A Lotka-Volterra competition model with cross-diffusions under homogeneous Dirichlet boundary condition is considered, where cross-diffusions are included in such a way that the two species run away from each other because of the competition between them. Using the method of upper and lower solutions, sufficient conditions for the existence of positive solutions are provided when the cross-diffusions are sufficiently small. Furthermore, the investigation of nonexistence of positive solutions is also presented.

1. Introduction

In this paper, we deal with the following Lotka-Volterra competition model with cross-diffusions:
(1)-Δ(u+αv)=u(a-u-cv),x∈Ω,-Δ(βu+v)=v(b-v-du),x∈Ω,u=v=0,x∈∂Ω,
where Ω is a bounded domain in RN with smooth boundary ∂Ω and all parameters a,b,c,d,α,β are positive constants. u and v stand for the densities of the two competitors; a and b are the intrinsic growth rates of u and v, respectively; c and d are the competitive parameters between the two species; Here α and β are referred to as cross-diffusions. Cross-diffusions express the two species run away from each other because of the competition between them. In this paper, the boundary condition is under homogeneous Dirichlet boundary condition which in biologically means that the boundary is not suitable for both species and they will all die on the boundary, and this is an ideal case.

In order to describe the meaning of cross-diffusions in this model (1) from the biological point, we give the general model with intrinsic diffusion and cross-diffusion:
(2)∂u∂t=div{k11(u,v)∇u+k12(u,v)∇v}+f(u,v),∂v∂t=div{k21(u,v)∇u+k22(u,v)∇v}+g(u,v),
where u and v stand for the densities of the two species, intrinsic diffusion parameters k11(u,v), k22(u,v)>0, cross-diffusion parameters k12(u,v),k21(u,v),
(3)Ju=-{k11(u,v)∇u+k12(u,v)∇v},Jv=-{k21(u,v)∇u+k22(u,v)∇v}
can be seen as the out-flux vector of u and v at x. The cross-diffusion parameters k12(u,v),k21(u,v)≥0 imply that the two competitors u and v diffuse in the direction of lower contrary of their competitor to avoid each other. f(u,v),g(u,v) are response function and in this paper the classical Logistic Type is considered and α,β≥0. More biological meaning of the system can be seen in [1–3].

The method of upper and lower solutions is a useful tool to study the existence of solutions of elliptic systems. However, there are many difficulties in investigating the existences of positive solutions of strongly coupled elliptic systems. Recently, by changing general strongly coupled elliptic systems into weakly coupled ones, the author in paper [4] gives the method to judge the solutions existence of elliptic systems by using the Schauder theorem. Furthermore, the method can be used to solve the existence of solutions of strongly coupled elliptic systems. In [5] Ko and Ryu investigate Lotka-Volterra prey-predator model with cross-diffusion:
(4)-Δu=u(a1-u-b12v),x∈Ω,-DΔu-Δv=v(a2+b21u-v),x∈Ω,u=v=0,x∈∂Ω.
Here D may be positive or negative. Using the developing method of upper and lower solutions in [4], the author gave a sufficient conditions for the existence of positive solutions to (4). Inspired by the paper [5], we investigate the existence and nonexistence of positive solutions to (1).

The main goal of this paper is to provide sufficient conditions for the existence of positive solutions to (1) when the cross-diffusions α and β are small. More precisely, we have the following theorem. Let λ1>0 be the principal eigenvalue of -Δ under homogeneous Dirichlet boundary condition. It is well known that the principal eigenfunction ϕ corresponding to λ1 does not change sign in Ω and ||ϕ||∞=1.

Theorem 1.

If min{a-cb,b-da}>λ1, then there exist positive constants α-=α-(a,b,c,d,Ω),β-=β-(a,b,c,d,Ω), when α<α-,β<β-, (1) has at least one positive solution.

For α=β=0, (1) is the Lotka-Volterra competition model under homogeneous Dirichlet boundary condition. In [6, 7], the authors use different methods to prove the existence of positive solutions, a sufficient condition for the existence is min{a-cb,b-da}>λ1. The conclusion implies that weakly cross-diffusion does not affect the existence of positive solution.

This paper is organized as follows. In Section 2, the existence theorem of solutions for a general class of strongly coupled elliptic systems is presented using the method of upper and lower solutions. In Section 3, sufficient conditions for the existence and nonexistence of positive solutions of (1) are investigated. Moreover, we give the corresponding results simply if the competitive system only has one cross-diffusion.

2. The Existence Theorem of Solutions for a Class of Strongly Coupled Elliptic Systems

In this section, we presented the existence theorem of solutions for a general class of strongly coupled elliptic systems:
(5)-ΔA(u,v)=f1(u,v),x∈Ω,-ΔB(u,v)=f2(u,v),x∈Ω,u=v=0,x∈∂Ω.
Here let A,B,f1,f2 satisfy the following hypotheses conditions.

U,V are domain in R2, (0,0)∈U. (A,B) is a C2 function about (u,v) from U to V, A(0,0)=B(0,0)=0, and have a continuous inverse (A*,B*)∈C2(V,U). Then for all (u,v)∈U, let
(6)w=A(u,v),z=B(u,v).
There exists only one (w,z)∈V, satisfying
(7)u=A*(w,z),v=B*(w,z).

The function A* is increasing in w and decreasing in z; B* is decreasing in w and increasing in z.

The functions f1(u,v),f2(u,v) are Lipschitz continuous in U, and there exist positive constants M1,M2 such that for all (u,v)∈U, the function f1(u,v)+M1A(u,v) is increasing in u; the function f2(u,v)+M2B(u,v) is increasing in v.

According to the hypothesis (H1), (5) can be rewritten as the following equal PDE equations:
(8)-Δw+M1w=f1(u,v)+M1A(u,v),x∈Ω,-Δz+M2z=f2(u,v)+M2B(u,v),x∈Ω,u=A*(w,z),v=B*(w,z),x∈Ω,w=z=0,x∈∂Ω.

Remark 2.

According to the hypothesis (H1), (5) can also be equal to the following weakly coupled elliptic equations:
(9)-Δw=f1(A*(w,z),B*(w,z)):=g1(w,z),x∈Ω,-Δz=f2(A*(w,z),B*(w,z)):=g2(w,z),x∈Ω,w=z=0,x∈∂Ω.

In its pure form, (9) is simpler than (8). However, due to the complicity of mixed functions g1(w(x),z(x)) and g2(w(x),z(x)), it is difficult to find the solutions of (9) directly. Therefore, we discuss (8).

Assume functions u-,v-,u_,v_∈C(Ω-),w-,z-,w_,z_∈Cα(Ω-)⋂C2(Ω), the values of functions (u-,v-) and (u_,v_) are in V and the values of functions (w-,z-) and (w_,z_) are in U. To describe easily, let
(10)U={u∈C(Ω-):u_(x)≤u(x)≤u-(x)},V={u∈C(Ω-):v_(x)≤v(x)≤v-(x)}.

According the definition of upper and lower solutions in [4] and conditions (H1)–(H3), we give the definition of upper and lower solutions of (5).

Definition 3.

A pair of functions ((u-,v-,w-,z-),(u_,v_,w_,z_)) are called upper and lower solutions of (9) provided that they satisfy the relation (u-,v-,w-,z-)≥(u_,v_,w_,z_), and for all (u,v)∈U×V, satisfy the following inequalities:
(11)-Δw-+M1w-≥f1(u-,v)+M1A(u-,v),x∈Ω,-Δz-+M2z-≥f2(u,v-)+M2B(u,v-),x∈Ω,-Δw_+M1w_≤f1(u_,v)+M1A(u_,v),x∈Ω,-Δz_+M2z_≤f2(u,v_)+M2B(u,v_),x∈Ω,u-≥A*(w-,z_),v-≥B*(w_,z-),x∈Ω,u_≤A*(w_,z-),v_≤B*(w-,z_),x∈Ω,w-≥0≥w_,z-≥0≥z_,x∈∂Ω.

We can have the following conclusion from [4, Theorem 2.1].

Proposition 4.

Assume that (8) has coupled upper and lower solutions ((u-,v-,w-,z-),(u_,v_,w_,z_)), then there exists at least one solution (u,v,w,z), satisfying the relation
(12)(u_,v_,w_,z_)≤(u,v,w,z)≤(u-,v-,w-,z-).
Furthermore, (u,v) is the solution of (5).

Next, if u-,v-,u_,v_ satisfy
(13)u-=A*(w-,z_),v-=B*(w_,z-),u_=A*(w_,z-),v_=B*(w-,z_),
then
(14)w-=A(u-,v_),z-=B(u_,v-),w_=A(u_,v-),z_=B(u-,v_),
(11) can be rewritten as
(15)-ΔA(u-,v_)+M1A(u-,v_)≥f1(u-,v)+M1A(u-,v),x∈Ω,-ΔB(u_,v-)+M2B(u_,v-)≥f2(u,v-)+M2B(u,v-),x∈Ω,-ΔA(u_,v-)+M1A(u_,v-)≤f1(u_,v)+M1A(u_,v),x∈Ω,-ΔB(u-,v_)+M2B(u-,v_)≤f2(u,v_)+M2B(u,v_),x∈Ω,A(u-,v_)≥0≥A(u_,v-),B(u_,v-)≥0≥B(u-,v_),x∈∂Ω.
Synthetically, we have the following result.

Theorem 5.

If there is a pair of functions ((u-,v-),(u_,v_)), satisfying
(16)(u-,v-,A(u-,v_),B(u_,v-))≥(u_,v_,A(u_,v-),B(u-,v_)),
and for all (u,v)∈U×V, (15) is satisfied, then (5) has at least one solution (u,v), satisfying the relation (u_,v_)≤(u,v)≤(u-,v-).

To make sure the upper and lower solutions reasonable, we give the following two lemmas; more details can be found in [8, 9].

Lemma 6.

If the functions u,v∈C1(Ω-) satisfy u|∂Ω=v|∂Ω=0,u|Ω>0, (∂u/∂ν)|∂Ω<0, ν is the outer unit normal vector of ∂Ω, then there exists positive constant ε, such that u(x)>εv(x), for all x∈Ω.

For the equation:
(17)-Δu=u(a-u),x∈Ω,u=0,x∈∂Ω.

Lemma 7.

If a>λ1, then (17) has a unique positive solution θa satisfying θa≤a. In addition, θa is increasing with respect to a.

3. A Lotka-Volterra Competition Model with Two Cross-Diffusions

In this section, the existence of positive solutions of (1) corresponding to α≥0,β≥0, is investigated by applying Theorem 5 to prove Theorem 1.

Proof.

We seek some positive constants R,K,δ,R,K>λ1 sufficiently large and δ sufficiently small, Lemma 6 may guarantee the existence of θR and θK. It can be easily known from Hopf boundary lemma:
(18)∂ϕ∂ν(x)<0,∂θR∂ν(x)<0,∂θK∂ν(x)<0,∀x∈∂Ω.
Observe that min{a-cb,b-da}>λ1, using Lemma 7, we can have R,K,δ, a<R<(b-λ1)/d,b<K<(a-λ1)/c, satisfying the following three conditions:

δϕ(x)<θR(x),δϕ(x)<θK(x), for all x∈Ω;

(∂(θR-δϕ)/∂ν)(x)<0,(∂(θK-δϕ)/∂ν)(x)<0;

δ<min{a-λ1-cK,b-λ1-dR}.

Let M1=2R+cK,M2=2K+dR. Using Lemma 7 again, there exist α-=α-(a,b,c,d,Ω)<1,β-=β-(a,b,c,d,Ω)<1, for all (ρ,τ)∈[0,α-)×[0,β-), for all x∈Ω, satisfying

We will verify α-,β- satisfying Theorem 5. Suppose that (α,β)∈[0,α-)×[0,β-). Then we construct a pair of upper and lower solutions of the form
(19)(u-,v-)=(θR,θK),(u_,v_)=(δϕ,δϕ),
where δ satisfies conditions (i)–(iii). Let
(20)A(u,v)=u+αv,B(u,v)=βu+v.
Then
(21)A*(w,z)=w-αz1-αβ,B*(w,z)=z-βw1-αβ.
By simply computing, (H1) and (H2) are satisfied, where U=[0,R]×[0,K],V=[0,R+αK]×[0,K+βR].

Note
(22)f1(u,v)=u(a-u-cv),f2(u,v)=v(b-v-du).
And for all (u,v)∈U, we have
(23)[f1(u,v)+M1A(u,v)]u=a-2u-cv+M1≥-2R-cK+M1=0,[f2(u,v)+M2B(u,v)]v=b-2v-du+M2≥-2K-dR+M2=0.
So (H3) is satisfied; observer that u-|∂Ω=v-|∂Ω=u_|∂Ω=v_|∂Ω=0, (u-,v-)≥(u_,v_) and (iv) and (15) and the boundary conditions of (16) can be checked. Therefore, if we want to obtain the existence of solutions through [4, Theorem 2.1], we should only verify for all (u,v)∈U×V,
(24)-ΔA(u-,v_)+M1A(u-,v_)≥f1(u-,v)+M1A(u-,v),x∈Ω,-ΔB(u_,v-)+M2B(u_,v-)≥f2(u,v-)+M2B(u,v-),x∈Ω,-ΔA(u_,v-)+M1A(u_,v-)≤f1(u_,v)+M1A(u_,v),x∈Ω,-ΔB(u-,v_)+M2B(u-,v_)≤f2(u,v_)+M2B(u,v_),x∈Ω.
Because f1 is decreasing in v, f2 is decreasing in u, and A(u,v) is increasing in v, B(u,v) is increasing in u, only to verify the following inequations:
(25)-ΔA(u-,v_)+M1A(u-,v_)≥f1(u-,v_)+M1A(u-,v-),x∈Ω,-ΔB(u_,v-)+M2B(u_,v-)≥f2(u_,v-)+M2B(u-,v-),x∈Ω,-ΔA(u_,v-)+M1A(u_,v-)≤f1(u_,v-)+M1A(u_,v_),x∈Ω,-ΔB(u-,v_)+M2B(u-,v_)≤f2(u-,v_)+M2B(u_,v_),x∈Ω.
It is easy to check (25) by (v), (vi), and (vii). So from [4, Theorem 2.1], (1) has a solution (u,v), in addition (u-,v-)≥(u,v)≥(u_,v_)>(0,0).

In the end, before investigating the nonexistence of positive solutions of (1), we give its priori bound of positive solutions.

Theorem 8.

Any positive solutions (u,v) of (1) have a priori bound; that is
(26)u(x)≤bd,v(x)≤ac.

Proof.

Let w=u+αv,z=βu+v; then
(27)u=w-αz1-αβ,v=z-βw1-αβ.
Equation (1) can be rewritten as
(28)-Δw=w-αz1-αβ(a-w-αz1-αβ-cz-βw1-αβ),x∈Ω,-Δz=z-βw1-αβ(b-z-βw1-αβ-dw-αz1-αβ),x∈Ω,(w,z)=(0,0),x∈∂Ω.
Since (u,v)>(0,0), it easily follows that w-αz>0,z-βw>0. Assume that z(x) attains its positive maximum at x0∈Ω, then
(29)a(1-αβ)-w(x0)+αz(x0)-cz(x0)+cβw(x0)>0a(1-αβ)-cz(x0)+cβαz(x0)>0,z(x)≤z(x0)≤ac
so that
(30)v=z-βu≤z(x0)≤ac.
Similarly, we can obtain the desired result
(31)u≤bd.

Theorem 9.

If one of the following conditions:

b≤ad,λ1≥(b+cβ(b/d))/(1-αβ);

(1-(α+β)/2)λ1≥max{a,b};

is satisfied, then (1) with α<α-,β<β- has no positive solution.
Proof.

Multiplying u and v to the first and second equations in (1), and integrating these equations on Ω, we have
(32)∫Ω|∇u|2dx+α∫Ω∇u∇vdx=∫Ωu2(a-u-cv)dx,α∫Ω|∇v|2dx+∫Ω∇u∇vdx=∫Ωuv(a-u-cv)dx,β∫Ω|∇u|2dx+∫Ω∇u∇vdx=∫Ωuv(b-v-du)dx,∫Ω|∇v|2dx+β∫Ω∇u∇vdx=∫Ωv2(b-v-du)dx.

(i) Suppose, by contradiction that (1) has a positive solution (u,v), then the second and fourth equations in (32) yield
(33)∫Ωv3dx+β∫Ωuv(a-u)dx=-(1-αβ)∫Ω|∇v|2dx+β∫Ωcuv2dx+∫Ωv2(b-du)dx.
Since u≤b/d by Theorem 8, the left-hand side of (33) must be positive. On the other hand, the Poincare inequality, ∥∇v∥L22≥λ1∥v∥L22, for v∈W21(Ω) and the given assumption shows the following contradiction:
(34)-(1-αβ)∫Ω|∇v|2dx+β∫Ωcuv2dx+∫Ωv2(b-du)dx≤-[(1-αβ)λ1-cβbd-b]∫Ωv2dx≤0.

(ii) A contraction argument is also used assuming that (1) has a positive solution (u,v). Adding the first equation to the fourth equation, and then subtracting a∫Ωu2dx+b∫Ωv2dx from the both sides, the following identity is obtained:
(35)∫Ω|∇u|2dx+(α+β)∫Ω∇u∇vdx+∫Ω|∇v|2dx-a∫Ωu2dx-b∫Ωv2dx=-∫Ωu2(u+cv)dx-∫Ωv2(v+du)dx.
Since 2∇u∇v=|∇(u+v)|2-|∇u|2-|∇v|2 and (1-(α+β)/2)λ1≥max{a,b}, the Poincare inequality shows that the left-hand side of (35) must be nonnegative, more precisely,
(36)∫Ω|∇u|2dx+(α+β)∫Ω∇u∇vdx+∫Ω|∇v|2dx-a∫Ωu2dx-b∫Ωv2dx=(1-α+β2)∫Ω|∇u|2dx+α+β2∫Ω|∇(u+v)|2dx+(1-α+β2)∫Ω|∇v|2dx-a∫Ωu2dx-b∫Ωv2dx≥[(1-α+β2)λ1-a]∫Ωu2dx+α+β2·∫Ω|∇(u+v)|2dx+[(1-α+β2)λ1-b]∫Ωv2dx≥[(1-α+β2)λ1-a]∫Ωu2dx+[(1-α+β2)λ1-b]∫Ωv2dx≥0.
However, this results in a contradiction since the right-hand side of (35) is clearly strictly negative by the positivity of u and v.

Remark 10.

Before closing this section, more sufficient conditions of the nonexistence of positive solutions of (1) with α+β>0, αβ=0 are investigated. Take α=0,β>0 for example, then (1) may be reduced as
(37)-Δu=u(a-u-cv),x∈Ω,-Δ(βu+v)=v(b-v-du),x∈Ω,(u,v)=(0,0),x∈∂Ω.
Using the same method, we can obtain that (37) has no positive solution, if one of the following conditions is satisfied:

λ1≥b+βca;

λ1≥a;

(1-β/2)λ1≥max{a,b};

c<1<a/b and (1-d)/β≤λ1/(b+βa)≤1.

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