DOI: 10.1515/auom-2016-0004
An. Şt. Univ. Ovidius Constanţa
Vol. 24(1),2016, 83–94
Existence results for a class of Kirchhoff type
systems with Caffarelli-Kohn-Nirenberg
exponents
G.A. Afrouzi∗ , H. Zahmatkesh and S. Shakeri
Abstract
This paper is concerned with the existence of positive solutions for
a class of infinite semipositone kirchhoff type systems with singular
weights. Our aim is to establish the existence of positive solution for
λ large enough. The arguments rely on the method of sub-and supersolutions.
1
Introduction
In this article, we are interested in the existence of positive solutions for the
following Kirchhoff type system
R

1
 −M1 RΩ |x|−ap |∇u|p dx div(|x|−ap |∇u|p−2 ∇u) = λ|x|−(a+1)p+c1 (f (v) − uα ) in Ω,
1
−bq
q
−bq
q−2
−(b+1)q+c
2
−M2 Ω |x|
|∇v| dx div(|x|
|∇v|
∇v) = λ|x|
(g(u) − vβ ) in Ω,

u = v = 0 on ∂Ω,
(1)
where Ω is a bounded smooth domain of RN with 0 ∈ Ω, 1 < p, q < N , 0 ≤
a, b < N p−p , c1 , c2 > 0, α, β ∈ (0, 1), λ is a parameter, f, g : (0, ∞) → (0, ∞)
are C 2 nondecreasing functions and M1 , M2 : Ω → R satisfy the following
condition
(H1) Mi : R0+ → R+ , i = 1, 2, are two continuous and increasing functions
and 0 < mi ≤ Mi (t) ≤ mi,∞ for all t ∈ R0+ , where R0+ := [0, +∞).
Key Words: Nonlocal problems; Singular weights; infinite semipositone systems; Sub
and supersolutions method.
2010 Mathematics Subject Classification: Primary 35J55, Secondary 35J65.
Received: 17.08.2014
Accepted: 29.10.2014.
83
Kirchhoff type systems with Caffarelli-Kohn-Nirenberg exponents
84
System (1) is related to the stationary problem of a model introduced by
Kirchhoff [11]. More precisely, Kirchhoff proposed a model given by the equation
Z L 2 ! 2
∂u P0
∂2u
E
dx ∂ u = 0,
(2)
ρ 2 −
+
∂t
h
2L 0 ∂x ∂x2
where ρ, P0 , h, E are all constants. This equation extends the classical
D’Alembert wave equation. A distinguishing feature of equation (2) is that
R L ∂u 2
E
dx which depends on the
the equations a nonlocal coefficient Ph0 + 2L
0 ∂x
R
2
L
1
∂u dx; hence the equation is no longer a pointwise identity.
average 2L
0 ∂x
Nonlocal problems can be used for modeling, for example, physical and biological systems for which u describes a process which depends on the average
of itself, such as the population density.
On the other hand, elliptic problems involving more general operator, such
as the degenerate quasilinear elliptic operator given by −div(|x|−ap |∇u|p−2 ∇u),
were motivated by the following Caffarelli, Kohn and Nirenberg’s inequality
(see [4, 19]). The study of this type of problems motivated by its various applications, for example, in fluid mechanics, in newtonian fluids, in flow through
porous media and in glaciology (see [3, 7]). So, the study of singular elliptic
problems has more practical meanings. We refer to [9, 16] for additional results on elliptic problems. Let F (h, k) = f (k) − h1α , and G(h, k) = g(h) − k1β .
Then lim(h,k)→(0,0) F (h, k) = −∞ = lim(h,k)→(0,0) G(h, k) and hence we refer
to (1.1) as an infinite semipositone problem. For regular case, that is, when
M1 (t) = M2 (t) ≡ 1, a = b = 0, c1 = p and c2 = q the quasilinear elliptic
equation has been studied by several authors (see [12, 13]). For the single
equation case when M1 (t) = M2 (t) ≡ 1, a = 0, c1 = p = 2, see [17]. In [12],
the authors extended the study of [17], to the corresponding systems, including p-Laplacian. Here we focus on further extending the study in [2, 15] for
infinite semipositone Kirchhoff type systems involving singularity. Our approach is based on the sub- and super-solution method, see [1, 2, 6, 8, 10, 15].
The concepts of sub- and super-solution were introduced by Nagumo [20] in
1937 who proved, using also the shooting method, the existence of at least
one solution for a class of nonlinear Sturm-Liouville problems. In fact, the
premises of the sub- and super-solution method can be traced back to Picard.
He applied, in the early 1880s, the method of successive approximations to argue the existence of solutions for nonlinear elliptic equations that are suitable
perturbations of uniquely solvable linear problems. This is the starting point
of the use of sub- and super-solutions in connection with monotone methods.
Picard’s techniques were applied later by Poincaré [21] in connection with
problems arising in astrophysics.
We make the following assumptions:
Kirchhoff type systems with Caffarelli-Kohn-Nirenberg exponents
85
(H2) f, g : (0, ∞) → (0, ∞) are C 2 nondecreasing functions , f (0), g(0) > 0.
(H3) limt→∞ g(t) = ∞ and for all M > 0
1
f (M [g(t)] q−1 )
= 0.
t→∞
tp−1
lim
Our main result is given by the following theorem.
Theorem 1.1. Under the conditions (H1)-(H3), there exists a positive constant λ∗ such that system (1) has a positive solution when λ ≥ λ∗ .
2
Preliminary results
In this paper, we denote W01,p (Ω, |x|−ap ), the completion of C0∞ (Ω), with
respect to the norm
Z
kukp =
−ap
|x|
p1
|∇u| dx
.
p
Ω
In order to precisely state our main result we first consider the following eigenvalue problem for the p-Laplace operator −∆p u, see [14, 18]:
−div(|x|−ar |∇u|r−2 ∇u) = λ|x|−(a+1)r+c |u|r−2 u in Ω,
(3)
u = 0 on x ∈ ∂Ω.
Let φ1,r ∈ C 1 (Ω) be the eigenfunction corresponding to the first eigenvalue
λ1,r of (3) such that φ1,r > 0 in Ω and kφ1,r k∞ = 1. It can be shown that
∂φ1,r
< 0 on ∂Ω and hence, depending on Ω, there exist positive constants
∂η
, δ, σ such that
|x|−ar |∇φ1,r |r − λ1,r |x|−(a+1)r+c φr1,r ≥ on Ωδ ,
(4)
φ1,r ≥ σ on x ∈ Ω\Ωδ ,
where Ωδ := {x ∈ Ω : d(x, ∂Ω) ≤ δ} and r = p, q.
We will also consider the unique solution e ∈ W01,p (Ω) of the boundary
value problem

 −div(|x|−ap |∇ep |p−2 ∇ep ) = |x|−(a+1)p+c1 in Ω,
(5)
div(|x|−bq |∇eq |p−2 ∇eq ) = |x|−(b+1)q+c2 in Ω,

ep = eq = 0 on x ∈ ∂Ω,
to discuss our result. It is known that er > 0 in Ω and
r = p, q, see [14].
∂er
∂η
< 0 on ∂Ω for
Kirchhoff type systems with Caffarelli-Kohn-Nirenberg exponents
86
We shall establish our existence result via the method of sub and supersolutions. A pair of nonnegative functions (ψ1 , ψ2 ), (z1 , z2 ) are called a subsolution
and a supersolution of system (1) if they satisfy (ψ1 , ψ2 ) = (0, 0) = (z1 , z2 ) on
∂Ω and
Z
Z
−ap
p
M1
|x|
|∇ψ1 | dx
|x|−ap |∇ψ1 |p−2 ∇ψ1 ∇w dx
Ω
Ω
Z
1
−(a+1)p+c1
≤λ
|x|
f (ψ2 ) − α w dx
ψ1
Ω
Z
Z
M2
|x|−bq |∇ψ2 |q dx
|x|−bq |∇ψ2 |q−2 ∇ψ2 ∇w dx
Ω
Ω
Z
1
−(b+1)q+c2
g(ψ1 ) − β w dx
≤λ
|x|
ψ2
Ω
and
Z
Z
M1
|x|
|∇z1 | dx
|x|−ap |∇z1 |p−2 ∇z1 ∇w dx
Ω
Ω
Z
1
−(a+1)p+c1
≥λ
|x|
f (z2 ) − α w dx
z1
Ω
Z
Z
M2
|x|−bq |∇z2 |q dx
|x|−bq |∇z2 |q−2 ∇z2 ∇w dx
Ω
Ω
Z
1
−(b+1)q+c2
≥λ
|x|
g(z1 ) − β w dx
z2
Ω
−ap
p
for all w ∈ W = {w ∈ C0∞ (Ω) : w ≥ 0 for all x ∈ Ω}.
A key role in our arguments will be played by the following auxiliary result.
Its proof is similar to those presented in [6], the reader can consult further the
papers[1, 2, 10].
Lemma 2.1. Assume that M : R0+ → R+ is continuous and increasing, and
there exists m0 > 0 such that M (t) ≥ m0 for all t ∈ R0+ . If the functions
u, v ∈ W01,p (Ω, |x|−ap ) satisfy
Z
Z
−ap
p
M
|x|
|∇u| dx
|x|−ap |∇u|p−2 ∇u · ∇ϕ dx
Ω
Ω
Z
Z
(6)
−ap
p
−ap
p−2
≤M
|x|
|∇v| dx
|x|
|∇v| ∇v · ∇ϕ dx
Ω
Ω
for all ϕ ∈ W01,p (Ω, |x|−ap ), ϕ ≥ 0, then u ≤ v in Ω.
Kirchhoff type systems with Caffarelli-Kohn-Nirenberg exponents
87
From Lemma 2.1 we can establish the basic principle of the sub- and supersolutions method for nonlocal systems. Indeed, we consider the following
nonlocal system

R
−ap
|∇u|p dx div(|x|−ap |∇u|p−2 ∇u) = |x|−(a+1)p+c1 h(x, u, v) in Ω,
 −M1 RΩ |x|
−M2 Ω |x|−bq |∇v|q dx div(|x|−bq |∇v|p−2 ∇v) = |x|−(b+1)q+c2 k(x, u, v) in Ω,

u = v = 0 on x ∈ ∂Ω,
(7)
where Ω is a bounded smooth domain of RN and h, k : Ω × R × R → R satisfy
the following conditions
(HK1) h(x, s, t) and k(x, s, t) are Carathéodory functions and they are bounded
if s, t belong to bounded sets.
(KH2) There exists a function g : R → R being continuous, nondecreasing, with
g(0) = 0, 0 ≤ g(s) ≤ C(1 + |s|min{p,q}−1 ) for some C > 0, and applications s 7→ h(x, s, t) + g(s) and t 7→ k(x, s, t) + g(t) are nondecreasing, for
a.e. x ∈ Ω.
If u, v ∈ L∞ (Ω), with u(x) ≤ v(x) for a.e. x ∈ Ω, we denote by [u, v] the
set {w ∈ L∞ (Ω) : u(x) ≤ w(x) ≤ v(x) for a.e. x ∈ Ω}. Using Lemma 2.1 and
the method as in the proof of Theorem 2.4 of [14] (see also Section 4 of [5]),
we can establish a version of the abstract lower and upper-solution method
for our class of the operators as follows.
Proposition 2.2. Let M1 , M2 : R0+ → R+ be two functions satisfying the
condition (H1). Assume that the functions h, k satisfy the conditions (HK1)
and (HK2). Assume that (u, v), (u, v), are respectively, a weak subsolution
and a weak supersolution of system (7) with u(x) ≤ u(x) and v(x) ≤ v(x)
for a.e. x ∈ Ω. Then there exists a minimal (u∗ , v∗ ) (and, respectively, a
maximal (u∗ , v ∗ )) weak solution for system (7) in the set [u, u] × [v, v]. In
particular, every weak solution (u, v) ∈ [u, u] × [v, v] of system (7) satisfies
u∗ (x) ≤ u(x) ≤ u∗ (x) and v∗ (x) ≤ v(x) ≤ v ∗ (x) for a.e. x ∈ Ω.
3
Proof of the main result
Proof. Choose η > 0 such that η ≤ min{|x|−(a+1)p+c1 , |x|−(b+1)q+c2 } in Ωδ .
1
1
1
1
For fixed r1 ∈ ( p−1+α
, p−1
) and r2 ∈ ( q−1+β
, q−1
), we shall verify that
p
q
1
1
p − 1 + α p−1+α
q − 1 + β q−1+β
r2
r1
p−1
q−1
(ψ1,λ , ψ2,λ ) = λ η
(
)φ1,p , λ η
(
)φ1,q
,
p
q
is a subsolution of (1). Let w ∈ W , Then a calculation shows that
1
1−α
p−1+α
∇ψ1,λ = λr1 η p−1 φ1,p
∇φ1,p
Kirchhoff type systems with Caffarelli-Kohn-Nirenberg exponents
88
1−β
1
q−1+β
∇ψ2,λ = λr2 η q−1 φ1,q
∇φ1,q
and
Z
Z
|x|−ap |∇ψ1,λ |p dx
|x|−ap |∇ψ1,λ |p−2 ∇ψ1,λ ∇w dx
Ω
Ω
Z
Z
(1−α)(p−1)
|x|−ap φ1,pp−1+α |∇φ1,p |p−2 ∇φ1,p ∇w dx
|x|−ap |∇ψ1,λ |p dx
= λr1 (p−1) ηM1
Ω
Ω
Z
Z
|x|−ap |∇φ1,p |p−2 ∇φ1,p ×
|x|−ap |∇ψ1,λ |p dx
= λr1 (p−1) ηM1
Ω
Ω
(1−α)(p−1)
(1−α)(p−1)
× ∇(φ1,pp−1+α w) − (∇φ1,pp−1+α )w dx


Z
p(p−1)
p
(1
−
α)(p
−
1)
|∇φ
|
1,p
−(a+1)p+c
r1 (p−1)
−ap
p−1+α
1
 w dx.
φ1,p
≤λ
ηm1,∞ λ1,p |x|
− |x|
αp
p−1+α
p−1+α
Ω
φ1,p
M1
Similarly,
Z
Z
|x|−bq |∇ψ2,λ |q dx
|x|−bq |∇ψ2,λ |q−2 ∇ψ2,λ ∇w dx
Ω
Ω


Z
q(q−1)
q
(1
−
β)(q
−
1)
|∇φ
|
1,q
−bq
r2 (q−1)
−(b+1)q+c
p−1+β
2
 w dx.
− |x|
φ1,q
≤λ
ηm2,∞ λ1,q |x|
βq
q−1+β
q−1+β
Ω
φ1,q
M2
First we consider the case when x ∈ Ωδ . We have |x|−ap |∇ψ1,p |p ≥ and
|x|−bq |∇ψ1,q |q ≥ on Ωδ . Since r1 (p − 1) ≥ 1 − αr1 and r2 (q − 1) ≥ 1 − βr2 ,
we can find λ̂ > 0 such that
− λr1 (p−1) ηm1,∞ |x|−ap
(1 − α)(p − 1) |∇φ1,p |p
αp
p − 1 + α φ p−1+α
1,p

≤ λ|x|−(a+1)p+c1 −

1
p
1
p−1+α α
[λr1 η p−1 ( p−1+α
)φ1,p
]
p
,
and
− λr2 (q−1) ηm2,∞ |x|−bq
(1 − β)(q − 1) |∇φ1,q |q
βq
q−1+β
q−1+β
φ1,q

≤ λ|x|−(b+1)p+c2 −

1
[λr2 η
1
q−1
q
q−1+β β
]
( q−1+β
)φ1,q
q
,
89
Kirchhoff type systems with Caffarelli-Kohn-Nirenberg exponents
for all x ∈ Ωδ and for all λ ≥ λ̂. Also since r1 (p − 1) < 1 and r2 (q − 1) < 1,
we can choose λ̌ > 0 so that
p(p−1)
λ
r1 (p−1)
−(a+1)p+c1
ηm1,∞ λ1,p |x|
−(a+1)p+c1
p−1+α
φ1,p
≤ λ|x|
−(a+1)p+c1
≤ λ|x|
f (0)
q
1
q−1+β
r
q−1+β
,
f λ 2 η q−1 (
)φ1,q
q
and
q(q−1)
r2 (q−1)
λ
−(b+1)q+c2
ηm2,∞ λ1,q |x|
q−1+β
φ1,q
≤ λ|x|
−(b+1)q+c2
≤ λ|x|
−(b+1)q+c2
g(0)
p
1
p − 1 + α p−1+α
r
g λ 1 η p−1 (
,
)φ1,p
p
for all x ∈ Ωδ and for all λ ≥ λ̌.
Let λ0 = max{λ̂, λ̌}. Hence, for all x ∈ Ωδ and for all λ ≥ λ0 ,
Z
Z
−ap
p
M1
|x|−ap |∇ψ1,λ |p−2 ∇ψ1,λ ∇w dx
|x|
|∇ψ1,λ | dx
Ωδ
Ω
Z
≤λ
|x|−(a+1)p+c1
Ωδ


q
1
1
q−1+β


q−1+β
× f λr2 η q−1
φ1,q
−
α  w dx
p
1
q
p−1+α
p−1+α
r
p−1
1
λ η
( q )φ1,p
!
Z
1
|x|−(a+1)p+c1 f (ψ2,λ ) − α
w dx,
=λ
ψ1,λ
Ωδ
and similarly
Z
Z
−bq
q
M2
|x| |∇ψ2,λ | dx
Z
≤λ
Ωδ
|x|−bq |∇ψ2,λ |q−2 ∇ψ2,λ ∇w dx
!
1
g(ψ1,λ ) − β
w dx.
ψ2,λ
Ωδ
Ω
|x|−(b+1)q+c2
On the other hand, on Ω \ Ωδ , we have φ1,r ≥ σ, for some 0 < σ < 1, and for
r = p, q. Since r1 (p − 1) < 1and r2 (q − 1) < 1 we can find λ̃ > 0 such that
p(p−1)
p−1+α
λr1 (p−1) ηm1,∞ λ1,p |x|−(a+1)p+c1 φ1,p

q

1
q − 1 + β q−1+β
r2 q−1
≤ λ|x|−(a+1)p+c1
f
λ
(
)σ
−
η
q

q

1
1
p
p−1+α
λr1 η p−1 ( p−1+α
)σp
p

α 
,
90
Kirchhoff type systems with Caffarelli-Kohn-Nirenberg exponents
and
q(q−1)
q−1+β
λr2 (q−1) ηm2,∞ λ1,q |x|−(b+1)p+c2 φ1,q


 
p
1
p − 1 + α p−1+α
1


≤ λ|x|−(b+1)p+c2g λr1 η p−1 (
)σp
−
β  ,
q
p


1
λr2 η q−1 ( q−1+β
)σqq−1+β
p
for all x ∈ Ω \ Ωδ and for all λ ≥ λ̃. Hence

r1 (p−1)
λ
Z
λ1,p |x|
ηm1,∞
−(a+1)p+c1
p(p−1)
p−1+α
φ1,p
− |x|
−ap (1
Ω\Ωδ
≤ λr1 (p−1) ηm1,∞

− α)(p − 1) |∇φ1,p |p 
w dx
αp
p−1+α
p−1+α
φ1,p
p(p−1)
Z
p−1+α
λ1,p |x|−(a+1)p+c1 φ1,p
w dx
Ω\Ωδ
Z
λ1,p |x|−(a+1)p+c1
≤λ
Ω\Ωδ


q

1
q−1+β
r2 q−1
q−1+β
η
×
f
λ
σ
−
q

q
Z
|x|
≤λ
−(a+1)p+c1
f (ψ2,λ ) −
Ω\Ωδ
1
λr1 η
1
p−1
p
p−1+α
( p−1+α
)σp
p

α 
 w dx
!
1
α
ψ1,λ
w dx.
and similarly

λ
r2 (q−1)
Z
λ1,q |x|
ηm2,∞
−(b+1)q+c2
q(q−1)
q−1+β
φ1,q
− |x|
Ω\Ωδ
Z
≤λ
|x|
Ω\Ωδ
−(b+1)q+c2
g(ψ1,λ ) −
1
β
ψ2,λ
−bq

(1 − β)(q − 1) |∇φ1,q |q 
w dx
βq
q−1+β
q−1+β
φ1,q
!
w dx.
Let λ∗ = max{λ0 , λ̃}. Hence
Z
Z
−ap
p
M1
|x|
|∇ψ1,λ | dx
|x|−ap |∇ψ1,λ |p−2 ∇ψ1,λ ∇w dx
Ω
Ω
!
Z
1
−(a+1)p+c1
w dx,
≤
|x|
f (ψ2,λ ) − α
ψ1,α
Ω
Kirchhoff type systems with Caffarelli-Kohn-Nirenberg exponents
91
and
Z
−bq
|x|
M2
Ω
Z
|∇ψ2,λ | dx
|x|−bq |∇ψ2,λ |q−2 ∇ψ2,λ ∇w dx
Ω
!
Z
1
−(b+1)p+c2
≤
g(ψ1,λ ) − β
w dx,
|x|
ψ2,λ
Ω
q
i.e., (ψ1,λ , ψ2,λ ) is a subsolution of (1.1) for all λ ≥ λ∗ .
Now, we will prove there exists a N large enough so that
1
(z1 , z2 ) = N ep (x), [λg(N lp )] q−1 eq (x) ,
is a supersolution of (1.1), where er is defined by (2.3) and lr = ker k∞ ; r = p, q.
A calculation shows that:
Z
Z
−ap
p
M1
|x|
|∇z1 | dx
|x|−ap |∇z1 |p−2 ∇z1 ∇w dx
Ω
Ω
Z
≥ m0 N p−1
|x|−ap |∇ep |p−2 ∇ep ∇w dx
ZΩ
p−1
= m0 N
|x|−(a+1)p+c1 w dx.
Ω
By monotonicity condition on f and (H1)-(H3), we can choose N large enough
so that
1
m0 N p−1 ≥ λf [λg(N lp )] q−1 lq
1
≥ λf [λg(N lp )] q−1 eq (x)
= λf (z2 )
1
≥ λ f (z2 ) − α .
z1
Hence
Z
Z
|x|−ap |∇z1 |p dx
|x|−ap |∇z1 |p−2 ∇z1 ∇w dx
Ω
Ω
Z
1
−(a+1)p+c1
≥λ
|x|
f (z2 ) − α w dx.
z1
Ω
M1
Kirchhoff type systems with Caffarelli-Kohn-Nirenberg exponents
92
Next, we have
Z
Z
|x|−bq |∇z2 |q dx
|x|−bq |∇z2 |q−2 ∇z2 ∇w dx
Ω
Ω
Z
≥ m0 λg(N lp )
|x|−bq |∇eq |q−2 ∇eq ∇w dx
Ω
Z
= m0 λg(N lp )
|x|−(b+1)q+c2 w dx
Ω
Z
−(b+1)q+c2
≥λ
g(N ep (x))w dx
|x|
ZΩ
=λ
|x|−(b+1)q+c2 g(z1 )w dx
Ω
Z
1
−(b+1)q+c2
≥λ
|x|
g(z1 ) − β w dx
z2
Ω
M2
This relations show that (z1 , z2 ) is a supersolution of (1.1). Moreover, zi ≥ ψi,λ
for N large, i = 1, 2. Thus, by Proposition 2.2 there exists a positive solution
(u, v) of (1.1) such that (ψ1,λ , ψ2,λ ) ≤ (u, v) ≤ (z1 , z2 ). This completes the
proof of Theorem 1.1.
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G.A. Afrouzi,
Department of Mathematics,
Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran.
Email: afrouzi@umz.ac.ir
H. Zahmatkesh,
Department of Mathematics,
Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran.
Email: h.zahmatkesh@stu.umz.ac.ir
S. Shakeri,
Department of Mathematics,
Ayatollah Amoli Branch,
Islamic Azad University, amol, Iran.
Email: s.shakeri@iauamol.ac.ir