Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 69, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS FOR SINGULAR (p, q)-KIRCHHOFF
TYPE SYSTEMS WITH MULTIPLE PARAMETERS
SAYYED HASHEM RASOULI
Abstract. This article concerns the existence of positive solutions for singular
(p, q)-Kirchhoff type systems with multiple parameters. Our approach is based
on the method of sub- and super-solutions.
1. Introduction
In this article, we are interested in the existence of positive solutions for the
singular (p, q)-Kirchhoff type system
Z
h 1
1 i
−M1
|∇u|p dx ∆p u = a(x) α1 f (v) − η + β1 h(u) − η , x ∈ Ω,
u
u
Ω
h Z
1 i
1
−M2
|∇v|q dx ∆q v = b(x) α2 g(u) − θ + β2 k(v) − θ , x ∈ Ω,
v
v
Ω
u = v = 0, x ∈ ∂Ω,
(1.1)
where Mi : R+ → R+ , i = 1, 2 are two continuous and increasing functions such
that Mi (t) ≥ mi > 0 for all t ∈ R+ , ∆r z = div(|∇z|r−2 ∇z), for r > 1 denotes the
r-Laplacian operator, α1 , α2 , β1 , β2 are positive parameters, Ω is a bounded domain
in Rn , n ≥ 1 with sufficiently smooth boundary and η, θ ∈ (0, 1). Here a(x), b(x) ∈
C(Ω) are weight functions such that a(x) ≥ a0 > 0, b(x) ≥ b0 > 0 for all x ∈ Ω,
f, g, h, k ∈ C([0, ∞) are nondecreasing functions and f (0), g(0),
R h(0), k(0) > 0.
Problem (1.1) is called nonlocal because of the term −M ( Ω |∇u|r dx) which implies that the first two equations in (1.1) are no longer pointwise equalities. This
phenomenon causes some mathematical difficulties which makes the study of such
a class of problem particularly interesting. Also, such a problem has physical motivation. Moreover, system (1.1) is related to the stationary version of the Kirchhoff
equation
Z L
∂2u
E
∂u
∂ 2 u P0
+
| |2 dx
=0
(1.2)
ρ 2 −
∂t
h
2L 0 ∂x
∂x2
presented by Kirchhoff [13]. This equation extends the classical d’Alembert’s wave
equation by considering the effects of the changes in the length of the strings during
the vibrations. The parameters in (1.2) have the following meanings: L is the length
2010 Mathematics Subject Classification. 35J55, 35J65.
Key words and phrases. (p, q)-Kirchhoff type system; subsolution; supersolution.
c
2016
Texas State University.
Submitted Mar 18, 2015. Published March 14, 2016.
1
2
S. H. RASOULI
EJDE-2016/69
of the string, h is the area of cross section, E is the Youngs modulus of the material,
ρ is the mass density, and P0 is the initial tension.
When an elastic string with fixed ends is subjected to transverse vibrations, its
length varies with the time: this introduces changes of the tension in the string. This
induced Kirchhoff to propose a nonlinear correction of the classical D’Alembert’s
equation. Later on, Woinowsky-Krieger (Nash-Modeer) incorporated this correction in the classical Euler-Bernoulli equation for the beam (plate) with hinged ends.
See, for example, [5, 6] and the references therein.
Nonlocal problems also appear in other fields: for example, biological systems
where u and v describe a process which depends on the average of itself (for instance,
population density). See [3, 4, 11, 19, 20] and the references therein. In recent years,
problems involving Kirchhoff type operators have been studied in many papers, we
refer to [1, 9, 17, 12, 22, 21, 8], in which the authors have used different methods
to prove the existence of solutions.
Let F (s, t) = (f (t)− s1η )+(h(s)− s1η ), and G(s, t) = (g(s)− t1η )+(k(t)− t1η ). Then
lim(s,t)→(0,0) F (s, t) = −∞ = lim(s,t)→(0,0) G(s, t), and hence we refer to (1.1) as an
infinite semipositone problem. See [2], where the authors studied the corresponding non-singular finite system when M1 (t) = M2 (t) ≡ 1, and a(x) = b(x) ≡ 1.
It is well documented that the study of positive solutions to such semipositone
problems is mathematically very challenging [7], [18]. In this paper, we study
the even more challenging semipositone system with lim(s,t)→(0,0) F (s, t) = −∞ =
lim(s,t)→(0,0) G(s, t). We do not need the boundedness of the Kirchhoff functions
M1 , M2 , as in [10]. Using the sub and supersolutions techniques, we prove the existence of positive solutions to the system (1.1). To our best knowledge, this is an
interesting and new research topic for singular (p, q)-Kirchhoff type systems. One
can refer to [14, 15, 16] for some recent existence results of infinite semipositone
systems.
To precisely state our existence result we consider the eigenvalue problem
−∆r φ = λ |φ|r−2 φ,
φ = 0,
x ∈ Ω,
x ∈ ∂Ω.
(1.3)
Let φ1,r be the eigenfunction corresponding to the first eigenvalue λ1,r of (1.3) such
that φ1,r (x) > 0 in Ω, and kφ1,r k∞ = 1 for r = p, q. Let m, σ, δ > 0 be such that
r
r−1+s
σ ≤ φ1,r
≤ 1,
x ∈ Ω − Ωδ ,
r
|∇φ1,r | ≥ m,
x ∈ Ωδ ,
(1.4)
(1.5)
for r = p, q, and s = η, θ, where Ωδ := {x ∈ Ω|d(x, ∂Ω) ≤ δ}. (This is possible
since |∇φ1,r |r 6= 0 on ∂Ω while φ1,r = 0 on ∂Ω for r = p, q). We will also consider
the unique solution ζr ∈ W01,r (Ω) of the boundary-value problem
−∆r ζr = 1,
ζr = 0,
It is known that ζr > 0 in Ω and
∂ζr
∂n
x ∈ Ω,
x ∈ ∂Ω.
< 0 on ∂Ω.
EJDE-2016/69
(p, q)-KIRCHHOFF TYPE SYSTEMS
3
2. Existence of solutions
In this section, we shall establish our existence result via the method of subsuper-solution [2]. For the system
Z
−M1
|∇u|p dx ∆p u = h1 (x, u, v), x ∈ Ω,
Ω
Z
−M2
|∇v|q dx ∆q v = h2 (x, u, v), x ∈ Ω,
Ω
u = v = 0,
x ∈ ∂Ω,
1,p
a pair of functions (ψ1 , ψ2 ) ∈ W ∩ C(Ω) × W 1,q ∩ C(Ω) and (z1 , z2 ) ∈ W 1,p ∩
C(Ω) × W 1,q ∩ C(Ω) are called a subsolution and supersolution if they satisfy
(ψ1 , ψ2 ) = (0, 0) = (z1 , z2 ) on ∂Ω,
Z
Z
Z
p
p−2
M1
|∇ψ1 | dx
|∇ψ1 | ∇ψ1 · ∇w dx ≤
h1 (x, ψ1 , ψ2 )w dx,
Ω
Ω
Ω
Z
Z
Z
|∇ψ2 |q−2 ∇ψ2 · ∇w dx ≤
h2 (x, ψ1 , ψ2 )w dx
M2
|∇ψ2 |q dx
Ω
Ω
Ω
and
Z
Z
|∇z1 |p dx
|∇z1 |p−2 ∇z1 · ∇w dx ≥
h1 (x, z1 , z2 )w dx,
Ω
Ω
Ω
Z
Z
Z
M2
|∇z2 |q dx
|∇z2 |q−2 ∇z2 · ∇w dx ≥
h2 (x, z1 , z2 )w dx,
M1
Z
Ω
for all w ∈ W = {w ∈
Ω
∞
C0 (Ω)|w
Ω
≥ 0, x ∈ Ω}. Then the following result holds.
Lemma 2.1 ([8]). Suppose there exist sub- and super-solutions (ψ1 , ψ2 ) and (z1 , z2 )
respectively of (1.1) such that (ψ1 , ψ2 ) ≤ (z1 , z2 ). Then (1.1) has a solution (u, v)
such that (u, v) ∈ [(ψ1 , ψ2 ), (z1 , z2 )].
We use the following hypotheses:
(H1) f, g, h, k ∈ C([0, ∞) are nondecreasing functions such that f (0) > 0, g(0) >
0, h(0) > 0, k(0) > 0,
lim f (s) = lim h(s) = lim g(s) = lim k(s) = +∞,
s→+∞
s→+∞
s→+∞
s→+∞
h(s)
k(s)
lim
= lim q−1 = 0.
s→+∞ sp−1
s→+∞ s
(H2) for all A > 0,
1
f (Ag(s) q−1 )
= 0.
s→∞
sp−1+η
Our main result read as follows.
lim
Theorem 2.2. Let (H1)–(H2) hold. Then (1.1) has a large positive solution (u, v)
provided α1 + β1 and α2 + β2 are large.
Proof. Let γ0 = min{f (0), h(0)} > 0 and
n 1
b0 q−1
q−1+θ γ1 = min f (α2 + β2 )r2 (
)
(
)σ ,
m2
q
1
a0 p−1
p − 1 + η o
h (α1 + β1 )r1 (
)
(
)σ .
m1
p
4
S. H. RASOULI
EJDE-2016/69
1
1
1
1
For fixed r1 ∈ ( p−1+η
, p−1
) and r2 ∈ ( q−1+θ
, q−1
), we shall verify that with
p
1
(α1 + β1 )r1 (p − 1 + η) a0 p−1
p−1+η
φ1,p
,
(
)
p
m1
q
1
(α2 + β2 )r2 (q − 1 + θ) b0 q−1
q−1+θ
ψ2 =
,
(
)
φ1,q
q
m2
ψ1 =
and (ψ1 , ψ2 )is a sub-solution of (1.1). Let w ∈ W . Then a calculation shows that
∇ψ1 = (α1 + β1 )r1 (
1−η
1
a0 p−1
p−1+η
φ1,p
∇φ1,p ,
)
m1
and we have
Z
Z
M1
|∇ψ1 |p dx
|∇ψ1 |p−2 ∇ψ1 · ∇w dx
Ω
Ω
Z
a0 (α1 + β1 )(p−1)r1
M1
|∇ψ1 |p dx
=
m1
Ω
Z
ηp
1− p−1+η
p−2
×
φ1,p
|∇φ1,p | ∇φ1,p ∇wdx
Ω
=
Z
a0 (α1 + β1 )(p−1)r1
M1
|∇ψ1 |p dx
m1
Ω
Z
ηp
1− p−1+η
1− ηp
p−2
|∇φ1,p | ∇φ1,p ∇(φ1,p
w) − w∇(φ1,p p−1+η ) dx
Ω
Z
a0 (α1 + β1 )(p−1)r1
=
M1
|∇ψ1 |p dx
m1
Ω
o
nZ ηp
1− ηp
p− p−1+η
− |∇φ1,p |p−2 ∇φ1,p ∇(φ1,p p−1+η ) wdx
λ1,p φ1,p
×
Ω
Z
a0 (α1 + β1 )(p−1)r1
=
M1
|∇ψ1 |p dx
m1
Ω
nZ o
ηp
p− p−1+η
− ηp ηp
×
λ1,p φ1,p
)φ1,pp−1+η wdx
− |∇φ1,p |p (1 −
p−1+η
Ω
nZ o
p(p−1)
(1 − η)(p − 1) |∇φ1,p |p p−1+η
≤ a0 (α1 + β1 )(p−1)r1
λ1,p φ1,p
−
wdx .
ηp
p−1+η
p−1+η
Ω
φ1,p
Similarly
Z
Z
M2
|∇ψ2 | dx
|∇ψ2 |q−2 ∇ψ2 · ∇w dx
Ω
Ω
nZ o
q(q−1)
(1 − θ)(q − 1) |∇φ1,q |q q−1+θ
(q−1)r2
≤ b0 (α2 + β2 )
λ1,q φ1,q
−
wdx .
θq
q−1+θ
q−1+θ
Ω
φ1,q
q
Thus (ψ1 , ψ2 ) is a sub-solution if
p(p−1)
(1 − η)(p − 1) |∇φ1,p |p p−1+η
a0 (α1 + β1 )r1 λ1,p φ1,p
−
ηp
p−1+θ
p−1+η
φ1,p
h 1 1 i
≤ a(x) α1 f (ψ2 ) − η + β1 h(ψ1 ) − η .
ψ1
ψ1
EJDE-2016/69
(p, q)-KIRCHHOFF TYPE SYSTEMS
5
and
q(q−1)
(1 − θ)(q − 1) |∇φ1,q |q q−1+θ
b0 (α2 + β2 )r2 λ1,q φ1,q
−
θq
q−1+θ
q−1+θ
φ1,q
h 1 i
1 ≤ b(x) α2 g(ψ1 ) − θ + β2 k(ψ2 ) − θ .
ψ2
ψ2
First we consider the case when x ∈ Ωδ . Since 1 − (p − 1)r1 − r1 η < 0, for
α1 + β1 1, we have
− (α1 + β1 )(p−1)r1
(1 − η)(p − 1) |∇φ1,p |p
pη
p−1+η
φ p−1+η
1,p
1
h
≤ (α1 + β1 ) − p
p−1+η
1
a0 p−1 p−1+η
(α1 + β1 )r1 ( m
)
( p )φ1,p
1
i
η .
Also in Ω̄δ (in fact in Ω), since (p − 1)r1 < 1, if α1 + β1 1 and α2 + β2 1,
p(p−1)
p−1+η
(α1 + β1 )(p−1)r1 λ1,p φ1,p
≤ (α1 + β1 )γ0
= α1 γ0 + β1 )γ0
q
1
q − 1 + θ q−1+θ
b0 q−1
)φ1,q
)
(
≤ α1 f (α2 + β2 )r2 (
m2
q
p
1
a
p
−
1 + η p−1+η
0 p−1
+ β1 h (α1 + β1 )r1 (
)
(
)φ1,p
.
m1
p
It follows that in Ωδ for α1 + β1 1 and α2 + β2 1, we have
h
p(p−1)
(1 − η)(p − 1) |∇φ1,p |p i
p−1+η
−
a0 (α1 + β1 )r1 λ1,p φ1,p
ηp
p−1+η
p−1+η
φ1,p
h
p(p−1)
(1 − η)(p − 1) |∇φ1,p |p i
p−1+η
− (α1 + β1 )r1
= a0 (α1 + β1 )r1 λ1,p φ1,p
ηp
p−1+η
p−1+η
φ1,p
h
q
1
b0 q−1
q − 1 + θ q−1+θ
≤ a(x) α1 f (α2 + β2 )r2 (
(
)
)φ1,q
+ β1 h (α1
m2
q
p
1
a0 p−1 p − 1 + η p−1+η
)
(
)φ1,p
+ β 1 ) r1 (
m1
p
i
(α1 + β1 )
−
p
1
p−1+η η
a0 p−1
(α1 + β1 )r1 ( m
)
( p−1+η
)φ1,p
p
1
h 1 i
1 = a(x) α1 f (ψ2 ) − η + β1 h(ψ1 ) − η .
ψ1
ψ1
r
r−1+s
On the other hand, on Ω − Ωδ , we have σ ≤ φ1,r
≤ 1, for r = p, q and s = η, θ.
Also, since (p − 1)r1 < 1, for α1 + β1 1 and α2 + β2 1,
h
p(p−1)
(1 − η)(p − 1) |∇φ1,p |p i
p−1+η
a0 (α1 + β1 )r1 λ1,p φ1,p
−
ηp
p−1+η
p−1+η
φ1,p
p(p−1)
p−1+η
≤ a(x)(α1 + β1 )r1 λ1,p φ1,p
6
S. H. RASOULI
≤ a(x)(α1 + β1 ) γ1 − EJDE-2016/69
1
1
η
a0 p−1 p−1+η
(α1 + β1 )r1 ( m
)
( p )σ
1
h 1
η
= a(x) α1 γ1 − 1
a0 p−1 p−1+η
(α1 + β1 )r1 ( m
)
( p )σ
1
i
1
+ β1 γ 1 − η
1
a0 p−1
(α1 + β1 )r1 ( m
)
( p−1+η
)σ
p
1
n h 1
b0 q−1
q−1+θ ≤ a(x) α1 f (α2 + β2 )r2 (
)
(
)σ
m2
q
i
1
η
−
1
a0 p−1 p−1+η
)
(α1 + β1 )r1 ( m
( p )σ
1
h 1
p−1+η a0 p−1
(
)σ
)
+ β1 h (α1 + β1 )r1 (
m1
p
io
1
η
−
1
a0 p−1 p−1+η
( p )σ
(α1 + β1 )r1 ( m
)
1
n h q
1
q − 1 + θ q−1+θ
b0 q−1
≤ a(x) α1 f (α2 + β2 )r2 (
(
)φ1,q
)
m2
q
i
1
−
p
1
p−1+η η
a0 p−1
( p−1+η
(α1 + β1 )r1 ( m
)
)φ1,p
p
1
h p
1
p − 1 + η p−1+η
a0 p−1
(
)φ1,p
+ β1 h (α1 + β1 )r1 (
)
m1
p
io
1
−
η
p
1
p−1+η
a0 p−1 p−1+η
(α1 + β1 )r1 ( m
)
( p )φ1,p
1
h 1 1 i
= a(x) α1 f (ψ2 ) − η + β1 h(ψ1 ) − η .
ψ1
ψ1
Hence, if α1 + β1 1 and α2 + β2 1, we see that
Z
Z
p
M1
|∇ψ1 | dx
|∇ψ1 |p−2 ∇ψ1 · ∇w dx
Ω
Ω
Z
h 1 i
1 ≤
a(x) α1 f (ψ2 ) − η + β1 h(ψ1 ) − η w dx.
ψ1
ψ1
Ω
Similarly, for α1 + β1 1 and α2 + β2 1, we get
Z
Z
M2
|∇ψ2 |q dx
|∇ψ2 |q−2 ∇ψ2 · ∇w dx
Ω
Z Ω h 1 i
1 ≤
b(x) α2 g(ψ1 ) − θ + β2 k(ψ2 ) − θ w dx.
ψ2
ψ2
Ω
This means that, (ψ1 , ψ2 ) is a positive subsolution of (1.1).
Now, we construct a supersolution (z1 , z2 ) ≥ (ψ1 , ψ2 ). By (H1) and (H2) we can
choose C 1 so that
1
1
kbk∞ (α2 +β2 ) q−1
q−1 kζ k
α
f
[
]
[g(Ckζ
k
)]
+ β1 h Ckζp k∞
1
p
∞
q
∞
m2
m1
≥
.
kak∞
C p−1
EJDE-2016/69
(p, q)-KIRCHHOFF TYPE SYSTEMS
7
Let
1
h kbk (α + β ) i q−1
1
∞
2
2
(z1 , z2 ) = Cζp ,
[g(Ckζp k∞ )] q−1 ζq .
m2
We shall show that (z1 , z2 ) is a supersolution of (1.1). Then
Z
Z
M1
|∇z1 |p dx
|∇z1 |p−2 ∇z1 · ∇w dx
Ω
Ω
Z
Z
p−1
p
=C
M1
|∇z1 | dx
|∇ζp |p−2 ∇ζp · ∇w dx
Ω
Ω
Z
Z
= C p−1 M1
|∇z1 |p dx
w dx
Ω
Ω
Z
≥ m1 C p−1
w dx
Ω
Z h
1
h kbk (α + β ) i q−1
1
∞
2
2
[g(Ckζp k∞ )] q−1 kζq k∞
≥ kak∞
α1 f
m2
Ω
i
+ β1 h Ckζp k∞ wdx
Z
h 1
1 i
≥
a(x) α1 f (z2 ) − η + β1 h(z1 ) − η w dx.
z1
z1
Ω
Again by (H2) for C large enough we have
1
h kbk (α + β ) i q−1
1
∞
2
2
[g(Ckζp k∞ )] q−1 kζq k∞ .
g(Ckζ(x)k∞ ) ≥ k
m2
Hence
Z
Z
M2
|∇z2 |q dx
|∇z2 |q−2 ∇z2 · ∇w dx
Ω
Ω
Z
Z
kbk∞ (α2 + β2 )
=
M2
|∇z2 |q dx
g(Ckζp k∞ )w dx
m2
Ω
Ω
Z
n
o
≥
b(x) α2 g(Ckζp k∞ ) + β2 g(Ckζp k∞ ) w dx
ZΩ
n
≥
b(x) α2 g(Ckζp k∞ )
Ω
1
h kbk (α + β ) i q−1
o
1
∞
2
2
[g(Ckζp k∞ )] q−1 kζq k∞ w dx
+ β2 k
m2
Z
≥
b(x) α2 g(z1 ) + β2 k(z2 ) w dx
ZΩ
h 1 i
1
≥
b(x) α2 g(z1 ) − θ + β2 k(z2 ) − θ w dx.
z2
z2
Ω
i.e., (z1 , z2 ) is a supersolution of (1.1). Furthermore, C can be chosen large enough
so that (z1 , z2 ) ≥ (ψ1 , ψ2 ). Thus, there exists a positive solution (u, v) of (1.1) such
that (ψ1 , ψ2 ) ≤ (u, v) ≤ (z1 , z2 ). This completes the proof.
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Sayyed Hashem Rasouli
Department of Mathematics, Faculty of Basic Sciences, Babol University of Technology, Babol, Iran
E-mail address: s.h.rasouli@nit.ac.ir