The purpose of this section is to investigate the stability and multiplicity of coexistence states of (2) by means of eigenvalue perturbation theory and the fixed point index theory.

Set *X*_{1} = [*C*^{2,α}(*Ω̄*) × *C*^{2,α}(*Ω̄*) ∩ *X*], *Y* = [*C*^{α}(*Ω̄*) × *C*^{α}(*Ω̄*), here 0 < *α* < 1. *i* : *X*_{1} → *Y* is the inclusion mapping. Since *L*_{1} is the linearized operator at (*ã*; 0, *θ*_{c}) of (2), according to the proof of Theorem 3.1, we get *N*(*L*_{1}) = span{(*Φ*, *Ψ*)}, Codim *R*(*L*_{1}) = 1, and *R*(*L*_{0}) = {(*u*, *v*) ∈ *X* : ∫_{Ω}*u**Φ* *dx* = 0}. Due to *i*(*Φ*, *Ψ*) ∈ *R*(*L*_{1}), it follows from [31] that 0 is an *i*−simple eigenvalue of *L*_{1}.

#### Lemma 4.1

0 *is the eigenvalue of* *L*_{1} *with the largest real part*, *and all the other eigenvalues of* *L*_{1} *lie in the left half complex plane*.

Let *L*(*u*(*s*), *v*(*s*), *a*(*s*)) and *L*(*a*; 0, *θ*_{c}) be the linearized operators of (2) at (*u*(*s*), *v*(*s*), *a*(*s*)), (*a*; 0, *θ*_{c}), respectively. Applying the linearized stability theory (see [31, 32]), we can obtain the following result.

#### Lemma 4.2

*There exists* *C*^{1}−*function*: *a* → (*M*(*a*), *γ*(*a*)) *and* *s* → (*N*(*s*), *π*(*s*)), *which are defined by the mapping from the neighborhood of* *ã* *and* 0 *into* *X*_{1} × *R*, *respectively*, *satisfying the forms* *γ*(*ã*) = *π*(0) = 0, *M*(*ã*) = *N*(0) = (*Φ*, *Ψ*) *and*

$$\begin{array}{}{\displaystyle L(a;0,{\theta}_{c})M(a)=\gamma (a)M(a),\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}|a-\stackrel{~}{a}|\ll 1,}\end{array}$$

$$\begin{array}{}{\displaystyle L(u(s),v(s),a(s))N(s)=\pi (s)N(s),\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}|s|\ll 1,}\end{array}$$

where *M*(*a*) = (*ϕ*_{1}(*a*), *ϕ*_{2}(*a*)), *N*(*s*) = (*φ*_{1}(*a*)*φ*_{2}(*a*)). Meanwhile, *γ*′(*ã*) ≠ 0, furthermore, for |*s*| ≪ 1, *π*(*s*) ≠ 0, *π*(*s*) and −*sa*′(*s*)*γ*′(*ã*) have the same sign, where *γ*′(*ã*) is the derivative of *γ*(*a*) at *a* = *ã*, and *a*′(*s*) is the derivative of *a*(*s*) on the point *s*.

#### Proof

By calculating *L*(*a*; 0, *θ*_{c})M(*a*) = *γ*(*a*)*M*(*a*), for |*a*−*ã*| ≪ 1, we have

$$\begin{array}{}{\displaystyle \{\begin{array}{ll}\mathit{\Delta}{\varphi}_{1}+(a-\frac{b{\theta}_{c}}{1+k{\theta}_{c}}){\varphi}_{1}=\gamma (a){\varphi}_{1},& x\in \mathit{\Omega},\\ \mathit{\Delta}{\varphi}_{2}+(c-2{\theta}_{c}){\varphi}_{2}+\frac{d{\theta}_{c}}{1+k{\theta}_{c}}{\varphi}_{1}=\gamma (a){\varphi}_{2},& x\in \mathit{\Omega},\\ {\varphi}_{1}={\varphi}_{2}=0,& x\in \mathrm{\partial}\mathit{\Omega}.\end{array}}\end{array}$$

Note that |*a*−*ã*| ≪ 1, then | *γ*(*a*)| ≪ 1. Obviously, *ϕ*_{1} ≢ 0, otherwise, *ϕ*_{1} ≡ 0, then *ϕ*_{2} ≡ 0, a contradiction. Thus *γ*(*a*) is an eigenvalue of $\begin{array}{}(\mathit{\Delta}+(a-\frac{b{\theta}_{c}}{1+k{\theta}_{c}})I)\end{array}$. So *Φ* > 0, implies *ϕ*_{1} = *ϕ*_{1}(*a*) > 0 as |*a*−*ã*| ≪ 1. It follows that *γ*(*a*) is the principal eigenvalue of $\begin{array}{}(\mathit{\Delta}+(a-\frac{b{\theta}_{c}}{1+k{\theta}_{c}})I)\end{array}$, and *γ*(*a*) is increasing along with *a* for |*a*−*ã*| ≪ 1. Meanwhile, *γ*′(*ã*) ≠ 0. Thus *γ*′(*ã*) > 0. □

#### Lemma 4.4

*a*′(0) *satisfies*

$$\begin{array}{}{\displaystyle {a}^{\prime}(0)\underset{\mathit{\Omega}}{\int}{\mathit{\Phi}}^{2}dx=\underset{\mathit{\Omega}}{\int}(1-\frac{bm{\theta}_{c}}{(1+k{\theta}_{c}{)}^{2}}){\mathit{\Phi}}^{3}dx+\underset{\mathit{\Omega}}{\int}\frac{b\mathit{\Psi}}{(1+k{\theta}_{c}{)}^{2}}{\mathit{\Phi}}^{2}dx.}\end{array}$$

#### Proof

By substituting (*u*(*s*), *v*(*s*), *a*(*s*)) into (2), differentiating on *s*, and let *s* = 0, we get

$$\begin{array}{}{\displaystyle -\mathit{\Delta}{\varphi}^{\prime}(0)=(\stackrel{~}{a}-\frac{b{\theta}_{c}}{1+k{\theta}_{c}}){\varphi}^{\prime}(0)+[{a}^{\prime}(0)-\mathit{\Phi}-b\frac{\mathit{\Psi}-m{\theta}_{c}\mathit{\Phi}}{(1+k{\theta}_{c}{)}^{2}}]\mathit{\Phi},}\end{array}$$

where *ϕ*′(0) is the derivative of *ϕ* on the point *s* = 0.

Multiplying the above equation by *Φ*, and applying Green’s formula and the definition of *Φ*, we obtain

$$\begin{array}{}{\displaystyle {a}^{\prime}(0)\underset{\mathit{\Omega}}{\int}{\mathit{\Phi}}^{2}dx=\underset{\mathit{\Omega}}{\int}(1-\frac{bm{\theta}_{c}}{(1+k{\theta}_{c}{)}^{2}}){\mathit{\Phi}}^{3}dx+\underset{\mathit{\Omega}}{\int}\frac{b\mathit{\Psi}}{(1+k{\theta}_{c}{)}^{2}}{\mathit{\Phi}}^{2}dx.}\end{array}$$ □

Taking the advantage of Lemma 4.1-Lemma 4.4, we directly derive the following theorem.

#### Theorem 4.5

*Let* $\begin{array}{}\sigma ={\int}_{\mathit{\Omega}}(1-\frac{bm{\theta}_{c}}{(1+k{\theta}_{c}{)}^{2}}){\mathit{\Phi}}^{3}dx+{\int}_{\mathit{\Omega}}\frac{b\mathit{\Psi}}{(1+k{\theta}_{c}{)}^{2}}{\mathit{\Phi}}^{2}dx.\end{array}$ *If* *σ* > 0, *then local bifurcation solution* (*u*(*s*), *v*(*s*)) *is stable; if* *σ* < 0, *then local bifurcation solution* (*u*(*s*), *v*(*s*)) *is unstable*.

In the following, we shall investigate the multiplicity of coexistence states in the gap.

#### Theorem 4.7

*Suppose that* $\begin{array}{}c>{\lambda}_{1}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\int}_{\mathit{\Omega}}(1-\frac{bm{\theta}_{c}}{(1+k{\theta}_{c}{)}^{2}}){\mathit{\Phi}}^{3}dx<0.\end{array}$ *If there exists a sufficiently small* *ε* > 0 *and d* ≪1, *then coexistence state* (*u*(*s*), *v*(*s*)) *is non*-*degenerate and unstable for* *a* ∈ (*ã* − *ε*, *ã*), *moreover*, (2) *has at least two coexistence states*.

#### Proof

Firstly, we will demonstrate the coexistence state (*u*(*s*), *v*(*s*)) is non-degenerate and unstable. To this end, we only need to prove that there exists a sufficiently small *ε* > 0 such that any coexistence state (*u*(*s*), *v*(*s*)) of (2) is non-degenerate for *a* ∈ (*ã* − *ε*, *ã*), and the linearized eigenvalue problem

$$\begin{array}{}{\displaystyle \{\begin{array}{ll}-\mathit{\Delta}\xi -[a(s)-2u(s)-\frac{bv(s)(1-u(s{)}^{2}+kv(s))}{(1+mu(s)+u(s{)}^{2}+kv(s){)}^{2}}]\xi +\frac{bu(s)(1+mu(s)+u(s{)}^{2})}{(1+mu(s)+u(s{)}^{2}+kv(s){)}^{2}}\eta =\mu \xi ,& x\in \mathit{\Omega},\\ -\mathit{\Delta}\eta -[c-2v(s)+\frac{du(s)(1+mu(s)+u(s{)}^{2})}{(1+mu(s)+u(s{)}^{2}+kv(s){)}^{2}}]\eta -\frac{dv(s)(1-u(s{)}^{2}+kv(s))}{(1+mu(s)+u(s{)}^{2}+kv(s){)}^{2}}=\mu \eta ,& x\in \mathit{\Omega},\\ \xi =\eta =0,& x\in \mathrm{\partial}\mathit{\Omega}\end{array}}\end{array}$$(9)

has a unique eigenvalue *μ*_{*} and Re(*μ*_{*}) < 0 with algebra multiplicity one.

Let {*ε*_{i} > 0} and {*d*_{i} > 0} be sequences which converge to 0 by *i* → ∞. Owing to *a* = *ã* + *a*′(0)*s* + *O*(*s*^{2}), we can get the sequence {*ε*_{i} > 0} and {*a*_{i}} yield *a*_{i} ∈ (*ã* − *ε*_{i}, *ã*) and *s*_{i} → 0 as *i* → ∞. It is easy to see that (*u*_{i}, *v*_{i}) is one solution of (2). Hence, the corresponding linearized problem (9) can be written in the following form:

$$\begin{array}{}{\displaystyle {L}_{i}\left(\begin{array}{l}{\xi}_{i}\\ {\eta}_{i}\end{array}\right)={\mu}_{i}\left(\begin{array}{l}{\xi}_{i}\\ {\eta}_{i}\end{array}\right){\textstyle \text{and}}\phantom{\rule{thinmathspace}{0ex}}{L}_{i}=\left(\begin{array}{l}{M}_{i}^{11}\phantom{\rule{1em}{0ex}}{M}_{i}^{12}\\ {M}_{i}^{21}\phantom{\rule{1em}{0ex}}{M}_{i}^{22}\end{array}\right),}\end{array}$$

where (*ξ*_{i}, *η*_{i}) ≢ (0, 0) and

$$\begin{array}{}\phantom{\rule{2em}{0ex}}{M}_{i}^{11}=-\mathit{\Delta}-[{a}_{i}-2{u}_{i}-\frac{b{v}_{i}(1-{u}_{i}^{2}+k{v}_{i})}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}}],\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{M}_{i}^{12}=\frac{b{u}_{i}(1+m{u}_{i}+{u}_{i}^{2})}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}},\\ {M}_{i}^{21}=-\frac{d{v}_{i}(1-{u}_{i}^{2}+k{v}_{i})}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}},\phantom{\rule{1em}{0ex}}{M}_{i}^{22}=-\mathit{\Delta}-[c-2{v}_{i}+\frac{d{u}_{i}(1+m{u}_{i}+{u}_{i}^{2})}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}}].\end{array}$$

Observe that, as *i* → ∞,$\begin{array}{}{L}_{i}\left(\begin{array}{l}{\xi}_{i}\\ {\eta}_{i}\end{array}\right)\end{array}$ converges to

$$\begin{array}{}{\displaystyle {L}_{0}\left(\begin{array}{l}\xi \\ \eta \end{array}\right)=\left(\begin{array}{l}-\mathit{\Delta}\xi -(\stackrel{~}{a}-\frac{b{\theta}_{c}}{1+k{\theta}_{c}})\xi \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}0\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-\mathit{\Delta}\eta -(c-2{\theta}_{c})\eta \end{array}\right).}\end{array}$$

It follows that 0 is a simple eigenvalue of *L*_{0} with the corresponding eigenfunction (*ξ*, *η*)^{T} = (*Φ*, 0)^{T}. Meanwhile, all the other eigenvalues of *L*_{0} are positive and keep away from 0. Furthermore, using perturbation theory [31, 33], for large *i*, the operator *L*_{i} has a unique eigenvalue *μ*_{i} which is close to zero. Moreover, all other eigenvalues of *L*_{i} have positive real parts and keep away from 0. Because *μ*_{i} is simple real eigenvalue which tends to zero, we denote the corresponding eigenfunction by (*ξ*_{i}, *η*_{i})^{T} which satisfies (*ξ*_{i}, *η*_{i}) → (*Φ*, 0) as *i* → ∞.

Now we claim that *Re**μ*_{i} < 0 for large *i*. Multiplying *Φ* on the first equation of *L*_{i}(*ξ*_{i},*η*_{i})^{T} = *μ*_{i}(*ξ*_{i},*η*_{i})^{T} and integrating over *Ω*, we obtain

$$\begin{array}{}{\displaystyle -\underset{\mathit{\Omega}}{\int}\mathit{\Phi}\mathit{\Delta}{\xi}_{i}-\underset{\mathit{\Omega}}{\int}({a}_{i}-2{u}_{i}-\frac{b{v}_{i}(1-{u}_{i}^{2}+k{v}_{i})}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}})\mathit{\Phi}{\xi}_{i}+\underset{\mathit{\Omega}}{\int}\frac{b{u}_{i}(1+m{u}_{i}+{u}_{i}^{2})\mathit{\Phi}{\eta}_{i}}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}}=\underset{\mathit{\Omega}}{\int}{\mu}_{i}\mathit{\Phi}{\xi}_{i}.}\end{array}$$(10)

Multiplying two sides of the first equation of (2) with (*a*, *u*, *v*) = (*a*_{i}, *u*_{i}, *v*_{i}) by *ξ*_{i} and integrating, we obtain

$$\begin{array}{}{\displaystyle -\underset{\mathit{\Omega}}{\int}{\xi}_{i}\mathit{\Delta}{u}_{i}-\underset{\mathit{\Omega}}{\int}({a}_{i}-{u}_{i}-\frac{b{v}_{i}}{1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}}){u}_{i}{\xi}_{i}=0.}\end{array}$$

Taking *u*_{i} = *s*_{i}*Φ* + $\begin{array}{}O({s}_{i}^{2})\end{array}$ into the above equation we have

$$\begin{array}{}{\displaystyle -\underset{\mathit{\Omega}}{\int}\mathit{\Phi}\mathit{\Delta}{\xi}_{i}-\underset{\mathit{\Omega}}{\int}{\xi}_{i}\mathit{\Phi}({a}_{i}-{u}_{i}-\frac{b{v}_{i}}{1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}})+O({s}_{i}^{2})=0.}\end{array}$$(11)

By combining (10) and (11), we obtain

$$\begin{array}{}{\displaystyle \underset{\mathit{\Omega}}{\int}[1-\frac{b{v}_{i}(m+2{u}_{i})}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}}]{u}_{i}\mathit{\Phi}{\xi}_{i}+\underset{\mathit{\Omega}}{\int}\frac{b{u}_{i}(1+m{u}_{i}+{u}_{i}^{2})\mathit{\Phi}{\eta}_{i}}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}}=\underset{\mathit{\Omega}}{\int}{\mu}_{i}\mathit{\Phi}{\xi}_{i}.}\end{array}$$

Notice that
(*u*_{i}, *v*_{i}) = $\begin{array}{}({s}_{i}\mathit{\Phi}+O({s}_{i}^{2}),{\theta}_{c}+{s}_{i}{\mathit{\Psi}}_{{d}_{i}}+O({s}_{i}^{2})),\end{array}$ where *Ψ*_{di} stands for *Ψ* defined in Remark 3.2. Thus, dividing the above equation by *s*_{i} and taking the limit, we get

$$\begin{array}{}{\displaystyle \underset{i\to \mathrm{\infty}}{lim}\frac{{\mu}_{i}}{{s}_{i}}=\underset{\mathit{\Omega}}{\int}(1-\frac{bm{\theta}_{c}}{(1+k{\theta}_{c}{)}^{2}}){\mathit{\Phi}}^{3}/\underset{\mathit{\Omega}}{\int}{\mathit{\Phi}}^{2}<0,}\end{array}$$

which results in *Re**μ*_{i} < 0 for large *i*. Hence, our claim is illustrated.

Finally, using similar method as in [25], we will demonstrate the remaining part of Theorem 4.7. In order to apply reduction to absurdity, we suppose that there exists a unique coexistence state (*ũ*,*ṽ*) for (2), then (*ũ*, *ṽ*) must be bifurcated from (0, *θ*_{c}), since there exists a coexistence state near *ã*. Thus, (*ũ*, *ṽ*) is non-degenerate, and the corresponding linearized problem has a unique eigenvalue *μ̃* with algebra multiplicity one which satisfies *Re**μ̃* < 0. The above fact implies that *I* − *F*′ (*ũ*, *ṽ*) is invertible and does not have property *α* on *W*_{(ũ,ṽ)}, hence index(*F*, (*ũ*, *ṽ*)) = (−1)^{1} = −1 by Lemma 2.2(ii). Thus, according to Lemma 2.3-2.4 and the fixed point index theory, we get

$$\begin{array}{}{\displaystyle 1={{\textstyle \text{index}}}_{W}(F,D))={{\textstyle \text{index}}}_{W}(F,(0,0))+{{\textstyle \text{index}}}_{W}(F,({\theta}_{a},0))}\\ \\ {\displaystyle +{{\textstyle \text{index}}}_{W}(F,(0,{\theta}_{c}))+{{\textstyle \text{index}}}_{W}(F,(\stackrel{~}{u},\stackrel{~}{v}))=0+0+1-1=0,}\end{array}$$

which leads to a contradiction. Thus, the proof is completed. □

Finally, we study the stability of any positive solutions (if exists) as the parameter *a*, *b*, *c* belong to some domain.

#### Theorem 4.9

*If* *a* > λ_{1}, *c* > λ_{1}, *then there exists some sufficiently small* *B̃* > 0 *such that any coexistence state of* (2) *(if exists) is non*-*degenerate and stable for* *b* ≤ *B̃*.

#### Proof

Let *ṽ* be a positive solution of the following equation

$$\begin{array}{}{\displaystyle -\mathit{\Delta}v=v(b-v-\frac{d{\theta}_{a}}{1+m{\theta}_{a}+l{\theta}_{a}^{2}+kv})\phantom{\rule{thinmathspace}{0ex}}x\in \mathit{\Omega},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}v{|}_{\mathrm{\partial}\mathit{\Omega}}=0.}\end{array}$$

Hence, we need only demonstrate that the linearized problem of (2) has no the real part of eigenvalue. By reduction to absurdity, we suppose that the positive solution (*u*_{i}, *v*_{i}) of (2) is either non-degenerate or unstable. Suppose the sequence *b*_{i} meets *b*_{i} → 0 with *i* ≥ 1, then there exists *μ*_{i} such that *Re* *μ*_{i} ≤ 0, and the corresponding eigenfunction (*ξ*_{i}, *η*_{i}) ≠ (0, 0) with ∥*ξ*_{i}∥^{2} + ∥*η*_{i}∥^{2} = 1 satisfying the following linearized problem

$$\begin{array}{}{\displaystyle \{\begin{array}{ll}-\mathit{\Delta}{\xi}_{i}-[a-2{u}_{i}-\frac{b{v}_{i}(1-{u}_{i}^{2}+k{v}_{i})}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}}]{\xi}_{i}+\frac{b{u}_{i}(1+m{u}_{i}+{u}_{i}^{2})}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}}{\eta}_{i}=\mu {\xi}_{i},& x\in \mathit{\Omega},\\ -\mathit{\Delta}{\eta}_{i}-[c-2{v}_{i}+\frac{d{u}_{i}(1+m{u}_{i}+{u}_{i}^{2})}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}}]{\eta}_{i}-\frac{d{v}_{i}(1-{u}_{i}^{2}+k{v}_{i})}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}}=\mu {\eta}_{i},& x\in \mathit{\Omega},\\ {\xi}_{i}={\eta}_{i}=0,& x\in \mathrm{\partial}\mathit{\Omega}\end{array}}\end{array}$$(12)

Multiplying two sides of (12) with *ξ*_{i},*η*_{i}), respectively, and integrating on *Ω*, it follows from Divergence theorem that

$$\begin{array}{}{\displaystyle {\mu}_{i}=\underset{\mathit{\Omega}}{\int}|\mathrm{\nabla}{\xi}_{i}{|}^{2}dx-\underset{\mathit{\Omega}}{\int}[a-2{u}_{i}-\frac{b{v}_{i}(1-{u}_{i}^{2}+k{v}_{i})}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}}]|{\xi}_{i}{|}^{2}dx+\underset{\mathit{\Omega}}{\int}\frac{b{u}_{i}(1+m{u}_{i}+{u}_{i}^{2})}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}}{\eta}_{i}\overline{{\xi}_{i}}dx}\\ \\ {\displaystyle \phantom{\rule{1em}{0ex}}+\underset{\mathit{\Omega}}{\int}|\mathrm{\nabla}{\eta}_{i}{|}^{2}dx-\underset{\mathit{\Omega}}{\int}[c-2{v}_{i}+\frac{d{u}_{i}(1+m{u}_{i}+{u}_{i}^{2})}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}}]|{\eta}_{i}{|}^{2}dx-\underset{\mathit{\Omega}}{\int}\frac{d{v}_{i}(1-{u}_{i}^{2}+k{v}_{i})}{(1+m{u}_{i}+{u}_{i}^{2}+k{v}_{i}{)}^{2}}{\xi}_{i}\overline{{\eta}_{i}}dx,}\end{array}$$

where *ξ̄*_{i} and *η̄*_{i} are the complex conjugates of *ξ*_{i} and *η*_{i}. According to the above equation, we can deduce that *Re*(*μ*_{i}) and Im(*μ*_{i}) are bounded, so we suppose that *μ*_{i} → *μ* with *Re*(*μ*) ≤ 0. Meanwhile, (*ξ*_{i}, *η*_{i}) → (*ξ*, *η*), *b*_{i} → 0 as *i* → ∞. Then (12) converges to

$$\begin{array}{}{\displaystyle \{\begin{array}{ll}-\mathit{\Delta}\xi -(a-2{\theta}_{a})\xi =\mu \xi ,& x\in \mathit{\Omega},\\ -\mathit{\Delta}\eta -[c-2\stackrel{~}{v}+\frac{d{\theta}_{a}(1+m{\theta}_{a}+{\theta}_{a}^{2})}{(1+m{\theta}_{a}+{\theta}_{a}^{2}+k\stackrel{~}{v}{)}^{2}}]{\eta}_{i}-\frac{d\stackrel{~}{v}(1-{\theta}_{a}^{2}+k\stackrel{~}{v})}{(1+m{\theta}_{a}+{\theta}_{a}^{2}+k\stackrel{~}{v}{)}^{2}}=\mu \eta ,& x\in \mathit{\Omega}.\\ \xi =\eta =0,& x\in \mathrm{\partial}\mathit{\Omega}\end{array}}\end{array}$$(13)

It follows that *μ* must be real number with *μ* ≤ 0. Suppose *ξ* ≢ 0, one can deduce that *μ* is an eigenvalue of the equation

$$\begin{array}{}{\displaystyle -\mathit{\Delta}\varphi +(-a+2{\theta}_{a})\varphi =\mu \varphi ,\phantom{\rule{thinmathspace}{0ex}}x\in \mathit{\Omega},\phantom{\rule{thinmathspace}{0ex}}\varphi {|}_{\mathrm{\partial}\mathit{\Omega}}=0,}\end{array}$$

Then *μ* ≥ λ_{1}(−*a* + 2*θ*_{a} ), since λ_{1}(−*a* + 2*θ*_{a} ) > λ_{1}(−*a* + *θ*_{a} ) = 0, hence *μ* ≤ 0, a contradiction.

Suppose *ξ* ≡ 0, then *η* ≢ 0. By the second equation of (13), we get

$$\begin{array}{}{\displaystyle -\mathit{\Delta}\eta -[c-2\stackrel{~}{v}+\frac{d{\theta}_{a}(1+m{\theta}_{a}+{\theta}_{a}^{2})}{(1+m{\theta}_{a}+{\theta}_{a}^{2}+k\stackrel{~}{v}{)}^{2}}]{\eta}_{i}=\mu \eta ,x\in \mathit{\Omega}.\eta {|}_{\mathrm{\partial}\mathit{\Omega}}=0.}\end{array}$$

Observe that *η* ≢ 0, we obtain $\begin{array}{}\mu ={\lambda}_{1}[-c+2\stackrel{~}{v}-\frac{d{\theta}_{a}(1+m{\theta}_{a}+{\theta}_{a}^{2})}{(1+m{\theta}_{a}+{\theta}_{a}^{2}+k\stackrel{~}{v}{)}^{2}}].\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{Set}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}g(x,v)=c-v+\frac{d{\theta}_{a}}{(1+m{\theta}_{a}+{\theta}_{a}^{2}+k\stackrel{~}{v})},\end{array}$ then *g*_{v} < 0. Hence,

$$\begin{array}{}{\displaystyle {\lambda}_{1}(-g(x,\stackrel{~}{v}))<{\lambda}_{1}(-g(x,\stackrel{~}{v})-{\lambda}_{1}(-\stackrel{~}{v}{g}_{v}(x,\stackrel{~}{v})))={\lambda}_{1}[-c+2\stackrel{~}{v}-\frac{d{\theta}_{a}(1+m{\theta}_{a}+{\theta}_{a}^{2})}{(1+m{\theta}_{a}+{\theta}_{a}^{2}+k\stackrel{~}{v}{)}^{2}}]=\mu .}\end{array}$$

Notice that λ_{1}(−*g*(*x*, *ṽ*)) = 0, so *μ* > 0, a contradiction. □

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