R E S E A R C H
Open Access
Existence and multiplicity of weak
solutions for a nonlinear impulsive
(
q
,
p
)
-Laplacian dynamical system
Xiaoxia Yang
**Correspondence:
[email protected] School of Mathematics and Statistics, Central South University, Changsha, Hunan 410075, P.R. China
Abstract
In this paper, we investigate the existence and multiplicity of nontrivial weak solutions for a class of nonlinear impulsive (q,p)-Laplacian dynamical systems. The key
contributions of this paper lie in (i) Exploiting the least action principle, we deduce that the system we are interested in has at least one weak solution if the potential function has sub-(q,p) growth or (q,p) growth; (ii) Employing a critical point theorem due to Ding (Nonlinear Anal. 25(11):1095-1113, 1995), we derive that the system involved has infinitely many weak solutions provided that the potential function is even.
MSC: 34C25; 58E50
Keywords: (q,p)-Laplacian; existence; multiplicity; nontrivial solution; variational methods
1 Introduction and main results
ForN∈N, let (RN,·,·,| · |) be theN-dimensional Euclidean space. For fixedl,k∈N, set B:={, , . . . ,l}andC:={, , . . . ,k}. Iff :Rn→Ris a smooth function, let∇f stand for the gradient operator. For a smooth functionf :Rn×Rn→R, denote by∇
xf and∇xf
the gradient operator with respect to the first component and the second component, respectively. For a mappingf:R+→R,f(t+) andf(t–) mean the right-hand side limit and
the left-land side limit att, respectively. For functionsf :Rn→Rnandg:R
+→Rn, let
(f(g(t))) =f(g(t+)) –f(g(t–)).
In this paper, we consider a nonlinear system with impulsive effects onHN:=RN×RN for anyp,q> ,λ> ,j∈B, andm∈C,
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
–d(q(u˙(t)))
dt +q(u(t)) =λ∇uF(t,u(t),u(t)), a.e.t∈[,T],
–d(p(u˙(t)))
dt +p(u(t)) =λ∇uF(t,u(t),u(t)), a.e.t∈[,T],
(q(u˙(tj))) =∇Ij(u(tj)), (p(u˙(sm))) =∇Km(u(sm)),
(.)
with the initial condition (u˙(),u˙()) = (u(),u())∈HN and the terminal condition
(u˙(T),u˙(T)) = (, )∈HN, where μ(z) :=|z|μ–z for anyμ> andz∈RN;F:R+× HN →R; (tj)j∈Band (sm)m∈Care impulsive times with =t
=s<s<s<· · ·<sk<sk+=T, and forj∈Bandm∈C,Ij:RN→RandKm:RN→R are continuously differentiable.
For the nonlinear termF: [,T]×HN →R, we assume that
(A) For fixedt∈[,T]andx∈HN,F(·,x)is measurable andF(t,·)is continuously differentiable;
(A) There exista,a∈C(R+;R+)andb∈L([,T];R+)such that F(t,x,x)≤
a
|x| +a
|x| b(t), ∇F(t,x,x)≤
a
|x| +a
|x| b(t), Ij(x)≤a
|x| , ∇Ij(x)≤a
|x| , j∈B, Km(x)≤a
|x| , ∇Km(x)≤a
|x| , m∈C
for all(x,x)∈RN×RN and a.e.t∈[,T].
ForN= ,p=q= ,F(t,x,x) =F(t,x), andIj≡ (j∈B), system (.) reduces to the following second order impulsive differential equation:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
u(t) +u(t) =λ∇uF(t,u(t)), a.e.t∈[, +∞),
u() =u(),
u(T) = ,
(u(tj)) =Ij(u(tj)), j∈B.
(.)
Recently, Chen and Sun [] investigated the following second order impulsive differential equation:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
u(t) +u(t) =λf(t,u(t)), a.e.t∈[, +∞),
u(+) =g(u ()),
u(+∞) = ,
(u(tj)) =Ij(u(tj)), j∈B,
(.)
wheref ∈C(R+×R;R),g,Ij∈C(R;R). In [], the authors not only established the vari-ational structure of equation (.) but also obtained that (.) enjoys three solutions by using an abstract critical point theorem taken from []. More precisely, they obtained the following theorem.
Theoremm A([], Theorem .) Suppose that
(H) g(u),Ij(u)are nondecreasing,andg(u)u≥,Ij(u)u≥for anyu∈R; (H) There exista> ,l∈(, ),b∈L(R+;R+),andc∈L(R+;R+)such that
F(t,u)≤b(t)a+|u|l , f(t,u)≤c(t)|u|l–,
for a.e.t≥andu∈R,whereF(t,u) :=uf(t,s)ds; (H) There existd,m,M> such that
d
M <m +
l
j= me–tj
Ij(s)ds+
m
(H)
M+∞
max|ξ|≤dF(t,ξ)dt
d <
+∞
F(t,me–t)dt
m+ l j=
me–tj
Ij(s)ds+ m
g(s)ds
.
Then,for each
λ∈
m
+ l
j= me–tj
Ij(s)ds+ m
g(s)ds +∞
F(t,me–t)dt
, d
M+∞
max|ξ|≤dF(t,ξ)dt
,
(.)has at least three classical solutions.
Also, Dai and Zhang [] showed by using the least action principle that (.) has at least one solution if the potential function has subquadratic growth and, by taking advantage of the fountain theorem due to [], that (.) has infinitely many solutions if the potential function is even.
To be precise, they obtained the following theorems.
Theoremm B([], Theorem .) Suppose that
(S) (Ij)j∈BandgsatisfyuIj(s)ds≥andug(s)ds≥,u∈R,respectively; (S) There exista> ,α∈(, ),andb∈L(R+;R+)such that
F(t,u)≤b(t)a+|u|α
for a.e.t≥and allu∈R.
Then,forλ> , (.)has at least one classical solution.
Theoremm C([], Theorem .) Besides(S)above,for a.e.t≥and all u∈R,assume that
(S) There existα∈(, )andd∈L–α(R+;R+)such that
F(t,u)≥d(t)|u|α;
(S) There existγ ∈(, )andh,h∈L(R+;R+)such that
f(t,u)≤h(t)|u|γ+h(t);
(S) There existγj>α– ,θ>α– ,andqj,q> ,j∈B,such that
Ij(u)≤qj|u|γj, g(u)≤q|u|θ;
(S) f(t,u),Ij(u),andg(u)are odd aboutu.
Recently, by applying the least action principle and saddle point theorem, [–] inves-tigated the existence of periodic solutions for the following dynamical systems:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
d
dtq(u˙(t)) =∇uF(t,u(t),u(t)), a.e.t∈[,T],
d
dtp(u˙(t)) =∇uF(t,u(t),u(t)), a.e.t∈[,T],
u() –u(T) =u˙() –u˙(T) = ,
u() –u(T) =u˙() –u˙(T) =
(.)
and
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
d
dtq(u˙(t)) +∇uF(t,u(t),u(t)) = , a.e.t∈[,T],
d
dtp(u˙(t)) +∇uF(t,u(t),u(t)) = , a.e.t∈[,T],
u() –u(T) =u˙() –u˙(T) = ,
u() –u(T) =u˙() –u˙(T) = ,
(.)
respectively. Subsequently, by variational approach, Yang and Chen [, ] discussed the existence and multiplicity of periodic solutions for the following two classes of nonlinear (q,p)-Laplacian dynamical systems with impulsive effects:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
d(ρ(t)q(u˙(t)))
dt –ρ(t)λ(u(t)) +∇uF(t,u(t),u(t)) = , a.e.t∈[,T],
d(γ(t)p(u˙(t)))
dt –γ(t)η(u(t)) +∇uF(t,u(t),u(t)) = , a.e.t∈[,T],
u() –u(T) =u˙() –u˙(T) = ,
u() –u(T) =u˙() –u˙(T) = ,
(ρ(tj)q(u˙(tj))) =∇Ij(u(tj)), j∈B,
(γ(sm)p(u˙(sm))) =∇Km(u(sm)), m∈C,
(.)
and
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
d(q(u˙(t)))
dt =∇uF(t,u(t),u(t)), a.e.t∈[,T],
d(p(u˙(t)))
dt =∇uF(t,u(t),u(t)), a.e.t∈[,T],
u() –u(T) =u˙() –u˙(T) = ,
u() –u(T) =u˙() –u˙(T) = ,
(q(u˙(tj))) =∇Ij(u(tj)), j∈B,
(p(u˙(sm))) =∇Km(u(sm)), m∈C,
(.)
respectively, wherep,q,λ,η> andρ,ρ,γ,γ∈C([,T];R+).
Motivated by [, , –], in this paper, we are interested in the existence and multiplic-ity of a nontrivial weak solution for system (.) by using the least action principle and a critical point theorem due to Ding []. To be precise, we obtain the following results.
Theorem . Suppose that (HIK) Forx,x∈RN,
l
j=
Ij(x)≥,
k
m=
(HF) There existα∈[,q),α∈[,p),a> ,andd∈L([,T];R+)such that
F(t,x,x)≤d(t)
a+|x|α+|x|α , ∀(x,x)∈RN×RN.
Then,for eachλ> ,system(.)has at least one weak solution in Xq×Xp,where,for s> , Xs=
u: [,T]→RN|u is absolutely continuous andu˙∈Ls[,T].
Remark . There exist examples satisfying Theorem .. For example, letq= ,p= ,
Ij(x) =
|x|+c ξ
(j∈B), Km(x) =ln
|x|+c ξ
(m∈C), (.) wherec,c,ξ,ξ> , and for allt∈[,T],
F(t,x,x) =sint|x|+cost|x|,
or
F(t,x,x) =tln
+|x| +
et|x
|
+|x|
.
Theorem . In addition to(HIK),we assume that (HF) There exista> andd∈L([,T];R+)such that
F(t,x,x)≤d(t)
a+|x|q+|x|p , ∀(x,x)∈RN×RN.
Then,for each <λ<min{q(D
(q))q,
p(D(p))p}, (.)has at least one weak solution in Xq×Xp,
where
D(s) :=
T–s– + (s– )T
s–s ·(s– )
s–
s
, s=p,q.
Remark . There exist examples satisfying Theorem .. For example, letq= ,p= , andIj,Kmdefined by (.). For allt∈[,T], let
F(t,x,x) =et|x|+sint|x|,
or
F(t,x,x) =t|x|ln
+|x| +
t+ |x|.
Theorem . Along with(HIK)and(HF),for x,x∈RN,j∈B,m∈C,and t∈[,T],
we suppose that
(HIK) There existν≥q,ν≥q,andδ> such that
Ij(x)≤d|x|ν, Km(x)≤d|x|ν, |x| ≤δ;
(HF) There existμ∈(,q),μ∈(,p),d> ,andδ> such that
F(t,x,x)≥d
|x|μ+|x|μ , |x| ≤δ,|x| ≤δ;
(HF) F(t,x,x) =F(t, –x, –x),F(t, , )≡.
Then,for each <λ<min{q(D
(q))q,
p(D(p))p}, (.)has infinitely many weak solutions in
Xq×Xp.
Remark . There exist examples satisfying Theorem .. For example, letq= ,p= , and
Ij(x) =c|x| (j∈B), Km(x) =c|x| (m∈C),
wherec,c> . For allt∈[,T], let
F(t,x,x) =
et+ |x|+
t+ |x|.
If we takeν= .,ν= .,μ= ., andμ= ., it is easy to see that the example satisfies
Theorem ..
2 Variational structure and some preliminaries Foru∈Xswiths=q,p, define
uXs=
T
u(t)sdt+
T
u˙(t)sdt
/s .
Set
us:=
T
u(t)sdt
/s
and u∞:= max
t∈[,T] u(t).
Set X:=Xq×Xp and define the norm(u,u)X =uXq +uXp. Obviously,X is a
reflexive Banach space. Let
C=u: [,T]→RN|uis continuous.
Xsembeds intoCcontinuously and, according to [], Lemma .,
u∞≤D(s)uXs for anyu∈Xs. (.)
Lemma .([], Proposition .) If ukconverges to u weakly,then ukuniformly converges to u on[,T].
Ifu∈Xs, thenuis absolutely continuous, whereasu˙need not be continuous. Hence, it is possible thats(u˙(t)) =s(u˙(t+)) –s(u˙(t–))= , which leads to impulse effects.
Following the idea [], multiplying byv∈Xqon both sides of the first equation in (.) and integrating from toTyields that
T
–d dtu˙(t)
q–
˙
u(t) +u(t)
q–
u(t) –λ∇xF
Sincevis continuous,v(tj–) =v(t+j) =v(tj). Combiningu˙(T) = withu˙() =u()
im-which, together with (.), further leads to
With the two equalities above in hand, we present the notion of weak solutions for (.).
–λ
Definition . shows that the critical point ofϕis the weak solution of system (.). The following lemma plays a crucial role in achieving the critical point ofϕ.
Lemma .([]) Assume thatϕ∈C(E,R)is bounded from below(above)and satisfies
is a critical value ofϕ.
Lemma .([]) Let E be an infinite dimensional Banach space,and letϕ∈C(E,R)with
thenϕpossesses infinitely many nontrivial critical points.
3 Proofs of theorems
Proof of Theorem. It follows from (HIK), (HF), and (.) that
Owing toα∈[,q) andα∈[,p), we readily obtain thatϕ(u)→+∞asuX→ ∞, i.e., ϕsatisfies the coercive condition onX. Soϕis bounded below onX.
Hereinafter, we claim thatϕsatisfies the (PS) condition. If{ϕ(un,un)}is bounded and
ϕ(un,un) → asn→ ∞, then there exists a positive constantDsuch that ϕ(un,un)≤D, ϕ(un,un)≤D, ∀n∈N.
Sinceϕsatisfies a coercive condition onX, we infer thatunXqandunXpis bounded.
Next, in light of the reflexive property ofXs, there exists a subsequence, still denoted by
{un= (un,un)}, such that
unu onXq, unu onXp.
Thus, Lemma . gives that
un→u inC
,T;RN and un→u inC
,T;RN .
Following the argument in [–], we can derive thatun–uX→, whereu= (u,u).
Consequently,ϕsatisfies the (PS) condition. Thus, with the help of Lemma ., we deduce thatϕ has at least one critical point on X. Hence system (.) has at least one solution
onX.
Proof of Theorem. By (HIK), (HF), and (.), it follows that
ϕ(u) =ϕ(u,u)
= q
T
u˙(t)
q dt+
p
T
u˙(t)
p dt–λ
T
Ft,u(t),u(t) dt
+ q
T
u(t)qdt+
p
T
u(t)pdt
+ l
j=
Ij
u(tj) + k
m=
Km
u(sm)
+ qu()
q +
pu() p
≥
qu q Xq+
pu
p Xp–λ
T
d(t)
a+u(t)q+u(t)p dt
≥
qu q Xq+
pu
p
Xp–λu
q ∞
T
d(t)dt
–λup∞
T
d(t)dt–λa T
d(t)dt
≥
qu q Xq+
pu
p Xp–λ
D(q) quqXq
T
d(t)dt
–λD(p) pupXp
T
d(t)dt–λa T
In view ofλ<min{q(D
(q))q,
p(D(p))p}, one hasϕ(u)→+∞asuX→ ∞, that is,ϕsatisfies
the coercive condition onX. Henceϕ is bounded below onX. By carrying out a similar argument to derive Theorem ., we get that system (.) has at least one solution inX.
Proof of Theorem. Keep in mind thatϕand –ϕhave the same critical points. Let= –ϕ. In the sequel, we aim at verifying that all conditions in Lemma . are fulfilled by. In fact, from (HIK) and (HF), we find thatis even and() = . Taking (HIK), (HF),
Thus it follows from (HIK), (HF), (.), and Hölder’s inequality that
Observing thatμ∈(,q) andμ∈(,p), we take sufficiently smallρ∈(,ρ) such that
(u)≥ onBρ∩ ˜Xand(u) > on∂Bρ∩ ˜X. Therefore,satisfies (ϕ) in Lemma ..
Then, according to Lemma .,has infinitely many critical points inXso that (.) has
infinitely many solutions inX.
Competing interests
The author declares that she has no competing interests.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (NO: 61304011) and by the Hunan Provincial Natural Science Foundation of China (NO: 2016JJ3139).
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 4 December 2016 Accepted: 17 March 2017
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