# Uniqueness and Multiplicity of a Prey-Predator Model with Predator Saturation and Competition

Authors: Xiaozhou Feng; Lifeng Li
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We investigate positive solutions of a prey-predator model with
predator saturation and competition under homogeneous Dirichlet boundary conditions. First,
the existence of positive solutions and some sufficient and necessary conditions is established by
using the standard fixed point index theory in cones. Second, the changes of solution branches,
multiplicity, uniqueness, and stability of positive solutions are obtained by virtue of bifurcation
theory, perturbation theory of eigenvalues, and the fixed point index theory. Finally, the exact
number and type of positive solutions are proved whenk  or m  converges to infinity.


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## Body

## 1. Introduction

Considering the destabilizing force of predator saturation and the stabilizing force of competition for prey, Bazykin [1] proposed the function response f(u,v)=1/(1+mu)(1+kv) in the prey-predator model instead of the classical Holling-type II functional response. For this functional response, the prey-predator model is taken as the following form:(1.1)ut-Δu=u(a-u-bv(1+mu)(1+kv)),x∈Ω,t>0,vt-Δv=v(c-v+du(1+mu)(1+kv)),x∈Ω,t>0,u=v=0,x∈∂Ω,t>0,u(x,0)=u0(x)≥0,v(x,0)=v0(x)≥0,x∈Ω.In this paper, we are concerned with the positive solution of the boundary value problem of the following elliptic system corresponding to the system (1.1):(1.2)-Δu=u(a-u-bv(1+mu)(1+kv)),x∈Ω,-Δv=v(c-v+du(1+mu)(1+kv)),x∈Ω,u=v=0,x∈∂Ω,
where Ω is a bounded domain in RN(N≥1) with smooth boundary ∂Ω;  a,b,c are positive constants; d is constant; m and k are nonnegative constants.Ifm=k=0, then (1.2) is reduced to the classical Lotka-Volterra prey-predator model which has received extensive study in the last decade, see [2–8]. In particular, the existence of positive solutions for this case was completely understood, see Dancer [8]. It has been conjectured that there is at most one positive solution, but this was shown only for the case the space dimension n is one, see [9]. For space dimension greater than one, this is still an open problem; we also refer to [10, 11] for some partial results on uniqueness. The stability of positive solutions was studied in [10, 11], but the results are still far from being complete.The case whenm>0 and k=0 was first studied by Blat and Brown [12]. In this case, the term f(u,v) is known as the Holling-Tanner interaction term, and we refer to [5, 12–17] for more discussion on this model. In [12], Blat and Brown studied the existence of positive solutions to (1.2) by making use of both local and global bifurcation theories. The case when m goes to infinity was extensively studied by Du and Lou in [13, 14, 18]. They gave a good understanding of the existence, stability, and number of positive solutions for large m.However, the case whenm>0 and k>0 was first studied by Bazykin in the paper [1], more detailed background on this case, we can refer to [1]. And more works can refer to [19], Wang studies the existence, multiplicity, and stability of positive solutions of (1.2). However, Our work is more specific and meticulous than theirs. In particular. Firstly, the changes of solution branches, uniqueness, and stability of positive solutions are obtained by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory. Secondly, the exact number and type of positive solutions are proved when k or m is large.This paper is organized as follows: in Section2, we give sufficient and necessary conditions for the existence of coexistence states of (1.2) by using index theory. In Section 3, by using a as a main bifurcation parameter, the multiplicity of coexistence stats to (1.2) is investigated in the gap between the sufficient and necessary conditions for the existence of coexistence states which are found in Section 2. In Section 4, the multiplicity, uniqueness, and stability of coexistence states of (1.2) are investigated when k or m converges to infinity.


## 2. Existence and Nonexistence of Coexistence States

In this section, we will obtain existence and nonexistence of coexistence states. Firstly, we present some basic results which will be used in this paper.Letλ1(q)<λ2(q)≤λ3(q)≤⋯ be all eigenvalues of the following problem:
(2.1)-Δϕ+q(x)ϕ=λϕ,ϕ|∂Ω=0,
where q(x)∈C(Ω¯). It is easy to see that λ1(q) is simple and λ1(q) is strictly increasing in the sense that q1≤q2 and q1≢q2 imply λ1(q1)<λ1(q2). When q(x)≡0, we denote by λiλi(0). Moreover, we denote by Φ1 the eigenfunction corresponding to λ1 with normalization ∥Φ1∥∞=1 and positive in Ω.DefineC0(Ω¯)={u∈C(Ω¯)∣u=0on∂Ω}. It is well known that for any a>λ1, the problem
(2.2)-Δu=au-u2,u|∂Ω=0
has a unique positive solution which we denote by θa. It is well known that the mapping a→θa is strictly increasing, continuously differentiable from (λ1,∞) to C2(Ω)∩C0(Ω¯) and that θa→0 uniformly on Ω¯ as a→λ1. Moreover, we have 0<θa<a in Ω. It follows that (1.2) has two semitrivial solutions (θa,0) and (0,θc) if a,c>λ1.Next, we give an a priori estimate based on maximum principle. Its proof will be omitted.Lemma 2.1.
Any coexistence state(u,v) of (1.2) has an a priori boundary, that is,
(2.3)u≤a,v≤B∶=c+ad1+am.In the following, we set up the fixed point index theory for later use. LetE be a Banach space. W⊂E is called a wedge if W is a closed convex set and βW⊂W for all β≥0. For y∈W, we define Wy={x∈E:∃r=r(x)>0,s.t.y+rx∈W}, Sy={x∈W¯y:-x∈W¯y}: we always assume that E=W-W¯. Let T:Wy→Wy be a compact linear operator on E. We say that T has property α on W¯y if there exists t∈(0,1) and ω∈W¯y∖Sy, such that ω-tTω∈Sy.For anyδ>0 and y∈W, we denote Bδ+(y)=Bδ(y)∩W. Assume that F:Bδ+(y)→W is a compact operator and y is an isolated fixed point of F. If F' is Fréchet differentiable at y, then the derivative F'(y) has the property that F'(y):W¯y→W¯y. We denote by indexW(F,y) the fixed point index of F at y relative to W.We state a general result of Dancer [20] on the fixed point index with respect to the positive cone W (see also [6]).Lemma 2.2.
Suppose thatI-L is invertible on W¯y.(i)
IfL has property α on W¯y, then indexW(F,y)=0.(ii)
IfL does not have property α on W¯y, then indexW(F,y)=(-1)σ, where σ is the sum of algebra multiplicities of the eigenvalues of L which are greater than 1.We introduce some notations as follows:X=C01(Ω¯)⨁C01(Ω¯), where C01(Ω¯)={ω∈C1(Ω¯):ω∣∂Ω=0},W=K⨁K, where K={φ∈C(Ω¯):φ(x)≥0},D∶={(u,v)∈X:u≤a+1,v≤B+1}, D′∶=(intD)⋂‍W.DefineFt:D′→W by
(2.4)Ft(u,v)=(-Δ+P)-1(tu(a-u-bv(1+mu)(1+kv))+Putv(c-v+du(1+mu)(1+kv))+Pv),
where t∈[0,1] and P>max{a+bB,c+da/(1+ma)}. It follows from Maximum Principle that (-Δ+P)-1 is a compact positive operator, Ft is complete continuous and Fréchet differentiable. Denote F1=F, observe that (1.2) has a positive solution in W if and only if Ft=F1 has a positive fixed point in D′.Ifa>λ1 and c>λ1, then (0,0), (θa,0), and (0,θc) are the only nonnegative fixed points of F. Then indexW(F,(0,0)), indexW(F,(θa,0)), and indexW(F,(0,θc)) are well defined. We calculate the Fréchet operator of F as follows:
(2.5)(-Δ+P)-1(a-2u-bv(1+mu)2(1+kv)+P-bu(1+mu)(1+kv)2dv(1+mu)2(1+kv)c-2v+du(1+mu)(1+kv)2+P).We can obtain the following lemmas by similar methods to those in the proofs of Lemmas 1 and 2 in [19].Lemma 2.3.
Suppose thata>λ1, one has(i)
degW(I-F,D')=1, where degW(I-F,D') is the degree of T-F in D′ relative to W(ii)
ifc≠λ1, then indexW(F,(0,0))=0;(iii)
ifc>λ1(-dθa/(1+mθa)), then indexW(F,(θa,0))=0;(iv)
ifc<λ1(-dθa/(1+mθa)), then indexW(F,(θa,0))=1.Lemma 2.4.
Suppose thatc>λ1, one has(i)
ifa>λ1(bθc/(1+kθc)), then indexW(F,(0,θc))=0;(ii)
ifa<λ1(bθc/(1+kθc)), then indexW(F,(0,θc))=1.Next, we will show some results of existence and nonexistence of positive solutions of (1.2).Theorem 2.5.
(
i
) If a≤λ1, then (1.2) has no positive solution; if a≤λ1 and c≤λ1, then (1.2) has no nonnegative nonzero solution.
(
i
i
) If c≤λ1 and (1.2) has a positive solution, then a>λ1,c+da/(1+ma)>λ1.
(
i
i
i
) If c>λ1 and (1.2) has a positive solution, then a>λ1(bθc/(1+mθa)(1+kθc)).Proof.
(i) First assume that(u,v) is a positive solution of (1.2), then (u,v) satisfies
(2.6)-Δu=u(a-u-bv(1+mu)(1+kv)),x∈Ω,u=0,x∈∂Ω,
and so a=λ1(u+bv/(1+mu)(1+kv)) by the eigenvalue problem. Due to the comparison principle for eigenvalues, we have a>λ1, a contradiction. Next, assume that (u,v) is a nonnegative nonzero solution of (1.2). If u≢0 and v≡0, then a>λ1 by the previous proof. We can also similarly derive c>λ1 when u≡0 and v≢0, which is a contradiction again.
(ii) Assume that(u,v) is a positive solution of (1.2). Then a>λ1 by (i), and so the positive semitrivial solution θa exists. Since
(2.7)-Δu=u(a-u-bv(1+mu)(1+kv))≤u(a-u),x∈Ω,u=0,x∈∂Ω,u is a lower solution of (1.2). By the uniqueness of θa, u≤θa. Furthermore, since v satisfies the equation
(2.8)-Δv=v(c-v+du(1+mu)(1+kv)),x∈Ω,v=0,x∈∂Ω,
one has 0=λ1(-c+v-du/(1+mu)(1+kv))>λ1(-c-da/(1+ma)), which implies the result.
(iii) Let(u,v) be a positive solution of (1.2); then θa exists with u≤θa as in (ii). Similarly, the given assumption c>λ1 implies the existence of positive solution θc of (1.2) with θc≤v. Similar to the proof of (i), we have a=λ1(u+bv/(1+mu)(1+kv))>λ1(bθc/(1+mθa)(1+kθc)). This follows since the function bv/(1+mu)(1+kv) has a minimum at u=θa and v=θc (for u≤θa and v≥θc).Theorem 2.6.
(
i
) If c>λ1 and a>λ1(bθc/(1+kθc)). Then (1.2) has at least a positive solution.
(
i
i
) Suppose that c<λ1. Then (1.2) has positive solution if and only if a>λ1 and c>λ1(-dθa/(1+mθa)).Proof.
(i) By Lemmas2.3 and 2.4, we have
(2.9)degW(I-F,D)-indexW(f,(0,0))-indexW(f,(θa,0))-indexW(f,(0,θc))=1.
So (1.2) has at least one positive solution.
(ii) We first prove the sufficiency. Sincec<λ1, (1.2) has no solution taking the form (0,v) with v>0. If a>λ1 and c>λ1(-dθa/(1+mθa)), note that c<λ1; from Lemma 2.4, we have
(2.10)degW(I-F,D)-indexW(f,(0,0))-indexW(f,(θa,0))=1.
Hence (1.2) has at least one positive solution.
Conversely, suppose that(u,v) is a positive solution of (1.2). Then a>λ1, and u<θa. Since (u,v) satisfies
(2.11)-Δv=v(c-v+du/(1+mu)(1+kv)),x∈Ω,v=0,x∈∂Ω.
It follows that 0=λ1(-c+v-du/(1+mu)(1+kv))>λ1(-c-da/(1+ma)).Theorem 2.7.
If one of the following conditions holds, then (1.2) has no positive solutions:(i)
b≥(1+ma)(1+kB) and a≤c,(ii)
b<(1+ma)(1+kB) and c-a≥(1-ba/(1+ma)(1+kB))B.Proof.
Since the proof of Theorem2.7 is similar to the proof of Theorem 3 of [19], we omit it.


## 3. Global Bifurcation and Stability of Positive Solution

In this section, we consider a positive solution bifurcates from the semitrivial nonnegative branch{(0,θc,a)} by taking a as a bifurcation parameter and fixing c>λ1. Furthermore, we show that the existence of global bifurcation of (1.2) with respect to parameter a and its stability. Moreover, the multiplicity, uniqueness, and stability of positive solutions are obtained by means of perturbation theory of eigenvalues and the fixed point theory.Leta~ be the principal eigenvalue of the following problem:
(3.1)-Δϕ+bθc1+kθcϕ=aϕ,ϕ|∂Ω=0,
and Φ is the corresponding eigenfunction with ∥Φ∥∞=1.Letω=u,χ=v-θc; then 0≤ω≤θa,χ≥0, and ω,χ satisfies
(3.2)-Δω=(a-bθc1+kθc)ω+F1(ω,χ),x∈Ω,-Δχ=(c-2θc)χ+dθc1+kθcω+F2(ω,χ),x∈Ω,ω=χ=0,x∈∂Ω,
where
(3.3)F1(ω,χ)=bωθc1+kθc-bω(χ+θc)(1+mω)(1+k(χ+θc))-ω2,F2(ω,χ)=dω(χ+θc)(1+mω)(1+k(χ+θc))-dωθc1+kθc-χ2.Clearly,F=(F1,F2) is continuous, F(0,0)=0, and the Fréchet derivative D(ω,χ)F∣(0,0)=0. Let K be the inverse of -Δ with Dirichlet boundary condition. Then, we have
(3.4)ω=aKω-bK(ωθc1+kθc)+KF1(ω,χ),x∈Ω,χ=cKχ-2K(χθc)+dK(ωθc1+kθc)+KF2(ω,χ),x∈Ω,ω=χ=0,x∈∂Ω.Define the operatorT:R+×X→X as follows:
(3.5)T(a;ω,χ)=(aKω-bK(ωθc1+kθc)+KF1(ω,χ)cKχ-2K(χθc)+dK(ωθc1+kθc)+KF2(ω,χ)),
then T(a;ω,χ) is a compact operator on X. Let G(a;ω,χ)=(ω,χ)T-T(a;ω,χ); then G is continuous, and G(a;0,0)=0. G(a;ω,χ)=0 with 0≤ω≤θa,χ≥0 if and only if (ω,χ+θc,a) is a nonnegative solution of (1.2).Lemma 3.1.
Assume thatc>λ1. Then (a;u,v)=(a~;0,θc) is a bifurcation point of (3.2), and there exist positive solutions of (3.2) in the neighborhood of (a~;0,θc), where a~=λ1(bθc/(1+kθc)).Remark 3.2.
The proof of Lemma3.1 is similar to the proof of Theorem 9 in [19]. The proof of Lemma 3.1 shows that there exist δ>0 and C1 continuous curve (a(s);ϕ(s),φ(s)):(-δ,δ)→R×Z such that a(0)=a~,ϕ(0)=0,φ(0)=0, and (a(s);ω(s),χ(s))=(a(s);s(Φ+ϕ(s)),s(Ψ+φ(s))) satisfies G(a(s);ω(s),χ(s))=0, where X=Z⊕span{(Φ,Ψ)}. Hence (a(s);U(s),V(s))(|s|<δ) is a bifurcation solution of (3.2), where U(s)=s(Φ+ϕ(s)),V(s)=θc+s(Ψ+φ(s)), Ψ=(-Δ-c+2θc)-1(dθc/(1+kθc)Φ).
If we take0<s<δ, then the nontrivial nonnegative solutions of (1.2) close to (a~;0,θc) are either on the branch {(a;0,θc):a∈R+} or the branch {(a(s);U(s),V(s)):0<s<δ}.
LetT:X×R→X be a compact continuously differentiable operator such that T(0,a)=0. Suppose that we can write T as T(u,a)=K(a)u+R(u,a), where K(a) is a linear compact operator and the Fréchet derivative Ru(0,0)=0. If x0 is an isolated fixed point of T, then we can define the index if T at x0 as index(T,x0)=deg(I-T,Uδ(x0),x0), where Uδ(x0) is a ball with center at x0 such that x0 is the only fixed point of T in Uδ(x0). If I-T′(x0) is invertible, then x0 is an isolated fixed point of T and index(T,x0)=deg(I-T,Uδ(x0),x0)=deg(I-T'(x0),Uδ(x0),0). If x0=0, then it is well known that the Leray-Schauder degree deg(I-K(a),Uδ(x0),0)=(-1)σ, where σ is equal to the algebraic multiplicities of the eigenvalue of K which is greater than one.
Next, we will extend the local bifurcation solution{(a(s);U(s),V(s)):0<s<δ} given by Lemma 3.1 to the global bifurcation.
LetP1={u∈C01(Ω-):u(x)>0,x∈Ω,∂u/∂n<0,x∈∂Ω},P={(u,v,a)∈X×R+:u,v∈P1}.Theorem 3.3.
Suppose thatc>λ1; then the global bifurcation C of bifurcating branch of positive solutions of (1.2) becomes unbounded by a going to infinity in P.Proof.
Let(3.6)T′(a)·(ω,χ)=D(ω,χ)T(a;0,0)·(ω,χ)=(aKω-bK(ωθc1+kθc),cKχ-2K(χθc)+dKωθc1+kθc).
Suppose that μ≥1 is an eigenvalue of T′(a). Then we have
(3.7)-μΔω=(a-bθc1+kθc)ω,x∈Ω,-μΔχ=(c-2θc)χ+dθc1+kθcω,x∈Ω,ω=χ=0,x∈∂Ω.
Clearly ω≢0, otherwise, ω≡0, since all eigenvalue of the operator (-μΔ-c+2θc) is greater than 0, so χ≡0, a contradiction. Therefore, for some i such that a=ai(μ) is the eigenvalue of the following problem:
(3.8)-μΔω+bθc1+kθcω=aω,ω|∂Ω=0.
It is well known that ai(μ) is increasing with respect to μ on [1,+∞) and can be ordered as
(3.9)0<a1(μ)<a2(μ)≤a3(μ)≤⋯→∞,a1(1)=a~.
On the other hand, if μ≥1, then all eigenvalues of (-μΔ-c+2θc) are greater than 0; furthermore, χ=(-μΔ-c+2θc)-1(dθc/(1+kθc)ω). Thus, μ≥1 is the eigenvalue of T'(a) if and only if there exists some i, such that a=ai(μ).
Suppose thata<a~. Then for any μ≥1,i≥1,a<a1(1)≤ai(μ). Hence, T'(a) has no eigenvalue greater than 1, and index(T(a;·),0)=1 as a<a~.
Suppose thata~<a<a2(1). Then for any μ≥1,i≥2,a<ai(μ). Since a1(1)=a~, limμ→∞a1(μ)=+∞, and a1(μ) is increasing with respect to μ. Hence, there exists a unique μ1>1, such that a=a1(μ1). So N(μ1I-T'(a))=span{(ω¯,χ¯)},dimN(μ1I-T'(a))=1, where ω¯>0 is the principal eigenvalue of the following problem:
(3.10)μ1Δω¯+(a-bθc1+kθc)ω¯=0,ω¯|∂Ω=0,
where χ¯=(-μ1Δ-c+2θc)-1(dθc/(1+kθc)ω¯).
In the following, we will prove thatR(μ1I-T'(a))∩N(μ1I-T'(a))=0. In fact, if the assertion is false, we may assume that (ω¯,χ¯)∈R(μ1I-T'(a)). Then there exists (ω,χ)∈X, such that (μ1I-T'(a))(ω,χ)=(ω¯,χ¯), that is,
(3.11)μ1Δω+(a-bθc1+kθc)ω=Δω¯,ω|∂Ω=0.
Multiplying the equation by ω¯, integrating over Ω, and using Green's formula, we obtain
(3.12)∫Ω‍ω¯Δω¯=∫Ω‍(μ1Δω+aω-bωθc1+kθc)ω¯=∫Ω‍(μ1Δω¯+aω¯-bω¯θc1+kθc)ω=0,
which leads to ∫Ω‍(1/μ1)(a-bθc/(1+kθc))ω¯2=0, a contradiction. This proves the assertion, so it is verified that the multiplicity of μ1 is one and index(T(a;·),0)=-1 for a~<a<a2(1). According to global bifurcation theory [12], there exists a continuum C0 of zeros of G(a;ω,χ)=0 in R+×X bifurcating from (a~;0,0), and all zeros of G(a;ω,χ) close to (a~;0,0) lie on the curve whose existence was proved by Lemma 3.1. Let C1 be the maximal continuum defined by C1=C0-{(a(s);s(Φ+ϕ(s)),s(Ψ+φ(s))):-δ<s<0}. Then, C1 consists of the curve {(a(s);s(Φ+ϕ(s)),s(Ψ+φ(s))):-δ<s<0} in the neighborhood of the bifurcation point (a~;0,0). Let C={(a;u,v):U=ω,V=θc+χ,(ω,χ)∈C1}. Then C is the solution branch of (1.2) which bifurcates from (a~;0,θc) and remains positive in a small neighborhood of (a~;0,θc) and C⊂P. Thus the continuum C-{(a~;0,θc)} must satisfy one of the following three alternatives.(i)
C contains in its closure points (a~;0,θc) and (a-;0,θc), where I-T′(a-) is not invertible, and a~≠a-.(ii)
C is joining up from (a~;0,θc) to ∞ in R×X.(iii)
C is containing points of the form (a;u,θc+v) and (a;-u,θc-v), where (u,v)≠(0,0).
Next, we prove thatC-{(a~;0,θc)}⊂P. Assume that C-{(a~;0,θc)}⊈P. Then there exists (a^;u^,v^)∈(C-{(a~;0,θc)})∩∂P and sequence {(an;un,vn)}⊂C∩P,un>0,vn>0 such that (an;un,vn)→(a^;u^,v^) when n→∞. It is easy to get that u^∈∂P1 or v^∈∂P1. Suppose u^∈∂P1, then u^≥0,x∈Ω¯. Hence, we find either x0∈Ω such that u^(x0)=0 or x0∈∂Ω such that ∂u^/∂n∣x0=0. Since u^ satisfies
(3.13)-Δu^=(a^-u^-bv^(1+mu^)(1+kv^))u^,u^|∂Ω=0.
It follows from the maximum principle that u^≡0. Similarly, we can show that v^≡0 for v^∈∂P1.
Thus we only need the following three cases:(3.14)(i)(u^,v^)≡(θa^,0),(ii)(u^,v^)≡(0,θc),(iii)(u^,v^)≡(0,0).
Suppose that(u^,v^)≡(θa^,0). Then (an;un,vn)→(a^;θa^,0) when n→∞. Let Vn=vn/||vn||∞; then Vn satisfies
(3.15)-ΔVn=(c-vn+dun(1+mun)(1+kvn))Vn,Vn|∂Ω=0.
Thanks to LP estimates and Sobolev embedding theorem, there exists a convergent subsequence of Vn, which we still denote by Vn, such that Vn→V in C01(Ω¯) as n→∞, and V≥0,≢0,x∈Ω because of ∥V∥=1. So taking the limit in (3.15) as n→∞, we get(3.16)-ΔV=(c+dθa^1+mθa^)V,V|∂Ω=0.
It follows from the maximum principle that V>0,x∈Ω, which implies c=λ1(-dθa^/(1+mθa^)). This contradicts c>λ1.
Suppose that(u^,v^)≡(0,θc). Then (an;un,vn)→(a^;0,θc) as n→∞. Let Un=un/||un||∞; then Un satisfies
(3.17)-ΔUn=(an-un-bvn(1+mun)(1+kvn))Un,Un|∂Ω=0.
Similarly, By LP estimates and and Sobolev embedding theorem, there exists a convergent subsequence of Un, which we still denote by Un, such that Un→U in C01(Ω¯) as n→∞, and U≥0,≢0,x∈Ω because of ∥V∥=1. So taking limit in (3.17) as n→∞, we obtain(3.18)-ΔU=(a^-bθc1+kθc)U,U|∂Ω=0.
It follows from the maximum principle that U>0,x∈Ω. Hence a^=λ1(bθc/(1+kθc)). A contradiction with a^≠a~.
Suppose that(u^,v^)≡(0,0). Similar to the previously mentioned, we can get contradiction.
ThusC-{(a~;0,θc)}⊂P. By Lemma 2.1, we have 0≤U≤a,θc≤V≤c+da/(1+αa). Thanks to LP estimates and and Sobolev embedding theorem, then there exists a constant M>0 such that ||U||C1,||V||C1≤M. Hence the global bifurcation C of positive solutions of (1.2) bifurcating at (a~;0,θc) contains points with a is arbitrarily large in P.In the following, we will study the stability of the bifurcation solution. LetX1=[C2,α(Ω-)×C2,α(Ω-)∩X],Y=[Cα(Ω-)×Cα(Ω-)], where 0<α<1.i:X1→Y is the inclusion mapping. Since L1 is the linearized operator at (a~;0,θc) for (1.2). By the proof of Lemma 3.1, we have N(L1)= span{(Φ,Ψ)}, CodimR(L1)=1, and R(L0)={(u,v)∈X:∫Ω‍uΦdx=0}. Since i(Φ,Ψ)∈R(L1), so it follows from [21] that 0 is an i-simple eigenvalue of L1.Lemma 3.4.
0 is the eigenvalue of L1 with the largest real part, and all the other eigenvalue of L1 lie in the left half complex plane.Proof.
We assume thatμ0 is the eigenvalue of L1 with the largest real part Reμ0>0, and (ξ,η) is the corresponding eigenfunction; then L1(ξ,η)=μ0(ξ,η), equivalently,
(3.19)Δξ+(a~-bθc1+kθc)ξ=μ0ξ,x∈Ω,Δη+(c-2θc)η+dθc1+kθcξ=μ0η,x∈Ω,η=ξ=0,x∈∂Ω.
Suppose thatξ≡0. Then μ0 is an eigenvalue of the operator (Δ+(c-2θc)I), and then μ0∈R and μ0<0, a contradiction. Thus Φ≡0. It follows that μ0 is an eigenvalue of the operator (Δ+(a~-bθc/(1+kθc))I). Since a~=λ1(bθc/(1+kθc)),0 is the principal eigenvalue of the operator (Δ+(a~-bθc/(1+kθc))I); furthermore, μ0≤0. This contradicts the assumption, so assumption does not hold. Which proves our conclusion.We will use the linearized stability theory from [22]. Let L(u(s),v(s),a(s)), and L(a;0,θc) be the linearized operators of (1.2) at (u(s),v(s),a(s)), and (a;0,θc), respectively. It follows that from Lemma 7, Corollary 1.13, and Theorem 1.16 [22, 23] that Lemma 3.5 holds.Lemma 3.5.
There existsC1-function: a→(M(a),γ(a)), and s→(N(s),π(s)), defined from the neighborhood of a~ and 0 into X1×R, respectively, such that γ(a~)=π(0)=0,M(a~)=N(0)=(Φ,Ψ) and
(3.20)L(a;0,θc)M(a)=γ(a)M(a),for|a-a~|≪1,L(u(s),v(s),a(s))N(s)=π(s)N(s),for|s|≪1,
where M(a)=(ϕ1(a),ϕ2(a)),N(s)=(φ1(a)φ2(a)). Moreover γ′(a~)≠0, whence π(s)≠0,π(s) and -sa′(s)γ′(a~) have the same sign for |s|≪1. Where γ′(a~) is the derivative of γ(a) with respect to a at a=a~, and a′(s) is the derivative of a(s) with respect to s.Lemma 3.6.
The derivative ofγ(s) with respect to a at a~ is positive.Proof.
It follows fromL(a;0,θc)M(a)=γ(a)M(a),for,|a-a~|≪1, that is,
(3.21)Δϕ1+(a-bθc1+kθc)ϕ1=γ(a)ϕ1,x∈Ω,Δϕ2+(c-2θc)ϕ2+dθc1+kθcϕ1=γ(a)ϕ2,x∈Ω,ϕ1=ϕ2=0,x∈∂Ω.
Since |a-a~|≪1, so |γ(a)|≪1. Clearly, ϕ1≢0, otherwise, ϕ1≡0, then ϕ2≡0, a contradiction. Hence γ(a) is an eigenvalue of the operator (Δ+(a-bθc/(1+kθc))I). We consider Φ>0, then ϕ1=ϕ1(a)>0, as |a-a~|≪1. So γ(a) is the principal eigenvalue of the operator (Δ+(a-bθc/(1+kθc))I), and γ(a) is increasing with respect to a as |a-a~|≪1. Moreover, γ'(a~)≠0. Hence γ'(a~)>0.Lemma 3.7.
The derivative ofa(s) with respect to s at s=0 satisfies
(3.22)a′(0)∫Ω‍Φ2dx=∫Ω‍(1-bmθc1+kθc)Φ3dx+∫Ω‍bΨ(1+kθc)2Φ2dx.Proof.
By substituting(u(s),v(s),a(s)) into (1.2), differentiating with respect to s, and then setting s=0, we find that
(3.23)-Δϕ'(0)=(a~-bθc1+kθc)ϕ'(0)+[a′(0)-Φ-bΨ-mθcΦ(1+kθc)(1+kθc)2]Φ,
where ϕ′(0) is the derivative of ϕ with respect to s at s=0.
Taking the inner product withΦ, using Green's formula, and noting the definition of Φ, we have
(3.24)a′(0)∫Ω‍Φ2dx=∫Ω‍(1-bmθc1+kθc)Φ3dx+∫Ω‍bΨ(1+kθc)2Φ2dx.It follows from Lemmas3.4–3.7 that we obtain the following Theorem.Theorem 3.8.
Letσ=∫Ω‍(1-bmθc/(1+kθc))Φ3dx+∫Ω‍(bΨ/(1+kθc)2)Φ2dx. If σ>0, then bifurcation solution (u(s),v(s)) is stable; If σ<0, then bifurcation solution (u(s),v(s)) is unstable.In Section2, from Theorems 2.5–2.6, we can obtain the sufficient condition and necessary condition on the existence of positive solution and find that there exists a gap between a>λ1(bθc/(1+kθc)) and a>λ1(bθc/(1+mθa)(1+kθc)) when c>λ1. Next, we will consider the multiplicity, stability, and uniqueness of positive solutions in the gap.Theorem 3.9.
Assumec>λ1 and ∫Ω‍(1-bmθc/(1+kθc))Φ3dx<0. Then there exist ε>0 such that bifurcation positive solution (u(s),v(s)) is nondegenerate and unstable for a∈(a~-ε,a~) and d≪1. Moreover, Problem (1.2) has at least two positive solutions.Proof.
We first prove that bifurcation positive solution(u(s),v(s)) is nondegenerate and unstable. To this end, it suffices to show that there exists a sufficiently small ε>0 such that for a∈(a~-ε,a~), any positive solution (u(s),v(s)) of (1.2) is nondegenerate and the linearized eigenvalue problem:
(3.25)-Δξ-[a(s)-u(s)-bv(s)(1+mu(s))2(1+kv(s))]ξ+bu(s)(1+mu(s))(1+kv(s))2η=μξ,x∈Ω,-Δη-[c-2v(s)+du(s)(1+mu(s))(1+kv(s))2]η-dv(s)(1+mu(s))2(1+kv(s))=μη,x∈Ω,ξ=η=0,x∈∂Ω,
has a unique eigenvalue μ* and Re(μ*)<0 with algebra multiplicity one.
Let{εi>0} and {di>0} be sequences which approach 0 as i→∞. Due to a=a~+a′(0)s+O(s2), we can set sequences {εi>0} and {ai} such that ai∈(a~-εi,a~) and si→0 as i→∞. It follows that (ui,vi) is a solution of (1.2). Then the corresponding linearized problem (3.25) can become the following form:
(3.26)Li(ξiηi)=μi(ξiηi),Li=(Mi11Mi12Mi21Mi22),
where (ξi,ηi)≢(0,0) and
(3.27)Mi11=-Δ-[ai-2ui-bvi(1+mui)2(1+kvi)],Mi12=bui(1+mui)(1+kvi)2,Mi21=-dvi(1+mui)2(1+kvi),Mi22=-Δ-[c-2vi+dui(1+mui)(1+kvi)2].
Observe that as i→∞,  Li(ξiηi) converges to(3.28)L0(ξη)=(-Δξ-(a~-bθc1+kθc)ξ00-Δη-(c-2θc)η).
It is easy to get that 0 is a simple eigenvalue of the operator L0 with corresponding eigenfunction (ξ,η)T=(Φ,0)T. Moreover, all the other eigenvalues of L0 are positive and stand apart from 0. Therefore, using perturbation theory [24], we get that for large i,Li has a unique eigenvalue μi which is close to zero. In addition, all the other eigenvalues of Li have positive real parts and stand apart from 0. Note that μi is simple real eigenvalue which converges to zero, and we can take the corresponding eigenfunction (ξi,ηi)T such that (ξi,ηi)→(Φ,0) as i→∞. If we show that Reμi<0 for large i, then the result follows. By multiplying Φ to the first equation of Li(ξi,ηi)T=μi(ξi,ηi)T and integrating over Ω, we obtain
(3.29)-∫Ω‍ΦΔξi-∫Ω‍(ai-2ui-bvi(1+mui)2(1+kvi))Φξi+∫Ω‍buiΦηi(1+mui)(1+kvi)2=∫Ω‍μiΦξi.
Multiplying the first equation of (1.2) with (a,u,v)=(ai,ui,vi) by ξi and integrating, we have
(3.30)-∫Ω‍ξiΔui-∫Ω‍(ai-ui-bvi(1+mui)(1+kvi))uiξi.
Due to ui=siΦ+O(si2), the previous equation becomes
(3.31)-∫Ω‍ΦΔξi-∫Ω‍ξiΦ(ai-ui-bvi(1+mui)(1+kvi))+O(si2).
Using (3.29) and (3.31), we have
(3.32)∫Ω‍[1-bmvi(1+mui)2(1+kvi)]uiΦξi+∫Ω‍buiΦηi(1+mui)(1+kvi)2=∫Ω‍uiΦξi.
Recall that (ui,vi)=(siΦ+O(si2),θc+siΨdi+O(si2)), here Ψdi is Ψ defined in Remark 3.2, and so dividing the previous equation by si and taking the limit, we have
(3.33)limi→∞μisi=∫Ω‍(1-bmθc/(1+kθc))Φ3∫Ω‍Φ2<0,
which implies that Reμi<0 for large i. This proved our claim.
Next, we apply the method in [25] to show the remaining part of Theorem 3.9. A contradiction argument will be used; we assume that (1.2) has a unique positive solution (u~,v~), then this solution must be bifurcated from (0,θc). Since there exists a positive solution near a~ by the local bifurcation theory. So (u~,v~) is nondegenerate, and the corresponding linearized eigenvalue problem has a unique eigenvalue μ~ with algebra multiplicity one such that Reμ~<0. Due to these facts, it is easy to show that I-F'(u~,v~) is invertible and does not have property α on W¯(u~,v~); it follows that index(F,(u~,v~))=(-1)1=-1 by Lemma 2.2 (ii). Finally, Using Lemmas 2.3–2.4 and the additivity property of the index, we obtain
(3.34)1=indexW(F,D)=indexW(F,(0,0))+indexW(F,(θa,0))+indexW(F,(0,θc))+indexW(F,(u~,v~))=0+0+1-1=0,
which derives a contradiction. Hence the Proof is complete.Remark 3.10.
In Theorem3.9, the multiplicity can be shown easily when m≥0. Note that a'(0)<0 for a sufficiently small d, since ∫Ω‍(1-bmθc/(1+kθc))Φ3<0, and so a=a(s)∈(λ1,a~). Since there is no positive solution of (1.2) if a≤λ1(bθc/(1+mθa)(1+kθc)) by Theorem 2.5 (i) when m≥0 and c>λ1. Therefore we easily see that there must be at least two positive solutions for a∈(a*,a~) and some a*∈(λ1(bθc/(1+mθa)(1+kθc)),λ1(bθc/(1+kθc))).


## 4. Multiplicity, Uniqueness and Stability ask or m Is Large

In this section, takingk or m as a parameter, we investigate the multiplicity, stability, and uniqueness of positive solutions of (1.2) as k or m is large. In the following, we will always assume that c>λ1, and let b,c,d be fixed, unless otherwise specified.Firstly, we consider the case thatk is large and a is bounded away from λ1. Hence the upper solution θa for u and the lower solution θc for v do not depend on k when a>λ1 and c>λ1.Lemma 4.1.
For any smallε>0, there exists K(ε) such that for k≥K(ε), (1.2) has at least one positive solution (u,v) which satisfies
(4.1)θ(a-ε)≤u≤θa,θc≤v≤θ(c+ε).Proof.
Since the proof of Lemma4.1 is similar to the proof of Lemma 3 in [19], we omit it.Lemma 4.2.
(
i
) If k→∞, then any positive solution of (1.2) approaches (θa,θc).
(
i
i
) There exists a large K(ϵ) such that any positive solution of (1.2) is nondegenerate and linearly stable when k>K(ϵ).Proof.
(i) Letk→∞, we show that the compact operator  F(u,v) converges to  F^(u,v), where
(4.2)F^(u,v)=(-Δ+P)-1(u(a-u)+Puv(c-v)+Pv).
It follows that any positive solution of (1.2) converges to the fixed point of  F^(u,v) in this case. It is easy to see that (θa,θc) is a unique fixed point of  F^(u,v), so the positive solutions of (1.2) could not converge to semitrivial solutions when k→∞. Thus the conclusion is complete.
(ii) We use a contradiction method. Assume that there existski→∞, μi with Reμi≤0 and (ξi,ηi)≢(0,0) with ∥ξi∥L22+∥ηi∥L22=1 such that
(4.3)-Δξi-[a-2ui-bvi(1+mui)2(1+kivi)]ξi+bui(1+mui)(1+kivi)2ηi=μiξi,x∈Ω,-Δηi-[c-2vi+dui(1+mui)(1+kivi)2]ηi-dvi(1+mui)2(1+kivi)ξi=μiηi,x∈Ω,ξi=ηi=0,x∈∂Ω,
where (ui,vi) is a positive solution of (1.2) with k=ki. By computing, we have
(4.4)μi=∫Ω‍|∇ξi|2-∫Ω‍(a-2ui-bvi(1+mui)2(1+kivi))|ξi|2+∫Ω‍buiηiξ¯i(1+mui)(1+kivi)2+∫Ω‍|∇ηi|2-∫Ω‍dviξiη¯i(1+mui)2(1+kivi)-∫Ω‍(c-2vi+dui(1+mui)(1+kivi)2)|ηi|2,
where ξ¯i,η¯i are the complex conjugates of  ξi,ηi. From Lemma 2.1, we know that 0≤un≤θa and θc≤vi≤c+da/(1+ma). It follows that Imμi is bounded and Reμi is bounded from the following. Thus μi is bounded as we assume Reμi≤0. Hence we can suppose that μi→μ and Reμ≤0. Thanks to Lp estimate, we obtain ∥ξi∥W2,2 and ∥ηi∥W2,2 are bounded. Hence we may assume that  ξi→ξ and ηi→η in H01 strongly, here (ξ,η)≢(0,0). Setting i→∞ in (3.1) and (3.2), we know that ξ,η satisfy the following two single equations weakly (then strongly):(4.5)-Δξ-(a-2θa)ξ=μξ,x∈Ω,-Δζ-(c-2θc)η=μη,x∈Ω,ξ=η=0,x∈∂Ω.
Clearly μ∈R. If ξ≢0, then μ=λ1(-a+2θa)>λ1(-a+θa)=0. However, Reμ≤0. Hence ξ≡0. Similarly, we have η≡0, a contradiction; hence the proof is complete.Theorem 4.3.
(
i
) Assume that λ1<a<λ1(bθc/(1+kθc)). Then (1.2) has no positive solution when k is sufficiently large.
(
i
i
) Assume that a>λ1(bθc/(1+kθc)) and k is sufficiently large. Then (1.2) has a unique positive solution, and it is asymptotically stable.Proof.
(i) Assume that there exists a positive solution(u*,v*) of (1.2) for sufficiently large k. It is easy to show that indexW(F,(u*,v*))=1 by Lemma 4.2. By Lemmas 2.3–2.4 and the additivity property of the index, we obtain
(4.6)1=indexW(F,D′)=indexW(F,(0,0))+indexW(F,(θa,0))+indexW(F,(0,θc))+indexW(F,(u*,v*))=0+0+1+1=2,
which gives a contradiction.
(ii) From Theorem2.6, the existence is trivial. Since a>λ1(bθc/(1+kθc)) and c>λ1, for k sufficiently large, the nondegenerate positive solutions may not converge to semitrivial solutions by the proof of Lemma 4.2 (i). We need only to show the uniqueness. It follows from compactness and nondegeneracy that F has at most finitely many positive fixed points in the region D′. We denote them by (ui,vi) for i=1,2,…,l. From the proof of (i), we obtain indexW(F,(ui,vi))=1. Applying Lemmas 2.3–2.4 and the additivity property of the index again, we have
(4.7)1=indexW(F,D')=indexW(F,(0,0))+indexW(F,(θa,0))+indexW(F,(0,θc))+∑i=1l‍indexW(F,(ui,vi))=0+0+0+l=l.
Hence the uniqueness is obtained. The stability has been given in Lemma 4.2.In the following, we investigate the case whenm is large. This part is motivated by the work of Du and Lou in [13, 14, 18], and many of our methods used nextly come from their work.Theorem 4.4.
For anyϵ>0 to be small, there that exists M=M(ϵ) is large such that for m≥M,(i)
ifa∈[λ1+ϵ,λ1(bθc/(1+kθc))), then (1.2) has at least two positive solutions;(ii)
ifa≥λ1(bθc/(1+kθc)), then (1.2) has a unique positive solution and it is asymptotically stable.Theorem4.4 is the main result we will prove that when m is large. The cases a∈[λ1+ϵ,λ1(bθc/(1+kθc))) and a≥λ1(bθc/(1+kθc)) will be treated separately. First, we deal with the multiplicity in Theorem 4.4. Similar to the method of [19], if a∈[λ1+ϵ,∞), and m is sufficiently large, then we can get the following result on (1.2).Lemma 4.5.
For any given smallϵ, there exists that M=M(ϵ) such that if a≥λ1+ϵ and m≥M, (1.2) has a positive solution (u~,v~), which satisfy
(4.8)θ(a-ϵ/2)≤u~≤θa,θc≤v~≤θ(c+ϵ/2).Lemma 4.6.
For anyϵ>0 to be small and any A^>λ1, there exists that M=M(ϵ,A^)>0 is large such that if a∈(λ1+ϵ,A^] and m≥M, then any positive solution which satisfies (4.8) is nondegenerate and linearly stable.Proof.
Assume thata∈(λ1+ϵ,A^] and m is large, then we can easily get that (1.2) is a regular perturbation of
(4.9)-Δu-(a-u)u=0,x∈Ω,-Δv-(c-v)v=0,x∈Ω,u=v=0,x∈∂Ω.
It is well know that (4.9) has a unique positive solution (θa,θc) which is linearly stable. So the positive solutions cannot bifurcate from semitrivial ones. Hence, Lemma 4.6 is proved by a standard regular perturbation argument. We omit the details.Proof of (i) of Theorem4.4.
For anyϵ>0 to be small, let
(4.10)M=max{M(ϵ),M(ϵ,λ1(bθc1+kθc))},
where M(ϵ) and M(ϵ,λ1(bθc/(1+kθc))) are defined in Lemmas 4.5 and 4.6, respectively. Assume that for some m≥M and some a∈[λ1+ϵ,λ1(bθc/(1+kθc))), Thus the unique positive solution (u~,v~) must be the one found in Lemma 4.5. It follows from Lemma 4.5 that I-F′(u~,v~) is invertible in X and F′(u~,v~) has no eigenvalue greater than one. Hence indexW(F,(u~,v~))=(-1)0=1. Applying Lemmas 2.3–2.4 and the additivity property of the index, we obtain
(4.11)1=indexW(F,D')=indexW(F,(0,0))+indexW(F,(θa,0))+indexW(F,(0,θc))+indexW(F,(u~,v~))=0+0+1+1=2,
a contradiction. Hence the proof if complete.Part (i) in Theorem4.4 implies that when a∈[λ1+ϵ,λ1(bθc/(1+kθc))) and m is large, (1.2) has at least two positive solutions. In the following, we will show that (1.2) has only two types of positive solutions in this case, one of which is close to (θa,θc) and asymptotically stable and the other is close to (0,θc) which is not stable.Theorem 4.7.
For anyϵ,δ>0 to be small, there exists that  M=M(ϵ,δ)>0 is large such that if m≥M and a∈[λ1+ϵ,λ1(bθc/(1+kθc))), one obtains either (i)∥u-θa∥C1+∥v-θc∥C1≤δ or (ii)∥u∥C1+∥v-θc∥C1≤δ, where (u,v) is any positive solution of (1.2). In particular, if (ii) occurs, by choosing M(ϵ,δ) suitably larger, one gets ∥mu-w∥C1≤δ, where w is a positive solution of the following equation:
(4.12)-Δw-(a-bθc(1+w)(1+kθc))w=0,w|∂Ω=0.Proof.
Suppose that the conclusion does not hold. Then there existmi→∞ai∈[λ1+ϵ,λ1(bθc/(1+kθc))) and a positive solution (ui,vi) of (1.2) with (a,m)=(ai,mi) such that (ui,vi) is bounded away from (θa,θc) and (0,θc). We may assume that ai→a∈[λ1+ϵ,λ1(bθc/(1+kθc))] and (1/(1+miui)(1+kvi))⇀L2h with 0<h<1. Since θc≤vi≤θ(c+d/mi), we have vi→C1θc as i→∞. Thanks to Lp estimate and Sobolev embedding theorems, we may assume that ui→C1u and u satisfies
(4.13)Δu+(a-u-bθch)u=0,u≥0,u|∂Ω=0.
If u≡0, then (ui,vi)→C1(u,θc)≡(0,θc), which contradicts our assumption that (ui,vi) is bounded away from (0,θc). If u≥0,≢0, from maximum principle, we have u>0 in Ω. Hence h≡0 and u≡θa, which also contradicts our assumption. Thus, the first part of the proof is complete.
To complete the proof, it suffices to prove that ifmi→∞,ai∈[λ1+ϵ,λ1(bθc/(1+kθc))) and ∥ui∥C1+∥vi-θc∥C1→0, then miui approaches some positive solution of (4.12) with a=ai in the C1 norm. It is easy to see that (4.12) has a positive solution when a∈[λ1+ϵ,λ1(bθc/(1+kθc))). First we claim that mi∥ui∥∞ is uniformly bounded. If this is not true, we may assume that mi∥ui∥∞→∞. Let u~i=ui/∥ui∥∞. Then we have
(4.14)Δu~i+(ai-ui-bvi(1+miui)(1+kvi))u~i=0,∥u~i∥∞=1,u~i|∂Ω=0.
Thanks to standard elliptic regularity theory, we may assume that u~i→C1u~, (1/((1+miui)(1+kvi)))⇀L2h and ai→a∈[λ1+ϵ,λ1(bθc/(1+kθc))]. By taking the limit in (4.12), we know that u~ satisfies the following equation weakly:
(4.15)Δu~+(a-bθch)u~=0,∥u~∥∞=1,u~|∂Ω=0.
By Harnack's inequality, we have u~>0 in Ω. Since mi∥ui∥∞→∞ and u~i→u~, then 1/(1+miui)(1+kvi)=1/(1+mi∥ui∥∞u~i)(1+kvi)→0 in any compact subset of Ω. Hence h≡0 and a=λ1, which contradicts the assumption that a≥λ1+ϵ. Therefore mi∥ui∥∞ is uniformly bounded.
Letwi=miui. Then wi satisfies
(4.16)Δwi+(ai-ui-bvi(1+wi)(1+kvi))wi=0,wi|∂Ω=0.
Since ∥wi∥∞=mi∥ui∥∞ is bounded, due to standard elliptic regularity theory and Sobolev embedding theorems, we may assume that wi→C1w. By letting i→∞ in (4.16), we know that w is a nonnegative solution of (4.12). There are two possibilities as follows.(i)
a=λ1(bθc/(1+kθc)). For this case, miui=wi→w≡0. Since any positive solution of (4.12) with a=ai approaches zero when ai→a, miui is certainly close to positive solutions of (4.12) with a=ai.(ii)
λ1+ϵ≤a<λ1(bθc/(1+kθc)). In this case, we will prove that w is a positive solution of (4.12). If not, by Harnack's inequality, we have w≡0. Let w~i=wi/∥wi∥∞. Then we have
(4.17)Δw~i+(ai-ui-bvi(1+wi)(1+kvi))w~i=0,w~i|∂Ω=0.Hence we may assume that w~i→C1w. By taking the limit in (4.17), we find
(4.18)Δw~+(a-bθc1+kθc)w~=0,w~|∂Ω=0.
Since a<λ1(bθc/(1+kθc)), we must have w~≡0, which contradicts ∥w~∥∞=limi→∞∥w~i∥∞=1. The proof is complete.The proof of (ii) in Theorem4.4 is more difficult. To this end, we need several lemmas. First we show that there is no positive solution of (1.2) with small u component if a≥λ1(bθc/(1+kθc)) and m is large.Lemma 4.8.
There exists a largeM such that if m≥M, then for all a≥λ1(bθc/(1+kθc)), any positive solution (u,v) of (1.2) satisfies u≥θλ~, where λ~=(λ1+λ1(bθc/(1+kθc)))/2.Proof.
Suppose that our conclusion is not true, then there existmi→∞,ai≥λ1(bθc/(1+kθc)), and a positive solution sequence {(ui,vi)} of (1.2) with (a,m)=(ai,mi) such that ui≥θλ~ does not hold.
First, letai→a∈(λ1(bθc/(1+kθc)),∞]. Since θc≤vi≤θ(c+d/mi), for large i, we obtain
(4.19)-Δui≥(a+λ1(bθc/(1+kθc))2-ui-bθ(c+δ)1+kθ(c+δ))ui,
where δ>0 is small such that λ1(bθ(c+δ)/(1+kθ(c+δ)))<(a+λ1(bθc/(1+kθc)))/2, which is possible when a>λ1(bθc/(1+kθc)). Due to the super- and subsolution method, we obtain ui≥w~, where w~ is a unique positive solution of
(4.20)-Δw~=(a+λ1(bθc/(1+kθc))2-bθ(c+δ)1+kθ(c+δ)-w~)w~,w~|∂Ω=0.
Hence for large i, we get
(4.21)-Δui≥(λ1(bθc1+kθc)-ui-bmisupΩθ(c+d/mi)w~)ui≥(λ~-ui)ui.
Applying the super- and subsolution method again, we obtain ui≥θλ~, which contradicts the assumption.
Secondly, we consider the case thatai→a=λ1(bθc/(1+kθc)). Thanks to standard elliptic regularity theory, we may assume that ui→u,vi→v in C1 and 1/(1+miui)(1+kvi) weakly converge to h in L2 with 0≤h≤1. Hence u satisfies the following equation weakly:
(4.22)-Δu=(λ1(bθc1+kθc)-bvh-u)u,u|∂Ω=0.
If u≥0,≢0, by Harnack's inequality, we have u>0 in Ω. Thus h≡0 and ui→C1θλ1(bθc/(1+kθc)). Since θλ1(bθc/(1+kθc))>θλ~, we have ui≥θλ~ for large i. This contradicts our assumption at the beginning of the proof. If u≡0, by letting u^i=ui/||ui||∞, then we know that u^i satisfies
(4.23)Δu^i+(ai-ui-bvi(1+miui)(1+kvi))u^i=0,u^i|∂Ω=0.
Thanks to standard elliptic regularity theory, we may assume that u^i→C1u^ with ∥u^∥∞=1 and u^>0 in Ω. Since ui→C1u≡0, we have vi→C1θc. By taking the weak limit in (4.23), we have
(4.24)Δu^+(λ1(bθc1+kθc)-bθch)u^=0,u^|∂Ω=0.
Let Φ>0 be the positive solution to
(4.25)-ΔΦ+bθc1+kθcΦ=λ1(bθc1+kθc)Φ,Φ|∂Ω=0,∥Φ∥∞=1.
Multiplying (4.24) by Φ and integrating, we obtain
(4.26)b∫Ω‍u^Φθc(11+kθc-h)=0.
Since 0≤h≤1/(1+kθc), we must have h≡1/(1+kθc) and u^≡Φ. Investigate the following equation for ui:
(4.27)Δui+(ai-ui-bvi(1+miui)(1+kvi))ui=0,ui|∂Ω=0.
Multiplying (4.27) by Φ and using (4.25), we obtain
(4.28)(ai-λ1(bθc1+kθc))∫Ω‍uiΦ=∫Ω‍ui2Φ+∫Ω‍buiviΦ(1+miui)(1+kvi)-∫Ω‍bθcuiΦ1+kθc.
After some rearrangements, we obtain
(4.29)(ai-λ1(bθc1+kθc))∫Ω‍uiΦ=∫Ω‍ui2Φ+b∫Ω‍ui(vi-θc)Φ(1+miui)(1+kvi)(1+kθc)-bmi∫Ω‍θcui2Φ(1+miui)(1+kθc).
Set wi=(vi-θc)/||ui||∞. By the definition of vi and θc, we have
(4.30)-Δwi+(-c+2θc)wi=(θc-vi)wi+dviu^i(1+miui)(1+kvi).
Multiplying (4.30) by wi and integrating, we obtain
(4.31)λ1(-c+2θc)∫Ω‍wi2≤∫Ω‍(|∇wi|2+(-c+2θ[c])wi2)≤∥vi-θc∥∞∫Ω‍wi2+d∥u^i∥∞∥vi∥∞∫Ω‍wi.
Since ||vi-θc||∞→0, ∥u^i∥∞ and ∥vi∥∞ are bounded, we know that ∥wi∥2 is bounded by Hölder inequality. Therefore from Lp estimate and Sobolev embedding theorem, it is easy to see that ||wi||∞ is bounded. Dividing (4.29) by ∥ui∥∞2, we show that
(4.32)ai-λ1(bθc/(1+kθc))∥ui∥∞∫Ω‍u^iΦ=∫Ω‍u^i2Φ+b∫Ω‍u^iwiΦ(1+miui)(1+kvi)(1+kθc)-bmi∫Ω‍θcu^i2Φ(1+miui)(1+kθc).
Since u^i→C1Φ>0,1/(1+miui)(1+kvi)⇀1/(1+kθc) weakly in L2 and wi are uniformly bounded, we obtain
(4.33)ai-λ1(bθc/(1+kθc))∥ui∥∞→-∞
as i→∞. Hence ai<λ1(bθc/(1+kθc)) if i is large enough, which contradicts our assumption that ai≥λ1(bθc/(1+kθc)) for all i. Thus, the proof is complete.By Lemma4.8 and a simple variant of the proof of Lemma 4.6, we immediately get the following result.Lemma 4.9.
For any givenA->λ1(bθc/(1+kθc)), there exists that M=M(A-)>0 is large such that if a∈[λ1(bθc/(1+kθc)),A-] and m≥M; then any positive solution of (1.2) is nondegenerate and linearly stable.Next we investigate the case thata is large.Lemma 4.10.
For anyϵ>0, there exists that A~=A~(ϵ)>0 is large such that if m≥ϵ,c≤1/ϵ and a>A~, then any positive solution of (1.2) is nondegenerate and linearly stable.Proof.
Suppose the conclusion is not true. Then there exist someϵ0>0,mi≥ϵ0,ci≤1/ϵ0,ai→∞,Reηi≤0, and (ϕi,φi)≢(0,0) with ∥ϕi∥22+∥φi∥22=1 such that
(4.34)Δϕi+(ai-2ui-bvi(1+miui)2(1+kvi))ϕi-bui(1+miui)(1+kvi)2φi+ηiϕi=0,(4.35)Δφi+(ci-2vi+dui(1+miui)(1+kvi)2)φi+dvi(1+miui)2(1+kvi)ϕi+ηiφi=0,
where (ui,vi) is a positive solution of (1.2) with (a,c,m)=(ai,ci,mi). Since ci>λ1-d/mi, we may assume that ci→c∈[λ1-d/ϵ0,(1/ϵ0)],mi→m∈[ϵ0,∞]. Let δ=b(1+d)/(ϵ0+k(1+d)). From vi≤θ(ci+d/mi)≤(1+d)/ϵ0, we have
(4.36)-Δui≥(ai-ui-bvi1+kvi)ui≥(ai-δ-ui).
Hence ui≥θ(ai-δ). Due to Kato's inequality, we have
(4.37)-Δ|ϕi|≤-Re(ϕ¯i|ϕi|Δϕi)≤(ai-2ui-bvi(1+miui)2(1+kvi))|ϕi|+bui|φi|(1+miui)(1+kvi)2+Reηi|ϕi|≤(ai-2ui)|ϕi|+b|φi|mi.
Multiplying (4.37) by |ϕi| and integrating by parts, we obtain
(4.38)λ1(-ai+2θ(ai-δ))∫Ω‍|ϕi|2≤λ1(-ai+2ui)∫Ω‍|ϕi|2≤∫Ω‍(|∇|ϕi||2+(-ai+2ui)|ϕi|2)≤bmi∫Ω‍|ϕi||φi|≤C.
By the method of Lemma 2.2 in [18], there exists k0∈(1,22/3) such that
(4.39)λ1(-ai+2θ(ai-δ))=-ai+λ1(2θ(ai-δ))≥-ai+k0(ai-δ)=(k0-1)ai-k0δ→+∞.
It follows from (4.38) and (4.39) that ∥ϕi∥2→0. Multiplying (4.34) by ϕ¯i and integrating, we have
(4.40)∫Ω‍|∇ϕi|2=∫Ω‍[ai-2ui-bvi(1+miui)2(1+kvi)]|ϕi|2-b∫Ω‍uiφiϕ¯i(1+miui)(1+kvi)2+ηi∫Ω‍|ϕi|2.
Multiplying (4.35) by φ¯i and integrating, we have
(4.41)∫Ω‍|∇φi|2=∫Ω‍[ci-2vi+dui(1+miui)(1+kvi)2]|φi|2+d∫Ω‍viϕiφ¯i(1+miui)2(1+kvi)+ηi∫Ω‍|φi|2.
Adding the previous two identities, we have
(4.42)ηi=∫Ω‍|∇ϕi|2-∫Ω‍[ai-2ui-bvi(1+miui)2(1+kvi)]|ϕi|2+b∫Ω‍uiφiϕ¯i(1+miui)(1+kvi)2+∫Ω‍|∇φi|2-∫Ω‍[ci-2vi+dui(1+miui)(1+kvi)2]|φi|2-d∫Ω‍viϕiφ¯i(1+miui)2(1+kvi).
It is easy to show that the imaginary part of the right-hand side of the previous identity is bounded. On the other hand, due to (4.38), (4.39) and the fact that ∫Ω‍|(∇|ϕi|)|2≤∫Ω‍|∇ϕi|2, we have Reηi is bounded nextly. Hence ηi is bounded as we assume that Reηi≤0. Thus we may assume that ηi→η with Reη≤0. Thanks to (4.35) and standard elliptic regularity theory, ∥φi∥W2,2 is bounded. Therefore we may assume that φi→φ in H01. Since ai→∞,θ(ai-δ)≤ui≤θ(ai) and θ(ai)/ai→1, we have ui→∞ and vi→θ(c+d/m) in C1. By letting i→∞ in (4.35), we know that φ satisfies the following equation weakly:(4.43)Δφ+(c+dm-2θ(c+d/m))φ+ηφ=0
with Reη≤0. The self-disjointness of the previous problem implies η∈R. Since λ1(θ(c+d/m)-(c+d/m))=0, we have
(4.44)η=λ1(2θ(c+d/m)-(c+dm))≥λ1(θ(c+d/m)-(c+dm))=0.
Hence η=0, which implies c+d/m=λ1,θ(c+d/m)≡0 and φ=βΦ1/∥Φ1∥2,|β|=1. Using Kato's inequality again, we have
(4.45)-Δ|φi|≤-Re(φ¯i|φi|Δφi)≤[ci-2vi+dui(1+miui)(1+kvi)2]|φi|+dvi|ϕi|(1+miui)2(1+kvi)+Reηi|φi|.
It follows that vi satisfies
(4.46)Δvi+(ci-vi+dui(1+miui)(1+kvi))vi=0,vi|∂Ω=0.
Multiplying (4.45) by vi, integrating by parts, and using (4.46), we obtain
(4.47)∫Ω‍vi2|φi|≤d∫Ω‍vi2|ϕi|(1+miui)2(1+kvi)+Reηi∫Ω‍vi|φi|≤d∫Ω‍vi2|ϕi|.
Set v^i=vi/||vi||∞. Then v^i satisfies
(4.48)Δv^i+(ci-vi+dui(1+miui)(1+kvi))v^i=0.
Thanks to standard elliptic regularity theory, we may assume that v^i→C1v^. Then v^ satisfies
(4.49)Δv^+(c+dm-θ(c+d/m))v^=0.
Since c+d/m=λ1, we have v^≡Φ1. Dividing both sides of (4.47) by ||vi||∞2, we know that
(4.50)∫Ω‍(vi∥vi∥∞)2|φi|≤d∫Ω‍(vi∥vi∥∞)2|ϕi|.
Since ∥ϕi∥2→0, the right-hand side of (4.50) converges to 0 by Hölder inequality. However, the previous discussion implies that ∫Ω‍(vi/||vi||∞)2|φi|→∫Ω‍(Φ13/||Φ1||2)(i→∞). The contradiction completes the proof.Proof of (ii) of Theorem4.4.
It suffices to prove the uniqueness. Investigate the following system witht∈[0,1]:
(4.51)Δu+u(a-u-tbv(1+mu)(1+kv))=0,x∈Ω,Δv+v(c-v+tdu(1+mu)(1+kv))=0,x∈Ω,u=v=0,x∈∂Ω.
Set S={(u,v)∈X∣θλ~/2<u<a,θc/2<v<c+ad} and Bt:S¯→W by
(4.52)Bt(u,v)=(-Δ+P)-1(u(a+P-u-tbv(1+mu)(1+kv))v(c+P-v+tdu(1+mu)(1+kv))),
where P=max{ad,b(c+ad)}. Thanks to standard regularity results, we can prove that Bt is a completely continuous operator. Clearly, (u,v) is a positive solution of (4.51) if and only if it is a positive fixed point of Bt in S. For m≥M and a≥λ1(bθc/(1+kθc)), we first show that Bt has no fixed point on ∂S.
We claim that any positive solution(u,v) of (4.51) satisfies u≥θλ~, where λ~=(λ1+λ1(bθc/(1+kθc)))/2. Suppose that this claim is not true. Then there exist mi→∞,ai≥λ1(bθc/(1+kθc)),ti∈[0,1], and a positive solution (ui,vi) of (4.51) with(a,m,t)=(ai,mi,ti) such that ui≥θλ~ fails. Since the case that ti→t0≡1 is considered in Lemma 4.8, it remains to discuss the case that ti→t0∈[0,1). Since vi≤θ(c+d/mi), we have
(4.53)-Δui≥(λ1(bθc1+kθc)-tibθ(c+d/mi)1+kθc-ui)ui≥(λ1(t-bθc1+kθc)-t0bθc1+kθc-ui)ui,
for large i, where t-∈(t0,1). Therefore ui is a supersolution to
(4.54)-Δω=(λ1(t-bθc1+kθc)-t0bθc1+kθc-ω)ω,ω|∂Ω=0.
Due to the choice of t-, (4.54) has a unique positive solution ω. Thus we have ui≥ω for all large i. Hence,(4.55)-Δui≥(λ1(bθc1+kθc)-bmisupΩθ(c+d/mi)ω-ui)ui≥(λ~-ui)ui.
By the super- and subsolution method again, we have ui≥θλ~ for large i. This is a contradiction.
Our claim implies thatBt has no fixed point on ∂S. Hence indexW(Bt,S)≡constant. In particular,
(4.56)indexW(B1,S)=indexW(B0,S).
Since B0 has a unique fixed point (θa,θc) in S and indexW(B0,(θa,θc))=1, we have indexW(B1,S)=1. From Lemmas 4.8–4.10, we see that for m≥M, all fixed points of B1 fall into S, and they are nondegenerate and linearly stable. Then by compactness, there are at most finitely many fixed points of B1, which we denote by {(ui,vi)}i=1l. As shown in the proof of part (i), we have indexW(B1,(ui,vi))=1. Using the additivity property of the index, we know that
(4.57)1=indexW(B1,S)=∑i=1l‍indexW(B1,(ui,vi))=l.
Hence for m≥M and a≥λ1(bθc/(1+kθc)), (1.2) has a unique positive solution, and it is stable.Our final task is to establish the exact multiplicity and stability results for largem and a close to λ1+ε or λ1(bθc/(1+kθc)). Firstly we consider the elliptic equation (4.12), which acts as a limiting problem of (1.2) when m→∞. Applying the similar method to Lemma 2.7 in paper [18], we have the following conclusion.Lemma 4.11.
The problem (4.12) has a positive solution if and only if λ1<a<λ1(bθc/(1+kθc)). Moreover, all positive solutions of (4.12) are unstable. Furthermore, there exists some ϵ1>0 such that if a∈(λ1,λ2]∪[λ1(bθc/(1+kθc))-ϵ1,λ1(bθc/(1+kθc))), then (4.12) has at most one positive solution and it is nondegenerate (if it exists).
Defineϵ0=min{λ2,λ1(bθc/(1+kθc))-ϵ1}-λ1, where ϵ1 is defined by Lemma 4.11.Theorem 4.12.
For anyϵ∈(0,ϵ0), one can find that M=M(ϵ) is large such that if a∈(λ1+ϵ,λ2+ϵ0]∪[λ1(bθc/(1+kθc))-ϵ1,λ1(bθc/(1+kθc))), and m>M(ϵ), then (1.2) has exactly two positive solutions, one asymptotically stable and the other unstable.To verify Theorem4.12, we need some intermediate results. Theorem 4.7 has shown that (1.2) has only two types of positive solutions for when m large and a∈[λ1(bθc/(1+kθc))-ϵ1,λ1(bθc/(1+kθc))). In the following lemma, we will prove another result.Lemma 4.13.
There exist thatϵ2>0 is small and  M1>0 is large; both depend only on b,c,d, and k, such that if a∈[λ1(bθc/(1+kθc))-ϵ2,λ1(bθc/(1+kθc))), and m>M1, then (1.2) has exactly two positive solutions, one asymptotically stable and the other unstable.Proof.
First we prove that for largem, (1.2) has a unique asymptotically stable positive solution of type (i) in Theorem 4.7. In fact, if we choose δ small enough in Theorem 4.7, then any positive solution of (1.2) of type (i) satisfies (4.8). Hence by Lemma 4.6, they are nondegenerate and linearly stable. Now by a simple variant of the proof of part (ii) of Theorem 4.4, we find that there is only one positive solution of (1.2) satisfying type (i), and it is asymptotically stable.
Next we show that (1.2) has a unique unstable positive solution of type (ii). If we can prove this, then by Theorem 4.7, our proof of Lemma 4.13 is complete. Due to Theorem 4.7 and Lemmas 4.11, if any solution (u;v) of (1.1) is close to (0,θc), then mu must be close to w, where w is the unique positive solution of (4.12). Hence to prove uniqueness, it suffices to show that for a∈[λ1(bθc/(1+kθc))-ϵ2,λ1(bθc/(1+kθc))), and m>M1, there is a unique pair (mu,v), (u;v) being a positive solution of (1.2), close to (w,θc) for certain ϵ2 and M1. Set U=mu, ρ=1/m, and discuss
(4.58)-ΔU=U(a-ρU-bv(1+U)(1+kv)),x∈Ω,-Δv=v(c-v+ρdU(1+U)(1+kv)),x∈Ω,U=v=0,x∈∂Ω.
Clearly (u,v) solves (1.2) if and only if (mu,v) solves (4.58) with ρ=1/m. Thus it suffices to prove uniqueness for (4.58). For fixed ρ≥0, regarding a as a parameter, we know that (λ1(bθc/(1+kθc)),0,θc) is a simple bifurcation point of (4.58). Due to a variant of Theorem 1 in [21], there exist δ1>0 and C1 curves
(4.59)Γρ={(a(ρ,s),U(ρ,s),v(ρ,s)):0<s≤δ1},0≤ρ≤δ1,
such that, if 0≤ρ≤δ1, then all positive solutions of (4.58) are close to
(4.60)(λ1(bθc1+kθc),0,θc)=(a(0,0),U(0,0),v(0,0))∈Γρ.
Hence we need only to prove that these curves uniformly cover a-Range[λ1(bθc/(1+kθc))-ϵ2,λ1(bθc/(1+kθc))), for suitably chosen ϵ2, and for fixed ϵ2 and ρ,Γρ cover the range only once. It is easy to obtain (see Theorem 3.9)
(4.61)∂a∂s(0,0)<0,0≤ρ≤δ1,0<s≤δ1.
Hence
(4.62)λ1(bθc1+kθc)-a(0,δ1)=a(0,0)-a(0,δ1)>0.
By the continuity of a(ρ,s), there exist δ∈(0,δ1] such that
(4.63)ϵ2=min0≤ρ≤δ[λ1(bθc1+kθc)-a(ρ,δ1)]>0.
Therefore, if a≥λ1(bθc/(1+kθc))-ϵ2, then for any ρ∈(0,δ], a(ρ,δ1)≤a. This implies that for ρ∈(0,δ], Γρ covers the a-range[λ1(bθc/(1+kθc))-ϵ2,λ1(bθc/(1+kθc))). Moreover, since ∂a/∂s≠0 for 0≤ρ,s≤δ1, each curve covers the range only once. By choosing M1=1/δ, we get that for m>M1 and a∈([λ1(bθc/(1+kθc))-ϵ2,λ1(bθc/(1+kθc))), (1.2) has exactly one positive solution of type (ii) in Theorem 4.7.
It remains to show that the positive solution of (1.2) close to (0,θc) is unstable. In fact, when m is sufficiently large, applying the method of the proof in Theorem 3.9, we can show that the positive solution of (1.2) close to (0,θc) is unstable. we omit the proof procedure.Proof of Theorem4.12.
By Lemma4.13, it suffices to establish the exact multiplicity and stability when a∈I=[λ1+ϵ,λ1+ϵ0]∪[λ1(bθc/(1+kθc))-ϵ1,λ1(bθc/(1+kθc))-ϵ2) for any given ϵ∈(0,ϵ0), where ϵ2 is defined in Lemma 4.13
From Theorem4.7 we know that the solutions of (1.2) for a∈[λ1+ϵ,λ1(bθc/(1+kθc))) and m large are of two types, that is, types (i) or (ii). As in the proof of Lemma 4.13, we can prove that there is a unique asymptotically stable positive solution of type (i). Thus to end the proof, we need only to show that there is a unique unstable positive solution of (1.2) close to (0,θc) if a∈I and m is large. Again by Lemma 4.11, it suffices to prove that there is a unique unstable positive solution (u,v) of (1.2) such that (mu,v) is close to (wa,θc), where wa is the unique positive solution of (4.12) as shown in Lemma 4.11. In this connection, we investigate (4.58) with a∈I and ρ small. Let a*∈I. Since the unique solution wa* of (4.12) with a=a* is nondegenerate, then (wa*,θc) is a nondegenerate solution of (4.58) with (a,ρ)=(a*,0). Clearly, (4.58) with ρ>0 small is a regular perturbation of (4.58) with ρ=0, and the perturbation is uniform for a in the compact set I. Thus it follows from the implicit function theorem that there exist δ,ϵ->0 small such that for any a∈I,0≤ρ≤ϵ-, (4.58) possesses a unique positive solution (u-a,va) which satisfies
(4.64)∥u-a-wa∥+∥va-θc∥≤δ.
Set M=max{1/ϵ,M(ϵ,δ)}, where M(ϵ,δ) is defined in Lemma 4.11. It is easy to see that for any ϵ∈(0,ϵ0), there exists M=M(ϵ) such that if m≥M and a∈I, then (1.2) has a unique positive solution of type (ii).
It remains to prove the instability for the unique positive solution of (1.2) of type (ii). Define T and T0:C02,α(Ω-)×C02,α(Ω-)→Cα(Ω-)×Cα(Ω-) by
(4.65)T(ξη)=(Δξ+(a-2u-bv(1+mu)2(1+kv))ξ-bu(1+mu)(1+kv)2ηΔη+(c-2v+du(1+mu)(1+kv)2)η+dv(1+mu)2(1+kv)ξ),T0(ξη)=(Δξ+(a-bθc(1+wa)2(1+kθc))ξΔη+(c-2θc)η+dθc(1+wa)2(1+kθc)ξ).
It is easy to show that, as m→∞, T→T0 in the operator norm uniformly for (u,v) approaches (0,θc) with mu close to wa and a∈[λ1+ϵ,λ1+ϵ0]. Since 0 belongs to the resolvent set of T0 and
(4.66)μ0=λ1(-a+bθc(1+wa)2(1+kθc))<0
is an eigenvalue of T0. Due to standard perturbation theory, it is easy to get that 0 also belongs to the resolvent set of T and that T has an eigenvalue μ close to μ0. In particular,  Reμ<0. This shows that for all large m, the positive solution of (1.2) close to (0,θc) is nondegenerate and unstable. Thus, the proof of Theorem 4.12 is complete.



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*Source: 627419-2012-12-31.xml*