In this section, we offer a rigorous study of interior solutions of the corresponding strongly coupled elliptic model
$$ \textstyle\begin{cases} -d_{1}\Delta u-d_{2}\Delta v=u (a-u- \frac{v}{1+\alpha u+\beta v} ), & x\in \Omega , t>0, \\ d_{3}\Delta u-d_{4}\Delta v=v (b-\frac{v}{\gamma u} ), & x\in \Omega , t>0, \\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=0, & x\in \partial \Omega , t>0, \\ u(x,0)=u_{0}(x)>0, \qquad v(x,0)=v_{0}(x)\geq (\not \equiv )\ 0, & x\in \Omega . \end{cases} $$
(3.1)
Meanwhile, we will also investigate the nonconstant positive steady states of model (1.1). The existence of an interior solution of a linear cross-diffusion model has been studied by utilizing the approach proposed by [11] (upper and lower solutions) and many others.
Lemma 3.1
Any interior solution \((u,v)\) for system (1.1) fulfills \(u(x),v(x)\leq \max \{a,ab\gamma \}\) in Ω̅.
Proof
In view of the equation of u for model (3.1), we obtain
$$ -\Delta (d_{1}u+d_{2}v)=u \biggl(a-u- \frac{v}{1+\alpha u+\beta v} \biggr)\leq u(a-u). $$
By the maximum principle proposed by [12] we have \(u(a-u)\geq 0\). Hence \(u\leq a\). Together with the equation for u and the equation for v for system (3.1), we get
$$ -\Delta (-d_{3}u+d_{4}v)=v \biggl(b- \frac{v}{\gamma u} \biggr)\leq bv-\frac{v^{2}}{a\gamma }. $$
Applying the maximum principle in [12], we have \(bv-\frac{v^{2}}{a\gamma }\geq 0\). Thus we obtain \(v\leq ab\gamma \). This ends the proof of the lemma. □
Next, we introduce the compact map \(CM\in C^{2}(\overline{\Omega })\oplus C^{2}(\overline{\Omega }) \rightarrow C^{1}(\overline{\Omega })\oplus C^{1}(\overline{\Omega })\) as follows:
Here we can choose a sufficiently large constant \(c>0\) guaranteeing that the functions \(au-u^{2}-\frac{uv}{1+\alpha u+\beta v}+cu\) and \(bv-\frac{v^{2}}{\gamma u}+cv\) are increasing for u and v, respectively. We can easily observe that system (3.1) is equivalent to \((u,v)=CM(u,v)\), which implies that there exists a nonconstant interior solution of system (3.1) representing a nonconstant interior fixed point of CM in \(\mathcal{D}'\). When \((a,0)\) and \((u^{*},v^{*})\) exist, we suppose that they stand for an isolated fixed point of CM. On the contrary, there must exist a nonconstant fixed point in \(\operatorname{Int}\mathcal{D}'\) (the interior of \(\mathcal{D}'\)). Therefore the associate indexes in \(\mathcal{Q}\) are well defined. Based on the method introduced by [13], we can directly calculate the fix point index of CM with respect to \(\mathcal{Q}\) over \(\operatorname{Int}\mathcal{D}'\).
Lemma 3.2
If \(\frac{b\gamma u^{*}[1+(2\alpha +b\beta \gamma )u^{*}]}{[1+(\alpha +b\beta \gamma )u^{*}]^{2}}< a+b\) and \(a> \frac{b\gamma u^{*}(\alpha +b\beta \gamma )u^{*}}{[1+(\alpha +b\beta \gamma )u^{*}]^{2}}\), then \(\operatorname{index}_{\mathcal{Q}}(CM,\operatorname{Int}\mathcal{D}')=1\).
Proof
We introduce the homotopic map, for \(\eta \in (0,1)\), \(CM_{\eta }\triangleq HM\in C^{2}(\overline{\Omega })\oplus C^{2}( \overline{\Omega })\rightarrow C^{1}(\overline{\Omega })\oplus C^{1}( \overline{\Omega })\) by
Utilizing a similar discussion as in Lemma 3.1, we can easily obtain that any interior fixed point \((u,v)\) for HM fulfills \(u(x),v(x)\leq \max \{a,ab\gamma \}\) in Ω̅. Thus we can easily draw a conclusion that each fixed point for HM belongs to \(\operatorname{Int}\mathcal{D}'\). In addition, the corresponding index of \(CM_{\eta }\) over the interior of \(\mathcal{D}'\) regarding \(\mathcal{Q}\) is well defined. In view of the homotopy invariance theorem, we can obtain that \(\operatorname{index}_{\mathcal{Q}}(CM_{0},\operatorname{Int}\mathcal{D}')= \operatorname{index}_{\mathcal{Q}}(CM_{1},\operatorname{Int}\mathcal{D}') = \operatorname{index}_{\mathcal{S}}(B,\operatorname{Int}\mathcal{D})\), where
Based on the above discussion, we argue that \(\operatorname{index}_{\mathcal{S}}(B,\operatorname{Int}\mathcal{D})=1\). To prove this, we propose the homotopic map
for \(\eta \in [0,1]\). Then we can directly obtain that \(\operatorname{index}_{\mathcal{S}}(B_{0},\operatorname{Int}\mathcal{D})= \operatorname{index}_{\mathcal{S}}(B_{1},\operatorname{Int}\mathcal{D})\). Our concern is the eigenvalue issue
$$ B_{0}'(0,0) (\Phi ,\Psi )^{T}=\rho (\Phi ,\Psi )^{T},\quad (\Phi ,\Psi )\neq (0,0), $$
(3.2)
for \(\rho >0\), where
The corresponding eigenfunction formulas for Φ and Ψ can be performed as follows:
$$ \Phi =\sum_{k=0}^{\infty }\sum _{j=1}^{\mathcal{M}_{k}} \phi _{kj} \mathcal{E}_{kj} \quad \mbox{and}\quad \Psi =\sum_{k=0}^{ \infty } \sum_{j=1}^{\mathcal{M}_{k}}\psi _{kj} \mathcal{E}_{kj}, $$
where \(\phi _{kj},\psi _{kj}\in \mathbb{R}\). We easily rewrite system (3.2) as follows:
$$ \textstyle\begin{cases} \sum_{k=0}^{\infty }\sum_{j=1}^{\mathcal{M}_{k}}[(d_{1} \lambda _{k}\rho +c\rho -c)\phi _{kj}+d_{2}\lambda _{k}\rho \psi _{kj}] \mathcal{E}_{kj}=0, \\ \sum_{k=0}^{\infty }\sum_{j=1}^{\mathcal{M}_{k}}[-d_{3}\lambda _{k} \rho \phi _{kj}+(d_{4}\lambda _{k}\rho +c\rho -c)\psi _{kj}] \mathcal{E}_{kj}=0. \end{cases} $$
(3.3)
As we know, \(\{\mathcal{E}_{kj}:1\leq j\leq \mathcal{M}_{k}\}\) in \(L^{2}(\Omega )\) stands for a complete orthonormal basis. By multiplying the above two equations by \(\varphi _{kj}\) and integrating over Ω we get the equation
Therefore
Denoting
$$ \rho _{\pm }= \frac{\overline{\Lambda }\pm \sqrt{\overline{\Lambda }^{2}-4[(d_{1} \lambda _{k}+c)(d_{4}\lambda _{k}+c)+d_{2}d_{3}\lambda _{k}^{2}]c^{2}}}{2[(d_{1} \lambda _{k}+c)(d_{4}\lambda _{k}+c)+d_{2}d_{3}\lambda _{k}^{2}]}, $$
where \(\overline{\Lambda }=c(d_{1}\lambda _{k}+d_{4}\lambda _{k}+2c)\), we have
$$ \begin{aligned} \rho _{+}&< \frac{\overline{\Lambda }+\sqrt{\overline{\Lambda }^{2}-4 (d_{1}\lambda _{k}+c)(d_{4}\lambda _{k}+c)c^{2}}}{2(d_{1}\lambda _{k}+c)(d_{4}\lambda _{k}+c)} \\ &=\frac{c[(d_{1}\lambda _{k}+c)+(d_{4}\lambda _{k}+c)+ \vert (d_{1}\lambda _{k}+c)- (d_{4}\lambda _{k}+c) \vert ]}{2(d_{1}\lambda _{k}+c)(d_{4}\lambda _{k}+c)} \leq 1. \end{aligned} $$
Based on Lemma 13.1 of [14], we get \(\operatorname{index}_{\mathcal{S}}(B_{0},\operatorname{Int}\mathcal{D})=1\). Hence \(\operatorname{index}_{\mathcal{S}}(B,\operatorname{Int}\mathcal{D})= \operatorname{index}_{\mathcal{S}}(B_{1},\operatorname{Int}\mathcal{D})=1\). This finishes the proof of the lemma. □
Lemma 3.3
\(\operatorname{index}_{\mathcal{Q}}(CM,(a,0))=0\).
Proof
Denote \(\overline{\mathcal{Q}}_{(a,0)}=C_{\nu }^{1}(\overline{\Omega }) \oplus \mathcal{S}\), \(\mathcal{W}_{(a,0)}=C_{\nu }^{1}(\overline{\Omega })\oplus \{ 0 \} \), and
Set \(CM'(a,0)(\Phi ,\Psi )^{T}=(\Phi ,\Psi )^{T}\in \overline{\mathcal{Q}}_{(a,0)}\). Then we get
$$ \textstyle\begin{cases} d_{1}\Delta \Phi +d_{2}\Delta \Psi =a\Phi + \frac{a}{1+a\alpha }\Psi & \mbox{in } \Omega , \\ d_{3}\Delta \Phi -d_{4}\Delta \Psi =b\Psi & \mbox{in } \Omega , \\ \frac{\partial \Phi }{\partial \nu }= \frac{\partial \Psi }{\partial \nu }=0 & \mbox{on } \partial \Omega . \end{cases} $$
(3.4)
From the first equation of system (3.4) we have \(\Phi =\Psi =0\) in Ω. Thus \(I-CM'(a,0)\) represents an invertible matrix on \(\overline{\mathcal{Q}}_{(a,0)}\). On the other hand, we can easily check that \(CM'(a,0)\) possesses property γ, which means that for \((\Phi ,\Psi )\equiv (0,1)\in \overline{\mathcal{Q}}_{(a,0)} \setminus \mathcal{W}_{(a,0)}\) and \(s=\frac{c}{b+c}\in (0,1)\), \([(\Phi ,\Psi )^{T}-sCM'(a,0)(\Phi ,\Psi )^{T}]\in \mathcal{W}_{(a,0)}\). Hence, applying Lemma 4.1(i) in [13], we obtain a precise result. This finishes the proof of the lemma. □
Denote
In the rest of the paper, we consider \(\operatorname{index}_{\mathcal{Q}}(CM,(u^{*},v^{*}))\) in the following three situations: (1) For all \(\lambda >0\), \(\operatorname{Det}(\lambda )>0\); (2) There exists, with multiplicity one, precisely a simple interior solution for \(\operatorname{Det}(\lambda )=0\); (3) There exist, with multiplicity one, two interior solutions for \(\operatorname{Det}(\lambda )=0\). Now we present our main theoretical result.
Theorem 3.1
-
(1)
If \(\operatorname{Det}(\lambda )>0\) for all \(\lambda >0\), then \(\operatorname{index}_{\mathcal{Q}}(CM,(u^{*},v^{*}))=1\);
-
(2)
If \(\operatorname{Det}(\lambda )=0\) has exactly a multiplicity-one simple positive solution \(\lambda ^{*}\) in an open interval \((\lambda _{k^{*}},\lambda _{k^{*}+1})\) for some nonnegative integer \(k^{*}\), then
$$ \operatorname{index}_{\mathcal{Q}}\bigl(CM,\bigl(u^{*},v^{*} \bigr)\bigr)= \textstyle\begin{cases} -1, & \sum_{k=0}^{k^{*}}\mathcal{M}_{k} \textit{ is odd}, \\ 1, & \sum_{k=0}^{k^{*}}\mathcal{M}_{k} \textit{ is even}. \end{cases} $$
In addition, when \(\sum_{k=0}^{k^{*}}\mathcal{M}_{k}\) is odd, system (3.1) possesses at least one nonconstant interior solution;
-
(3)
Suppose that \(\operatorname{Det}(\lambda )=0\) has two interior solutions \(\lambda _{+}^{*}\) and \(\lambda _{-}^{*}\) in two open intervals \((\lambda _{k_{1}^{*}},\lambda _{k_{1}^{*}+1})\) and \((\lambda _{k_{2}^{*}},\lambda _{k_{2}^{*}+1})\), respectively, where \(k_{1}^{*}>k_{2}^{*}\geq 0\). Then
$$ \operatorname{index}_{\mathcal{Q}}\bigl(CM,\bigl(u^{*},v^{*} \bigr)\bigr)= \textstyle\begin{cases} -1, & \sum_{k=k_{1}^{*}+1}^{k_{2}^{*}}\mathcal{M}_{k} \textit{ is odd}, \\ 1, & \sum_{k=k_{1}^{*}+1}^{k_{2}^{*}}\mathcal{M}_{k} \textit{ is even}. \end{cases} $$
In addition, when \(\sum_{k=k_{1}^{*}+1}^{k_{2}^{*}}\mathcal{M}_{k}\) is odd, system (3.1) possesses at least one nonconstant interior solution;
-
(4)
Suppose that \(a< \frac{b\gamma u^{*}[1+(2\alpha +b\beta \gamma )u^{*}]}{[1+(\alpha +b\beta \gamma )u^{*}]^{2}}\) and \(\frac{F_{u}^{*}}{d_{1}}\in (\lambda _{k^{*}},\lambda _{k^{*}+1})\) for some nonnegative integer \(k^{*}\). Then there is a constant \(d_{4}'>0\) such that for \(d_{4}'< d_{4}\),
$$ \operatorname{index}_{\mathcal{Q}}\bigl(CM,\bigl(u^{*},v^{*} \bigr)\bigr)= \textstyle\begin{cases} -1, & \sum_{k=0}^{k^{*}}\mathcal{M}_{k} \textit{ is odd}, \\ 1, & \sum_{k=0}^{k^{*}}\mathcal{M}_{k} \textit{ is even}. \end{cases} $$
In addition, when \(\sum_{k=0}^{k^{*}}\mathcal{M}_{k}\) is odd, system (3.1) possesses at least one nonconstant interior solution.
Proof
By applying the elliptic PDE theory and the eigenfunction theory, we prove that \(\operatorname{index}_{\mathcal{Q}}(CM,(u^{*},v^{*}))=1\).
(1) First, note that \(\overline{\mathcal{Q}}_{(u^{*},v^{*})}=\mathcal{W}_{(u^{*},v^{*})}= \mathcal{A}\) and
Set \(CM'(u^{*},v^{*})(\Phi ,\Psi )^{T}=(\Phi ,\Psi )^{T}\in \overline{\mathcal{Q}}_{(u^{*},v^{*})}\). Using the corresponding eigenfunction formulas in Lemma 3.2 for Φ and Ψ, we get
Hence \(\operatorname{Det}(\lambda _{k})=(d_{1}d_{4}+d_{2}d_{3})\lambda _{k}^{2} -[d_{1}G_{v}^{*}-d_{2}G_{u}^{*}+d_{3}F_{v}^{*}+d_{4}F_{u}^{*}] \lambda _{k} +F_{u}^{*}G_{v}^{*}-F_{v}^{*}G_{u}^{*}>0\) since \(\operatorname{Det}(\lambda )>0\). This implies that \(I-CM'(u^{*},v^{*})\) is an invertible matrix on \(\overline{\mathcal{Q}}_{(u^{*},v^{*})}\). Therefore \(CM'(u^{*},v^{*})\) does not possess property γ on \(\overline{\mathcal{Q}}_{(u^{*},v^{*})}\).
Now we offer a rigorous proof to guarantee that \(\delta =0\). Here δ is described in Lemma 4.1 of [13]. For \(\rho >0\), we study the eigenvalue issue \((CM'(u^{*},v^{*})-I)(\Phi ,\Psi )^{T}=\rho (\Phi ,\Psi )^{T}\), \((\Phi ,\Psi )\neq (0,0)\), which means that for some positive constant ρ, the model
$$ \textstyle\begin{cases} (\rho +1)(-d_{1}\Delta \Phi -d_{2}\Delta \Psi )=(F_{u}^{*}- \rho c)\Phi +F_{v}^{*}\Psi & \mbox{in } \Omega , \\ (\rho +1)(d_{3}\Delta \Phi -d_{4}\Delta \Psi )=G_{u}^{*}\Phi +(G_{v}^{*}- \rho c)\Psi & \mbox{in } \Omega , \\ \frac{\partial \Phi }{\partial \nu }= \frac{\partial \Psi }{\partial \nu }=0 & \mbox{on } \partial \Omega , \\ \Phi \Psi \neq 0 & \mbox{in } \Omega , \end{cases} $$
(3.5)
has a nontrivial solution if and only if ρ satisfies the quadratic equation \(\operatorname{Det}(Q(\rho ,\lambda _{k}))=0\), where
For \(\lambda _{k}>0\), we obtain \(\operatorname{Det}(\lambda _{k})>0\). This implies that \(\operatorname{Det}(Q(\rho ,\lambda _{k}))=0\) may have either precisely a multiplicity-two simple interior solution, or two multiplicity-one interior solutions, or no interior solution. Thus we can conclude that for \(k\geq 0\), the total algebraic multiplicity for \(\operatorname{Det}(Q(\rho ,\lambda _{k}))=0\) equals zero or two, which means that \(\delta =0\). Finally, we obtain that \(\operatorname{index}_{\mathcal{Q}}(CM,(u^{*},v^{*}))=1\).
(2) In view of the assumption introduced by (2), we can observe that for a nonnegative integer k, \(\operatorname{Det}(\lambda _{k})\neq 0\). Clearly, \(I-CM'(u^{*},v^{*})\) is an invertible matrix on \(\overline{\mathcal{Q}}_{(u^{*},v^{*})}\). Meanwhile, we can conclude that \(CM'(u^{*},v^{*})\) cannot possess property γ on \(\overline{\mathcal{Q}}_{(u^{*},v^{*})}\). Next, we focus on the total algebraic multiplicity δ for each eigenvalue of \(CM'(u^{*},v^{*})-I\), which is greater than zero. When \(\lambda _{0}=0\) (\(k=0\)), we get \(\operatorname{Det}(Q(\rho ,0))=c^{2}\rho ^{2}-c(F_{u}^{*}+G_{v}^{*})\rho +F_{u}^{*}G_{v}^{*}-F_{v}^{*}G_{u}^{*}=0\). By the hypothesis of (2) we get \(F_{u}^{*}G_{v}^{*}-F_{v}^{*}G_{u}^{*}\leq 0\). Obviously, \(\operatorname{Det}(Q(\rho ,0))=0\) has exactly a simple interior solution. Together with \(1\leq k\leq k^{*}\) and the hypothesis in (2), wee get \(\operatorname{Det}(\lambda _{k})<0\), and hence \(\operatorname{Det}(Q(\rho ,\lambda _{k}))=0\) possesses precisely one positive simple solution. If \(k\geq k^{*}+1\), then \(\operatorname{Det}(\lambda _{k})>0\), and hence \(\operatorname{Det}(Q(\rho ,\lambda _{k}))=0\) may have either precisely one multiplicity-two simple interior solution, or two multiplicity-one interior solutions, or no interior solution. This discussion yields that \(\sum_{k=0}^{k^{*}}\mathcal{M}_{k}+t=\delta \), where t is an even number or 0. In addition, suppose on the contrary that model (3.1) possesses no nonconstant interior solution. By Lemma 3.2, Lemma 3.3, and the last discussion we get
$$ \begin{aligned} 1&=\operatorname{index}_{\mathcal{Q}}\bigl(CM, \operatorname{Int} \mathcal{D}'\bigr)=\operatorname{index}_{\mathcal{Q}} \bigl(CM,(a,0)\bigr)+\operatorname{index}_{\mathcal{Q}}\bigl(CM, \bigl(u^{*},v^{*}\bigr)\bigr) \\ &=0+(-1)=-1. \end{aligned} $$
This contradiction tells us that system (3.1) possesses at least one nonconstant interior solution. The above argument derives the ideal outcome.
(3) The proof of this part is similar to that of part (2), and we omit it.
(4) Due to \(a< \frac{b\gamma u^{*}[1+(2\alpha +b\beta \gamma )u^{*}]}{[1+(\alpha +b\beta \gamma )u^{*}]^{2}}\), we get \(F_{u}^{*}>0\). Let \(\operatorname{Det}(\lambda )=0\), which means that
$$ \operatorname{Det}(\lambda )=(d_{1}d_{4}+d_{2}d_{3}) \lambda ^{2}- \Theta \lambda +F_{u}^{*}G_{v}^{*}-F_{v}^{*}G_{u}^{*}=0, $$
(3.6)
where
$$ \Theta =d_{1}G_{v}^{*}-d_{2}G_{u}^{*}+d_{3}F_{v}^{*}+d_{4}F_{u}^{*}. $$
(3.7)
By direct calculation we get that
$$\begin{aligned}& \lim_{d_{4}\rightarrow \infty }\lambda _{+}^{*}= \lim_{d_{1}\rightarrow \infty } \frac{\Theta +\sqrt{\Theta ^{2}-4(d_{1}d_{4}+d_{2}d_{3})(F_{u}^{*}G_{v}^{*}-F_{v}^{*}G_{u}^{*})}}{2(d_{1}d_{4}+d_{2}d_{3})}= \frac{F_{u}^{*}}{d_{1}}, \\& \lim_{d_{4}\rightarrow \infty }\lambda _{-}^{*}=\lim _{d_{1}\rightarrow \infty } \frac{\Theta -\sqrt{\Theta ^{2}-4(d_{1}d_{4}+d_{2}d_{3})(F_{u}^{*}G_{v}^{*}-F_{v}^{*}G_{u}^{*})}}{2(d_{1}d_{4}+d_{2}d_{3})}=0. \end{aligned}$$
Hence we can find a constant \(d_{4}'>0\) guaranteeing \(\lambda _{+}^{*}>\lambda _{k^{*}}\) and \(\lambda _{-}^{*}<\lambda _{1}\) if \(d_{4}'< d_{4}\) due to \(\frac{F_{u}^{*}}{d_{1}}\in (\lambda _{k^{*}},\lambda _{k^{*}+1})\). Therefore, for a nonnegative integer k, \(\operatorname{Det}(\lambda _{k})\neq 0\). We can observe that \(I-CM'(u^{*},v^{*})\) is an invertible matrix on \(\overline{\mathcal{Q}}_{(u^{*},v^{*})}\). Meanwhile, we can conclude that \(CM'(u^{*},v^{*})\) cannot possess property γ on \(\overline{\mathcal{Q}}_{(u^{*},v^{*})}\). On the other hand, note that \(\operatorname{Det}(\lambda _{1})<0\) when
$$ d_{4}>\sigma _{1}:= \frac{d_{2}d_{3}\lambda _{1}^{2}+(-d_{1}G_{v}^{*} +d_{2}G_{u}^{*}-d_{3}F_{v}^{*})\lambda _{1}F_{u}^{*}G_{v}^{*} -F_{v}^{*}G_{u}^{*}}{F_{u}^{*}\lambda _{1}-d_{1}\lambda _{1}^{2}} $$
and \(\operatorname{Det}(\lambda _{k^{*}})<0\) when
$$ d_{4}>\sigma _{2}:= \frac{d_{2}d_{3}\lambda _{k^{*}}^{2}+(-d_{1}G_{v}^{*}+d_{2}G_{u}^{*} -d_{3}F_{v}^{*})\lambda _{k^{*}}F_{u}^{*}G_{v}^{*}-F_{v}^{*}G_{u}^{*}}{F_{u}^{*}\lambda _{k^{*}}-d_{1}\lambda _{k^{*}}^{2}}. $$
We can take a constant \(d_{4}'>\max \{\sigma _{1},\sigma _{2}\}\). We easily verify that for each \(1\leq k\leq k^{*}\), \(\operatorname{Det}(Q(\rho ,\lambda _{k}))=0\) has exactly one simple interior solution due to \(\operatorname{Det}(\lambda _{k})<0\). Meanwhile, together with \(k\geq 1+k^{*}\) and \(\operatorname{Det}(\lambda _{k})>0\), \(\operatorname{Det}(Q(\rho ,\lambda _{k}))=0\) has either precisely one multiplicity-two simple interior solution, or two multiplicity-one interior solutions, or no interior solution. According to the above discussion, we derive that \(\sum_{k=0}^{k^{*}}\mathcal{M}_{k}+0=\delta \). In addition, suppose on the contrary that model (3.1) possesses no nonconstant interior solution. By Lemmas 3.2 and 3.3 and the last argument we get
$$ \begin{aligned} 1&=\operatorname{index}_{\mathcal{Q}}\bigl(CM, \operatorname{Int} \mathcal{D}'\bigr)=\operatorname{index}_{\mathcal{Q}} \bigl(CM,(a,0)\bigr)+\operatorname{index}_{\mathcal{Q}}\bigl(CM, \bigl(u^{*},v^{*}\bigr)\bigr) \\ &=0+(-1)=-1. \end{aligned} $$
This contradiction tells us that system (3.1) possesses at least one nonconstant interior solution. The above argument derives the ideal outcome. This ends the proof of Theorem 3.1. □
Example 1
Choosing \(a=2\), \(b=2\), \(\alpha =1\), \(\beta =1\), and \(\gamma =2\), we obtain \((u^{*},v^{*})= (\frac{5+\sqrt{65}}{10}, \frac{10+2\sqrt{65}}{5} )\approx (1.3062,5.2249)\). Thus \(\frac{b\gamma u^{*}[1+(2\alpha +b\beta \gamma )u^{*}]}{[1+(\alpha +b\beta \gamma )u^{*}]^{2}} \approx 0.8141< a+b=4\) and \(a=2> \frac{b\gamma u^{*}(\alpha +b\beta \gamma )u^{*}}{[1+(\alpha +b\beta \gamma )u^{*}]^{2}} \approx 0.6017\). Therefore system (1.1) possesses at least one nonconstant interior solution.