantsμ>p,G> and the functiond(t)∈L([,T],R)such that
μF(t,u) –∇F(t,u),u≤d(t)|u|p
for allu∈RN,|u| ≥Gand a.e.t∈[,T],and
lim sup
|u|→∞
μF(t,u) – (∇F(t,u),u)
|u|p ≤
uniformly for a.e.t∈[,T];
(H) There exists a subsetEof[,T]withmeas(E) > such that
lim inf
|u|→∞ F(t,u)
|u|p >
uniformly for a.e.t∈E.
Then problem()has at least one nonconstant T -periodic solution.
On the other hand, thep(t)-Laplacian system has been studied by many authors in the last two decades, see, for example, [–] and the references cited therein. In [], by us-ing linkus-ing methods, the authors obtained an existence result under theARcondition as follows.
Theorem C([], Theorem .) Suppose that conditions(P)and(F)hold and F(t,u)
sat-isfies the following conditions:
(A) F(, ) = andF(t,u)≥for allt∈[,T]andu∈RN;
(A) There are constantsμ>P+andG> such that
μF(t,u)≤∇F(t,u),u
for allu∈RN,|u| ≥Gand a.e.t∈[,T];
(A) There existν>P+andg∈C([,T],R)such that
lim sup
|u|→∞ F(t,u)
|u|ν ≤g(t).
Then problem()has at least one periodic solution.
Moreover, in [–], the authors studied a superlinear elliptic equation with p(x)-Laplacian without theARcondition and obtained some existence results.
Theorem Suppose that conditions(P)and(F)hold and,in addition,F(t,u)satisfies the following conditions:
(F) F(t,u)≥for allt∈[,T]andu∈RN;
(F) lim inf|u|→F|u(t|p,u+)= uniformly for a.e.t∈[,T];
(F) There are constantsμ>p+,G> and the functiond(t)∈L([,T],R)such that
μF(t,u) –∇F(t,u),u≤d(t)|u|p–
for allu∈RN,|u| ≥Gand a.e.t∈[,T],and
lim sup
|u|→∞
μF(t,u) – (∇F(t,u),u)
|u|p– ≤
uniformly for a.e.t∈[,T];
(F) There exists a subsetof[,T]withmeas() > such that
lim inf
|u|→∞
F(t,u)
|u|p+ >
uniformly for a.e.t∈.
Then problem () has at least one nonconstantT-periodic solution.
Example Setp(t) =+sin(π
Tt–
π
), thenp(t) satisfies condition (P) andp –=
,p +=
. Let
ψ(t) =
⎧ ⎨ ⎩
sin(π
Tt), t∈[, T ], , t∈[T,T],
φ(u) =
⎧ ⎨ ⎩
|u|, |u| ≤,u∈RN,
|u|
, |u|> ,u∈RN,
F(t,u) =ψ(t)|u|+φ(u), t∈[,T],u∈RN.
It is clear that (F) and (F) hold. Letμ= ,G= , then (F) holds. And take= [T,T], then (F) holds fort∈. Therefore,Fsatisfies all the conditions of our Theorem . More-over, it is easy to verify that the functionF(t,u) does not satisfy theARconditionA in Theorem C fort∈[T,T].
2 Preliminaries
For the reader’s convenience, we first give some necessary background knowledge and propositions concerning the generalized Lebesgue-Sobolev spaces. We can refer the reader to [, –]. In the following, we use| · |to denote the Euclidean norm inRN.
Letp(t) satisfy condition (P) and define
Lp(t)[,T];RN=
u∈L[,T];RN: T
|u|p(t)dt<∞
with the norm
|u|Lp(t)=|u|p(t)=inf
λ> : T
uλp(t)dt≤
.
Define
CT∞=CT∞R;RN=u∈C∞R;RN:uisT-periodic.
Foru∈L([,T];RN), if there existsv∈L([,T];RN) satisfying
T
vϕdt= – T
uϕdt, ∀ϕ∈CT∞,
thenvis called theT-weak derivative ofuand is denoted byu. Define
WT,p(t)[,T];RN=u∈Lp(t)[,T];RN:u∈Lp(t)[,T];RN
with the norm
u
WT,p(t)=u=|u|p(t)+u
p(t).
Foru∈WT,p(t)([,T];RN), let
¯
u= T
T
x(s)ds, u(t) =˜ u(t) –u¯
and
WT,p(t)[,T];RN=
x∈WT,p(t)[,T];RN: T
x(s)ds=
,
then
WT,p(t)[,T];RN=WT,p(t)[,T];RN⊕RN.
In the following we useLp(t),W,p(t) T ,W
,p(t)
T to denoteLp(t)([,T];RN),W ,p(t)
T ([,T];RN),
WT,p(t)([,T];RN), respectively.
Proposition ([]) For u∈Lp(t),one has
() |u|p(t)< (= ; > ) ⇔ T
u(t)p(t)dt< (= ; > );
() |u|p(t)> ⇒ |u|p
–
p(t)≤ T
u(t)p(t)dt≤ |u|pp+(t);
|u|p(t)< ⇒ |u|p
+
p(t)≤ T
() |u|p(t)→ ⇔ T
u(t)p(t)dt→;
|u|p(t)→ ∞ ⇔ T
u(t)p(t)dt→ ∞.
Proposition ([]) The spaces Lp(t)and W,p(t)
T are separable and reflexive Banach spaces when p–> .
Proposition ([]) There is a continuous embedding WT,p(t)→C([,T];RN);when p–> ,
it is a compact embedding.
Proposition ([]) For every u∈WT,p(t),there is a constant C independent of u such that
u∞≤Cup(t).
Proposition ([]) Let u=u¯+u˜∈WT,p(t),then the norm|u|p(t)is an equivalent norm on
WT,p(t)and|¯u|+|u|p(t)is an equivalent norm on WT,p(t).
To prove the main theorem of the paper, we need the following generalized mountain pass theorem.
Lemma ([]) Let E be a real Banach space with E=V⊕X,where V= is finite
dimen-sional.Supposeϕ∈C(E,R)satisfies the(PS)condition,and
(a) There existρ,α> such thatϕ|∂Bρ∩X≥α,whereBρ={u∈E|uE≤ρ},∂Bρ denotes the boundary ofBρ;
(b) There existe∈∂B∩Xandr>ρsuch that ifQ≡(Br¯ ∩V)⊕ {se|≤s≤r},then ϕ|∂Q≤α.
Thenϕpossesses a critical value c≥αwhich can be characterized as
c≡inf h∈maxu∈Qϕ
h(u),
where={h∈C(Q,¯ E)|h=id on∂Q},and id denotes the identity operator.
3 Proof of Theorem 1
Define a functionalϕonWT,p(t)by
ϕ(u) = T
p(t)u
(t)p(t) dt–
T
Ft,u(t)dt
for eachu∈WT,p(t). It follows from assumption (F) that the functionalϕis continuously differentiable onWT,p(t). Moreover, we have
ϕ(u),v= T
u(t)p(t)–
u(t),v(t)dt– T
∇Ft,u(t),v(t)dt
We shall apply Lemma toϕto prove Theorem .
For the convenience to verify the (PS) condition, we need the following lemma. The proof can be found in [] or [].
Lemma Let J(u) =Tp(t)|u(t)|p(t)dt for u∈W,p(t)
T .ThenJ(u),v=
T
|(|u(t)|p(t)–× u(t),v(t))dt for all u,v∈WT,p(t). And J is a mapping of type(S+), i.e.,if unu and lim supn→∞J(un) –J(u),un–u ≤,then{un}has a convergent subsequence in WT,p(t).
In the following lemma we will show thatϕsatisfies the (PS) condition.
Lemma The functionalϕ satisfies the (PS)condition, i.e.,for every sequence {un} ∈
WT,p(t),{un}has a convergent subsequence if
ϕ(un) is bounded and ϕ(un)→ as n→ ∞. () Proof First we prove that{un}is a bounded sequence inWT,p(t). Otherwise,{un}would be unbounded. Passing to a subsequence, we may assume thatun → ∞. Letwn= un
un, so
thatwn= . By Proposition , also passing to a subsequence, we can suppose that
wnw weakly inWT,p(t),
wn→w strongly inC
[,T];RN
asn→ ∞. Moreover, we have
¯
wn= T
T
wn(t)dt→ T
T
w(t)dt=w¯ ()
asn→ ∞. By () there exists a constantC> such that
T
μ p(t)–
un(t)p(t)dt
=μϕ(un) –ϕ(un),un+ T
μFt,un(t)–∇Ft,un(t),un(t)dt
≤C
+un+ T
μFt,un(t)–∇Ft,un(t),un(t)dt.
Notice thatun → ∞, we have
μ p+–
T
wn(t)p(t)dt=
μ p+–
T
|un(t)|p(t)
unp(t) dt
≤ T
μ p(t)–
|un(t)|p(t)
unp– dt.
So, we obtain
μ p+–
T
wn(t)p(t)dt
≤C( +un)
unp
– +
T
(μF(t,un(t)) – (∇F(t,un(t)),un(t)))
unp
In view of (F) and (F), there exists⊂[,T] withmeas() = such that
F(t,u)≤a|u|b(t), ∇F(t,u)≤a|u|b(t) ()
for allu∈RNandt∈[,T]\ and
lim sup
|u|→∞
μF(t,u) – (∇F(t,u),u)
|u|p– ≤
uniformly fort∈[,T]\. This yields
lim sup n→∞
μF(t,un(t)) – (∇F(t,un(t)),un(t))
unp– ≤ ()
fort∈[,T]\. Otherwise, there existt∈[,T]\and a subsequence ofun, still denoted byun, such that
lim sup n→∞
μF(t,un(t)) – (∇F(t,un(t)),un(t))
unp– > . ()
If{un(t)}is bounded, then there exists a positive constantCsuch that|un(t)| ≤Cfor alln∈N. By () we find
μF(t,un(t)) – (∇F(t,un(t)),un(t))
unp
–
≤(μ+C)max≤s≤Ca(s)b(t)
unp– →
asn→ ∞, which contradicts (). So, there is a subsequence ofun(t), still denoted by un(t), such that|un(t)| → ∞asn→ ∞.
lim sup n→∞
μF(t,un(t)) – (∇F(t,un(t)),un(t))
unp–
=lim sup n→∞
μF(t,un(t)) – (∇F(t,un(t)),un(t))
|un(t)|p–
wn(t) p–
=lim sup n→∞
μF(t,un(t)) – (∇F(t,un(t)),un(t))
|un(t)|p–
lim n→∞wn(t)
p–
≤.
This contradicts (). Thus, () holds. From () and () we obtain
lim sup n→∞
μ p+–
T
wn(t)p(t)dt≤.
Sinceμ>p+, we get
lim n→∞
μ p+–
T
Combining with (), this yields
wn→ ¯w asn→ ∞,
which means that
w=w¯ and T| ¯w|=w= .
Then we have
un(t)→ ∞ asn→ ∞
uniformly for a.e.t∈[,T]. And we get from (F), (F) and Fatou’s lemma that
lim inf n→∞
T
F(t,un(t))
unp+ dt
≥ T
lim inf
n→∞
F(t,un(t))
unp+ dt
= T
lim inf
n→∞
F(t,un(t))
|un(t)|p+ wn(t)
p+
dt
≥
lim inf n→∞
F(t,un(t))
|un(t)|p
+ |w|p
+
dt> . ()
On the other hand, we have
T
F(t,un(t))
unp
+ dt=
T
p(t)
|un(t)|p(t)
unp
+ dt–
ϕ(un)
unp
+
≤
p– T
un(t)
un
p(t)dt– ϕ(un)
unp+
= p–
T
wn(t)p(t)dt– ϕ(un)
unp+.
Therefore, combining () and (), we obtain that
lim inf n→∞
T
F(t,un(t))
unp+ dt≤,
which contradicts (). Hence,{un}is a bounded sequence inWT,p(t).
By Proposition and Proposition ,{un}has a subsequence, again denoted by{un}, such that
unu weakly inWT,p(t),
un→u strongly inC
[,T];RN.
()
It follows from Proposition thatmax≤t≤T|un(t)| ≤Cun, which implies
un(t)≤C for allt∈[,T]. ()
From (), () and (F), we get
T∇Ft,un(t),un(t) –u(t)dt
≤ T
∇Ft,un(t)un(t) –u(t)dt
≤ un–u∞
T
aun(t)b(t)dt
≤Cun–u∞
T
b(t)dt.
Thus, from (), we obtain
T
∇Ft,un(t),un(t) –u(t)dt→. ()
By () and (), we have
ϕ(un),un–u→. ()
Then it follows from () and () that
J(un),un–u
= T
un(t)p(t)–un(t),un(t) –u(t)dt
=ϕ(un),un–u+ T
∇Ft,un(t),un(t) –u(t)dt
→ asn→ ∞. ()
Moreover, sinceJ(u)∈(WT,p(t))∗, we getϕ(u),un–u →, which combined with () implies that
lim n→∞
J(un) –J(u),un–u≤.
Hence, from Lemma ,{un}has a subsequence convergent strongly touinWT,p(t). The
proof of the lemma is completed.
The following result establishes the generalized mountain pass geometry for the func-tionalϕ(u).
Lemma Let WT,p(t)=RN⊕W,p(t)
T , Br={u∈W ,p(t)
T |u ≤r}, Sr=W ,p(t)
T ∩∂Br.Then there existρ> andα> such that
inf u∈Sρ
And there exist r> ,r>ρand e∈WT,p(t)such that
sup u∈∂Q
ϕ(u)≤,
whereQ={u+se|u∈RN∩Br,s∈[,r]}.
Proof Firstly, we show that there existsρ> such that infu∈Sρϕ(u) > . LetC be the constant in Proposition . By condition (F), we know that for any positive constant <min{C,
p+TCp+}, there existsδ∈(,) such that
F(t,u)≤|u|p+ ()
for all |u| ≤δ and a.e. t∈[,T]. Let <ρ ≤ Cδ and by Proposition set Sρ ={u∈
WT,p(t)||u|p(t)=ρ}. By Proposition , we get|u(t)| ≤C|u|p(t)=Cρ=δ. Sinceρ< , then it follows from Proposition and () that
ϕ(u) = T
p(t)u
(t)p(t) dt–
T
Ft,u(t)dt
≥
p+ T
u(t)p(t)dt– T
u(t)p+dt
≥
p+u
p+ p(t)–TC
p+up+ p(t)
=
p+–TC
p+
ρp+=α> .
Secondly, we prove that there existr> ,r>ρande∈WT,p(t)such thatsupu∈∂Qϕ(u)≤
. By (F) and (F) there exist constantsC>max{,G} ,η> and a subset of, still denoted by, with||=meas() > such that
μF(t,u) –∇F(t,u),u≤η|u|p– ()
and
F(t,u) > η μ–p–|u|
p+
()
for all|u| ≥Candt∈. Foru∈RN\ {}andt∈, letf(s) =F(t,su) for alls≥C|u|. We deduce from () that
f(s) = s
∇F(t,su),su
≥μ
sF(t,su) –ηs p––|u|p–
= μ
sf(s) –ηs
p––|u|p–,
which yields
g(s) =f(s) –μ
sf(s) +ηs
for alls≥C
|u|. From () we have
f(s) = s
C
|u|
g(r) –ηrp––|u|p– rμ dr+
|u| C
μ
f
C
|u|
sμ ()
for alls≥C
|u|. It follows from () and () that
f(s)≥
| u| C μ f C
|u|
+ η|u| p–
(μ–p–)sμ–p– –
η|u|μ
(μ–p–)Cμ–p– sμ ≥ F
t,C
|u|u
– ηC p–
μ–p–
|u| C
μ
sμ.
Combining this with () yields
F(t,u) =f()≥
F
t,C
|u|u
– ηC p–
μ–p–
| u| C μ ≥
ηCp+–μ μ–p– –
ηCp––μ μ–p–
|u|μ
≥C|u|μ
for all|u| ≥Candt∈, whereC=μ–ηp–(
Cμ–p+ –
Cμ–p–) > . So, notice thatF(t,u)≥, we get
F(t,u)≥C
|u|μ–Cμ=C|u|μ–C ()
for allu∈RNandt∈.
Choosee(t)∈WT,p(t)withe(t)= such thate(t) = for allt∈[,t]\. Therefore, one has
e(t)dt= T
e(t)dt– [,t]\
e(t)dt= ,
which implies that
u,e(t)dt= T
u,e(t)dt– [,t]\
u,e(t)dt= ()
for allu∈RN. LetW,p(t)
T =RN⊕span{e(t)}. Sincedim(W ,p(t)
T ) <∞, all the norms are equiv-alent. For anyv=u+se(t)∈WT,p(t), there exists a positive constantKsuch that
DenotingE=
T
|e(t)|p(t)dt,E=
|e(t)|dt, by (), (), () and (F), we get
ϕ(u+se) = T
p(t)se
(t)p(t) dt–
T
Ft,u+se(t)dt
≤
p– T
se(t)p(t)dt–
Ft,u+se(t)dt
≤
p– T
se(t)p(t)dt–C
u+se(t)μdt+C||
≤
p– T
se(t)p(t)
dt–CKμ
u+se(t)dt
μ
+C||
= p–
T
se(t)p(t)dt–CKμ
|u|+se(t)dt
μ
+C||
≤
p– T
se(t)p(t)dt–CKμ|u|μ||
μ
–CKμsμ|E| μ
+C||.
Therefore, whens> , we have
ϕ(u+se)≤E p–s
p+–C Kμ|E|
μ sμ+C
||.
Sinceμ>p+, there existsr>max{,ρ}such that
ϕ(u+se)≤ for allu∈RNands=r. ()
Moreover, for allu∈RN and ≤s≤r
, we have
ϕ(u+se)≤Er p+ p– –CK
μ|
u|μ||μ +C
||.
This deduces that
ϕ(u+se)≤ when|u|μ≥Er p+
+C||p– CKμ||
μ p–
.
Letu∈RN,|u| ≥, from Proposition , we know that
|u|μT=
T
|u|μdt≥
T
|u|p(t)dt≥ |u|pp–(t).
So, letrsatisfy
rp–≥max
,Er p+
+C||p– CKμ||
μ p–
T
,
then, whenu∈RN,u=|u|
p(t)=r, we obtain
On the other hand, ifs= , by (F), we get
ϕ(u+se) = – T
F(t,u)dt≤ for allu∈RN. ()
Setting Q ={u+se|u∈RN∩Br,s∈[,r]}, by (),() and (), we have
sup u∈∂Q
ϕ(u)≤. ()
The proof of Lemma is completed.
Proof of Theorem By Lemma and Lemma , applying Lemma , thenϕ possesses a critical pointu(t) whose critical valuecsatisfiesc≥α> . ByF, we can see that u(t) is nonconstant. Hence, problem () has at least one nonconstantT-periodic solution in
W,Tp(t).
4 Conclusions
In this work, we have established an existence result for nonconstant periodic solutions of a class of second-order systems withp(t)-Laplacian. Forp(x) is a constantp, it is easy to see that the conditions and conclusion in Theorem are the same as those in Theorem . in []. Thus Theorem generalizes Theorem . in [] and Theorem . in []. Further-more, obviously, conditions (F) and (F) of Theorem are weaker than (A) and (A) of Theorem . in []. Therefore, Theorem extends Theorem . in [].
Funding
This work is supported in part by the National Natural Foundation of China-NSAF (Grant No.11571092, No.11501165) and the Fundamental Research Funds for the Central Universities (2015B19414).
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The authors have contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
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1College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P.R. China.2College of Science,
China Pharmaceutical University, Nanjing, 211198, P.R. China. 3College of Science, Hohai University, Nanjing, 210098, P.R. China.
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