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Stability of Positive Weak Solution for Generalized Weighted p-Fisher-Kolmogoroff Nonlinear

Stationary-State Problem

Salah A. Khafagy∗, Hassan M. Serag
Department of Mathematics, Faculty of Science,

Al-Azhar University, Nasr City (11884), Cairo, Egypt
salahabdelnaby.211@azhar.edu.eg, serraghm@yahoo.com
∗Correspondence: salahabdelnaby.211@azhar.edu.eg

Abstract. In the present paper, we investigate the stability results of positive weak solution for thegeneralized Fisher–Kolmogoroff nonlinear stationary-state problem involving weighted p-Laplacianoperator −d∆P,pu = ka(x)u[ν−υu] in Ω, Bu = 0 on ∂Ω, where ∆P,p with p > 1 and P = P (x)is a weight function, denotes the weighted p-Laplacian defined by ∆P,pu ≡ div[P (x)|∇u|p−2∇u],the continuous function a(x) : Ω → R satisfies either a(x) > 0 or a(x) < 0 for all x ∈ Ω,d,k,νand υ are positive parameters and Ω ⊂ RN is a bounded domain with smooth boundary Bu =
δh(x)u + (1 −δ) ∂u

∂n
where δ ∈ [0, 1], h : ∂Ω → R+ with h = 1 when δ = 1.

1. Introduction:
In this paper we study the stability results of positive weak solution for the generalized weighted

p-Fisher–Kolmogoroff nonlinear stationary-state problem
−d∆P,pu = ka(x)f (u) = ka(x)u[ν −υu] in Ω,

Bu = 0 on ∂Ω,
} (1.1)

where ∆P,p with p > 1 and P = P (x) is a weight function, denotes the weighted p-Laplaciandefined by ∆P,pu ≡ div[P (x)|∇u|p−2∇u] (see for details [6]), the continuous function a(x) : Ω → Rsatisfies either a(x) > 0 or a(x) < 0 for all x ∈ Ω,d,k,ν,ν and υ are positive parameter and
Ω ⊂ RN is a bounded domain with smooth boundary Bu = δh(x)u + (1 − δ)∂u

∂n
where δ ∈ [0, 1],

h : ∂Ω → R+ with h = 1 when δ = 1. System (1.1) is the generalized weighted p-Fisher–Kolmogoroff nonlinear stationary-state problem [21], where d is the diffusion coefficient, k is theis the linear reproduction rate and u is the population density. Situations where d is space-dependent are arising in more and more modelling situations of biomedical importance from diffusionof genetically engineered organisms in heterogeneous environments to the effect of white and grey
Received: 12 Sep 2021.
Key words and phrases. stability; weak solution; p-Laplacian.1

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https://doi.org/10.28924/ada/ma.2.8


Eur. J. Math. Anal. 10.28924/ada/ma.2.8 2
matter in the growth and spread of brain tumours. Problem (1.1) arises from the population biologyof one species.Systems of type (1.1) have received considerable attention in the last decade (see, e.g., [18,19,24]and the references therein). It has been shown that for some certian values of ν,υ, system (1.1)has a rich mathematical structure. In [8, 23] the system (1.1) is considered under the hypothesis
P (x) = (k/d) = 1,p = 2 and f (u) = u.This corresponds to the Emden-Fowler stationary-stateproblem of polytropic index of order one. While in [9, 19], system (1.1) is considered under thehypothesis P (x) = (k/d) = 1,p = 2 and f (u) = u − u2,where u is the population denistyof degree two.This corresponds to the Logestic nonlinear stationary-state problem. Due to theappearance of weighted p-Laplacian operator in (1.1) and the particular cases; the extensions arechallenging and nontrivial.Many authors are interested in the study of stability and instability of nonnegative solutions oflinear [2] , semilinear (see [10,26]), semiposiotne (see [3,25]), nonlinear (see [1,16]) and singular (see[17]) systems, due to the great number of applications in reaction-diffusion problems, in autocatalyticreaction, in temperature on plasma, population dynamics, etc.; see [4, 23] and references therein.Also, in the recent past, many authors devoted their attention to study the weighted p-Laplaciannonlinear systems (see [11, 12, 14, 15]).Tertikas in [25] have been proved the stability and instability results of positive solutions for thesemilinear system

−∆u = λf (u) in Ω, Bu = 0 on ∂Ω,
under various choices of the function f . In [3], the authors have been studied the uniqueness andstability of nonnegative solutions for classes of nonlinear elliptic Dirichlet problems in a ball, whenthe nonlinearity is monotone, negative at the origin, and either concave or convex. In the case
P (x) = a(x) = 1, p = 2 and a function λf (u) instead of λuα + uβ, system (1.1) have been studiedby several authors (see [5, 7, 20]).Khafagy in [13] have been studied the stability and instability of positive weak solution for thenonlinear system

−∆P,pu + a(x)|u|p−2u = λb(x)uα in Ω,
Bu = 0 on ∂Ω.

} (1.2)
where 0 < α < p − 1. He proved that if 0 < α < p − 1 and b(x) > 0(< 0) for all x ∈ Ω, thenevery positive weak solution u of (1.2) is linearly stable (unstable) respectively.
Definition 1.1. We recall that, if u be any positive weak solution of (1.1), then the linearizedequation of (1.1) about u is given by

−(p− 1)div[P (x)|∇u|p−2∇φ] − (k/d)a(x)[ν − 2υu]φ = µφ,x ∈ Ω,
Bφ = 0, x ∈ ∂Ω,

} (1.3)
where µ is the eigenvalue corresponding to the eigenfunction φ.

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Eur. J. Math. Anal. 10.28924/ada/ma.2.8 3
Definition 1.2. [3] A solution u of (1.1) is called stable solution if all eigenvalues of (1.3) arestrictly positive, which can be implied if the principal eigenvalue µ1 > 0. Otherwise u unstable.

2. Main Results
The main goal of this section is to prove the stability and instability of the positive weak solution

u of (1.1). Our main results are formulate in the following theorems.
Theorem 2.1. If α + 1 < p < β + 1 and a(x) > 0 for all x ∈ Ω, then every positive weak solution
of ( 1.1) is linearly stable.

Proof. Let u0 be any positive weak solution of (1.1), then the linearized equation bout u0 is
−(p− 1)div[P (x)|∇u0|p−2∇φ] − (k/d)a(x)[ν − 2υu0]φ = µφ, x ∈ Ω

Bφ = 0, x ∈ ∂Ω.

} (2.1)
Let µ1 be the first eigenvalue of (2.1) and let ψ(x) ≥ 0 be the corresponding eigenfunction.Multiplying (1.1) by ψ and integrating over Ω, we have

−
∫
Ω

ψdiv[P (x)|∇u0|p−2∇u0]dx = (k/d)
∫
Ω

a(x)[νu0 −υu20 ]ψdx. (2.2)
The first term of the L.H.S. of (2.2) may be written in the form∫

Ω

ψdiv[P (x)|∇u0|p−2∇u0]dx =
∫
Ω

ψ∇u0∇[P (x)|∇u0|p−2]dx

+

∫
Ω

ψ[P (x)|∇u0|p−2]div(∇u0)dx.

Applying Green’s first identity, we have∫
Ω

ψdiv[P (x)|∇u0|p−2∇u0]dx =
∫
Ω

ψ∇u0∇[P (x)|∇u0|p−2]dx

−
∫
Ω

∇[ψ(P (x)|∇u0|p−2)∇u0dx

+

∫
∂Ω

ψ[P (x)|∇u0|p−2]
∂u0
∂n

ds,

= −
∫
Ω

∇ψ[P (x)|∇u0|p−2]∇u0dx

+

∫
∂Ω

ψ[P (x)|∇u0|p−2]
∂u0
∂n

ds. (2.3)

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Eur. J. Math. Anal. 10.28924/ada/ma.2.8 4
From (2.3) in (2.2), we have

(k/d)

∫
Ω

a(x)[νu0 −υu20 ]ψdx =
∫
Ω

∇ψ[P (x)|∇u0|p−2]∇u0dx

−
∫
∂Ω

ψ[P (x)|∇u0|p−2]
∂u0
∂n

ds

+

∫
Ω

a(x)ψ|u0|p−2u0]dx. (2.4)
Also, Multiplying (2.1) by (−u0) and integrating over Ω, we have

−µ1
∫
Ω

u0ψdx = (p− 1)
∫
Ω

u0div[P (x)|∇u0|p−2∇ψ]dx

−(p− 1)
∫
Ω

u0a(x)|u0|p−2ψ

+λ

∫
Ω

a(x)[ν − 2υu0]ψdx. (2.5)
The first term of the L.H.S. of (2.5) may be written in the form

∫
Ω

u0div[P (x)|∇u0|p−2∇ψ]dx =
∫
Ω

u0[P (x)|∇u0|p−2]∇·∇ψdx

+

∫
Ω

u0∇ψ∇[P (x)|∇u0|p−2]dx.

Using Green’s first identity, one have
∫
Ω

u0div[P (x)|∇u0|p−2∇ψ]dx = −
∫
Ω

∇[u0P (x)|∇u0|p−2]∇ψ

+

∫
Ω

u0∇[P (x)|∇u0|p−2]∇ψdx

+

∫
∂Ω

u0[P (x)|∇u0|p−2]
∂ψ

∂n
ds,

= −
∫
Ω

[P (x)|∇u0|p−2]∇u0∇ψ

+

∫
∂Ω

u0[P (x)|∇u0|p−2]
∂ψ

∂n
ds. (2.6)

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Eur. J. Math. Anal. 10.28924/ada/ma.2.8 5
From (2.6) in (2.5) we have

−µ1
∫
Ω

u0ψdx = (p− 1)[
∫
∂Ω

u0[P (x)|∇u0|p−2]
∂ψ

∂n
ds

−
∫
Ω

[P (x)|∇u0|p−2]∇u0∇ψ]

+(k/d)

∫
Ω

a(x)[νu0 − 2υu20 ]ψdx. (2.7)
Multiplying (2.4) by (p− 1) and adding with (2.7), we have

−µ1
∫
Ω

u0ψdx = (p− 1)[
∫
∂Ω

u0[P (x)|∇u0|p−2]
∂ψ

∂n
ds

−
∫
∂Ω

ψ[P (x)|∇u0|p−2]
∂u0
∂n

ds]

+(k/d)

∫
Ω

a(x)[νu0 − 2υu20 ]ψdx

−(p− 1)(k/d)
∫
Ω

a(x)[νu0 −υu20 ]ψdx.

Hence
−µ1

∫
Ω

u0ψdx = (p− 1)
∫
∂Ω

[P (x)|∇u0|p−2][u0
∂ψ

∂n
−ψ

∂u0
∂n

]ds

+(k/d)

∫
Ω

a(x)νu0[1 − (p− 1)]ψdx

+(k/d)

∫
Ω

a(x)υu20 [(p− 1) − 2]ψdx. (2.8)
Now, when δ = 1, we have Bu0 = u0 = 0 for s ∈ ∂Ω and also we have ψ = 0 for s ∈ ∂Ω. Then∫

∂Ω

[P (x)|∇u0|p−2][u0
∂ψ

∂n
−ψ

∂u0
∂n

]ds = 0. (2.9)
Also, when δ 6= 1, we have

∂u0
∂n

= −
δhu0
1 −δ

and ∂ψ
∂n

= −
δhψ

1 −δ
,

which implies again the result given by (2.9).Hence
−µ1

∫
Ω

u0ψdx = (k/d)

∫
Ω

a(x)[νu0[2 −p] + υu20 [p− 3]]ψdx. (2.10)

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Eur. J. Math. Anal. 10.28924/ada/ma.2.8 6
Since 2 < p < 3 and a(x) > 0 for all x, then (2.10) becomes

−µ1
∫
Ω

u0ψdx < 0, (2.11)
so µ1 > 0 and the result follows. �

Theorem 2.2. If 2 < p < 3 and a(x) < 0 for all x ∈ Ω, then every positive weak solution of (1.1)
is unstable.

Proof. As in the proof of Theorem 1., we have
−µ1

∫
Ω

u0ψdx > 0, (2.12)
so µ1 < 0 and the result follows. �

3. Applications and related results
Here we introduce some examples to demonstrate the effectiveness of our results.

Example 3.1. Consider the Emden-Fowler steady-state problem of polytropic index of order one [8],

−∆u = λa(x)u in Ω,
Bu = 0 on ∂Ω,

} (3.1)
with a(x) > 0 for all x ∈ Ω.Here P (x) = 1, (k/d) = λ,p = 2. Then according to Theorem 1.,
every positive weak solution of (3.1) is unstable.

Example 3.2. Consider the population denisty steady-state problem of degree two [19],

−∆pu = λa(x)[u −u2] in Ω,
Bu = 0 on ∂Ω,

} (3.2)
with a(x) > 0 for all x ∈ Ω.Hence, according to Theorem 1., every positive weak solution of (3.1)
is stable.

Example 3.3. Consider the chemotaxis steady-state problem of degree two [9, 19],

−∆pu = λa(x)[−u + u2] in Ω,
Bu = 0 on ∂Ω,

} (3.3)
with a(x) > 0 for all x ∈ Ω.Hence, according to Theorem 1., every positive weak solution of (3.1)
is unstable.

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Eur. J. Math. Anal. 10.28924/ada/ma.2.8 7
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	1. Introduction:
	2. Main Results
	3. Applications and related results
	References

