DISCRETE AND CONTINUOUS
DYNAMICAL SYSTEMS
Supplement 2013
Website: www.aimSciences.org
pp. 193–195
A UNIQUE POSITIVE SOLUTION TO A SYSTEM OF
SEMILINEAR ELLIPTIC EQUATIONS
Diane Denny
Department of Mathematics and Statistics
Texas A&M University - Corpus Christi
Corpus Christi, Texas 78412, USA
Abstract. We study a system of semilinear elliptic equations that arises from
a predator-prey model. Previous related work proved the existence of a unique
positive solution to this system of equations in the special case in which the
parameter α = 0 in this system of equations, provided that a positive parameter
κ in this system of equations is sufficiently large. We prove the existence of a
unique positive solution to this system of equations for any α ≥ 0 and for any
κ > 0.
1. Introduction. In this paper, we consider the system of equations
− ∆u = u(λ − αu − βv)
(1.1)
v
(1.2)
−∆v = κ (u − ξv)
u
for x ∈ Ω, where Ω ⊂ RN and N ≥ 2 under the boundary conditions ∇u · n = 0 and
∇v · n = 0 on ∂Ω, where n is the outward unit normal vector on ∂Ω. Here α ≥ 0,
β > 0, ξ > 0, λ > 0, and κ > 0 are constants.
Applications include predator-prey problems (see, e.g., [1], [2], [3]). In previous
related work, Zhou and Wei [3] considered the special case in which α = 0, and
proved that there exists a constant κ0 > 0, where κ0 depends on λ and on the
λ
domain Ω, such that a unique positive solution (u, v) = ( λξ
β , β ) to (1.1), (1.2) exists
for κ ≥ κ0 and α = 0.
We use the fact that (û, v̂) is a positive solution of the system of equations
(1.1), (1.2) if and only if ( βξ û, βv̂) is a positive solution of the equivalent system of
equations
ξα
− ∆u = u(λ −
u − v)
(1.3)
β
v
−∆v = κ (u − v)
(1.4)
u
We let γ = ξα
β , where γ ≥ 0. Then (1.3), (1.4) becomes
− ∆u =
u(λ − γu − v)
(1.5)
v
(1.6)
−∆v = κ (u − v)
u
The purpose of this paper is to prove the existence of a unique positive solution
λ
λ
(u, v) = ( 1+γ
, 1+γ
) to the system of equations (1.5), (1.6) for any κ > 0, where γ =
2010 Mathematics Subject Classification. Primary: 35A02.
Key words and phrases. Elliptic, existence, uniqueness, semilinear.
193
194
ξα
β
DIANE DENNY
≥ 0 and λ > 0, where x ∈ Ω ⊂ RN and N ≥ 2, under the boundary conditions
λξ
λ
∇u · n = 0, ∇v · n = 0 on ∂Ω. It immediately follows that (u, v) = ( β(1+γ)
, β(1+γ)
)
is the unique positive solution to the equivalent system of equations (1.1), (1.2). If
α = 0, then γ = 0, and this solution is the same solution found by Zhou and Wei
[3], but their proof requires that κ ≥ κ0 , while our proof holds for any κ > 0.
2. Existence and uniqueness for the system of equations. In this section,
we present the proof of the following theorem
Theorem 2.1. Let γ ≥ 0, λ > 0, κ > 0 be constants. There exists a unique positive
solution u ∈ H 2 (Ω), v ∈ H 2 (Ω) to the system of equations
− ∆u =
−∆v
=
u(λ − γu − v)
v
κ (u − v)
u
(2.1)
(2.2)
for x ∈ Ω ⊂ RN , N ≥ 2, under the boundary conditions ∇u · n = 0 and ∇v · n = 0
on ∂Ω, where ∂Ω is smooth, and where n is the outward unit normal vector on ∂Ω.
λ
The unique positive solution is u = v = 1+γ
.
R
Proof of Theorem 2.1. In the following proof, we let (f, g) = Ω f gdx denote the
L2 inner product of f and g on the domain Ω.
First, we take the L2 inner product of −∆u with (λ − γu − v), and then using
identity (2.1) for −∆u yields
− (∆u, (λ − γu − v)) = (u(λ − γu − v), (λ − γu − v))
(2.3)
We now consider the left-hand side of (2.3), and integrate by parts to obtain the
identity
− (∆u, (λ − γu − v)) = −(∇u, γ∇u) − (∇u, ∇v)
(2.4)
where we used the boundary condition ∇u · n = 0.
Substituting (2.4) into (2.3) and re-arranging terms yields
(∇u, γ∇u) + (∇u, ∇v) + (u(λ − γu − v), (λ − γu − v)) = 0
(2.5)
Next, we take the L2 inner product of −∆v with u − v, and then using identity
(2.2) for −∆v yields
v
− (∆v, u − v) = (κ (u − v), (u − v)))
(2.6)
u
We now consider the left-hand side of (2.6), and integrate by parts to obtain the
identity
− (∆v, u − v) = (∇v, ∇u) − (∇v, ∇v))
(2.7)
where we used the boundary condition ∇v · n = 0.
Substituting (2.7) into (2.6) and re-arranging terms yields
v
(∇v, ∇u) = (∇v, ∇v) + (κ (u − v), (u − v))
(2.8)
u
Substituting the identity for (∇v, ∇u) from (2.8) into (2.5) and re-arranging
terms yields the identity
v
(∇u, γ∇u) + (∇v, ∇v) + (κ (u − v), (u − v))
u
+(u(λ − γu − v), (λ − γu − v)) = 0
(2.9)
A UNIQUE POSITIVE SOLUTION TO A SYSTEM
195
Since each term in (2.9) is non-negative when u, v are positive, and the sum of
these terms is zero, it follows that each term in (2.9) must be zero. Therefore, any
positive solution (u, v) of (2.1), (2.2) must satisfy
(∇u, γ∇u) = 0,
(2.10)
(∇v, ∇v) = 0,
v
(κ (u − v), (u − v)) = 0,
u
(u(λ − γu − v), (λ − γu − v)) = 0.
(2.11)
(2.12)
(2.13)
It follows from (2.10)–(2.12) that u = v = C, for some positive constant C.
Substituting u = v = C, where C > 0, into (2.13) yields
(C(λ − γC − C), (λ − γC − C)) = 0
(2.14)
Solving (2.14) for C yields
λ
(2.15)
1+γ
λ
And by directly substituting u = v = C = 1+γ
into (2.1), (2.2), we verify that
λ
λ
(u, v) = ( 1+γ , 1+γ ) is a solution of (2.1), (2.2) for x ∈ Ω, under the boundary
conditions ∇u · n = 0 and ∇v · n = 0 on ∂Ω.
This solution is unique because (2.10)–(2.12) implies that any positive solution
(u, v) must satisfy u = v = C, where C is a positive constant, and (2.13) implies
λ
.
that C must satisfy C = 1+γ
C=
We remark that in the case N = 1, by repeating the above proof it follows that
the result of the theorem also holds.
REFERENCES
[1] Y.H. Du and S.B. Hsu, A diffusive predator-prey model in heterogeneous environment, Journal
of Differential Equations, 203 (2004), 331–364.
[2] Y.H. Du and M.X. Wang, Asymptotic behaviour of positive steady states to a predator-prey
model, Proceedings of the Royal Society of Edinburgh, 136A (2006), 759–778.
[3] W. Zhou and X. Wei, Uniqueness of positive solutions for an elliptic system, Electronic
Journal of Differential Equations, 2011 (2011), 1–6.
Received July 2012; revised November 2012.
E-mail address: [email protected]