# Non-synchronized solutions to nonlinear elliptic Schrödinger systems on a
closed Riemannian manifold

Saikat Mazumdar  Saikat Mazumdar, Department of Mathematics, Indian Institute
of Technology Bombay, Mumbai 400076, India<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Jérôme Vétois  Jérôme Vétois, Department of
Mathematics and Statistics, McGill University, 805 Sherbrooke Street West,
Montreal, Quebec H3A 0B9, Canada<EMAIL_ADDRESS>
(Date: June 22, 2021)

###### Abstract.

On a smooth, closed Riemannian manifold, we study the question of
proportionality of components, also called synchronization, of vector-valued
solutions to nonlinear elliptic Schrödinger systems with constant
coefficients. In particular, we obtain bifurcation results showing the
existence of branches of non-synchronized solutions emanating from the
constant solutions.

The second author was supported by the Discovery Grant RGPIN-2016-04195 from
the Natural Sciences and Engineering Research Council of Canada. This work was
partially conducted when the first author held a postdoctoral position at
McGill University under the co-supervision of Professors Pengfei Guan, Niky
Kamran and the second author, that was supported by the NSERC Discovery Grants
RGPIN-04443-2018, RGPIN-05490-2018 and RGPIN-04195-2016.

##  1\. Introduction

On a smooth, closed Riemannian manifold $\left(M,g\right)$ of dimension $n$,
we consider vector-valued solutions $\left(u_{1},u_{2}\right)\in
C^{2}\left(M\right)^{2}$ to elliptic systems of the form

$\left\\{\begin{aligned}
&\Delta_{g}u_{1}=F_{1}\left(u_{1},u_{2}\right)&&\text{in }M\\\
&\Delta_{g}u_{2}=F_{2}\left(u_{1},u_{2}\right),&&\text{in
}M\end{aligned}\right.$ (1.1)

where $F_{1}$ and $F_{2}$ are $C^{1}$ functions and
$\Delta_{g}:=-\operatorname{div}\nabla$ is the Laplace–Beltrami operator. In
particular, we are interested in the stationary nonlinear Schrödinger system

$\left\\{\begin{aligned}
&\Delta_{g}u_{1}+\lambda_{1}u_{1}=a_{11}u_{1}^{q-1}+a_{12}u_{2}^{q-2}u_{1}&&\text{in
}M\\\
&\Delta_{g}u_{2}+\lambda_{2}u_{2}=a_{21}u_{1}^{q-2}u_{2}+a_{22}u_{2}^{q-1}&&\text{in
}M\\\ &u_{1},u_{2}>0&&\text{in }M,\end{aligned}\right.$ (1.2)

where $\lambda_{1},\lambda_{2},a_{11},a_{12},a_{21},a_{22}\in\mathbb{R}$ and
$q\in\left(2,\infty\right)$. In the cubic case $q=4$, the system (1.2) arises
in particular in nonlinear optics (see for instance Akhmediev and Ankiewicz
[AA] and Kanna and Lakshmanan [KL]) and the Hartree–Fock theory for
Bose–Einstein condensates (see Esry, Greene, Burke and Bohn [EGBB] and
Timmermans [T]). Such systems have received considerable attention from
mathematicians in recent years. Among many others, let us refer for instance
to the work of Bartsch, Dancer and Wang [BDW], Clapp and Pistoia [CP1, CP2],
Gladiali, Grossi and Troestler [GGT2, GGT1], Guo, Li and Wei [GLW], Guo and
Liu [GL], Lin and Wei [LW1], Liu and Wang [LW2], Peng, Peng and Wang [PPW],
Sirakov [S], Soave and Zilio [SZ], Terracini and Verzini [TV] and Wei and Wu
[WW] in the case where $M=\mathbb{R}^{n}$ (note that when $\left(M,g\right)$
is the standard round sphere, $n\geq 3$,
$\lambda_{1}=\lambda_{2}=n\left(n-2\right)/4$ and $q=2n/\left(n-2\right)$, we
can use stereographic projection to write (1.2) as a system in
$\mathbb{R}^{n}$) and Chen and Zou [CZ], Clapp, Pistoia and Tavares [CPT],
Druet and Hebey [DH] and Druet, Hebey and Vétois [DHV] in the case of a more
general manifold.

In this paper, we are interested in the question of proportionality of
components, also called synchronization, of solutions to the system (1.2). A
solution $\left(u_{1},u_{2}\right)$ of (1.2) is said to be synchronized if
there exists a constant $\Lambda>0$ such that $u_{2}\equiv\Lambda u_{1}$ in
$M$. This question has been studied for instance by Montaru, Sirakov and
Souplet [MSS] and Quittner and Souplet [QS] in the case of systems in domains
of $\mathbb{R}^{n}$. It also naturally arises in the case of a closed
manifold.

It is easy to see that every synchronized solution
$\left(u_{1},u_{2}\right)=\left(u_{1},\Lambda u_{1}\right)$ of the system
(1.2) is such that $u_{1}$ is constant in the case where
$\lambda_{1}\neq\lambda_{2}$ and $u_{1}$ is a solution of the equation

$\left\\{\begin{aligned}
&\Delta_{g}u_{1}+\lambda_{1}u_{1}=\mu_{1}u_{1}^{q-2}&&\text{ in }M\\\
&u_{1}>0&&\text{ in }M,\end{aligned}\right.$ (1.3)

where $\mu_{1}:=a_{11}+a_{12}\Lambda^{q-2}=a_{21}+a_{22}\Lambda^{q-2}$, in the
case where $\lambda_{1}=\lambda_{2}$. We know from a result of Bidaut-Véron
and Véron [BV] that the equation (1.3) does not have any non-constant
solutions when

$2<q\leq 2^{*}\quad\text{and}\quad\left\\{\begin{aligned}
&\frac{n-1}{n}\left(q-2\right)\lambda_{1}g\leq\operatorname{Ric}_{g}&&\text{if
}q<2^{*}\\\
&\frac{n-1}{n}\left(q-2\right)\lambda_{1}g<\operatorname{Ric}_{g}&&\text{if
}q=2^{*},\end{aligned}\right.$ (1.4)

where $2^{*}:=\infty$ if $n\leq 2$, $2^{*}:=2n/\left(n-2\right)$ if $n\geq 3$,
$\operatorname{Ric}_{g}$ is the Ricci curvature of the manifold and the latter
inequalities are in the sense of bilinear forms. On the other hand, existence
results of non-constant solutions to equation (1.3) abound in the case where
(1.4) is not satisfied (see for instance Chen, Wei and Yan [CWY], Hebey and
Vaugon [HV], Hebey and Wei [HW], Micheletti, Pistoia and Vétois [MPV] and
Vétois and Wang [VW].)

For simplicity, in this introduction, we state our results in the case of the
sphere $\left(\mathbb{S}^{n},g_{0}\right)$, where $g_{0}$ is the standard
round metric. Furthermore, we assume that $\lambda_{1}=\lambda_{2}$ and
$a_{12}=a_{21}$, namely we consider the system

$\left\\{\begin{aligned} &\Delta_{g_{0}}u_{1}+\lambda
u_{1}=au_{1}^{q-1}+bu_{2}^{q-2}u_{1}&&\text{in }\mathbb{S}^{n}\\\
&\Delta_{g_{0}}u_{2}+\lambda u_{2}=bu_{1}^{q-2}u_{2}+cu_{2}^{q-1}&&\text{in
}\mathbb{S}^{n}\\\ &u_{1},u_{2}>0&&\text{in
}\mathbb{S}^{n},\end{aligned}\right.$ (1.5)

where $\lambda,a,b,c\in\mathbb{R}$ and $q\in\left(2,\infty\right)$. In the
Euclidean space, this case has been studied for instance by Clapp and Pistoia
[CP1] (via stereographic projection, the system studied in [CP1] matches with
(1.5) when $n=4$, $\lambda=2$ and the parameters $\alpha$ and $\beta$ in [CP1]
are equal to 2). We refer to Sections 2 and 3 for results applying to more
general systems and more general manifolds. For the system (1.5), we obtain
the following:

######  Theorem 1.1.

Let $\lambda,a,b,c\in\mathbb{R}$ and $q\in\left(2,\infty\right)$.

  1. (i)

If either $c\leq b\leq a$ or $a\leq b\leq c$ and at least one of the two
inequalities is strict, then the system (1.5) has no solutions.

  2. (ii)

If either [$a<b$ and $c<b$] or $a=b=c$, then every solution of (1.5) is
synchronized.

  3. (iii)

There exist real numbers $\lambda$, $a$, $b$ and $c$ such that $\lambda>0$,
$a=c>b>0$ and (1.5) has non-synchronized solutions. More precisely, we have
the following result: for every $\lambda,a,b\in
C^{1}\left(\left[-\delta,\delta\right]\right)$, $\delta>0$, if the following
conditions hold:

    1. (A1)

$\lambda\left(0\right)\left(a\left(0\right)+b\left(0\right)\right)>0$,

    2. (A2)

$\lambda\left(0\right)\not\in\big{\\{}\frac{2j\left(2j+n-1\right)}{q-2}:\
j\in\mathbb{N}\big{\\}}$, where $\mathbb{N}:=\left\\{1,2,\dotsc\right\\}$,

    3. (A3)

$\beta\left(0\right):=\lambda\left(0\right)\frac{a\left(0\right)-b\left(0\right)}{a\left(0\right)+b\left(0\right)}\in\big{\\{}\frac{j\left(j+n-1\right)}{q-2}:\
j\in\mathbb{N}\big{\\}}$ and $\beta^{\prime}\left(0\right)\neq 0$,

then there exists a $C^{1}$ branch (see Definition 2.1) of non-synchronized
solutions to (1.5) with $\lambda=\lambda\left(\alpha\right)$,
$c=a=a\left(\alpha\right)$ and $b=b\left(\alpha\right)$ emanating from the
constant solution at $\alpha=0$.

Theorem 1.1 (iii) extends a previous result obtained by Gladiali, Grossi and
Troestler [GGT1] for systems with Sobolev critical growth in $\mathbb{R}^{n}$,
which, via stereographic projection, corresponds to the case where $n\geq 3$,
$\lambda=n\left(n-2\right)/4$ and $q=2n/\left(n-2\right)$. Like in [GGT1], our
approach is based on the bifurcation theory at eigenvalues of odd
multiplicity. Unlike in [GGT1], by taking advantage of our closed manifold
setting, we perform our constructions in $C^{1,\theta}\left(M\right)$,
$\theta\in\left(0,1\right)$, instead of Sobolev spaces, which allows us to
treat the case of systems with supercritical growth.

Theorem 1.1 (iii) is proven in Section 2 (as a particular case of Theorem 2.3)
and Theorem 1.1 (i) and (ii) are proven in Section 3 (as a particular cases of
Theorem 3.1 (i) and (ii)).

##  2\. Bifurcation results

This section is devoted to bifurcation results showing the existence of
branches of non-synchronized solutions for systems like (1.2).

######  Definition 2.1.

Let $\left(M,g\right)$ be a smooth, closed Riemannian manifold, $\Omega$ be an
open set in $\mathbb{R}^{2}$, $I:=\left[-\delta,\delta\right]$, $\delta>0$,
and $F_{1},F_{2}\in C^{1}\left(I\times\Omega\right)$. Consider the system

$\left\\{\begin{aligned}
&\Delta_{g}u_{1}=F_{1}\left(\alpha,u_{1},u_{2}\right)&&\text{in }M\\\
&\Delta_{g}u_{2}=F_{2}\left(\alpha,u_{1},u_{2}\right)&&\text{in
}M,\end{aligned}\right.$ (2.1)

where $\alpha\in I$. Assume that for every $\alpha\in I$, there exists a
solution
$\left(\overline{u}_{1}\left(\alpha\right),\overline{u}_{2}\left(\alpha\right)\right)\in
C^{2}\left(M\right)^{2}$ of (2.1) such that
$\left(\overline{u}_{1}\left(\alpha\right),\overline{u}_{2}\left(\alpha\right)\right)\to\left(\overline{u}_{1}\left(0\right),\overline{u}_{2}\left(0\right)\right)$
in $C^{2}\left(M\right)^{2}$ as $\alpha\to 0$. Let $\mathcal{S}$ be the set of
all solutions $\left(\alpha,u_{1},u_{2}\right)\in I\times
C^{2}\left(M\right)^{2}$ to (2.1) such that
$\left(u_{1},u_{2}\right)\neq\left(\overline{u}_{1}\left(\alpha\right),\overline{u}_{2}\left(\alpha\right)\right)$.
We say that the solution
$\left(0,\overline{u}_{1}\left(0\right),\overline{u}_{2}\left(0\right)\right)$
is a _bifurcation point_ of (2.1) if
$\left(0,\overline{u}_{1}\left(0\right),\overline{u}_{2}\left(0\right)\right)\in\overline{\mathcal{S}}$,
where $\overline{\mathcal{S}}$ stands for the closure of $\mathcal{S}$ in
$I\times C^{2}\left(M\right)^{2}$. Furthermore, we say that a subset
$\mathcal{B}\subseteq\mathcal{S}$ is a $C^{1}$ _branch_ of solutions to (2.1)
emanating from
$\left(0,\overline{u}_{1}\left(0\right),\overline{u}_{2}\left(0\right)\right)$
if $\mathcal{B}\neq\emptyset$ and
$\mathcal{B}\cup\left\\{\left(0,\overline{u}_{1}\left(0\right),\overline{u}_{2}\left(0\right)\right)\right\\}$
is a $C^{1}$ curve in $I\times C^{2}\left(M\right)^{2}$.

In the case of the sphere, we obtain Theorem 1.1 (iii). In the case of a more
general manifold, we obtain the following:

######  Theorem 2.2.

Let $\left(M,g\right)$ be a smooth, closed Riemannian manifold,
$I:=\left[-\delta,\delta\right]$, $\delta>0$,
$\lambda_{1},\lambda_{2},a_{11},a_{12},a_{21},a_{22}\in C^{1}\left(I\right)$
and $q\in\left(2,\infty\right)$. Consider the system

$\left\\{\begin{aligned}
&\Delta_{g}u_{1}+\lambda_{1}\left(\alpha\right)u_{1}=a_{11}\left(\alpha\right)u_{1}^{q-1}+a_{12}\left(\alpha\right)u_{2}^{q-2}u_{1}&&\text{in
}M\\\
&\Delta_{g}u_{2}+\lambda_{2}\left(\alpha\right)u_{2}=a_{21}\left(\alpha\right)u_{1}^{q-2}u_{2}+a_{22}\left(\alpha\right)u_{2}^{q-1}&&\text{in
}M\\\ &u_{1},u_{2}>0&&\text{in }M,\end{aligned}\right.$ (2.2)

where $\alpha\in I$. Assume that the following conditions hold:

  1. (B1)

For every $\alpha\in I$, the system (1.2) has a unique constant solution
$\left(\overline{u}_{1}\left(\alpha\right),\overline{u}_{2}\left(\alpha\right)\right)$.

  2. (B2)

For every $\alpha\in I$, the matrix

$\displaystyle\mathcal{A}\left(\alpha\right):=\left(\begin{array}[]{cc}a_{11}\left(\alpha\right)\overline{u}_{1}\left(\alpha\right)^{q-2}&a_{12}\left(\alpha\right)\overline{u}_{2}\left(\alpha\right)^{q-3}\overline{u}_{1}\left(\alpha\right)\\\
a_{21}\left(\alpha\right)\overline{u}_{1}\left(\alpha\right)^{q-3}\overline{u}_{2}\left(\alpha\right)&a_{22}\left(\alpha\right)\overline{u}_{2}\left(\alpha\right)^{q-2}\end{array}\right)$

has two distinct, non-zero, real eigenvalues $\beta_{1}\left(\alpha\right)$
and $\beta_{2}\left(\alpha\right)$.

  3. (B3)

$\mathcal{H}^{*}_{1}\times\mathcal{H}^{*}_{2}$ has odd dimension, where

$\mathcal{H}^{*}_{i}:=\left\\{\varphi\in
C^{2}\left(M\right):\Delta_{g}\varphi=\left(q-2\right)\beta_{i}\left(0\right)\varphi\hbox{
in }M\right\\}\quad\forall i\in\left\\{1,2\right\\}.$

  4. (B4)

For every $i\in\left\\{1,2\right\\}$, if
$\mathcal{H}^{*}_{i}\neq\left\\{0\right\\}$, then
$\beta^{\prime}_{i}\left(0\right)\neq 0$ and either
$\lambda_{1}\left(0\right)\neq\lambda_{2}\left(0\right)$ or
$\beta_{i}\left(0\right)\neq\lambda_{1}\left(0\right)=\lambda_{2}\left(0\right)$.

Then the solution
$\left(0,\overline{u}_{1}\left(0\right),\overline{u}_{2}\left(0\right)\right)$
is a bifurcating point of the system (2.2). Furthermore, there exists a
neighborhood $\mathcal{N}$ of
$\left(0,\overline{u}_{1}\left(0\right),\overline{u}_{2}\left(0\right)\right)$
in $I\times C^{2}\left(M\right)^{2}$ such that for every solution
$\left(\alpha,u_{1},u_{2}\right)\in\mathcal{N}$ of (2.2), if
$\left(u_{1},u_{2}\right)\neq\left(\overline{u}_{1}\left(\alpha\right),\overline{u}_{2}\left(\alpha\right)\right)$,
then $\left(u_{1},u_{2}\right)$ is non-synchronized. If moreover
$\mathcal{H}^{*}_{1}\times\mathcal{H}^{*}_{2}$ has dimension one, then there
exists a $C^{1}$ branch of non-synchronized solutions to (2.2) emanating from
$\left(0,\overline{u}_{1}\left(0\right),\overline{u}_{2}\left(0\right)\right)$.

Both Theorem 1.1 (iii) and Theorem 2.2 follow from the following general
bifurcation result for systems of the form (2.1):

######  Theorem 2.3.

Let $\left(M,g\right)$ be a smooth, closed Riemannian manifold, $\Omega$ be an
open set in $\mathbb{R}^{2}$ such that $\left(0,0\right)\in\Omega$,
$I:=\left[-\delta,\delta\right]$, $\delta>0$, and $F_{1},F_{2}\in
C^{1}\left(I\times\Omega\right)$ such that
$\partial_{\alpha}\partial_{u_{i}}F_{j}$ exists and is continuous in
$I\times\Omega$ for all $i,j\in\left\\{1,2\right\\}$, where we denote by
$\left(\alpha,u_{1},u_{2}\right)$ a point in $I\times\Omega$. Assume that the
following conditions hold:

  1. (C1)

$F_{1}\left(\alpha,0,0\right)=F_{2}\left(\alpha,0,0\right)=0$ for all
$\alpha\in I$.

  2. (C2)

$\partial_{u_{2}}F_{1}\left(\alpha,0,0\right)=\partial_{u_{1}}F_{2}\left(\alpha,0,0\right)=0$
for all $\alpha\in I$.

  3. (C3)

There exist two closed subspaces $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ of
$C^{1,\theta}\left(M\right)$, $\theta\in\left(0,1\right)$, and two open
subsets $\mathcal{U}_{1}\subseteq\mathcal{H}_{1}$ and
$\mathcal{U}_{2}\subseteq\mathcal{H}_{2}$ which contain 0 and satisfy the
following conditions:

     * –

$\left(u_{1}\left(x\right),u_{2}\left(x\right)\right)\in\Omega$ for all $x\in
M$ and $\left(u_{1},u_{2}\right)\in{\mathcal{U}_{1}}\times{\mathcal{U}_{2}}$.

     * –

$\left(\Delta_{g}+1\right)^{-1}u_{i}\in\mathcal{H}_{i}$ for all
$u_{i}\in{{\mathcal{U}}_{i}}$ and $i\in\left\\{1,2\right\\}$.

     * –

$\left(\Delta_{g}+1\right)^{-1}F_{i}\left(\alpha,u_{1},u_{2}\right)\in\mathcal{H}_{i}$
for all $\left(\alpha,u_{1},u_{2}\right)\in
I\times{{\mathcal{U}}_{1}}\times{{\mathcal{U}}_{2}}$ and
$i\in\left\\{1,2\right\\}$.

  4. (C4)

$\mathcal{H}^{*}_{1}\times\mathcal{H}^{*}_{2}$ has odd dimension, where

$\mathcal{H}^{*}_{i}:=\left\\{\varphi\in\mathcal{H}_{i}:\
\Delta_{g}\varphi=\partial_{u_{i}}F_{i}\left(0,0,0\right)\varphi\text{ in
}M\right\\}\quad\forall i\in\left\\{1,2\right\\}.$

  5. (C5)

For every $i\in\left\\{1,2\right\\}$, if
$\mathcal{H}^{*}_{i}\neq\left\\{0\right\\}$, then
$\partial_{\alpha}\partial_{u_{i}}F_{i}\left(0,0,0\right)\neq 0$.

Then there exists a sequence of solutions
$\left(\left(\alpha_{m},u_{1,m},u_{2,m}\right)\right)_{m\in\mathbb{N}}$ to the
system (2.1) such that $\left(\alpha_{m},u_{1,m},u_{2,m}\right)\in
I\times\left(\left(\mathcal{U}_{1}\times\mathcal{U}_{2}\right)\backslash\left\\{\left(0,0\right)\right\\}\right)$
and $\left(\alpha_{m},u_{1,m},u_{2,m}\right)\to\left(0,0,0\right)$ in $I\times
C^{2}\left(M\right)^{2}$ as $m\to\infty$. Furthermore, every such sequence
$\left(\left(\alpha_{m},u_{1,m},u_{2,m}\right)\right)_{m\in\mathbb{N}}$ is
such that up to a subsequence,

$u_{i,m}=\varepsilon_{m}\varphi_{i}+\operatorname{o}\left(\varepsilon_{m}\right)\quad\hbox{in
}\mathcal{H}_{i}\quad\forall i\in\left\\{1,2\right\\}$ (2.3)

as $m\to\infty$ for some
$\left(\varphi_{1},\varphi_{2}\right)\in\left(\mathcal{H}^{*}_{1}\times\mathcal{H}^{*}_{2}\right)\backslash\left\\{\left(0,0\right)\right\\}$
and $\varepsilon_{m}>0$ such that $\varepsilon_{m}\to 0$. If moreover
$\mathcal{H}^{*}_{1}\times\mathcal{H}^{*}_{2}$ has dimension one and
$\partial_{u_{i}}\partial_{u_{j}}F_{k}$ exists and is continuous in
$I\times\Omega$ for all $i,j,k\in\left\\{1,2\right\\}$, then there exists a
neighborhood $\mathcal{N}$ of $\left(0,0,0\right)$ in
$I\times\mathcal{U}_{1}\times\mathcal{U}_{2}$ such that the set of solutions
$\left(\alpha,u_{1},u_{2}\right)\in\mathcal{N}\backslash\left(I\times\left\\{\left(0,0\right)\right\\}\right)$
to (2.1) is a $C^{1}$ branch emanating from $\left(0,0,0\right)$ whose tangent
line at $\left(0,0,0\right)$ is directed by some vector in
$\mathbb{R}\times\left(\left(\mathcal{H}^{*}_{1}\times\mathcal{H}^{*}_{2}\right)\backslash\left\\{\left(0,0\right)\right\\}\right)$.

###### Proof of Theorem 2.3.

Let

$\mathcal{H}:=\mathcal{H}_{1}\times\mathcal{H}_{2},\quad\mathcal{H}^{*}:=\mathcal{H}_{1}^{*}\times\mathcal{H}_{2}^{*}\quad\hbox{and}\quad\mathcal{U}:=\mathcal{U}_{1}\times\mathcal{U}_{2}.$

By replacing $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ by smaller sets if
necessary, we may assume that there exists a compact set $K\subset\Omega$ such
that $\left(u_{1}\left(x\right),u_{2}\left(x\right)\right)\in K$ for all $x\in
M$ and $\left(u_{1},u_{2}\right)\in\mathcal{U}$. The solutions
$\left(u_{1},u_{2}\right)\in\mathcal{U}$ to (2.1) are given by the zeros of
the function $\mathcal{T}:I\times{{\mathcal{U}}}\to\mathcal{H}$ defined by

$\mathcal{T}\left(\alpha,u_{1},u_{2}\right):=\mathcal{I}\left(u_{1},u_{2}\right)-\mathcal{K}_{\alpha}\left(u_{1},u_{2}\right),\quad\text{where}\quad\mathcal{I}\left(u_{1},u_{2}\right):=\left(\begin{array}[]{c}u_{1}\\\
u_{2}\end{array}\right)$

and

$\mathcal{K}_{\alpha}\left(u_{1},u_{2}\right):=\left(\begin{array}[]{c}\left(\Delta_{g}+1\right)^{-1}\left[F_{1}\left(\alpha,u_{1},u_{2}\right)+u_{1}\right]\\\
\left(\Delta_{g}+1\right)^{-1}\left[F_{2}\left(\alpha,u_{1},u_{2}\right)+u_{2}\right]\end{array}\right)$

for all $\left(\alpha,u_{1},u_{2}\right)\in I\times{{\mathcal{U}}}$. By the
assumption (C3), we have that the functions $\mathcal{K}_{\alpha}$ and
$\mathcal{T}$ are well-defined. In what follows, we write a point in
$I\times{{\mathcal{U}}}$ as $\left(\alpha,U\right)$.

Step 1. We begin with proving that $\mathcal{K}_{\alpha}$ is compact,
$\mathcal{T}\in C^{1}\left(I\times\mathcal{U}\right)$ and
$D_{U}\,\partial_{\alpha}\mathcal{T}$ exists and is continuous in
$I\times\mathcal{U}$.

Suppose that $\left(\left(u_{1,m},u_{2,m}\right)\right)_{m\in\mathbb{N}}$ is a
bounded sequence in ${{\mathcal{U}}}$. Then the sequences
$\left(u_{1,m}\right)_{m}$ and $\left(u_{2,m}\right)_{m}$ are bounded in
$C^{1,\theta}\left(M,K\right)$ and up to a subsequence,
$\left(u_{1,m},u_{2,m}\right)\to\left(u_{1,0},u_{2,0}\right)$ in
$C^{1}\left(M,K\right)^{2}$ as $m\to\infty$. Let $\alpha\in I$ and

$\left(\begin{array}[]{c}\tilde{u}_{1,m}\\\
\tilde{u}_{2,m}\end{array}\right):=\mathcal{K}_{\alpha}\left(u_{1,m},u_{2,m}\right)$

so that

$\left(\begin{array}[]{c}\Delta_{g}\tilde{u}_{1,m}+\tilde{u}_{1,m}\\\
\Delta_{g}\tilde{u}_{2,m}+\tilde{u}_{2,m}\end{array}\right)=\left(\begin{array}[]{c}F_{1}\left(\alpha,u_{1,m},u_{2,m}\right)+u_{1,m}\\\
F_{2}\left(\alpha,u_{1,m},u_{2,m}\right)+u_{2,m}\end{array}\right).$

Since $F_{1},F_{2}\in C^{1}\left(I\times\Omega\right)$, $u_{1,0},u_{2,0}\in
C^{1}\left(M,K\right)$ and $K\subset\Omega$, by standard elliptic estimates,
we obtain that there exists $\left(\tilde{u}_{1,0},\tilde{u}_{2,0}\right)\in
C^{2}\left(M\right)^{2}$ satisfying

$\left(\begin{array}[]{c}\Delta_{g}\tilde{u}_{1,0}+\tilde{u}_{1,0}\\\
\Delta_{g}\tilde{u}_{2,0}+\tilde{u}_{2,0}\end{array}\right)=\left(\begin{array}[]{c}F_{1}\left(\alpha,u_{1,0},u_{2,0}\right)+u_{1,0}\\\
F_{2}\left(\alpha,u_{1,0},u_{2,0}\right)+u_{2,0}\end{array}\right).$

For $i=1,2$, we then obtain

$\left(\Delta_{g}+1\right)\left(\tilde{u}_{i,m}-\tilde{u}_{i,0}\right)=F_{i}\left(\alpha,u_{1,m},u_{2,m}\right)-F_{i}\left(\alpha,u_{1,0},u_{2,0}\right)\\\
+u_{i,m}-u_{i,0}=\operatorname{o}\left(1\right)$ (2.4)

uniformly in $M$. Therefore,
$\left(\tilde{u}_{1,m},\tilde{u}_{2,m}\right)\to\left(\tilde{u}_{1,0},\tilde{u}_{2,0}\right)$
in $\mathcal{H}$ and so $\mathcal{K}_{\alpha}$ is compact.

Now, if
$\left(\alpha_{m},u_{1,m},u_{2,m}\right)\to\left(\alpha_{0},u_{1,0},u_{2,0}\right)$
in $I\times{\mathcal{U}}$, then, arguing as above, we obtain that
$\mathcal{K}_{\alpha_{m}}\left(u_{1,m},u_{2,m}\right)\to\mathcal{K}_{\alpha_{0}}\left(u_{1,0},u_{2,0}\right)$
in $\mathcal{H}$. This shows that $\mathcal{T}$ is continuous in
$I\times{\mathcal{U}}$.

For every $\left(\alpha,u_{1},u_{2}\right)\in I\times{\mathcal{U}}$, we have

$\displaystyle\partial_{\alpha}\mathcal{T}\left(\alpha,u_{1},u_{2}\right)=-\left(\begin{array}[]{c}\left(\Delta_{g}+1\right)^{-1}\left[\partial_{\alpha}F_{1}\left(\alpha,u_{1},u_{2}\right)\right]\\\
\left(\Delta_{g}+1\right)^{-1}\left[\partial_{\alpha}F_{2}\left(\alpha,u_{1},u_{2}\right)\right]\end{array}\right),\allowdisplaybreaks$
$\displaystyle
D_{U}\mathcal{T}\left(\alpha,u_{1},u_{2}\right)\left[\left(v_{1},v_{2}\right)\right]$
$\displaystyle=\left(\begin{array}[]{c}v_{1}-\left(\Delta_{g}+1\right)^{-1}\left[\partial_{u_{1}}F_{1}\left(\alpha,u_{1},u_{2}\right)v_{1}+\partial_{u_{2}}F_{1}\left(\alpha,u_{1},u_{2}\right)v_{2}+v_{1}\right]\\\
v_{2}-\left(\Delta_{g}+1\right)^{-1}\left[\partial_{u_{1}}F_{2}\left(\alpha,u_{1},u_{2}\right)v_{1}+\partial_{u_{2}}F_{2}\left(\alpha,u_{1},u_{2}\right)v_{2}+v_{2}\right]\end{array}\right)$

and

$\displaystyle
D_{U}\,\partial_{\alpha}\mathcal{T}\left(\alpha,u_{1},u_{2}\right)\left[\left(v_{1},v_{2}\right)\right]$
$\displaystyle\quad=-\left(\begin{array}[]{c}\left(\Delta_{g}+1\right)^{-1}\left[\partial_{u_{1}}\partial_{\alpha}F_{1}\left(\alpha,u_{1},u_{2}\right)v_{1}+\partial_{u_{2}}\partial_{\alpha}F_{1}\left(\alpha,u_{1},u_{2}\right)v_{2}\right]\\\
\left(\Delta_{g}+1\right)^{-1}\left[\partial_{u_{1}}\partial_{\alpha}F_{2}\left(\alpha,u_{1},u_{2}\right)v_{1}+\partial_{u_{2}}\partial_{\alpha}F_{2}\left(\alpha,u_{1},u_{2}\right)v_{2}\right]\end{array}\right).$

If
$\left(\alpha_{m},u_{1,m},u_{2,m}\right)\to\left(\alpha_{0},u_{1,0},u_{2,0}\right)$
in $I\times{\mathcal{U}}$, then for $i,j=1,2$, by using the regularity
assumptions on $F$, we obtain that

$\displaystyle\partial_{\alpha}F_{j}\left(\alpha_{m},u_{1,m},u_{2,m}\right)$
$\displaystyle\longrightarrow\partial_{\alpha}F_{j}\left(\alpha_{0},u_{1,0},u_{2,0}\right),\allowdisplaybreaks$
$\displaystyle\partial_{u_{i}}F_{j}\left(\alpha_{m},u_{1,m},u_{2,m}\right)$
$\displaystyle\longrightarrow\partial_{u_{i}}F_{j}\left(\alpha_{0},u_{1,0},u_{2,0}\right)$

and

$\partial_{u_{i}}\partial_{\alpha}F_{j}\left(\alpha_{m},u_{1,m},u_{2,m}\right)\longrightarrow\partial_{u_{i}}\partial_{\alpha}F_{j}\left(\alpha_{0},u_{1,0},u_{2,0}\right)$

uniformly in $M$. Then, arguing as in (2.4), we obtain that $\mathcal{T}\in
C^{1}\left(I\times\mathcal{U}\right)$ and
$D_{U}\,\partial_{\alpha}\mathcal{T}$ is continuous in $I\times\mathcal{U}$.

Step 2. We now establish the main bifurcation results.

We have $\mathcal{T}\left(\alpha,0,0\right)=\left(0,0\right)$ for all
$\alpha\in I$ by the assumption (C1). Furthermore, by the assumption (C2), we
have

$\displaystyle
D_{U}\mathcal{T}\left(0,0,0\right)\left[\left(v_{1},v_{2}\right)\right]=\left(\begin{array}[]{c}v_{1}-\left(\Delta_{g}+1\right)^{-1}\left[\partial_{u_{1}}F_{1}\left(0,0,0\right)v_{1}+v_{1}\right]\\\
v_{2}-\left(\Delta_{g}+1\right)^{-1}\left[\partial_{u_{2}}F_{2}\left(0,0,0\right)v_{2}+v_{2}\right]\end{array}\right).$

Then $\left(v_{1},v_{2}\right)\in\ker D_{U}\mathcal{T}\left(0,0,0\right)$ if
and only if $\left(v_{1},v_{2}\right)\in\mathcal{H}^{*}$. By the assumption
(C4), it follows that $D_{U}\mathcal{T}\left(0,0,0\right)$ has a nontrivial
kernel consisting of eigenfunctions of $\Delta_{g}$.

Now, $\ker D_{U}\mathcal{T}\left(0,0,0\right)$ has odd dimension by the
assumption (C4). This along with the results of Step 1 allows to apply Theorem
A of Westreich [W] (see also Theorems II.3.3 and II.4.4 and the statements
(II.4.29) and (II.4.31) in Kielhöfer’s book [K]), which gives that the
solution $\left(0,0,0\right)$ is a bifurcation point of the system (2.1) in
$I\times\left(\mathcal{H}\backslash\left\\{\left(0,0\right)\right\\}\right)$
provided the following condition holds:

$\Big{[}D_{U}\partial_{\alpha}\mathcal{T}\left(0,0,0\right)\left[\left(v_{1},v_{2}\right)\right]\in\operatorname{range}\left(D_{U}\mathcal{T}\left(0,0,0\right)\right)\\\
\text{and}\quad\left(v_{1},v_{2}\right)\in\ker\left(D_{U}\mathcal{T}\left(0,0,0\right)\right)\Big{]}\iff\left(v_{1},v_{2}\right)=\left(0,0\right).$
(2.5)

Remark that by standard elliptic estimates, the $C^{2}$ topology in Definition
2.1 can be replaced without loss of generality by the $C^{1,\theta}$ topology.
If moreover $\mathcal{H}^{*}$ has dimension one and
$\partial_{u_{i}}\partial_{u_{j}}F_{k}$ exists and is continuous in
$I\times\Omega$ for all $i,j,k\in\left\\{1,2\right\\}$, then the last part of
Theorem 2.3 follows from Theorem 1.7 of Crandall–Rabinowitz [CR] (see also
Kielhöfer [K]*Theorem I.5.1).

We now show that the condition (2.5) holds for the function $\mathcal{T}$
under our assumptions on $F$. Let $\left(v_{1},v_{2}\right)\in\ker
D_{U}\mathcal{T}\left(0,0,0\right)$ and
$\left(w_{1},w_{2}\right):=D_{U}\partial_{\alpha}\mathcal{T}\left(0,0,0\right)\left[\left(v_{1},v_{2}\right)\right]$.
Then

$\left(\begin{array}[]{c}\Delta_{g}v_{1}\\\
\Delta_{g}v_{2}\end{array}\right)=\left(\begin{array}[]{c}\partial_{u_{1}}F_{1}\left(0,0,0\right)v_{1}\\\
\partial_{u_{2}}F_{2}\left(0,0,0\right)v_{2}\end{array}\right).$

Furthermore, by the assumption (C2), we obtain

$\left(\begin{array}[]{c}\Delta_{g}w_{1}+w_{1}\\\
\Delta_{g}w_{2}+w_{2}\end{array}\right)=-\left(\begin{array}[]{c}\partial_{u_{1}}\partial_{\alpha}F_{1}\left(0,0,0\right)v_{1}\\\
\partial_{u_{2}}\partial_{\alpha}F_{2}\left(0,0,0\right)v_{2}\end{array}\right).$

So then

$\displaystyle\left(\begin{array}[]{c}\left(\partial_{u_{1}}F_{1}\left(0,0,0\right)+1\right)\left(\Delta_{g}w_{1}+w_{1}\right)\\\
\left(\partial_{u_{2}}F_{2}\left(0,0,0\right)+1\right)\left(\Delta_{g}w_{2}+w_{2}\right)\end{array}\right)$
$\displaystyle\qquad=-\left(\begin{array}[]{c}\partial_{u_{1}}\partial_{\alpha}F_{1}\left(0,0,0\right)\left(\Delta_{g}v_{1}+v_{1}\right)\\\
\partial_{u_{2}}\partial_{\alpha}F_{2}\left(0,0,0\right)\left(\Delta_{g}v_{2}+v_{2}\right)\end{array}\right),$

which gives

$\left(\begin{array}[]{c}\left(\partial_{u_{1}}F_{1}\left(0,0,0\right)+1\right)w_{1}\\\
\left(\partial_{u_{2}}F_{2}\left(0,0,0\right)+1\right)w_{2}\end{array}\right)=-\left(\begin{array}[]{c}\partial_{u_{1}}\partial_{\alpha}F_{1}\left(0,0,0\right)v_{1}\\\
\partial_{u_{2}}\partial_{\alpha}F_{2}\left(0,0,0\right)v_{2}\end{array}\right).$

Now, if we suppose that
$\left(w_{1},w_{2}\right)\in\operatorname{range}\left(D_{U}\mathcal{T}\left(0,0,0\right)\right)$,
then by the assumption (C5), we obtain that
$\left(v_{1},v_{2}\right)\in\operatorname{range}\left(D_{U}\mathcal{T}\left(0,0,0\right)\right)$
and so there exists $\left(\varphi_{1},\varphi_{2}\right)\in\mathcal{H}$ such
that

$\left(\begin{array}[]{c}\Delta_{g}\varphi_{1}-\partial_{u_{1}}F_{1}\left(0,0,0\right)\varphi_{1}\\\
\Delta_{g}\varphi_{2}-\partial_{u_{2}}F_{2}\left(0,0,0\right)\varphi_{2}\end{array}\right)=\left(\begin{array}[]{c}\Delta_{g}v_{1}+v_{1}\\\
\Delta_{g}v_{2}+v_{2}\end{array}\right).$

For $i=1,2$, straightforward integrations by parts then yield

$\displaystyle\int_{M}\left(\left|\nabla
v_{i}\right|^{2}+v_{i}^{2}\right)dv_{g}$
$\displaystyle=\int_{M}v_{i}\left(\Delta_{g}\varphi_{i}-\partial_{u_{i}}F_{i}\left(0,0,0\right)\varphi_{i}\right)dv_{g}$
$\displaystyle=\int_{M}\left(\Delta_{g}v_{i}-\partial_{u_{i}}F_{i}\left(0,0,0\right)v_{i}\right)\varphi_{i}dv_{g}=0,$

where $dv_{g}$ is the volume element with respect to the metric $g$. It
follows that $\left(v_{1},v_{2}\right)=\left(0,0\right)$. Hence condition
(2.5) is satisfied.

Step 3. Finally we prove the expansion (2.3).

Let $\left(\left(\alpha_{m},u_{1,m},u_{2,m}\right)\right)_{m\in\mathbb{N}}$ be
a sequence of solutions to (2.1), such that
$\left(\alpha_{m},u_{1,m},u_{2,m}\right)\in
I\times\left(\mathcal{U}\backslash\left\\{\left(0,0\right)\right\\}\right)$
and $\left(\alpha_{m},u_{1,m},u_{2,m}\right)\to\left(0,0,0\right)$ in $I\times
C^{2}\left(M\right)^{2}$ as $m\to\infty$. For $i=1,2$, consider the sequence

$w_{i,m}:=\varepsilon_{m}^{-1}u_{i,m},\quad\text{where}\quad\varepsilon_{m}:=\max\left(\left\|u_{1,m}\right\|_{C^{1,\theta}},\left\|u_{2,m}\right\|_{C^{1,\theta}}\right)$

so that

$\max\left(\left\|w_{1,m}\right\|_{C^{1,\theta}},\left\|w_{2,m}\right\|_{C^{1,\theta}}\right)=1.$

Since $\left(\alpha_{m},u_{1,m},u_{2,m}\right)$ satisfies (2.1), it follows by
our assumptions on $F$ that

$\displaystyle\Delta_{g}w_{i,m}$
$\displaystyle=\varepsilon_{m}^{-1}F_{i}\left(\alpha_{m},u_{1,m},u_{2,m}\right)=\partial_{u_{i}}F_{i}(\alpha_{m},0,0)w_{i,m}+\operatorname{o}\left(1\right)$

uniformly in $M$. Then, by standard elliptic theory, it follows that up to a
subsequence $w_{i,m}\to\varphi_{i}$ in $\mathcal{H}_{i}$ for some function
$\varphi_{i}\in\mathcal{H}_{i}$ satisfying

$\Delta_{g}\varphi_{i}=\partial_{u_{i}}F_{i}\left(0,0,0\right)\varphi_{i}\quad\hbox{in
}M.$

Hence $\varphi_{i}$ belongs to $\mathcal{H}_{i}^{*}$ and further
$\max\left(\left\|\varphi_{1}\right\|_{C^{1,\theta}},\left\|\varphi_{2}\right\|_{C^{1,\theta}}\right)=1$.
It follows that
$\left(\varphi_{1},\varphi_{2}\right)\in\mathcal{H}^{*}\backslash\left\\{\left(0,0\right)\right\\}$
and

$u_{i,m}=\varepsilon_{m}w_{i,m}=\varepsilon_{m}\left(\varphi_{i}+\operatorname{o}\left(1\right)\right)\quad\hbox{in
}\mathcal{H}_{i}.$

This completes the proof of Theorem 2.3. ∎

We can now prove Theorem 1.1 (iii) and Theorem 2.2 by using Theorem 2.3. We
start with proving Theorem 2.2.

###### Proof of Theorem 2.2.

First note that the system in (2.2) can be rewritten as

$\left(\begin{array}[]{c}\Delta_{g}u_{1}\\\
\Delta_{g}u_{2}\end{array}\right)=\left(\begin{array}[]{c}{\tilde{F}}_{1}\left(\alpha,u_{1},u_{2}\right)\\\
{\tilde{F}}_{2}\left(\alpha,u_{1},u_{2}\right)\end{array}\right),$

where

$\displaystyle{\tilde{F}}\left(\alpha,u_{1},u_{2}\right)$
$\displaystyle=\left(\begin{array}[]{c}{\tilde{F}}_{1}\left(\alpha,u_{1},u_{2}\right)\\\
{\tilde{F}}_{2}\left(\alpha,u_{1},u_{2}\right)\end{array}\right)$
$\displaystyle:=\left(\begin{array}[]{c}a_{11}\left(\alpha\right)\left|u_{1}\right|^{q-2}u_{1}+a_{12}\left(\alpha\right)\left|u_{2}\right|^{q-2}u_{1}-\lambda_{1}\left(\alpha\right)u_{1}\\\
a_{21}\left(\alpha\right)\left|u_{1}\right|^{q-2}u_{2}+a_{22}\left(\alpha\right)\left|u_{2}\right|^{q-2}u_{2}-\lambda_{2}\left(\alpha\right)u_{2}\end{array}\right).$

Now let’s transform this system so as to apply Theorem 2.3. For every
$\alpha\in I$, the unique constant solution to (2.2) (which existence follows
from the assumption (B1)) is given by

$\left\\{\begin{aligned}
&\overline{u}_{1}\left(\alpha\right)=\left(\frac{\lambda_{1}\left(\alpha\right)a_{22}\left(\alpha\right)-\lambda_{2}\left(\alpha\right)a_{12}\left(\alpha\right)}{a_{11}\left(\alpha\right)a_{22}\left(\alpha\right)-a_{21}\left(\alpha\right)a_{12}\left(\alpha\right)}\right)^{1/\left(q-2\right)}\\\
&\overline{u}_{2}\left(\alpha\right)=\left(\frac{\lambda_{2}\left(\alpha\right)a_{11}\left(\alpha\right)-\lambda_{1}\left(\alpha\right)a_{21}\left(\alpha\right)}{a_{11}\left(\alpha\right)a_{22}\left(\alpha\right)-a_{21}\left(\alpha\right)a_{12}\left(\alpha\right)}\right)^{1/\left(q-2\right)}\end{aligned}\right.$

and it satisfies

$\left\\{\begin{aligned}
&a_{11}\left(\alpha\right)\overline{u}_{1}\left(\alpha\right)^{q-2}+a_{12}\left(\alpha\right)\overline{u}_{2}\left(\alpha\right)^{q-2}=\lambda_{1}\left(\alpha\right)&&\hbox{
in }M\\\
&a_{21}\left(\alpha\right)\overline{u}_{1}\left(\alpha\right)^{q-2}+a_{22}\left(\alpha\right)\overline{u}_{2}\left(\alpha\right)^{q-2}=\lambda_{2}\left(\alpha\right)&&\hbox{
in }M.\end{aligned}\right.$

We look for solutions of (1.2) bifurcating from
$\left(\overline{u}_{1}\left(\alpha\right),\overline{u}_{2}\left(\alpha\right)\right)$.
By the assumption (B2), for every $\alpha\in I$, the matrix
$\mathcal{A}\left(\alpha\right)$ has two distinct, non-zero, real eigenvalues
$\beta_{1}\left(\alpha\right)$ and $\beta_{2}\left(\alpha\right)$ given by

$\left\\{\beta_{1}\left(\alpha\right),\beta_{2}\left(\alpha\right)\right\\}:=\left\\{\frac{a_{11}\left(\alpha\right)\overline{u}_{1}\left(\alpha\right)^{q-2}+a_{22}\left(\alpha\right)\overline{u}_{2}\left(\alpha\right)^{q-2}}{2}\pm\frac{\sqrt{D\left(\alpha\right)}}{2}\right\\},$

where

$D\left(\alpha\right):=\left(a_{11}\left(\alpha\right)\overline{u}_{1}\left(\alpha\right)^{q-2}-a_{22}\left(\alpha\right)\overline{u}_{2}\left(\alpha\right)^{q-2}\right)^{2}\\\
+4a_{12}\left(\alpha\right)a_{21}\left(\alpha\right)\overline{u}_{1}\left(\alpha\right)^{q-2}\overline{u}_{2}\left(\alpha\right)^{q-2}.$

Let $\mathcal{P}\left(\alpha\right)$ be the $2\times 2$ matrix such that

$\mathcal{A}\left(\alpha\right)=\mathcal{P}\left(\alpha\right)^{-1}\left(\begin{array}[]{cc}\beta_{1}\left(\alpha\right)&0\\\
0&\beta_{2}\left(\alpha\right)\end{array}\right)\mathcal{P}\left(\alpha\right).$

Consider $\left(u_{1},u_{2}\right)\in C^{2}\left(M\right)^{2}$ and let

$\left(\begin{array}[]{c}v_{1}\\\
v_{2}\end{array}\right):=P\left(\alpha\right)\left(\begin{array}[]{c}u_{1}-\overline{u}_{1}\left(\alpha\right)\\\
u_{2}-\overline{u}_{2}\left(\alpha\right)\end{array}\right).$

We then define

$F\left(\alpha,v_{1},v_{2}\right)=\left(\begin{array}[]{c}F_{1}\left(\alpha,v_{1},v_{2}\right)\\\
F_{2}\left(\alpha,v_{1},v_{2}\right)\end{array}\right):=P\left(\alpha\right)\tilde{F}\left(\alpha,u_{1},u_{2}\right),$
(2.6)

where

$\left(\begin{array}[]{c}u_{1}\\\
u_{2}\end{array}\right)=\left(\begin{array}[]{c}\overline{u}_{1}\left(\alpha\right)\\\
\overline{u}_{2}\left(\alpha\right)\end{array}\right)+P\left(\alpha\right)^{-1}\left(\begin{array}[]{c}v_{1}\\\
v_{2}\end{array}\right).$ (2.7)

We then obtain that the system (1.2) is equivalent to

$\displaystyle\left(\begin{array}[]{c}\Delta_{g}v_{1}\\\
\Delta_{g}v_{2}\end{array}\right)=\left(\begin{array}[]{c}F_{1}\left(\alpha,v_{1},v_{2}\right)\\\
F_{2}\left(\alpha,v_{1},v_{2}\right)\end{array}\right).$ (2.12)

Next we apply Theorem 2.3 to (2.12). Note that the condition (C1) of Theorem
2.3 is satisfied by (2.12). Furthermore, since
$\left(\overline{u}_{1}\left(\alpha\right),\overline{u}_{2}\left(\alpha\right)\right)\in\left(0,\infty\right)^{2}$
for all $\alpha\in I$, by continuity, we obtain that there exists
$\delta_{0}>0$ such that
$\left(u_{1},u_{2}\right)\in\left(0,\infty\right)^{2}$ for all
$\left(\alpha,v_{1},v_{2}\right)\in
I\times\left(-\delta_{0},\delta_{0}\right)^{2}$. In particular, letting
$\Omega:=\left(-\delta_{0},\delta_{0}\right)^{2}$, we then obtain that $F\in
C^{1}\left(I\times\Omega\right)$, $\partial_{u_{i}}\partial_{\alpha}F_{j}$ and
$\partial_{u_{i}}\partial_{u_{j}}F_{k}$ exist and are continuous in
$I\times\Omega$ for all $i,j,k\in\left\\{1,2\right\\}$ and the condition (C3)
is satisfied with
$\mathcal{H}_{1}=\mathcal{H}_{2}:=C^{1,\theta}\left(M\right)$ and
$\mathcal{U}_{1}=\mathcal{U}_{2}:=C^{1,\theta}\left(M,\left(-\delta_{0},\delta_{0}\right)\right)$.
The condition (C2) is also satisfied as we obtain differentiating

$\displaystyle D_{\left(v_{1},v_{2}\right)}F\left(\alpha,0,0\right)$
$\displaystyle=\mathcal{P}\left(\alpha\right)\left[\left(q-2\right)\mathcal{A}\left(\alpha\right)\right]\mathcal{P}^{-1}\left(\alpha\right)$
$\displaystyle=\left(q-2\right)\left(\begin{array}[]{cc}\beta_{1}\left(\alpha\right)&0\\\
0&\beta_{2}\left(\alpha\right)\end{array}\right).$

The assumptions (B3) and (B4) then imply that the conditions (C4) and (C5) of
Theorem 2.3 are also satisfied.

By applying Theorem 2.3 and reversing the above change of function, we then
obtain that the solution
$\left(0,\overline{u}_{1}\left(0\right),\overline{u}_{2}\left(0\right)\right)$
is a bifurcation point of the system (2.2). Furthermore, we obtain that for
every sequence
$\left(\left(\alpha_{m},u_{1,m},u_{2,m}\right)\right)_{m\in\mathbb{N}}$ of
solutions to (2.2), if
$\left(u_{1,m},u_{2,m}\right)\neq\left(\overline{u}_{1}\left(0\right),\overline{u}_{2}\left(0\right)\right)$
and
$\left(\alpha_{m},u_{1,m},u_{2,m}\right)\to\left(0,\overline{u}_{1}\left(0\right),\overline{u}_{2}\left(0\right)\right)$
in $I\times C^{2}\left(M\right)^{2}$ as $m\to\infty$, then up to a
subsequence,

$u_{i,m}=\overline{u}_{i}\left(\alpha_{m}\right)+\varepsilon_{m}\left(q_{i1}\varphi_{1}+q_{i2}\varphi_{2}+\operatorname{o}\left(1\right)\right)\quad\text{
in }\mathcal{H}_{i}\quad\forall i\in\left\\{1,2\right\\},$

where $\left(q_{ij}\right)_{1\leq i,j\leq 2}:=P\left(0\right)^{-1}$, for some
$\left(\varphi_{1},\varphi_{2}\right)\in\mathcal{H}^{*}\backslash\left\\{\left(0,0\right)\right\\}$
and $\varepsilon_{m}>0$ such that $\varepsilon_{m}\to 0$. By the assumptions
(B2) and (B4), we have that for $i=1,2$, either $\varphi_{i}\equiv 0$ or
$\varphi_{i}$ is not constant in $M$. Also by assumption (B4) we have that if
$\varphi_{i}\neq 0$ for $i=1,2$, then either
$\lambda_{1}\left(0\right)\neq\lambda_{2}\left(0\right)$ or
$\beta_{i}\left(0\right)\neq\lambda_{1}\left(0\right)=\lambda_{2}\left(0\right)$,
which implies
$\overline{u}_{2}\left(0\right)q_{1i}\neq\overline{u}_{1}\left(0\right)q_{2i}$.
In particular, we obtain that $\left(u_{1,m},u_{2,m}\right)$ is non-
synchronized. Therefore, we obtain that there exists a neighborhood
$\mathcal{N}$ of
$\left(0,\overline{u}_{1}\left(0\right),\overline{u}_{2}\left(0\right)\right)$
in $I\times C^{2}\left(M\right)^{2}$ such that for every solution
$\left(\alpha,u_{1},u_{2}\right)\in\mathcal{N}$ of (2.2), if
$\left(u_{1},u_{2}\right)\neq\left(\overline{u}_{1}\left(\alpha\right),\overline{u}_{2}\left(\alpha\right)\right)$,
then $\left(u_{1},u_{2}\right)$ is non-synchronized. If moreover
$\mathcal{H}^{*}$ has dimension one, then it follows from the last part of
Theorem 2.3 that there exists a $C^{1}$ branch of non-synchronized solutions
to (2.2) emanating from
$\left(0,\overline{u}_{1}\left(0\right),\overline{u}_{2}\left(0\right)\right)$.
This completes the proof of Theorem 2.2. ∎

###### Proof of Theorem 1.1 (iii).

We proceed as in the proof of Theorem 2.2. By the assumptions (A1) and (A3)
along with the continuity of $\lambda$, $a$ and $b$, letting $\delta$ be
smaller if necessary, we may assume that
$a\left(\alpha\right)>b\left(\alpha\right)$ and
$\lambda\left(\alpha\right)\left(a\left(\alpha\right)+b\left(\alpha\right)\right)>0$
for all $\alpha\in I$. Then the unique constant solution for the system (1.5)
is given by

$\left(\begin{array}[]{c}\overline{u}_{1}\left(\alpha\right)\\\
\overline{u}_{2}\left(\alpha\right)\end{array}\right):=\overline{u}\left(\alpha\right)\left(\begin{array}[]{c}1\\\
1\end{array}\right),\quad\text{where
}\overline{u}\left(\alpha\right):=\left(\frac{\lambda\left(\alpha\right)}{a\left(\alpha\right)+b\left(\alpha\right)}\right)^{{1}/{\left(q-2\right)}}$

and it satisfies

$\left\\{\begin{aligned}
&a\left(\alpha\right)\overline{u}_{1}\left(\alpha\right)^{q-2}+b\left(\alpha\right)\overline{u}_{2}\left(\alpha\right)^{q-2}=\lambda\left(\alpha\right)\\\
&b\left(\alpha\right)\overline{u}_{1}\left(\alpha\right)^{q-2}+a\left(\alpha\right)\overline{u}_{2}\left(\alpha\right)^{q-2}=\lambda\left(\alpha\right).\end{aligned}\right.$

We look for solutions of (1.5) bifurcating from
$\left(\overline{u}_{1}\left(\alpha\right),\overline{u}_{2}\left(\alpha\right)\right)$.
For (1.5), the eigenvalues $\beta_{1}\left(\alpha\right)$ and
$\beta_{2}\left(\alpha\right)$ of $\mathcal{A}\left(\alpha\right)$ are given
by

$\beta_{1}\left(\alpha\right)=\lambda\left(\alpha\right)\quad\text{and}\quad\beta_{2}\left(\alpha\right)=\lambda\left(\alpha\right)\frac{a\left(\alpha\right)-b\left(\alpha\right)}{a\left(\alpha\right)+b\left(\alpha\right)}\,.$

We let $F$ and $\Omega=\left(-\delta_{0},\delta_{0}\right)^{2}$ be defined
similarly as in the proof of Theorem 2.2, so that in particular, the
conditions (C1) and (C2) of Theorem 2.3 are satisfied. In this case, we find

$\mathcal{P}\left(\alpha\right)=\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right).$

Next, we choose the appropriate subspaces $\mathcal{H}_{1}$ and
$\mathcal{H}_{2}$ and open subsets $\mathcal{U}_{1}\subseteq\mathcal{H}_{1}$
and $\mathcal{U}_{2}\subseteq\mathcal{H}_{2}$. For this, we use an idea from
Gladiali, Grossi and Troestler [GGT1]. Consider the reflexion $\hat{v}$ across
the equator $\left\\{x_{n}=0\right\\}$ of the sphere $\mathbb{S}^{n}$ defined
by

$\hat{v}\left(x\right):=v\left(x_{1},\ldots,x_{n},-x_{n+1}\right)\quad\forall
x=\left(x_{1},\ldots,x_{n},x_{n+1}\right)\in\mathbb{S}^{n}$

for all functions $v:\mathbb{S}^{n}\to\mathbb{R}$. By stereographic projection
along with a conformal change of metric, this corresponds to the Kelvin
transform in $\mathbb{R}^{n}$. We let $N_{0}:=\left(0,\ldots,0,1\right)$ and
$j_{0}\in\mathbb{N}$ be such that

$\left(q-2\right)\beta_{2}\left(0\right)=\lambda_{j_{0}}:=j_{0}\left(j_{0}+n-1\right)$

i.e. the $j_{0}$-th eigenvalue of $\Delta_{g_{0}}$ on $\mathbb{S}^{n}$ (the
existence of $j_{0}$ is given by the assumption (A3)). We then define
$\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ as

$\mathcal{H}_{1}:=\left\\{v\in
C^{1,\theta}\left(\mathbb{S}^{n}\right):\hbox{$v$ is radial with respect to
$N_{0}$ and $\hat{v}=v$}\right\\}$

and

$\mathcal{H}_{2}:=\left\\{\begin{aligned} &\left\\{v\in
C^{1,\theta}\left(\mathbb{S}^{n}\right):\hbox{$v$ is radial w.r.t.~{}$N_{0}$
and $\hat{v}=v$}\right\\}\,\hbox{if $j_{0}$ is even}\\\ &\left\\{v\in
C^{1,\theta}\left(\mathbb{S}^{n}\right):\hbox{$v$ is radial w.r.t.~{}$N_{0}$
and $\hat{v}=-v$}\right\\}\,\hbox{if $j_{0}$ is odd}.\end{aligned}\right.$

For $i=1,2$, we take
$\mathcal{U}_{i}:=C^{1,\theta}\left(\mathbb{S}^{n},\left(-\delta_{0},\delta_{0}\right)\right)\cap\mathcal{H}_{i}$.
For every $\left(v_{1},v_{2}\right)\in\mathcal{U}$, letting
$\left(u_{1},u_{2}\right)$ be as in (2.7), we then obtain

$\displaystyle\left(\begin{array}[]{c}\hat{u}_{1}\\\
\hat{u}_{2}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{c}\overline{u}_{1}\left(\alpha\right)\\\
\overline{u}_{2}\left(\alpha\right)\end{array}\right)+\frac{1}{2}\left(\begin{array}[]{c}\hat{v}_{1}+\hat{v}_{2}\\\
\hat{v}_{1}-\hat{v}_{2}\end{array}\right)$
$\displaystyle=\left\\{\begin{aligned}
&\left(\begin{array}[]{c}\overline{u}_{1}\left(\alpha\right)\\\
\overline{u}_{2}\left(\alpha\right)\end{array}\right)+\frac{1}{2}\left(\begin{array}[]{c}v_{1}+v_{2}\\\
v_{1}-v_{2}\end{array}\right)=\left(\begin{array}[]{c}u_{1}\\\
u_{2}\end{array}\right)&&\text{if }j_{0}\text{ is even}\\\
&\left(\begin{array}[]{c}\overline{u}_{1}\left(\alpha\right)\\\
\overline{u}_{2}\left(\alpha\right)\end{array}\right)+\frac{1}{2}\left(\begin{array}[]{c}v_{1}-v_{2}\\\
v_{1}+v_{2}\end{array}\right)=\left(\begin{array}[]{c}u_{2}\\\
u_{1}\end{array}\right)&&\text{if }j_{0}\text{ is odd}.\end{aligned}\right.$

It follows that

$\displaystyle F\left(\alpha,\hat{v}_{1},\hat{v}_{2}\right)$
$\displaystyle=\left(\begin{array}[]{c}F_{1}\left(\alpha,\hat{v}_{1},\hat{v}_{2}\right)\\\
F_{2}\left(\alpha,\hat{v}_{1},\hat{v}_{2}\right)\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right)\left(\begin{array}[]{c}a\left(\alpha\right)\hat{u}_{1}^{q-1}+b\left(\alpha\right)\hat{u}_{2}^{q-2}\hat{u}_{1}-\lambda\left(\alpha\right)\hat{u}_{1}\\\
b\left(\alpha\right)\hat{u}_{1}^{q-2}\hat{u}_{2}+a\left(\alpha\right)\hat{u}_{2}^{q-1}-\lambda\left(\alpha\right)\hat{u}_{2}\end{array}\right)$
$\displaystyle=\left\\{\begin{aligned}
&F\left(\alpha,v_{1},v_{2}\right)&&\text{if }j_{0}\text{ is even}\\\
&\left(\begin{array}[]{c}F_{1}\left(\alpha,v_{1},v_{2}\right)\\\
-F_{2}\left(\alpha,v_{1},v_{2}\right)\end{array}\right)&&\text{if }j_{0}\text{
is odd}.\end{aligned}\right.$

This along with standard elliptic regularity and symmetry arguments gives that
the condition (C3) is satisfied.

Recall that the spherical harmonics $\varphi$ satisfying
$\Delta_{g_{0}}\varphi=\lambda_{j_{0}}\varphi$ are given by the restriction to
$\mathbb{S}^{n}$ of the harmonic polynomials of degree $j_{0}$ in
$\mathbb{R}^{n+1}$. In particular, up to a constant factor, the unique such
function $\varphi_{N_{0},j_{0}}$ that is radial with respect to $N_{0}$ is
given by the Jacobi polynomial

$\varphi_{N_{0},j_{0}}\left(x_{1},\dotsc,x_{n+1}\right):=\sum_{j=0}^{j_{0}}\left(\begin{array}[]{c}j_{0}+\left(n-2\right)/2\\\
j\end{array}\right)\left(\begin{array}[]{c}j_{0}+\left(n-2\right)/2\\\
j_{0}-j\end{array}\right)\\\
\times\left(\frac{x_{n+1}-1}{2}\right)^{j_{0}-j}\left(\frac{x_{n+1}+1}{2}\right)^{j}\quad\forall
x\in\mathbb{S}^{n}$

(see Gladiali [G]). So then the assumptions (A2) and (A3) give that
$\mathcal{H}^{*}_{1}=\left\\{0\right\\}$ and $\mathcal{H}^{*}_{2}$ has
dimension one. In particular, we obtain that the condition (C4) is satisfied.
Furthermore, the condition (C5) follows from (A3).

By applying Theorem 2.3, we then obtain that there exists a $C^{1}$ branch of
solutions to (2.2) emanating from
$\left(0,\overline{u}\left(0\right),\overline{u}\left(0\right)\right)$ whose
tangent line at
$\left(0,\overline{u}\left(0\right),\overline{u}\left(0\right)\right)$ is
directed by some vector of the form $\left(\mu,\varphi,-\varphi\right)$, where
$\mu\in\mathbb{R}$ and
$\varphi\in\mathcal{H}_{2}^{*}\backslash\left\\{0\right\\}$. In particular, we
obtain that near
$\left(0,\overline{u}\left(0\right),\overline{u}\left(0\right)\right)$, the
solutions on this branch are non-synchronized, which completes the proof of
Theorem 1.1 (iii). ∎

##  3\. Synchronization and non-existence results

In this section, we prove the following results, which extend Theorem 1.1 to
more general systems and manifolds:

######  Theorem 3.1.

(Case $\lambda_{1}=\lambda_{2}$) Let
$\lambda,a_{11},a_{12},a_{21},a_{22}\in\mathbb{R}$,
$q\in\left(2,\infty\right)$ and $\left(M,g\right)$ be a smooth, closed
Riemannian manifold. Consider the system (1.2) with
$\lambda_{1}=\lambda_{2}=\lambda$.

  1. (i)

If either [$a_{21}\leq a_{11}$ and $a_{22}\leq a_{12}$] or [$a_{11}\leq
a_{21}$ and $a_{12}\leq a_{22}$] and at least one of the two inequalities is
strict, then (1.2) has no solutions.

  2. (ii)

If either [$a_{11}<a_{21}$ and $a_{22}<a_{12}$] or [$a_{11}=a_{21}$ and
$a_{22}=a_{12}$], then every solution of (1.2) is synchronized.

  3. (iii)

Assuming that there exists at least one non-zero eigenvalue of $\Delta_{g}$
with odd multiplicity, Theorem 2.2 provides examples of real numbers
$\lambda,a_{11},a_{12},a_{21},a_{22}>0$ such that $a_{21}<a_{11}$,
$a_{12}<a_{22}$ and (1.2) has non-synchronized solutions.

######  Theorem 3.2.

(Case $\lambda_{1}<\lambda_{2}$) Let
$\lambda_{1},\lambda_{2},a_{11},a_{12},a_{21},a_{22}\in\mathbb{R}$,
$q\in\left(2,\infty\right)$ and $\left(M,g\right)$ be a smooth, closed
Riemannian manifold. Consider the system (1.2). Assume that
$\lambda_{1}<\lambda_{2}$.

  1. (i)

If [$a_{21}\leq a_{11}$ and $a_{22}\leq a_{12}$], then (1.2) has no solutions.

  2. (ii)

Assuming that there exists at least one non-zero eigenvalue of $\Delta_{g}$
with odd multiplicity, Theorem 2.2 provides examples of real numbers
$\lambda_{1},\lambda_{2},a_{11},a_{12},a_{21},a_{22}>0$ such that
$\lambda_{2}>\lambda_{1}>0$ and (1.2) has non-synchronized solutions in each
of the following cases: [$a_{21}<a_{11}$ and $a_{12}<a_{22}$],
[$a_{11}<a_{21}$ and $a_{22}<a_{12}$], [$a_{11}<a_{21}$ and $a_{12}<a_{22}$],
[$a_{11}=a_{21}$ and $a_{12}<a_{22}$] and [$a_{11}<a_{21}$ and
$a_{12}=a_{22}$].

###### Proof of Theorem 3.1 (i) and Theorem 3.2 (i).

Suppose that there exists a solution $\left(u_{1},u_{2}\right)$ of (1.2). We
define

$v\left(x\right):=\frac{u_{1}\left(x\right)}{u_{2}\left(x\right)}\qquad\forall
x\in M.$

We then obtain

$\displaystyle\Delta_{g}v=\frac{\Delta_{g}u_{1}}{u_{2}}+2\frac{\left<\nabla_{g}u_{1},\nabla_{g}u_{2}\right>_{g}}{u_{2}^{2}}-\frac{u_{1}\Delta_{g}u_{2}}{u_{2}^{2}}-2\frac{u_{1}\left|\nabla_{g}u_{2}\right|^{2}_{g}}{u_{2}^{3}}$
$\displaystyle=\left(a_{11}-a_{21}\right)u_{1}^{q-2}v+\left(a_{12}-a_{22}\right)u_{2}^{q-2}v+\left(\lambda_{2}-\lambda_{1}\right)v+2\frac{\left<\nabla_{g}v,\nabla_{g}u_{2}\right>_{g}}{u_{2}}$

in $M$. In the case where $a_{21}\leq a_{11}$, $a_{22}\leq a_{12}$,
$\lambda_{1}\leq\lambda_{2}$ and one of these inequalities is strict, we then
obtain

$\Delta_{g}v>2\left<\nabla_{g}v,\nabla_{g}\left[\ln
u_{2}\right]\right>_{g}\quad\text{in }M,$

which is in contradiction with the minimum principle. Similarly, in the case
where $a_{11}\leq a_{21}$, $a_{12}\leq a_{22}$, $\lambda_{2}\leq\lambda_{1}$
and one of these inequalities is strict, we obtain

$\Delta_{g}v<2\left<\nabla_{g}v,\nabla_{g}\left[\ln
u_{2}\right]\right>_{g}\quad\text{in }M,$

which is in contradiction with the maximum principle. This proves Theorem 3.1
(i) and Theorem 3.2 (i). ∎

###### Proof of Theorem 3.1 (ii).

Suppose first that $a_{11}<a_{21}$, $a_{22}<a_{12}$ and
$\lambda_{1}=\lambda_{2}$. Let $v$ be as in the previous proof and
$x_{1},x_{2}\in M$ be such that

$v\left(x_{1}\right)=\min\left\\{v\left(x\right):\ x\in
M\right\\}\quad\text{and}\quad
v\left(x_{2}\right)=\max\left\\{v\left(x\right):\ x\in M\right\\}.$

We then obtain

$0\geq\Delta_{g}v\left(x_{1}\right)=\left[\left(a_{12}-a_{22}\right)u_{2}\left(x_{1}\right)^{q-2}-\left(a_{21}-a_{11}\right)u_{1}\left(x_{1}\right)^{q-2}\right]v\left(x_{1}\right)$

and

$0\leq\Delta_{g}v\left(x_{2}\right)=\left[\left(a_{12}-a_{22}\right)u_{2}\left(x_{2}\right)^{q-2}-\left(a_{21}-a_{11}\right)u_{1}\left(x_{2}\right)^{q-2}\right]v\left(x_{2}\right),$

which imply

$v\left(x_{2}\right)\leq\left(\frac{a_{12}-a_{22}}{a_{21}-a_{11}}\right)^{1/\left(q-2\right)}\leq
v\left(x_{1}\right).$

It follows that $v$ is constant in $M$.

Now suppose that $a_{11}=a_{21}$, $a_{22}=a_{12}$ and
$\lambda_{1}=\lambda_{2}$. In this case, we have

$\Delta_{g}v=2\left<\nabla_{g}v,\nabla_{g}\left[\ln
u_{2}\right]\right>_{g}\quad\text{in }M.$

It then follows from the maximum principle that $v$ is constant in $M$. This
completes the proof of Theorem 3.1 (ii). ∎

###### Proof of Theorem 3.1 (iii) and Theorem 3.2 (ii).

We choose our examples of the form
$a_{11}\left(\alpha\right)=\lambda_{1}\left(\alpha\right)a$,
$a_{12}\left(\alpha\right)=\lambda_{1}\left(\alpha\right)b$,
$a_{21}\left(\alpha\right)=\lambda_{2}\left(\alpha\right)b$ and
$a_{22}\left(\alpha\right)=\lambda_{2}\left(\alpha\right)a$ for some $a,b>0$
and $\lambda_{1},\lambda_{2}\in C^{1}\left(I,\left(0,\infty\right)\right)$,
where $I:=\left[-\delta,\delta\right]$ for some $\delta>0$ to be chosen small.
As is easily seen, for every $\alpha\in I$, if

$a\neq b\quad\text{and}\quad
D\left(\alpha\right):=\left(\lambda_{1}\left(\alpha\right)-\lambda_{2}\left(\alpha\right)\right)^{2}a^{2}+4\lambda_{1}\left(\alpha\right)\lambda_{2}\left(\alpha\right)b^{2}>0,$

then the system (1.2) has a unique constant solution given by

$\left(\begin{array}[]{c}\overline{u}_{1}\left(\alpha\right)\\\
\overline{u}_{2}\left(\alpha\right)\end{array}\right)=\left(a+b\right)^{-1/\left(q-2\right)}\left(\begin{array}[]{c}1\\\
1\end{array}\right).$

and the matrix $A\left(\alpha\right)$ has two distinct, non-zero real
eigenvalues given by

$\left\\{\beta_{1}\left(\alpha\right),\beta_{2}\left(\alpha\right)\right\\}=\left\\{\frac{\left(\lambda_{1}\left(\alpha\right)+\lambda_{2}\left(\alpha\right)\right)a\pm\sqrt{D\left(\alpha\right)}}{2\left(a+b\right)}\right\\}.$

Now, we treat each case separately and choose $a,b,\lambda_{1}$ and
$\lambda_{2}$ in such a way that $a\neq b$, $D>0$,
$\left(q-2\right)\beta_{1}\left(0\right)\not\in\operatorname{spec}\left(\Delta_{g}\right)$,
$\left(q-2\right)\beta_{2}\left(0\right)=\lambda_{0}$,
$\beta^{\prime}_{2}\left(0\right)\neq 0$ and either
$\lambda_{1}\left(0\right)\neq\lambda_{2}\left(0\right)$ or
$\beta_{2}\left(0\right)\neq\lambda_{1}\left(0\right)=\lambda_{2}\left(0\right)$,
where $\operatorname{spec}\left(\Delta_{g}\right)$ stands for the spectrum of
$\Delta_{g}$ and $\lambda_{0}$ is a non-zero eigenvalue of $\Delta_{g}$ with
odd multiplicity (which existence is given by assumption). Choosing $\delta$
small enough, we can then apply Theorem 2.2.

Case $\lambda_{1}=\lambda_{2}$, $a_{21}<a_{11}$ and $a_{12}<a_{22}$. Choose
for instance $a:=\frac{\lambda_{0}+\lambda}{q-2}$,
$b:=\frac{\lambda-\lambda_{0}}{q-2}$ and
$\lambda_{1}\left(\alpha\right)=\lambda_{2}\left(\alpha\right):=\frac{\lambda\left(\alpha+1\right)}{q-2}$
for some
$\lambda\in\left(\lambda_{0},\infty\right)\backslash\operatorname{spec}\left(\Delta_{g}\right)$.
Then $\lambda_{1}=\lambda_{2}>0$,
$0<a_{21}=a_{12}<\frac{\lambda\lambda_{1}}{q-2}<a_{11}=a_{22}$, $a\neq b$,
$D=4\lambda_{1}^{2}b^{2}>0$, $\beta_{1}\left(0\right)=\frac{\lambda}{q-2}$ (so
that
$\left(q-2\right)\beta_{1}\left(0\right)\not\in\operatorname{spec}\left(\Delta_{g}\right)$)
and
$\beta_{2}\left(0\right)=\beta_{2}^{\prime}\left(0\right)=\frac{\lambda_{0}}{q-2}>\frac{\lambda}{q-2}=\lambda_{1}\left(0\right)=\lambda_{2}\left(0\right)$.

Case $\lambda_{1}<\lambda_{2}$, $a_{21}<a_{11}$ and $a_{12}<a_{22}$. Choose
for instance $a:=1$, $b:=\varepsilon$,
$\lambda_{1}\left(\alpha\right):=\frac{2\left(1+\varepsilon\right)\lambda_{0}\left(\alpha+1\right)}{(2+\varepsilon+\varepsilon\sqrt{5+4\varepsilon})\left(q-2\right)}$
and
$\lambda_{2}\left(\alpha\right):=\left(1+\varepsilon\right)\lambda_{1}\left(\alpha\right)$
for some small $\varepsilon>0$. Then $0<\lambda_{1}<\lambda_{2}$,
$0<a_{21}=\varepsilon\left(1+\varepsilon\right)\lambda_{1}<\lambda_{1}=a_{11}$,
$0<a_{12}=\varepsilon\lambda_{1}<\left(1+\varepsilon\right)\lambda_{1}=a_{22}$,
$a\neq b$, $D=\varepsilon^{2}\left(5+4\varepsilon\right)\lambda_{1}^{2}>0$,
$\beta_{1}\left(0\right)=\frac{(2+\varepsilon-\varepsilon\sqrt{5+4\varepsilon})\lambda_{0}}{(2+\varepsilon+\varepsilon\sqrt{5+4\varepsilon})\left(q-2\right)}<\frac{\lambda_{0}}{q-2}$,
$\beta_{1}\left(0\right)\to\frac{\lambda_{0}}{q-2}$ as $\varepsilon\to 0$ (so
that
$\left(q-2\right)\beta_{1}\left(0\right)\not\in\operatorname{spec}\left(\Delta_{g}\right)$)
and
$\beta_{2}\left(0\right)=\beta_{2}^{\prime}\left(0\right)=\frac{\lambda_{0}}{q-2}$.

Case $\lambda_{1}<\lambda_{2}$, $a_{11}<a_{21}$ and $a_{22}<a_{12}$. Choose
for instance $a:=1$, $b:=\sqrt{6}$,
$\lambda_{1}\left(\alpha\right):=\frac{(1+\sqrt{6})\lambda_{0}\left(\alpha+1\right)}{5\left(q-2\right)}$
and $\lambda_{2}\left(\alpha\right):=2\lambda_{1}\left(\alpha\right)$. Then
$0<\lambda_{1}<\lambda_{2}$,
$0<a_{11}=\lambda_{1}<2\sqrt{6}\lambda_{1}=a_{21}$,
$0<a_{22}=2\lambda_{1}<\sqrt{6}\lambda_{1}=a_{12}$, $a\neq b$,
$D=49\lambda_{1}^{2}>0$,
$\beta_{1}\left(0\right)=\frac{-2\lambda_{0}}{5\left(q-2\right)}<0$ (so that
$\left(q-2\right)\beta_{1}\left(0\right)\not\in\operatorname{spec}\left(\Delta_{g}\right)$)
and
$\beta_{2}\left(0\right)=\beta_{2}^{\prime}\left(0\right)=\frac{\lambda_{0}}{q-2}$.

Case $\lambda_{1}<\lambda_{2}$, $a_{11}<a_{21}$ and $a_{12}<a_{22}$. Choose
for instance $a:=1$, $b:=2$,
$\lambda_{1}\left(\alpha\right):=\frac{3\lambda_{0}\left(\alpha+1\right)}{(3+2\sqrt{6})\left(q-2\right)}$
and $\lambda_{2}\left(\alpha\right):=5\lambda_{1}\left(\alpha\right)$. Then
$0<\lambda_{1}<\lambda_{2}$, $0<a_{11}=\lambda_{1}<10\lambda_{1}=a_{21}$,
$0<a_{12}=2\lambda_{1}<5\lambda_{1}=a_{22}$, $a\neq b$,
$D=96\lambda_{1}^{2}>0$,
$\beta_{1}\left(0\right)=\frac{(3-2\sqrt{6})\lambda_{0}}{(3+2\sqrt{6})\left(q-2\right)}<0$
(so that
$\left(q-2\right)\beta_{1}\left(0\right)\not\in\operatorname{spec}\left(\Delta_{g}\right)$)
and
$\beta_{2}\left(0\right)=\beta_{2}^{\prime}\left(0\right)=\frac{\lambda_{0}}{q-2}$.

Case $\lambda_{1}<\lambda_{2}$, $a_{11}=a_{21}$ and $a_{12}<a_{22}$. Choose
for instance $a:=1+\varepsilon$, $b:=1$,
$\lambda_{1}\left(\alpha\right):=\frac{2\left(2+\varepsilon\right)\lambda_{0}\left(\alpha+1\right)}{((2+\varepsilon)(1+\varepsilon)+\sqrt{\left(1+\varepsilon\right)\left(4+\varepsilon^{2}+\varepsilon^{3}\right)})\left(q-2\right)}$
and
$\lambda_{2}\left(\alpha\right):=\left(1+\varepsilon\right)\lambda_{1}\left(\alpha\right)$
for some small $\varepsilon>0$. Then $0<\lambda_{1}<\lambda_{2}$,
$a_{11}=a_{21}=\left(1+\varepsilon\right)\lambda_{1}>0$,
$0<a_{12}=\lambda_{1}<\left(1+\varepsilon\right)^{2}\lambda_{1}=a_{22}$,
$a\neq b$,
$D=\left(1+\varepsilon\right)\left(4+\varepsilon^{2}+\varepsilon^{3}\right)\lambda_{1}^{2}>0$,
$\beta_{1}\left(0\right)=\frac{((2+\varepsilon)(1+\varepsilon)-\sqrt{\left(1+\varepsilon\right)\left(4+\varepsilon^{2}+\varepsilon^{3}\right)})\lambda_{0}}{((2+\varepsilon)(1+\varepsilon)+\sqrt{\left(1+\varepsilon\right)\left(4+\varepsilon^{2}+\varepsilon^{3}\right)})\left(q-2\right)}>0$,
$\beta_{1}\left(0\right)\to 0$ as $\varepsilon\to 0$ (so that
$\left(q-2\right)\beta_{1}\left(0\right)\not\in\operatorname{spec}\left(\Delta_{g}\right)$)
and
$\beta_{2}\left(0\right)=\beta_{2}^{\prime}\left(0\right)=\frac{\lambda_{0}}{q-2}$.

Case $\lambda_{1}<\lambda_{2}$, $a_{11}<a_{21}$ and $a_{12}=a_{22}$. Choose
for instance $a:=1$, $b:=3$,
$\lambda_{1}\left(\alpha\right):=\frac{2\lambda_{0}\left(\alpha+1\right)}{(1+\sqrt{7})\left(q-2\right)}$
and $\lambda_{2}\left(\alpha\right):=3\lambda_{1}\left(\alpha\right)$. Then
$0<\lambda_{1}<\lambda_{2}$, $0<a_{11}=\lambda_{1}<9\lambda_{1}=a_{21}$,
$a_{12}=a_{22}=3\lambda_{1}>0$, $a\neq b$, $D=112\lambda_{1}^{2}>0$,
$\beta_{1}\left(0\right)=\frac{(1-\sqrt{7})\lambda_{0}}{(1+\sqrt{7})\left(q-2\right)}<0$
(so that
$\left(q-2\right)\beta_{1}\left(0\right)\not\in\operatorname{spec}\left(\Delta_{g}\right)$)
and
$\beta_{2}\left(0\right)=\beta_{2}^{\prime}\left(0\right)=\frac{\lambda_{0}}{q-2}$.

The results then follow by applying Theorem 2.2. ∎

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