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Stability Results of Positive Weak Solution for a Class of Chemically Reacting Systems

Salah A. Khafagy1, A. Ezzat Mohamed2∗

1Department of Mathematics, Faculty of Science, Al-Azhar University (11884), Cairo, Egypt
salahabdelnaby.211@azhar.edu.eg

2Department of Mathematics, Faculty of Science, Fayoum University (63514), Fayoum, Egypt
aam35@fayoum.edu.eg

∗Correspondence: aam35@fayoum.edu.eg

Abstract. This paper aims to study the existence and non-existence results of positive weak solutionto the quasilinear elliptic system:
−∆pu = λa(x)[f (u, v)− 1

uα
], x ∈ Ω,

−∆qv = λb(x)[g(u, v)− 1
vβ

], x ∈ Ω,

u = 0 = v, x ∈ ∂Ω,

where ∆rw = div(|∇w |r−2∇w) is the r-Laplacian (r = p, q), r > 1, α, β ∈ (0, 1), Ω is a boundeddomain in RN(N > 1) with smooth boundary ∂Ω and λ is a positive parameter. Here f , g are C1

increasing functions such that f , g : R+ × R+ → R+; f (υ1, υ2) > 0, g(υ1, υ2) > 0 for υ1, υ2 > 0.With C1 sign-changing functions a(x), b(x) that perhaps have negative values nearby the boundary.We establish our results via the sub-supersolution method. In addition, we study the stability andinstability results of positive weak solution with different choices of f and g.

1. Introduction
This paper aims to study the existence and non-existence results of positive weak solution tothe quasilinear elliptic system:

−∆pu = λa(x)[f (u, v)− 1
uα ], x ∈ Ω,

−∆qv = λb(x)[g(u, v)− 1
vβ

], x ∈ Ω,

u = 0 = v , x ∈ ∂Ω,

(1)
where ∆rw = div(|∇w |r−2∇w) is the r-Laplacian (r = p, q), r > 1, α, β ∈ (0, 1), Ω is a boundeddomain in RN(N > 1) with smooth boundary ∂Ω and λ is a positive parameter. Here f , g are C1increasing functions such that f , g : R+ × R+ → R+; f (υ1, υ2) > 0, g(υ1, υ2) > 0 for υ1, υ2 > 0.With C1 sign-changing functions a(x), b(x) that perhaps have negative values nearby the boundary.

Received: 17 Jul 2024.
Key words and phrases. Positive weak solution; Sub-Supersolution method; Stability.1

https://adac.ee
https://doi.org/10.28924/ada/ma.5.1


Eur. J. Math. Anal. 10.28924/ada/ma.5.1 2General evolutionary problems are defined as follows:

ut = η∆pu + λa(x)[f (u, v)− 1

uα ], x ∈ Ω,

vt = δ∆qv + λb(x)[g(u, v)− 1
vβ

], x ∈ Ω,

u = 0 = v , x ∈ ∂Ω,

(2)
have stationary counterpart of systems of singular equations like (1), such that η and δ are positiveparameters. System (2) is an inspiration from major applications in chemically reacting systems,where the activator chemical substance’s density is denoted by u, while an inhibitor is denotedby v . The slow and fast diffusion of u and v , respectively, are turned into a small η and large
δ (see [1]). Furthermore, systems like (1) appear in many contexts in Engineering and Biology.It presents a simple model where u, v denote the density of two diffusing biological species fordescribing the interaction between these two species.Recently, similar problems have been discussed in [2–5]. The authors in [6] investigated thepositive weak solution of the system:−∆u = λ[f (u)− 1

uα ], x ∈ Ω,

u = 0, x ∈ ∂Ω,
(3)

where f ∈ C2(R+), f ′ > 0, f (0) ≥ 0, lim
ε→∞

f (ε)
ε = ∞ and Ω ⊂ RN(N ≥ 1). When N = 1, theyused the quadrature method to discuss the multiplicity and uniqueness results, while for N > 1they established their existence results using the sub-supersolution method. In [7], it was discussedthe existence of positive weak solution to the non-linear system:


−∆pu = λa(x)[f (v)− 1

uα ], x ∈ Ω,

−∆qv = λb(x)[g(u)− 1
vβ

], x ∈ Ω,

u = 0 = v , x ∈ ∂Ω,

(4)

where ∆rw = div(|∇w |r−2∇w) is the r-Laplacian (r = p, q), r > 1. Here f , g are C1 increasingfunctions such that f , g : R+ → R+; f (ω) > 0, g(ω) > 0 for ω > 0 and lim
ω→∞

f (Mg(ω)
1
q−1 )

ωp−1 =

0 ∀ M > 0. With C1 sign-changing functions a(x), b(x) that perhaps have negative valuesnearby the boundary. See [8], where system (4) studied by some authors when p = q = 2.Also, we studied in [9] the existence and non-existence results of positive weak solution of (1) incase p = q = 2, where f , g are C1 increasing functions, lim
ω→∞

f (ω,Mg(ω,ω))
ω = 0 ∀ M > 0 and

lim
ω→∞

g(ω,ω)
ω = 0. With C1 sign-changing functions a(x), b(x) that perhaps have negative valuesnearby the boundary.

https://doi.org/10.28924/ada/ma.5.1


Eur. J. Math. Anal. 10.28924/ada/ma.5.1 3Our first aim of this paper is to study system (1) as an extension of system (4) with C1 increasingfunctions f , g satisfying
lim
ξ→∞

f (ξ,M[g(ξ, ξ)]
1
q−1 )

ξp−1
= 0 ∀M > 0, lim

ξ→∞

g(ξ, ξ)

ξq−1
= 0.

On the other side, many authors have an interest in studying the stability and instability ofpositive solution to semiposiotne [10–12], linear [13], semilinear [14–17] and fractional [19,20] sys-tems, they are used in several applications such as Fluid mechanics, Newtonian fluids, Populationdynamics, Reaction-diffusion problems, Glaciology, etc.; see [20–22].
Shivaji and Brown in [11] discussed the stability properties of positive solution for the system:−∆u = λf (u), x ∈ Ω,

u = 0, x ∈ ∂Ω,
(5)

they proved that every positive solution of (5) is unstable when f (0) ≤ 0 and f ′′ ≥ 0. See [12], whereTertikas proved the non-monotone case. Maya and Shivaji in [16] overcame the non-monotone casethrough re-formulating f as a combination of a linear and monotone function. Simon and Karatsongave a direct proof of the result (see [14]). In summary, if f (0) ≥ 0 (≤ 0) and f ′′ < 0 (> 0), thenevery positive solution of (5) is stable (unstable). Also in [9], we studied the stability and instabilityproperties of system (1) in case p = q = 2, under certain conditions such that every weak solution isstable near the boundary; otherwise, it is unstable. In [23], some authors investigated the stabilityof non-negative weak solution for the nonlinear system:−∆pu = λf (x, u), x ∈ Ω,

Bu = 0, x ∈ ∂Ω,
(6)

where f : Ω× [0,∞)→ R be a continuous function. They discussed (6) when f (x, u) = w(x)f (u),where w(x) is a continuous weight function. They showed that every positive solution is unstable(stable) if f (x,u)
up−1 is strictly increasing (decreasing) function.

Our second aim of this paper is to extend these results to (1) with different choices of f , g. Forfurther stability and instability results on elliptic systems (see [10,17,18,24–26]).Let λ1,r > 0, r = p, q, be the principal eigenvalue of the following eigenvalue problem toaccurately state our existence results:−∆rϕ = λ|ϕ|r−2ϕ, x ∈ Ω,

ϕ = 0, x ∈ ∂Ω,
(7)

https://doi.org/10.28924/ada/ma.5.1


Eur. J. Math. Anal. 10.28924/ada/ma.5.1 4where ϕ1,r be the corresponding eigenfunction satisfying ϕ1,r (x) > 0 in Ω with ‖ϕ1,r‖∞ = 1.Suppose µ, δ,m > 0 be such that
r

sr

(
1−

r s

s + 1

)
|∇ϕ1,r |r ≥ m, x ∈ Ω̄δ, (8)

µ ≤ ϕ1,r ≤ 1, x ∈ Ω− Ω̄δ, (9)for s = α, β and sr = (s + 1)r−1, where Ω̄δ := {x ∈ Ω | d(x, ∂Ω) ≤ δ}. By Hopf’s lemma, we findthis available since ϕ1,r = 0 while |∇ϕ1,r | 6= 0 on ∂Ω. Furthermore, we suppose er ∈ W 1,r
0 (Ω) bethe unique solution of the problem: −∆rer = 1, x ∈ Ω,

er = 0, x ∈ ∂Ω,
(10)

where ∂
∂n is the outer normal derivative, er > 0 in Ω and ∂er

∂n < 0 on ∂Ω (see [27]). To be morespecific, we will split our results into two cases:
• Case(I): when x ∈ Ω̄δ; assume a(x), b(x) < 0 with

a0, a0, b0, b0 > 0 : −a0 ≤ a(x) ≤ −a0, −b0 ≤ b(x) ≤ −b0.

• Case(II): when x ∈ Ω− Ω̄δ; assume a(x), b(x) > 0 with
a1, a1, b1, b1 > 0 : a1 ≤ a(x) ≤ a1, b1 ≤ b(x) ≤ b1.

2. Existence and non-existence results
In this section, the results of the existence and non-existence are established by using thesub-supersolution method.

Definition 2.1. A pair of non-negative functions (u, v) is called a positive weak solution of (1) such
that (u, v) ∈ W 1,p

0 (Ω)×W 1,q
0 (Ω) if they satisfy∫

Ω

|∇u|p−2∇u · ∇ζ dx = λ

∫
Ω

a(x)[f (u, v)−
1

uα
]ζ dx,∫

Ω

|∇v |q−2∇v · ∇ζ dx = λ

∫
Ω

b(x)[g(u, v)−
1

vβ
]ζ dx,

∀ ζ ∈ W := {ζ ∈ C∞0 (Ω) | ζ ≥ 0, x ∈ Ω}.

Definition 2.2. A pair of non-negative functions (ψ1, ψ2) and (z1, z2) are called a positive weak
subsolution and supersolution of (1), respectively, such that (ψ1, ψ2), (z1, z2) ∈ W 1,p

0 (Ω)×W 1,q
0 (Ω)

if they satisfy ∫
Ω

|∇ψ1|p−2∇ψ1 · ∇ζ dx ≤ λ
∫

Ω

a(x)[f (ψ1, ψ2)−
1

ψα1
]ζ dx,∫

Ω

|∇ψ2|q−2∇ψ2 · ∇ζ dx ≤ λ
∫

Ω

b(x)[g(ψ1, ψ2)−
1

ψβ2
]ζ dx,

https://doi.org/10.28924/ada/ma.5.1


Eur. J. Math. Anal. 10.28924/ada/ma.5.1 5

and ∫
Ω

|∇z1|p−2∇z1 · ∇ζ dx ≥ λ
∫

Ω

a(x)[f (z1, z2)−
1

zα1
]ζ dx,∫

Ω

|∇z2|q−2∇z2 · ∇ζ dx ≥ λ
∫

Ω

b(x)[g(z1, z2)−
1

zβ2
]ζ dx,

∀ ζ ∈ W := {ζ ∈ C∞0 (Ω) | ζ ≥ 0, x ∈ Ω}.

Now, we state our results as follows:
Lemma 2.1. (see [2]): Let (ψ1, ψ2) and (z1, z2) be a subsolution and supersolution of (1), respec-
tively, with ψ1 ≤ z1 and ψ2 ≤ z2. Therefore, system (1) has a solution (u, v) with ψ1 ≤ u ≤ z1

and ψ2 ≤ v ≤ z2.

Our assumptions are as follows:
(S1) f , g : R+ × R+ → R+ are C1 increasing functions such that f (υ1, υ2) > 0, g(υ1, υ2) > 0 for
υ1, υ2 > 0 and lim

υ1,υ2→∞
f (υ1, υ2) = lim

υ1,υ2→∞
g(υ1, υ2) =∞,

(S2) lim
ξ→∞

f (ξ,M[g(ξ, ξ)]
1
q−1 )

ξp−1
= 0 ∀M > 0 and lim

ξ→∞

g(ξ, ξ)

ξq−1
= 0,

(S3) Let εo > 0 such that:
(i)

N = f

(
µε

1
p−1
o

po
,
µε

1
q−1
o

qo

)
−
(

po

µε
1
p−1
o

)α
> 0,

and
M = g

(
µε

1
p−1
o

po
,
µε

1
q−1
o

qo

)
−
(

qo

µε
1
q−1
o

)β
> 0,

(ii) f (ε
1
p−1
o ,ε

1
q−1
o )

m ≤ min
{

pαo αp

λ1,p pε
α
p−1
o

,
Nαpa1
λ1,p pa0

,
qβo βqb0

λ1,q ε
β
q−1
o pa0

,
Mβqb1
λ1,q pa0

},
(iii) g(ε

1
p−1
o ,ε

1
q−1
o )

m ≤ min
{

qβo βq

λ1,q qε
β
q−1
o

,
Nαpa1
λ1,p qb0

,
pαo αpa0

λ1,p ε
α
p−1
o qb0

,
Mβqb1
λ1,q qb0

},
with po = p

p−1 , qo = q
q−1 , αp = (α+ 1)p−1 and βq = (β + 1)q−1.

(S4) There exist f0, g0 > 0 where f (υ1, υ2) ≤ f0υ
γ1
1 υ

κ1
2 and g(υ1, υ2) ≤ g0υ

κ2
1 υ

γ2
2 such that

γ1, γ2, κ1, κ2 are positive parameters, γ1,γ2 ∈ (0, 1) and κ2 + γ2 < min{1, 1
κ1
}.To be more specific we consider λo(εo) and λo(εo) by the following

λo = min

{
mεo

pa0f (ε
1
p−1
o , ε

1
q−1
o )

,
mεo

qb0g(ε
1
p−1
o , ε

1
q−1
o )

}
,

and
λo = max

{
ε
α+p−1
p−1
o λ1,p

pαoαpa0

,
ε
β+q−1
q−1
o λ1,q

qβoβqb0

,
εo λ1,p

Nαpa1

,
εo λ1,q

Mβqb1

}
.

https://doi.org/10.28924/ada/ma.5.1


Eur. J. Math. Anal. 10.28924/ada/ma.5.1 6

Example 2.1. Assume f (υ1, υ2) = [υk1
2 + (υ1υ2)l1 − 1], g(υ1, υ2) = [υk2

1 + (υ1υ2)
l2
2 − 1] where

k1, k2, l1, l2 are positive parameters. Thus, f , g clearly satisfy (S1) and (S2) if max{k2, l2} k1
q−1 <

p − 1, max{k2, l2} < q − 1 and (max{k2, l2} 1
q−1 + 1)l1 < p − 1 such that

lim
ξ→∞

f (ξ,M[g(ξ, ξ)]
1
q−1 )

ξp−1
= 0 ∀M > 0, lim

ξ→∞

g(ξ, ξ)

ξq−1
= 0,

and lim
ξ→∞

g(ξ, ξ) =∞. We can take εo > 0 small enough that f , g satisfy (S3).

Remark 2.1. Note that (S3) implies λo < λo .

Here, we can establish our existence results.
Theorem 2.1. Suppose (S1)-(S3) hold, hence (1) has a positive weak solution for every λ ∈
[λo(εo), λo(εo)].

Proof. We shall verify that
(ψ1, ψ2) =

(
ε

1
p−1
o ϕ

po
α+1

1,p

po
,
ε

1
q−1
o ϕ

qo
β+1

1,q

qo

)
is a positive weak subsolution of (1). Then

∇ψ1 =
ε

1
p−1
o ∇ϕ

po
α+1

1,p

po

=
ε

1
p−1
o

1 + α
ϕ
po−1−α
α+1

1,p ∇ϕ1,p,and ∫
Ω

|∇ψ1|p−2∇ψ1 · ∇ζ dx

=
εo
αp

∫
Ω

ϕ
(1− αp

α+1
)

1,p |∇ϕ1,p|p−2∇ϕ1,p · ∇ζdx

=
εo
αp

∫
Ω

|∇ϕ1,p|p−2∇ϕ1,p

[
∇
(
ϕ

(1− αp
α+1

)

1,p · ζ
)
−
(

1−
αp

α+ 1

)
ϕ
−αp
α+1

1,p ∇ϕ1,p · ζ
]
dx

=
εo
αp

∫
Ω

[
|∇ϕ1,p|p−2∇ϕ1,p∇

(
ϕ

(1− αp
α+1

)

1,p · ζ
)
−
(

1−
αp

α+ 1

)
ϕ
−αp
α+1

1,p |∇ϕ1,p|p · ζ
]
dx

=
εo
αp

∫
Ω

[
λ1,p ϕ

p
α+1

1,p −
(

1−
αp

α+ 1

)
ϕ
−αp
α+1

1,p |∇ϕ1,p|p
]
ζ dx,

then, ∫
Ω

|∇ψ1|p−2∇ψ1 · ∇ζ dx =
εo
αp

∫
Ω

[
λ1,p ϕ

p
α+1

1,p −
(

1−
αp

α+ 1

)
ϕ
−αp
α+1

1,p |∇ϕ1,p|p
]
ζ dx. (11)

Similarly,∫
Ω

|∇ψ2|q−2∇ψ2 · ∇ζ dx =
εo
βq

∫
Ω

[
λ1,q ϕ

q
β+1

1,q −
(

1−
βq

β + 1

)
ϕ
−βq
β+1

1,q |∇ϕ1,q|q
]
ζ dx.

https://doi.org/10.28924/ada/ma.5.1


Eur. J. Math. Anal. 10.28924/ada/ma.5.1 7Case(I): when x ∈ Ω̄δ . Put s = α, r = p in (8), we have
−p
αp

(1−
αp

α+ 1
)|∇ϕ1,p|p ≤ −m.

Hence,
−εo ϕ

−αp
α+1

1,p

αp
(1−

αp

α+ 1
)|∇ϕ1,p|p ≤

−mεo
p

,

and since λ ≤ λo , then λ ≤ mεo

pa0f (ε
1
p−1
o , ε

1
q−1
o )

. Hence,
−mεo
p
≤ −λa0f (ε

1
p−1
o , ε

1
q−1
o ) ≤ −λa0f

(
ε

1
p−1
o ϕ

po
α+1

1,p

po
,
ε

1
q−1
o ϕ

qo
β+1

1,q

qo

)
,

so,
−εo ϕ

−αp
α+1

1,p

αp

(
1−

αp

α+ 1

)
|∇ϕ1,p|p ≤ −λa0f

(
ε

1
p−1
o ϕ

po
α+1

1,p

po
,
ε

1
q−1
o ϕ

qo
β+1

1,q

qo

)
, (12)

and since λ ≥ λo , then λ ≥ ε
α+p−1
p−1
o λ1,p

pαoαpa0

. Hence,
εo λ1,p ϕ

p
α+1

1,p

αp
≤
εo λ1,p

αp
≤

λa0

( ε
1
p−1
o
po

)α
≤

λa0(
ε

1
p−1
o ϕ

po
α+1
1,p

po

)α . (13)
Using (12) and (13) in (11), we see that

∫
Ω

|∇ψ1|p−2∇ψ1 · ∇ζ dx ≤
∫

Ω

λa0 ζ(
ε

1
p−1
o ϕ

po
α+1
1,p

po

)α dx − ∫
Ω

λa0f

(
ε

1
p−1
o ϕ

po
α+1

1,p

po
,
ε

1
q−1
o ϕ

qo
β+1

1,q

qo

)
ζdx

= −λ
∫

Ω

a0[f (ψ1, ψ2)−
1

ψα1
]ζ dx

≤ λ
∫

Ω

a(x)[f (ψ1, ψ2)−
1

ψα1
]ζ dx.

Case(II): when x ∈ Ω− Ω̄δ; µ ≤ ϕ1,p ≤ 1. Since λ ≥ λo , then
εo λ1,p

Nαpa1

≤ λ.

https://doi.org/10.28924/ada/ma.5.1


Eur. J. Math. Anal. 10.28924/ada/ma.5.1 8Hence,∫
Ω

|∇ψ1|p−2∇ψ1 · ∇ζ dx =
εo
αp

∫
Ω

[
λ1,p φ

p
α+1

1,p −
(

1−
αp

α+ 1

)
φ
−αp
α+1

1,p |∇φ1,p|p
]
ζdx

≤
εo
αp

∫
Ω

λ1,p φ
p

α+1

1,p ζ dx

≤ λ
∫

Ω

a1Nζ dx

= λ

∫
Ω

a1

[
f

(
µε

1
p−1
o

po
,
µε

1
q−1
o

qo

)
−
(

po

µε
1
p−1
o

)α]
ζ dx

≤ λ
∫

Ω

a1

[
f

(
ε

1
p−1
o φ

po
α+1

1,p

po
,
ε

1
q−1
o φ

qo
β+1

1,q

qo

)
−

1(
ε

1
p−1
o φ

po
α+1
1,p

po

)α]ζ dx
= λ

∫
Ω

a1[f (ψ1, ψ2)−
1

ψα1
]ζ dx

≤ λ
∫

Ω

a(x)[f (ψ1, ψ2)−
1

ψα1
]ζ dx.

Similarly, we can get also∫
Ω

|∇ψ2|q−2∇ψ2 · ∇ζ dx ≤ λ
∫

Ω

b(x)[g(ψ1, ψ2)−
1

ψβ2
]ζ dx.

Thus, (ψ1, ψ2) be a positive weak subsolution of (1).
On the other side, we will construct a positive weak supersolution of (1). Suppose

(z1, z2) =
(
c ep(x), [λµbg(cµp, cµp)]

1
q−1 eq(x)

)
where µa = ‖a(x)‖∞, µb = ‖b(x)‖∞ and µr = ‖er (x)‖∞ for r = p, q. Now by (S2), we can takec large enough such that

cp−1 ≥ λµaf
(
cµp, [λµbg(cµp, cµp)]

1
q−1µq

)
,

then, ∫
Ω

|∇z1|p−2∇z1 · ∇ζ dx = cp−1

∫
Ω

|∇ep|p−2∇ep · ∇ζ dx

= cp−1

∫
Ω

ζ dx

≥
∫

Ω

λµaf
(
cµp, [λµbg(cµp, cµp)]

1
q−1µq

)
· ζ dx

≥ λ
∫

Ω

a(x)f
(
c ep(x), [λµbg(cµp, cµp)]

1
q−1 eq(x)

)
· ζ dx

= λ

∫
Ω

a(x)f (z1, z2) · ζ dx

https://doi.org/10.28924/ada/ma.5.1


Eur. J. Math. Anal. 10.28924/ada/ma.5.1 9

≥ λ
∫

Ω

a(x)[f (z1, z2)−
1

zα1
]ζ dx.

Also, by (S2) we can take cµp ≥ µq[λµbg(cµp, cµp)]
1
q−1 . Then∫

Ω

|∇z2|q−2∇z2 · ∇ζ dx = λ

∫
Ω

µb g(cµp, cµp)|∇eq|q−2∇eq · ∇ζ dx

= λ

∫
Ω

µb g(cµp, cµp) · ζ dx

≥ λ
∫

Ω

b(x)g
(
c ep(x), [λµbg(cµp, cµp)]

1
q−1 eq(x)

)
· ζ dx

≥ λ
∫

Ω

b(x)g(z1, z2) · ζ dx

≥ λ
∫

Ω

b(x)[g(z1, z2)−
1

zβ2
]ζ dx.

Thus, (z1, z2) be a positive weak supersolution of (1) for c large with ψ1 ≤ z1 and ψ2 ≤ z2. Thus,there exists a positive weak solution (u, v) of (1) such that ψ1 ≤ u ≤ z1 and ψ2 ≤ v ≤ z2. �

Theorem 2.2. Let (S4) holds with (p − 1 − γ1)(q − 1 − γ2) = κ1κ2 and pκ1 = q(p − 1 − γ1).
Hence (1) has no positive weak solution if λ ∈ (λmax , λmin), where λmax = max

{λ1,p

−2t ,
λ1,q

−2t

}
and

λmin = min
{λ1,p

2s ,
λ1,q

2s

}
with t = min{f0a0, g0b0} and s = max{f0a1, g0b1}.

Proof. Let (u, v) be a positive weak solution of (1). Proof’s idea is that a contradiction will beobtained in the end. Multiplying the 1st and 2nd equation of (1) by u, v , respectively. ApplyingYoung’s inequality, so ∫
Ω

|∇u|pdx ≤ λ
∫
Ω

f0a(x)
(up
µ1

+
vq

µ2

)
dx, (14)

with µ1 = p
1+γ1

> 1 and µ2 = p
p−1−γ1

> 1. Similarly, we have∫
Ω

|∇v |qdx ≤ λ
∫
Ω

g0b(x)
(up
ϑ1

+
vq

ϑ2

)
dx, (15)

with ϑ1 = q
q−1−γ2

> 1 and ϑ2 = q
1+γ2

> 1. Note that
λ1,p

∫
Ω

updx ≤
∫
Ω

|∇u|pdx, λ1,q

∫
Ω

vqdx ≤
∫
Ω

|∇v |qdx. (16)
Combining (14)-(16), we obtain
λ1,p

∫
Ω

updx + λ1,q

∫
Ω

vqdx ≤ λ
[ ∫

Ω

(
f0a(x)

µ1
+
g0b(x)

ϑ1

)
updx +

∫
Ω

(
f0a(x)

µ2
+
g0b(x)

ϑ2

)
vqdx

]
.

(17)

https://doi.org/10.28924/ada/ma.5.1


Eur. J. Math. Anal. 10.28924/ada/ma.5.1 10Case(I): when x ∈ Ω̄δ; a(x) ≤ −a0, b(x) ≤ −b0, hence
(λ1,p + 2λt)

∫
Ω

updx + (λ1,q + 2λt)

∫
Ω

vqdx ≤ 0, (18)
where t = min{f0a0, g0b0}, that is a contradiction when λ > λmax .Case(II): when x ∈ Ω− Ω̄δ; a(x) ≤ a1, b(x) ≤ b1, hence

(λ1,p − 2λs)

∫
Ω

updx + (λ1,q − 2λs)

∫
Ω

vqdx ≤ 0, (19)
where s = max{f0a1, g0b1}, that is a contradiction when λ < λmin. �

3. Stability and instability results
Now, we study the stability and instability results of positive weak solution for (1) with differentchoices of f and g (see [28,29]).
Suppose (u, v) be any positive weak solution of (1), hence the linearized system associated with(1) is defined as follows:

−(p − 1)div(|∇u|p−2∇ϕ)− λa(x)
[(
fu + α

uα+1

)
ϕ+ fvψ

]
= µϕ, x ∈ Ω,

−(q − 1)div(|∇v |q−2∇ψ)− λb(x)
[
guϕ+

(
gv + β

vβ+1

)
ψ
]

= µψ, x ∈ Ω,

ϕ = 0 = ψ, x ∈ ∂Ω,

(20)
where subscripts refer to the partial derivative of f or g (see [30]). Let µ1 be the first eigenvalueand (ϕ1, ψ1) be the corresponding eigenfunction of (20) such that ϕ1, ψ1 > 0 in Ω.
Definition 3.1. We say (u, v) is a stable solution of (1) if all eigenvalues of (20) are strictly positive,
which can be implied if the first eigenvalue µ1 > 0. Otherwise (u, v) is unstable.

Our assumptions are as follows:
(T1): For u, v > 0, the functions fv , gu are positive.
(T2): For every v > 0, the function (f (u, v)− u−α

)
/up−1 is strictly increasing at u.

(T3): For every u > 0, the function (g(u, v)− v−β
)
/vq−1 is strictly increasing at v .

Theorem 3.1. Suppose that (T1)-(T3) are satisfied, hence every positive weak solution of (1) is
stable in Ω̄δ and unstable in Ω− Ω̄δ .

Proof. Let (uo , vo) be any positive weak solution of (1). Multiplying the 1st and 2nd equation of(1) by (p − 1)ϕ1, (q − 1)ψ1, respectively and integrating over Ω, so
− (p − 1)

∫
Ω

ϕ1(x)div(|∇uo |p−2∇uo)dx = (p − 1)λ

∫
Ω

ϕ1(x)a(x)
[
f (uo , vo)−

1

uoα
]
dx, (21)

https://doi.org/10.28924/ada/ma.5.1


Eur. J. Math. Anal. 10.28924/ada/ma.5.1 11and
− (q − 1)

∫
Ω

ψ1(x)div(|∇vo |q−2∇vo)dx = (q − 1)λ

∫
Ω

ψ1(x)b(x)
[
g(uo , vo)−

1

voβ
]
dx. (22)

Similarly, multiplying the 1st and 2nd equation of (20) by −uo , −vo , respectively and integratingover Ω, so
−µ1

∫
Ω

uoϕ1(x)dx =(p − 1)

∫
Ω

uo div(|∇uo |p−2∇ϕ1)dx

+ λ

∫
Ω

ϕ1(x)a(x)
[
uo fu +

α

uoα
]
dx

+ λ

∫
Ω

ψ1(x)a(x)fvuo dx,

(23)

and
−µ1

∫
Ω

voψ1(x)dx =(q − 1)

∫
Ω

vo div(|∇vo |q−2∇ψ1)dx

+ λ

∫
Ω

ϕ1(x)b(x)
[
vogv +

β

voβ
]
dx

+ λ

∫
Ω

ψ1(x)b(x)guvo dx.

(24)

Combining (21) to (24), we get
− (p − 1)

∫
Ω

[
ϕ1(x)div(|∇uo |p−2∇uo)− uo div(|∇uo |p−2∇ϕ1)

]
dx

− (q − 1)

∫
Ω

[
ψ1(x)div(|∇vo |q−2∇vo)− vo div(|∇vo |q−2∇ψ1)

]
dx

+ λ

∫
Ω

ϕ1(x)a(x)
[
uo fu +

α

uoα
]
dx + λ

∫
Ω

ψ1(x)b(x)
[
vogv +

β

voβ
]dx

− (p − 1)λ

∫
Ω

ϕ1(x)a(x)
[
f (uo , vo)−

1

uoα
]
dx

− (q − 1)λ

∫
Ω

ψ1(x)b(x)[g(uo , vo)−
1

voβ
]dx

+ λ

∫
Ω

a(x)ψ1(x)fvuo dx + λ

∫
Ω

b(x)ϕ1(x)guvo dx

= −µ1

∫
Ω

[uoϕ1(x) + voψ1(x)]dx.

(25)

Using Green’s first identity, then∫
Ω

uo div(|∇uo |p−2∇ϕ1)dx =

∫
Ω

ϕ1(x)div(|∇uo |p−2∇uo)dx, (26)
and ∫

Ω

vo div(|∇vo |q−2∇ψ1)dx =

∫
Ω

ψ1(x)div(|∇vo |q−2∇vo)dx. (27)

https://doi.org/10.28924/ada/ma.5.1


Eur. J. Math. Anal. 10.28924/ada/ma.5.1 12By using (26) and (27) in (25), then
λ

∫
Ω

ϕ1(x)a(x)

[
uo fu − (p − 1)f (uo , vo) +

α+ p − 1

uoα

]
dx

+ λ

∫
Ω

ψ1(x)b(x)

[
vogv − (q − 1)g(uo , vo) +

β + q − 1

voβ

]
dx

+ λ

∫
Ω

ψ1(x)a(x)fvuodx + λ

∫
Ω

ϕ1(x)b(x)guvodx

= −µ1

∫
Ω

[uoϕ1(x) + voψ1(x)]dx.

(28)

Also, since (f (uo , vo)− u−αo
)
/uo

p−1 is strictly increasing at uo ∀vo > 0, then for uo , vo > 0

uo fu − (p − 1)f (uo , vo) + (α+ p − 1)uo
−α

uop
> 0, (29)

and since (g(uo , vo)− vo−β
)
/vo

q−1 is strictly increasing at vo ∀uo > 0, then for uo , vo > 0

vogv − (q − 1)g(uo , vo) + (β + q − 1)vo
−β

voq
> 0. (30)

Case(I): when x ∈ Ω̄δ; a(x), b(x) < 0. Thus substituting (29)-(30) in (28), so
− µ1

∫
Ω

[uoϕ1(x) + voψ1(x)]dx < 0, (31)
then, µ1 > 0 and the solution is stable.Case(II): when x ∈ Ω− Ω̄δ; a(x), b(x) > 0. Thus substituting (29)-(30) in (28), so

− µ1

∫
Ω

[uoϕ1(x) + voψ1(x)]dx > 0, (32)
then µ1 < 0 and the solution is unstable. �

Remark 3.1. By replacing assumptions (T1)-(T3) with next:
(L1): For u, v > 0, the functions fv , gu are negative.
(L2): For every v > 0, the function

(
f (u, v)− u−α

)
/up−1 is strictly decreasing at u.

(L3): For every u > 0, the function
(
g(u, v)− v−β

)
/vq−1 is strictly decreasing at v .

We deduce the following:
Corollary 3.1. Suppose that (L1)-(L3) are satisfied, hence every positive weak solution of (1) is
unstable in Ω̄δ and stable in Ω− Ω̄δ .

Proof. In the same way that Theorem 3.1 is proved, the proof procedure is similar. �

Remark 3.2. As shown in the preceding theorem and corollary, the stability results of positive
weak solution are dependent on the domain, in addition to the provided assumptions.

Remark 3.3. If p = q = 2 in system (1), we get the results of the system which have been studied
in [9].

https://doi.org/10.28924/ada/ma.5.1


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	1. Introduction
	2. Existence and non-existence results
	3. Stability and instability results
	References

