# Global Bifurcation of Non-Radial Solutions for Symmetric Sub-linear Elliptic
Systems on the Planar Unit Disc

Ziad Ghanem  Department of Mathematical Sciences, University of Texas at
Dallas, Richardson, TX 75080, USA<EMAIL_ADDRESS>,  Casey Crane
Department of Mathematical Sciences, University of Texas at Dallas,
Richardson, TX 75080, USA<EMAIL_ADDRESS>and  Jingzhou Liu
Department of Mathematical Sciences, University of Texas at Dallas,
Richardson, TX 75080, USA<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove a global bifurcation result for the existence of non-
radial branches of solutions to the paramterized family of $\Gamma$-symmetric
problems $-\Delta u=f(\alpha,z,u)$, $u|_{\partial D}=0$ on the unit disc
$D:=\\{z\in{\mathbb{C}}:|z|<1\\}$ with $u(z)\in\mathbb{R}^{k}$, where
$\mathbb{R}^{k}$ is an orthogonal $\Gamma$-representation,
$f:\mathbb{R}\times\overline{D}\times\mathbb{R}^{k}\to\mathbb{R}^{k}$ is a
sub-linear $\Gamma$-equivariant continuous function, differentiable with
respect to $u$ at zero and satisfying the conditions
$f(\alpha,e^{i\theta}z,u)=f(\alpha,z,u)$ for all $\theta\in\mathbb{R}$ and
$f(z,-u)=-f(z,u)$.

Mathematics Subject Classification: Primary: 37G40, 35B06; Secondary: 47H11,
35J57, 35J91, 35B32, 47J15

Key Words and Phrases: global bifurcation, non-linear Laplace equation,
symmetric bifurcation, bifurcation invariant, non-radial solutions,
equivariant Brouwer degree.

##  1\. Introduction

We draw our motivation from natural phenomena that are modeled by
parameterized systems of autonomous Partial Differential Equations (PDEs),
where the possible states of each phenomenon are represented by solutions of
the associated system of equations at corresponding parameter values. These
solutions can be categorized as trivial or non-trivial. Trivial solutions are
so-named because they persist for all parameter values and are straightforward
to determine. In contrast, Non-trivial solutions may only exist for a
particular range of parameter values and often exhibit exotic properties which
enhance our understanding of the model. The classical bifurcation problem is
concerned with the existence and global behaviour of branches of non-trivial
solutions emerging from their trivial counterparts. Should the phenomena admit
the symmetries of a certain group $G$ (represented by the $G$-equivariance of
the associated equations), one may study the equivariant bifurcation problem,
which additionally considers the symmetric properties of these branches (cf.
[1, 3, 9, 11, 19, 21, 22, 23, 24, 25]). The application of topological methods
to the study of differential equations, as with most tools in the arsenal of
the nonlinear analyst, traces back to Maître Poincare [13]. More recently, the
Local/Equivariant Brouwer Degrees and their infinite dimensional
generalizations – the Local/Equivariant Leray-Schauder Degrees – have proven
prodigiously successful in obtaining existence results for solutions to a wide
class of nonlinear differential equations. Somewhat remarkably, these degree
theories have also been shown to be useful in solving classical and
equivariant bifurcation problems. The first use of the Leray-Schauder degree
for the detection of (local) bifurcation to a parameterized system of
nonlinear differential equations is attributed to the seminal work of M.A.
Krasnosel’skii, in which sufficient conditions for the existence of a branch
of non-trivial solutions are established (cf. [17]). Only a decade later, P.
Rabinowitz (cf. [20]) proposed his famous Rabinowitz alternative, which
provides sufficient conditions for a given branch of non-trivial solutions to
be unbounded. In this paper, equivariant analogues of Krasnosel’skii and
Rabinowitz type results are employed to solve an equivariant bifurcation
problem. Specifically, our objective is to study the equivariant bifurcation
problem for a symmetric system of parameterized nonlinear Laplace equations
subject to Dirichlet boundary conditions. The bifurcation and global behaviour
of solutions for such systems has been extensively studied, due to their
significance in applied mathematics and physics. In the case that the
nonlinearity arises as the gradient of some potential function, the
bifurcation problem can be studied as a parameterized variational problem and
appropriate topological methods such as Morse theory, index theory, and the
equivariant gradient degree (cf. [14]) can be administered (cf. [6, 7, 8, 10,
12, 11, 21, 22, 23, 24, 25, 26]). However, these approaches are of no use for
systems which do not admit variational structure. The degree theoretic methods
we employ in this paper impose no variational requirements on the equations
and, as such, are broadly applicable to a wider class of problems. Consider
the following parameterized family of symmetric Laplace equations:

(1) $\begin{cases}-\Delta u=f(\alpha,z,u),\quad\alpha\in\mathbb{R},\;z\in
D,\;u(z)\in V\\\ u|_{\partial D}=0,\end{cases}$

where $D:=\\{z\in{\mathbb{C}}:|z|<1\\}$ is the unit planar disc,
$V:=\mathbb{R}^{k}$ is an orthogonal representation of a finite group $\Gamma$
and $f:\mathbb{R}\times\overline{D}\times V\to V$ is a continuous, odd,
radially symmetric and $\Gamma$-equivariant family of functions of sub-linear
growth, which are differentiable with respect to $u$ at the origin in $V$. In
particular, we assume that $f$ satisfies the following conditions:

  1. ($A_{1}$)

$f(\alpha,e^{i\theta}z,u)=f(\alpha,z,u)$ for all $\alpha\in\mathbb{R},\;z\in
D$, $u\in V$ and $\theta\in\mathbb{R}$;

  2. ($A_{2}$)

$f(\alpha,z,\gamma u)=\gamma f(\alpha,z,u)$ for all
$\alpha\in\mathbb{R},\;z\in D$, $u\in V$ and $\gamma\in\Gamma$;

  3. ($A_{3}$)

$f(\alpha,z,-u)=-f(\alpha,z,u)$ for all $\alpha\in\mathbb{R}$, $z\in D$ and
$u\in V$;

  4. ($A_{4}$)

there exist continuous functions $A:\mathbb{R}\to L(k,\mathbb{R})$,
$c:\mathbb{R}\rightarrow(0,\infty)$ and a number $\beta>1$ such that for each
$\alpha\in\mathbb{R}$ one has

$\displaystyle|f(\alpha,z,u)-A(\alpha)u|\leq
c(\alpha)|u|^{\beta}\quad\text{for all}\;{z\in\overline{D}},\;{u\in V};$

  5. ($A_{5}$)

there exist continuous functions $a,b:\mathbb{R}\rightarrow(0,\infty)$ and a
number $\nu\in(0,1)$ such that for each $\alpha\in\mathbb{R}$ the following
inequality holds

$\displaystyle|f(\alpha,z,u)|<a(\alpha)|u|^{\nu}+b(\alpha)\quad\text{for
all}\;{z\in\overline{D}},{u\in V}.$

Conditions ($A_{1}$) ($A_{2}$) and ($A_{3}$) imply the
$O(2)\times\Gamma\times\mathbb{Z}_{2}$-symmetry of system (1) in the sense
that $f:\mathbb{R}\times\overline{D}\times V\to V$ is $O(2)$-invariant with
respect to the $O(2)$-action $O(2)\times D\rightarrow D$ given by
$(\theta,z)\rightarrow e^{i\theta}z$, $(\kappa,z)\rightarrow\overline{z}$ and
$\Gamma\times\mathbb{Z}_{2}$-equivariant with respect to the
$\Gamma\times\mathbb{Z}_{2}$ action $(\Gamma\times\mathbb{Z}_{2})\times
V\rightarrow V$ given by $(\gamma,\pm 1,u)\rightarrow\pm\gamma u$. On the
other hand, conditions ($A_{4}$) and ($A_{5}$) guarantee differentiability at
the origin and sublinearity, respectively.

######  Remark 1.1.

Since the zero function is a solution to (1) for all values
$\alpha\in\mathbb{R}$, all trivial solutions to the system are of the form
$(\alpha,0)$. Non-trivial solutions to (1), on the other hand, fall into two
categories, namely:

  * (i)

radial solutions, which depend only on $|z|$ for any $z\in D$ (although less
obvious than the trivial solution, radial solutions can often be identified
using classical methods, eg. by reducing (1) to a second order ODE);

  * (ii)

and non-radial solutions, which exhibit dependency on the angular variable.

Our objective is to identify branches of non-radial solutions to (1), describe
their possible symmetric properties, and characterize their global behavior.

The methods used in this paper are inspired by [2], where an existence result
was obtained for a similar sublinear elliptic system using equivariant degree
theory. For a more thorough exposition of these topics, we direct readers to
the recent monograph [4]. The equivariant degree is intimately connected with
the classical Brouwer degree. Although its application is relatively simple,
technical difficulties arise when dealing with algebraic computations related
to unfamiliar group structures. Many of these issues can be resolved with
usage of the G.A.P. system, and the G.A.P. package EquiDeg (created by Haopin
Wu), which is available online at https://github.com/psistwu/equideg (cf.
[28]). In the remainder of this paper, we employ tools from the equivariant
degree theory to determine $(i)$ under what conditions branches of non-trivial
solutions may bifurcate from a trivial solution, $(ii)$ under what conditions
these branches consist only of non-radial solutions and $(iii)$ under what
conditions these branches are unbounded. Subsequent sections are organized as
follows: In Section 2 the problem (1) is reformulated in an appropriate
functional setting. In Section 3 we recall the abstract equivariant
bifurcation results, including the equivariant analogues of the classical
Krasnosel’skii and Rabinowitz theorems. In Section 4, we apply the equivariant
degree theory methods to establish local and global bifurcation results for
(1). Finally, in Section 5 we present a motivating example of vibrating
membranes where our main results, Theorem 4.1 and 4.3, are applied to
demonstrate the existence of unbounded branches of non-radial solutions
admitting all possible maximal orbit types. For convenience, the Appendices
include an explanation of notations used, a summary of the spectral properties
of the Laplace operator on the unit disc, and a brief introduction to the
Brouwer equivariant degree theory. Acknowledgment: We are deeply grateful to
our advisor, Professor Krawcewicz, whose invaluable guidance and unwavering
support made this work possible. We would also like to acknowledge Professor
Garcia-Azpetia for insightful discussions regarding the estimation of the
behaviour of our system near a bifurcation point.

##  2\. Functional Space Reformulation and a priori Bounds

Consider the Sobolev space $\mathscr{H}:=H^{2}(D,V)\cap H^{1}_{0}(D,V)$
equipped with the usual norm

$\displaystyle\|u\|_{\mathscr{H}}:=\max\\{\|D^{s}u\|_{2}:|s|\leq 2\\},$

where $s=(s_{1},s_{2})$, $|s|:=s_{1}+s_{2}\leq 2$, and
$D^{s}\varphi=\frac{\partial^{|s|}\varphi}{\partial^{s_{1}}x\partial^{s_{2}}y}$.
It is well known that the Laplacian operator $\mathscr{L}:\mathscr{H}\to
L^{2}(D,V)$ given by

(2) $\displaystyle\mathscr{L}u:=-\Delta u,$

is a linear isomorphism. Let $\nu\in(0,1)$ be the scalar from condition
($A_{5}$). If one chooses $q>\max\\{1,2\nu\\}$ (for example, it is enough to
take $q:=2\beta$ (cf. assumption ($A_{4}$)), then there is the standard
Sobolev embedding $j:\mathscr{H}\to L^{q}(D,V)$

(3) $j(u)(z)=u(z),\quad u\in\mathscr{H},$

and also its associated Nemytski operator $N_{\alpha}:L^{q}(D,V)\rightarrow
L^{2}(D,V)$

(4) $N_{\alpha}(v)(z):=f(\alpha,z,v(z)),\quad z\in D,\;\alpha\in\mathbb{R}.$

######  Lemma 2.1.

Assume that the conditions ($A_{1}$)–($A_{5}$) are satisfied. The Nemystiki
operator (4) is well-defined.

###### Proof.

It suffices to demonstrate that for any $v\in L^{q}(D,V)$ and
$\alpha\in\mathbb{R}$ one has $N_{\alpha}(v)\in L^{2}(D,V)$. Indeed, combining
($A_{5}$) with the Hölder inequality, one has:

$\displaystyle\|N_{\alpha}(v)\|_{L^{2}}$
$\displaystyle=\|f(\alpha,z,v)\|_{L^{2}}$
$\displaystyle\leq\|\,a(\alpha)|v|^{\nu}\|_{L^{2}}+\|\,b(\alpha)\|_{L^{2}}$
$\displaystyle=a(\alpha)\|\,|v|^{\nu}\|_{L^{2}}+b(\alpha)\sqrt{\pi}$
$\displaystyle=a(\alpha)\left(\int_{D}|v|^{2\nu}\right)^{\frac{1}{2}}+b(\alpha)\sqrt{\pi}$
$\displaystyle\leq a(\alpha)\pi^{{1-2\nu/q}\over
2}\left(\int_{D}|v|^{\frac{2\nu
q}{2\nu}}\right)^{\frac{2\nu}{2}q}+b(\alpha)\sqrt{\pi}$ (5)
$\displaystyle=a(\alpha)\pi^{1/2-\nu/q}\|v\|^{\nu}_{L^{q}}+b(\alpha)\sqrt{\pi},$

where the result follows from the assumptions that
$b(\alpha),a(\alpha),\|v\|_{q}<\infty$. $\square$ Notice that the operator
equation

(6) $\mathscr{L}u=N_{\alpha}(ju),\quad\alpha\in\mathbb{R},u\in\mathscr{H},$

is equivalent to system (1) in the sense that $u\in\mathscr{H}$ is a solution
to (1) for some $\alpha\in\mathbb{R}$ if and only if
$(\alpha,u)\in\mathbb{R}\times\mathscr{H}$ satisfies (6). In turn, since
$\mathscr{L}$ is an isomorphism, the map
$\mathscr{F}:\mathbb{R}\times\mathscr{H}\to\mathscr{H}$ given by

(7) $\mathscr{F}(\alpha,u):=u-\mathscr{L}^{-1}N_{\alpha}(ju),$

is well-defined so that system (1) is also equivalent to the equation

(8) $\mathscr{F}(\alpha,u)=0,\quad\alpha\in\mathbb{R},\;u\in\mathscr{H}.$

In what follows, we will call (8) the operator equation associated with (1).

######  Lemma 2.2.

Let $f:\mathbb{R}\times\overline{D}\times V\rightarrow V$ be a continuous
function satisfying the assumption ($A_{5}$). For every $\alpha\in\mathbb{R}$,
there exists a constant $R(\alpha)>0$ such that, if
$(\alpha,u)\in\mathbb{R}\times\mathscr{H}$ is a solution to system (1), then
$\|u\|_{\mathscr{H}}<R(\alpha)$.

###### Proof.

Assume that $(\alpha,u)\in\mathbb{R}\times\mathscr{H}$ is a solution to system
(1). Combining (5) with $u=\mathscr{L}^{-1}N_{\alpha}(ju)$, one has

(9) $\|u\|_{\mathscr{H}}\leq
a(\alpha)\pi^{1/2-\nu/q}\|\mathscr{L}^{-1}\|\,\|u\|_{L_{2}}^{\nu}+b(\alpha)\sqrt{\pi}\|\mathscr{L}^{-1}\|,$

which, together with $\|u\|_{\mathscr{H}}\geq\|u\|_{L^{2}}$, implies

(10) $\displaystyle\|u\|_{L^{2}}\leq c\|u\|^{\nu}_{L^{2}}+d,$

where $c:=a(\alpha)\pi^{1/2-\nu/q}\|\mathscr{L}^{-1}\|$,
$d:=b(\alpha)\sqrt{\pi}\|\mathscr{L}^{-1}\|$. Finally, since $0<\nu<1$, there
exists $R_{0}(\alpha)>0$ such that $\psi(t):=t-ct^{\nu}-d>0$ for $t\geq
R_{0}(\alpha)$. Consequently, $\|u\|_{L^{2}}<R_{0}(\alpha)$, and by (9),

$\|u\|_{\mathscr{H}}\leq
c\|u\|^{\nu}_{L^{2}}+d<cR_{0}(\alpha)^{\nu}+d=:R(\alpha)$

is the required constant. $\square$

##  3\. Abstract Local and Global Equivariant Bifurcation

In this section we present a concise exposition, following [4], of an
equivariant Brouwer degree method to study symmetric bifurcation problems.
Given a compact Lie group $\mathcal{G}$, an isometric Banach
$\mathcal{G}$-representation $\mathcal{H}$ and a completely continuous
$\mathcal{G}$-equivariant field
$\mathcal{F}:\mathbb{R}\times\mathcal{H}\rightarrow\mathcal{H}$, we use the
Leray-Schauder Equivariant $\mathcal{G}$-Degree to describe local and global
properties of the solution set to the equation,

(11) $\mathcal{F}(\alpha,u)=0,\quad\alpha\in\mathbb{R},\;u\in\mathcal{H}.$

To simplify our exposition and for compatibility with the system of interest
(8), we make the following two assumptions.

  1. ($B_{1}$)

The set of trivial solutions to (11) is given by

$\displaystyle M:=\\{(\alpha,0)\in\mathbb{R}\times\mathcal{H}\\}.$

  2. ($B_{2}$)

There exists a continuous family of linear operators
$\mathcal{A}(\alpha):\mathcal{H}\rightarrow\mathcal{H}$ such that, for every
$\alpha\in\mathbb{R}$, the derivative $D_{u}\mathcal{F}(\alpha,0)$ exists and

$\displaystyle\mathcal{A}(\alpha):=D\mathcal{F}(\alpha,0).$

Solutions to (11) which do not belong to $M$ are called nontrivial. Let
$\mathcal{S}\subset\mathcal{F}^{-1}(0)$ denote the set of all non-trivial
solutions, i.e.

$\mathcal{S}:=\\{(\alpha,u)\in\mathbb{R}\times\mathcal{H}\;:\;\mathcal{F}(\alpha,u)=0\text{
and }u\neq 0\\}.$

Clearly, the set of non-trivial solutions
$\mathcal{S}\subset\mathbb{R}\times\mathcal{H}$ is $\mathcal{G}$-invariant.

###  3.1. The Local Bifurcation Invariant and Krasnosel’skii’s Theorem

Formulation of a Krasnosel’skii type local bifurcation result for equation
(11) necessitates the introduction of additional notations and terminology
(for more details, the reader is referred to [1, 4]). Our first definition
clarifies what is meant by a bifurcation of the equation (11).

######  Definition 3.1.

A trivial solution $(\alpha_{0},0)\in M$ is said to be a bifurcation point for
the equation (11) if every open neighborhood of the point $(\alpha_{0},0)$ has
non-trivial intersection with $\mathcal{S}$.

It is well-known that a necessary condition for any trivial solution
$(\alpha_{0},0)\in M$ to be a bifurcation point for the equation (11) is that
the linear operator
$\mathcal{A}(\alpha_{0}):\mathcal{H}\rightarrow\mathcal{H}$ is not an
isomorphism. This leads to the following definition.

######  Definition 3.2.

A trivial solution $(\alpha_{0},0)\in M$ is said to be a regular point for the
equation (11) if $\mathcal{A}(\alpha_{0})$ is an isomorphism and a critical
point otherwise. Moreover, a critical point $(\alpha_{0},0)\in M$ is said to
be isolated if there exists a deleted $\epsilon$-neighborhood
$0<|\alpha-\alpha_{0}|<\epsilon$ such that for all
$\alpha\in(\alpha_{0}-\epsilon,\alpha_{0}+\epsilon)\setminus\\{\alpha_{0}\\}$,
the point $(\alpha,0)\in M$ is regular.

The set of all critical points for equation (11), denoted $\Lambda$, is called
the critical set, i.e.

(12) $\displaystyle\Lambda:=\\{(\alpha,0):\text{
$\mathcal{A}(\alpha):\mathcal{H}\rightarrow\mathcal{H}$ is not an
isomorphism}\\}.$

The next definition concerns our interest in the continuation of non-trivial
solution emerging from a bifurcation point $(\alpha_{0},0)\in M$.

######  Definition 3.3.

A trivial solution $(\alpha_{0},0)\in M$ is said to be a branching point for
the equation (11) if there exists a non-trivial continuum
$K\subset\overline{S}$ with $K\cap M=\\{(\alpha_{0},0)\\}$ and the maximal
connected set $\mathcal{C}\subset\overline{S}$ containing the branching point
$(\alpha_{0},0)$ we call a branch of nontrivial solutions bifurcating from the
point $(\alpha_{0},0)$.

Whereas the classical Krasnosiel’skii bifurcation result is only concerned
with the existence of a branch of nontrivial solutions for the equation (11)
bifurcating from a given critical point $(\alpha_{0},0)\in\Lambda$, the
equivariant Krasnosiel’skii bifurcation result, which we employ in this paper,
is also concerned with the symmetric properties of such a branch.

######  Definition 3.4.

Let $H\leq\mathcal{G}$ be a given subgroup and denote by $\mathcal{S}^{H}$ the
corresponding $H$-fixed point space of non-trivial solutions. A branch of
solutions $\mathcal{C}$ is said to have symmetries at least $(H)$ if
$\mathcal{C}\cap\mathcal{S}^{H}\not=\emptyset$.

Take an isolated critical point $(\alpha_{0},0)\in\Lambda$ and choose
$\alpha^{\pm}_{0}\in(\alpha_{0}-\varepsilon,\alpha_{0}+\varepsilon)$ with
$\alpha^{-}_{0}\leq\alpha_{0}\leq\alpha^{+}_{0}$, where $\varepsilon>0$ was
chosen such that for all $0<|\alpha-\alpha_{0}|<\varepsilon$ the solution
$(\alpha,0)\in M$ is a regular point. For convenience, we adopt the notations
$\mathcal{F}_{\pm}(u):=\mathcal{F}(\alpha^{\pm}_{0},u)$ and
$B_{\delta}:=\\{u\in\mathscr{H}:\|u\|<\delta\\}$. By construction, the linear
operators $\mathcal{A}(\alpha^{\pm}):\mathscr{H}\rightarrow\mathscr{H}$ are
non-singular; so, from assumption ($B_{2}$), there exists $\delta>0$
sufficiently small such that $\mathcal{F}_{\pm}^{-1}(0)\cap\partial
B_{\delta}=\emptyset$ and $\mathcal{F}_{\pm}$ are $B_{\delta}$-admissibly
$\mathcal{G}$-homotopic to $\mathcal{A}(\alpha^{\pm})$. Moreover, since
$\mathcal{G}$ acts isometrically on $\mathscr{H}$, the ball $B_{\delta}$ is
clearly $\mathcal{G}$-invariant. It follows that
$(\mathcal{F}_{\pm},B_{\delta})$ are admissible $\mathcal{G}$-pairs in
$\mathscr{H}$ and also that
$\mathcal{G}\text{-deg}(\mathcal{F}_{\pm},B_{\delta})=\mathcal{G}\text{-deg}(\mathcal{A}(\alpha^{\pm}_{0}),B(\mathscr{H}))$,
where $B(\mathscr{H})$ is the open unit ball in $\mathscr{H}$. We define the
the local $G$-equivariant bifurcation invariant
$\omega_{\mathcal{G}}(\alpha_{0})\in A(\mathcal{G})$ at the isolated critical
solution $(\alpha_{0},0)\in\Lambda$ as follows,

$\omega_{\mathcal{G}}(\alpha_{0})=\mathcal{G}\text{-deg}(\mathcal{A}(\alpha^{-}_{0}),B(\mathscr{H}))-\mathcal{G}\text{-deg}(\mathcal{A}(\alpha^{+}_{0}),B(\mathscr{H})).$

The invariant $\omega_{\mathcal{G}}(\alpha_{0})$ does not depend on the choice
of $\alpha^{\pm}_{0}\in\mathbb{R}$ or the radius $\delta>0$. We refer to [4]
for the proof of the following local bifurcation result, which is a
consequence of the equivariant version of a classical result of K. Kuratowski
(cf. [18], Thm. 3, p. 170).

######  Theorem 3.1.

(M.A. Krasnosel’skii-Type Local Bifurcation) Let
$\mathcal{F}:\mathbb{R}\times\mathscr{H}\rightarrow\mathscr{H}$ be a
completely continuous $\mathcal{G}$-equivariant field satisfying the
assumptions ($B_{1}$) and ($B_{2}$) and with an isolated critical point
$(\alpha_{0},0)$. If $\omega_{\mathcal{G}}(\alpha_{0})\neq 0$, then

  * (i)

there exists a branch of nontrivial solutions $\mathcal{C}$ to system (11)
with branching point $(\alpha_{0},0)$;

  * (ii)

moreover, if $(H)\in\Phi_{0}(G)$ is an orbit type with

$\displaystyle\operatorname{coeff}^{H}(\omega_{\mathcal{G}}(\alpha_{0}))\neq
0,$

then there exists a branch of non-trivial solutions bifurcating from
$(\alpha_{0},0)$ with symmetries at least $(H)$.

###  3.2. Global Bifurcation and the Rabinowitz Alternative

In order to employ the Leray-Schauder $\mathcal{G}$-equivariant degree to
describe the global properties of a branch of non-trivial solutions
bifurcating from an isolated critical point of the equation (11), we need to
make an additional assumption.

  1. ($B_{3}$)

The critical set $\Lambda\subset M$ (given by (12)) is discrete.

Notice that the local bifurcation invariant $\omega_{\mathcal{G}}(\alpha_{0})$
at any critical point $(\alpha_{0},0)\in\Lambda$ is well-defined under
assumption ($B_{3}$). Moreover, if
$\mathcal{U}\subset\mathbb{R}\times\mathscr{H}$ is an open bounded
$\mathcal{G}$-invariant set, then its intersection with the critical set is
finite. These observations will be important for the statement of the
following global bifurcation result, the proof of which can be found in [4].

######  Theorem 3.2.

(The Rabinowitz Alternative) Suppose that
$\mathcal{F}:\mathbb{R}\times\mathscr{H}\rightarrow\mathscr{H}$ is a
completely continuous $\mathcal{G}$-equivariant field satisfying conditions
($B_{1}$),($B_{2}$) and ($B_{3}$). Let
$\mathcal{U}\subset\mathbb{R}\times\mathscr{H}$ be an open bounded
$\mathcal{G}$-invariant set with $\partial\mathcal{U}\cap\Lambda=\emptyset$
and $(\alpha_{0},0)\in\mathcal{U}\cap\Lambda$. If $\mathcal{C}$ is a branch of
nontrivial solutions to (11) bifurcating from the critical point
$(\alpha_{0},0)$, then one has the following alternative:

  1. $(a)$

either $\mathcal{C}\cap\partial\mathcal{U}\neq\emptyset$;

  2. $(b)$

or there exists a finite set

$\displaystyle\mathcal{C}\cap\Lambda=\\{(\alpha_{0},0),(\alpha_{1},0),\ldots,(\alpha_{n},0)\\},$

satisfying the following relation

$\displaystyle\sum\limits_{k=1}^{n}\omega_{\mathcal{G}}(\alpha_{k})=0.$

######  Remark 3.1.

Suppose that Theorem 3.1 is used to demonstrate the existence of a branch
$\mathcal{C}$ of nontrivial solutions to (11) bifurcating from a critical
point $(\alpha_{0},0)$. If, in addition, certain conditions are met such that
for any open bounded $\mathcal{G}$-invariant neighborhood
$\mathcal{U}\ni(\alpha_{0},0)$ with $\partial\mathcal{U}\cap\Lambda=\emptyset$
the alternative $(b)$ is impossible, then, according to Theorem 3.2, the
branch $\mathcal{C}$ must be unbounded.

##  4\. Local and Global Bifurcation of Non-radial Solutions in (1)

Returning to the functional reformulation of our original problem described in
Section 2, put $G:=O(2)\times\Gamma\times\mathbb{Z}_{2}$ and notice that the
Sobolev space $\mathscr{H}$ introduced in Section 2 is a natural Banach
$G$-representation with respect to the isometric $G$-action
$G\times\mathscr{H}\rightarrow\mathscr{H}$ given by

$\displaystyle(\theta,\gamma,\pm 1)u(z)$ $\displaystyle:=\pm\gamma
u(e^{i\theta}\cdot z),\quad z\in D,\;\;u\in\mathscr{H};$
$\displaystyle(\kappa,\gamma,\pm 1)u(z)$ $\displaystyle:=\pm\gamma
u(\overline{z}),\quad\gamma\in\Gamma,\;\kappa\in O(2),\;\theta\in SO(2),$

where $\overline{z}$ is the complex conjugation of $z\in D$ and
$e^{i\theta}\cdot z$ is the standard complex multiplication.

######  Remark 4.1.

Under assumptions ($A_{1}$)-($A_{5}$), the nonlinear operator
$\mathscr{F}:\mathbb{R}\times\mathscr{H}\rightarrow\mathscr{H}$ given by (7)
is a completely continuous $G$-equivariant field, differentiable at
$0\in\mathscr{H}$ with

$\displaystyle
D\mathscr{F}(\alpha,0)=\operatorname{Id}-\mathscr{L}^{-1}A(\alpha):\mathscr{H}\rightarrow\mathscr{H},$

where $A(\alpha):V\rightarrow V$ is the linearization of the map
$f(\alpha,z,u)$ from the equation (1) at the origin (see for example see [16,
2]). For convenience, we introduce the notation

(13) $\mathscr{A}(\alpha)(u):=u-\mathscr{L}^{-1}A(\alpha)u,\quad
u\in\mathscr{H}.$

Choose a value $\alpha\in\mathbb{R}$ such that
$\mathscr{A}(\alpha):\mathscr{H}\rightarrow\mathscr{H}$ is an isomorphism and
denote by $B(\mathscr{H}):=\\{u\in\mathscr{H}:\|u\|_{\mathscr{H}}<1\\}$ the
open unit ball in $\mathscr{H}$. Before approaching the question of
bifurcation in the equation (1), we must derive a workable formula for the
computation of the degree $G\text{\rm-
deg}(\mathscr{A}(\alpha),B(\mathscr{H}))$. Assuming that a complete list of
the irreducible $\Gamma$-representations $\\{\mathcal{V}_{j}\\}_{j=1}^{r}$ is
made available we denote by $\\{\mathcal{V}_{j}^{-}\\}_{j=1}^{r}$ the
corresponding list of irreducible
$\Gamma\times\mathbb{Z}_{2}$-representations, where the superscript is meant
to indicate that each irreducible $\Gamma$-representation has been equipped
with the antipodal $\mathbb{Z}_{2}$-action in the standard way (cf. [4]). Now,
as a $\Gamma\times\mathbb{Z}_{2}$-representation, $V=\mathbb{R}^{k}$ has the
$\Gamma\times\mathbb{Z}_{2}$-isotypic decomposition

$\displaystyle V=V_{1}\oplus V_{1}\oplus\cdots\oplus V_{r},$

where each $\Gamma\times\mathbb{Z}_{2}$-isotypic component component $V_{j}$
is modeled on the irreducible $\Gamma\times\mathbb{Z}_{2}$-representation
$\mathcal{V}_{j}^{-}$ in the sense that $V_{j}$ is equivalent to the direct
sum of some number of copies of $\mathcal{V}_{j}^{-}$, i.e.

$\displaystyle
V_{j}\simeq\mathcal{V}_{j}^{-}\oplus\cdots\oplus\mathcal{V}_{j}^{-}.$

The exact number of irreducible $\Gamma$-representations $\mathcal{V}_{j}^{-}$
‘contained’ in the $\Gamma\times\mathbb{Z}_{2}$-isotypic component $V_{j}$ is
called the $\mathcal{V}_{j}^{-}$-isotypic multiplicity of $V$ and is
calculated according to the ratio,

$\displaystyle m_{j}:=\dim V_{j}/\dim\mathcal{V}_{j}^{-},\quad
j\in\\{1,2,\ldots,r\\}.$

To simplify computations, we introduce an additional condition on the
linearization of (1):

  1. ($A_{0}$)

For each $j\in\\{1,2,\ldots,r\\}$ there exists a continuous map
$\mu_{j}:\mathbb{R}\rightarrow\mathbb{R}$ with

$A_{j}(\alpha):=A(\alpha)|_{V_{j}}=\mu_{j}(\alpha)\operatorname{Id}|_{V_{j}}.$

In pursuit of a $G$-isotypic decomposition of $\mathscr{H}$, let us consider
the spectrum of the Laplacian operator (2), understood in this context as an
unbounded operator in $L^{2}(D;V)$. Namely, one has

$\sigma(\mathscr{L})=\\{s_{nm}:n\in{\mathbb{N}},\;m=0,1,2,\dots\\},$

where $\sqrt{s_{nm}}$ denotes the $n$-th positive zero of the $m$-th Bessel
function of the first kind $J_{m}$. Corresponding to each eigenvalue
$s_{nm}\in\sigma(\mathscr{L})$, there is an associated eigenspace
$\mathscr{E}_{nm}\subset\mathscr{H}$ which can be expressed, using standard
polar coordinates $(r,\theta)$, as follows

$\displaystyle\mathscr{E}_{nm}:=\left\\{J_{m}(\sqrt{s_{nm}}r)\Big{(}\cos(m\theta)\vec{a}+\sin(m\theta)\vec{b}\Big{)}:\vec{a},\,\vec{b}\in
V\right\\}.$

Then, $\mathscr{H}$ admits the $O(2)$-isotypic decomposition

$\displaystyle\mathscr{H}:=\overline{\bigoplus\limits_{m=0}^{\infty}\mathscr{H}_{m}},\quad\mathscr{H}_{m}:=\overline{\bigoplus\limits_{n=1}^{\infty}\mathscr{E}_{nm}},$

where the closure is taken in $\mathscr{H}$. In particular, adopting the
notations

(14)
$\mathscr{E}_{nm}^{j}:=\left\\{J_{m}(\sqrt{s_{nm}}r)\Big{(}\cos(m\theta)\vec{a}+\sin(m\theta)\vec{b}\Big{)}:\vec{a},\,\vec{b}\in
V_{j}\right\\},$

and

$\displaystyle\mathscr{A}_{n,m}^{j}(\alpha):=\mathscr{A}(\alpha)|_{\mathscr{E}_{nm}^{j}},$

the $G$-isotypic of $\mathscr{H}$ is given by

$\displaystyle\mathscr{H}=\bigoplus_{j=1}^{r}\overline{\bigoplus\limits_{m=0}^{\infty}\mathscr{H}_{m,j}},\quad\mathscr{H}_{m,j}:=\overline{\bigoplus\limits_{n=1}^{\infty}\mathscr{E}_{nm}^{j}}.$

To be clear, each $G$-isotypic component $\mathscr{H}_{m,j}$ is modeled on the
irreducible $G$-representation
$\mathcal{V}_{m,j}:=\mathscr{H}_{m}\otimes\mathcal{V}_{j}^{-}$, where
$\mathscr{H}_{m}$ denotes the $m$-th irreducible $O(2)$-representation and
$\mathcal{V}_{j}^{-}$ is the $j$-th irreducible $\Gamma$-representation
$\mathcal{V}_{j}$ with antipodal $\mathbb{Z}_{2}$-action.

######  Lemma 4.1.

Under assumption ($A_{0}$), each eigenvalue of the linear operator
$\mathscr{A}(\alpha):\mathscr{H}\rightarrow\mathscr{H}$ is of the form

$\displaystyle\xi_{n,m,j}(\alpha):=1-\frac{\mu_{j}(\alpha)}{s_{nm}},$

where $n\in\mathbb{N}$, $m\in\mathbb{N}\cup\\{0\\}$, $j\in\\{1,2,\ldots,r\\}$
and $\mu_{j}(\alpha)\in\sigma(A_{j}(\alpha))$.

###### Proof.

Indeed, since $\mathscr{A}(\alpha)$ is $G$-equivariant, one has

$\displaystyle\mathscr{A}_{n,m}^{j}(\alpha):\mathscr{E}_{nm}^{j}\rightarrow\mathscr{E}_{nm}^{j}$

such that

$\displaystyle\sigma(\mathscr{A}(\alpha))=\bigcup\limits_{m=0}^{\infty}\bigcup\limits_{n=1}^{\infty}\bigcup\limits_{j=1}^{r}\sigma(\mathscr{A}^{j}_{n,m}(\alpha)),$

where one easily obtains (see for example [2, 4])

$\displaystyle\sigma(\mathscr{A}^{j}_{n,m}(\alpha))$
$\displaystyle=\left\\{1-\frac{\mu_{j}(\alpha)}{s_{nm}}\;:\;\mu_{j}(\alpha)\in\sigma(A_{j}(\alpha))\right\\}.$

$\square$ Assume that, for a given $\alpha\in\mathbb{R}$, the operator
$\mathscr{A}(\alpha):\mathscr{H}\to\mathscr{H}$ is an isomorphism. The product
property of the Leray-Schauder $G$-equivariant degree (cf. Appendix C) permits
us to express $G\text{\rm-deg}(\mathscr{A}(\alpha),B(\mathscr{H}))$ in terms
of a Burnside ring product of the Leray-Schauder $G$-equivariant degrees of
the various restrictions
$\mathscr{A}_{n,m}^{j}(\alpha):\mathscr{E}^{j}_{nm}\rightarrow\mathscr{E}^{j}_{nm}$
of the $G$-equivariant linear isomorphism
$\mathscr{A}(\alpha):\mathscr{H}\rightarrow\mathscr{H}$ to the
$G$-subrepresentations $\mathscr{E}^{j}_{nm}$ on their respective open unit
balls
$B(\mathscr{E}^{j}_{nm}):=\\{u\in\mathscr{E}^{j}_{nm}\;:\;\|u\|_{\mathscr{H}}<1\\}$
as follows

(15) $G\text{\rm-
deg}(\mathscr{A}(\alpha),B(\mathscr{H}))=\prod\limits_{j=1}^{r}\prod\limits_{m=0}^{\infty}\prod\limits_{n=1}^{\infty}G\text{\rm-
deg}(\mathscr{A}_{n,m}^{j}(\alpha),B(\mathscr{E}_{nm}^{j})).$

Notice that, under assumption ($A_{0}$), one has $G\text{\rm-
deg}(\mathscr{A}_{n,m}^{j}(\alpha),B(\mathscr{E}_{nm}^{j}))=(G)$ for almost
all indices $m$, $n$ and $j$, so that the product (15) is well-defined.
Indeed, each Leray-Schauder $G$-equivariant degree $G\text{\rm-
deg}(\mathscr{A}_{n,m}^{j}(\alpha),B(\mathscr{E}_{nm}^{j}))$ is fully
specified by the $\mathcal{V}_{j}^{-}$-isotypic multiplicities
$\\{m_{j}\\}_{j=1}^{r}$ together with the real spectra of
$\mathscr{A}(\alpha)$ according to formula,

(16) $G\text{\rm-
deg}(\mathscr{A}_{n,m}^{j}(\alpha),B(\mathscr{E}^{j}_{nm}))=\begin{cases}(\deg_{\mathcal{V}_{m,j}})^{m_{j}}\quad&\text{
if }s_{nm}<\mu_{j}(\alpha);\\\ (G)\quad&\text{ otherwise,}\end{cases}$

where $\deg_{\mathcal{V}_{m,j}}\in A(G)$ is the basic degree (cf. Appendix C)
associated with the irreducible $G$-representation $\mathcal{V}_{m,j}$ and
$(G)\in A(G)$ is the unit element of the Burnside Ring. In addition, since
each basic degree $\deg_{\mathcal{V}_{m,j}}$ is involutive in the Burnside
ring (cf. Appendix C), one has

(17)
$(\deg_{\mathcal{V}_{m,j}})^{m_{j}}=\begin{cases}\deg_{\mathcal{V}_{m,j}}\quad&\text{
if }2\nmid m_{j};\\\ (G)\quad&\text{ otherwise.}\end{cases}$

Putting together (15), (16) and (17), we introduce some notations to keep
track of the indices

$\displaystyle\Sigma:=\\{(n,m,j)\;:\;n\in\mathbb{N},\;m\in\mathbb{N}\cup\\{0\\},\;j\in\\{1,2,\ldots,r\\}\\},$

which contribute non-trivially to the Burnside Ring product (15). To begin,
the negative spectrum of
$\mathscr{A}(\alpha):\mathscr{H}\rightarrow\mathscr{H}$ is accounted for with
the index set

(18)
$\displaystyle\Sigma_{-}(\alpha):=\left\\{(n,m,j)\in\Sigma\;:\;1-\frac{\mu_{j}(\alpha)}{s_{nm}}<0\right\\}.$

As is well known (see, for example, [27], p. 486), $s_{1m}>m(m+2)$, from which
it follows that the set (18) is finite. Combining (18) with formulas (15) and
(16), one obtains

$\displaystyle G\text{\rm-
deg}(\mathscr{A}(\alpha),B(\mathscr{H}))\;=\prod\limits_{(n,m,j)\in\Sigma_{-}(\alpha)}(\deg_{\mathcal{V}_{m,j}})^{m_{j}}.$

Computation of (15) can be further reduced by accounting for the even
$\mathcal{V}_{j}$-isotypic multiplicities $m_{j}$, whose corresponding basic
degrees contribute trivially to the Burnside Product (15). We put,

(19)
$\displaystyle\Sigma(\alpha):=\left\\{(n,m,j)\in\Sigma_{-}(\alpha)\;:\;2\nmid
m_{j}\right\\},$

which, together with (17), yields

(20) $\displaystyle G\text{\rm-
deg}(\mathscr{A}(\alpha),B(\mathscr{H}))\;=\prod\limits_{(n,m,j)\in\Sigma(\alpha)}\deg_{\mathcal{V}_{m,j}}.$

###  4.1. Computation of the Local Bifurcation Invariant

Under the assumptions ($A_{1}$)–($A_{5}$), conditions ($B_{1}$) and ($B_{2}$)
are satisfied for the operator equation (8) (cf. Remarks 1.1, 4.1). Therefore,
the existence of a branch of non-trivial solutions to (1) bifurcating from an
isolated critical point $(\alpha_{0},0)\in\Lambda$ is reduced, by Theorem 3.1,
to computation of the local bifurcation invariant $\omega_{G}(\alpha_{0})$. In
addition, we assume ($A_{0}$), so that the local bifurcation invariant is
well-defined at each critical point in $\Lambda$. Adopting notations from
Section 3, choose
$\alpha^{\pm}_{0}\in(\alpha_{0}-\epsilon,\alpha_{0}+\epsilon)$ with
$\alpha^{-}_{0}\leq\alpha_{0}\leq\alpha^{+}_{0}$, where $\epsilon>0$ is chosen
such that, for all $0<|\alpha-\alpha_{0}|<\epsilon$, the solution
$(\alpha,0)\in M$ is a regular point and put
$\mathscr{A}(\alpha^{\pm}_{0}):=D_{u}\mathscr{F}(\alpha^{\pm}_{0},0)$. Then,
the local bifurcation invariant $\omega_{G}(\alpha_{0})$ at the isolated
critical point $(\alpha_{0},0)\in\mathbb{R}\times\mathscr{H}$ is given by

(21) $\omega_{G}(\alpha_{0})=G\text{\rm-
deg}(\mathscr{A}(\alpha^{-}_{0}),B(\mathscr{H}))-G\text{\rm-
deg}(\mathscr{A}(\alpha^{+}_{0}),B(\mathscr{H})),$

where $B(\mathscr{H}):=\\{u\in\mathscr{H}\;:\;\|u\|_{\mathscr{H}}<1\\}$ is the
open unit ball in $\mathscr{H}$. Notice that, since
$(\alpha^{\pm}_{0},0)\in\mathbb{R}\times\mathscr{H}$ are regular points of
(8), computation of the local bifurcation invariant $\omega_{G}(\alpha_{0})$
amounts to computation of the Leray-Schuader $G$-equivariant degree of the
$G$-equivariant linear isomorphism,

$\displaystyle\mathscr{A}(\alpha^{\pm}_{0}):=D_{u}\mathscr{F}(\alpha^{\pm}_{0},0)=\operatorname{Id}-\mathscr{L}^{-1}(A(\alpha^{\pm}_{0})):\mathscr{H}\rightarrow\mathscr{H}.$

######  Lemma 4.2.

Under the assumptions ($A_{1}$)-($A_{5}$) and ($A_{0}$) and using the notation
(19), the local bifurcation invariant at an isolated critical point
$(\alpha_{0},0)\in\Lambda$ with deleted regular neighborhood
$\alpha^{-}_{0}\leq\alpha_{0}\leq\alpha^{+}_{0}$ on which
$\mathscr{A}(\alpha):\mathscr{H}\rightarrow\mathscr{H}$ is an isomorphism is
given by

(22)
$\omega_{G}(\alpha_{0})=\prod\limits_{(n,m,j)\in\Sigma(\alpha^{-}_{0})}\deg_{\mathcal{V}_{m,j}}-\prod\limits_{(n,m,j)\in\Sigma(\alpha^{+}_{0})}\deg_{\mathcal{V}_{m,j}}.$

In order to formulate the main local equivariant bifurcation result, we must
first consider some additional properties of the basic degree. For a more
thorough exposition of these topics we refer the reader to Appendix C. Take
$s\in\mathbb{N}$ and define the $s$-folding map as the Lie group homomorphism,

$\displaystyle\psi_{s}(e^{i\theta},\gamma,\pm 1)=(e^{sl\theta},\gamma,\pm
1),\quad\psi_{s}(\kappa e^{i\theta},\gamma,\pm 1)=(\kappa
e^{is\theta},\gamma,\pm 1).$

Each $\psi_{s}:O(2)\times\Gamma\times\mathbb{Z}_{2}\rightarrow
O(2)\times\Gamma\times\mathbb{Z}_{2}$ induces a corresponding Burnside ring
homomorphism $\Psi_{s}:A(G)\rightarrow A(G)$ defined the generators
$(H)\in\Phi_{0}(G)$ by,

(23) $\displaystyle\Psi_{s}(H)=({}^{s}H),\quad{}^{s}H:=\psi^{-1}_{s}(H).$

Notice that, for $j\in\\{1,\ldots,r\\}$ and $m\geq 0$, there is the following
relation between basic degrees

(24)
$\displaystyle\Psi_{s}(\deg_{\mathcal{V}_{m,j}})=\deg_{\mathcal{V}_{sm,j}}.$

######  Remark 4.2.

An orbit type which is maximal in $\Phi_{0}(G;\mathscr{H}\setminus\\{0\\})$ is
also maximal in $\Phi_{0}(G;\mathscr{H}_{0}\setminus\\{0\\})$. Therefore, any
$u\in\mathscr{H}\setminus\\{0\\}$ with an isotropy $G_{u}\leq G$ such that
$(G_{u})$ is maximal in $\Phi_{0}(G;\mathscr{H}\setminus\\{0\\})$ must be
radially symmetric. It follows that in order to detect branches of solutions
to (7) corresponding to maps which are both non-trivial and non-radial, we
must restrict our focus to orbit types which are maximal in
$\Phi_{0}(G;\mathscr{H}_{m}\setminus\\{0\\})$ for some positive
$m\in\mathbb{N}$. Indeed, to demonstrate the existence of a branch of non-
radial solutions bifurcating from some isolated critical point
$(\alpha_{0},0)\in\Lambda$, it is sufficient to show that for some $m>0$ there
is an orbit type $(H)\in\Phi_{0}(G;\mathscr{H}_{m}\setminus\\{0\\})$ with
$\operatorname{coeff}^{H}(\omega_{G}(\alpha_{0}))\not=0$. Moreover, in such a
case one can conclude that for some $s\geq 1$ there exists a branch non-radial
solutions to (7) with symmetries at least $(^{s}H)$ bifurcating from
$(\alpha_{0},0)$.

For $m>0$, denote by $\mathfrak{M}_{m}$ the set of all maximal orbit types in
$\Phi_{0}(G;\mathscr{H}_{m}\setminus\\{0\\})$ and by $\mathfrak{M}_{m,j}$ the
set of orbit types
$\Phi_{0}(G;\mathscr{H}_{m,j}\setminus\\{0\\})\cap\mathfrak{M}_{m}$. Since
each $(H)\in\mathfrak{M}_{m}$ is also an orbit type in
$\Phi_{0}(G;\mathscr{H}_{m,j}\setminus\\{0\\})$ for at least one
$j\in\\{1,2,\ldots,r\\}$, one has

$\displaystyle\mathfrak{M}_{m}=\bigcup_{j=1}^{r}\mathfrak{M}_{m,j}.$

Notice that $\Psi_{s}(\mathfrak{M}_{m,j})=\mathfrak{M}_{sm,j}$ for any
$m>0,\;s\in\mathbb{N}$ and $j\in\\{1,2,\ldots,r\\}$ (in particular, one has
$\Psi_{s}(\mathfrak{M}_{1,j})=\mathfrak{M}_{s,j}$). Hence, any orbit type
$(H_{0})\in\mathfrak{M}_{m,j}$ can be recovered from an orbit type in
$\mathfrak{M}_{1,j}$ by the relation $(H):=\Psi_{s}^{-1}(H_{0})$.

######  Remark 4.3.

Take $m>0$ and $j\in\\{1,2,\ldots,r\\}$. For any basic degree
$\deg_{\mathcal{V}_{m,j}}\in A(G)$ and orbit type $(H)\in\mathfrak{M}_{1,j}$,
the recurrence formula for the Leray-Schauder $G$-equivariant degree (cf.
Appendix C) implies

$\displaystyle\deg_{\mathcal{V}_{m,j}}=(G)-y_{j}({}^{m}H)+a_{j},$

where $a_{j}\in A(G)$ is such that $\operatorname{coeff}^{{}^{s}H}(a_{j})=0$
for all $s\in{\mathbb{N}}$ and where the coefficient $y_{j}\in\mathbb{Z}$ is
determined by the rule

$\displaystyle y_{j}=\begin{cases}0\quad&\text{if
}\dim{\mathcal{V}_{1,j}^{H}}\text{ is even};\\\ 1\quad&\text{if
}\dim{\mathcal{V}_{1,j}^{H}}\text{ is odd and }|W(H)|=2;\\\ 2\quad&\text{if
}\dim{\mathcal{V}_{1,j}^{H}}\text{ is odd and }|W(H)|=1.\end{cases}$

equivalently

$\displaystyle y_{j}=\frac{x_{0}}{2}(1-(-1)^{\dim{\mathcal{V}_{1,j}^{H}}}),$

where

$\displaystyle x_{0}=\begin{cases}1\quad&\text{if }|W(H)|=2;\\\
2\quad&\text{if }|W(H)|=1.\end{cases}$

It follows that non-triviality of the coefficient associated with an orbit
type $({}^{m}H)\in\mathfrak{M}_{m,j}$ in the basic degree
$\deg_{\mathcal{V}_{m,j}}$ is characterized by the parity of
$\dim{\mathcal{V}_{1,j}^{H}}$ in the following way

$\displaystyle\operatorname{coeff}^{{}^{m}H}(\deg_{\mathcal{V}_{m,j}})\neq
0\iff 2\nmid\dim{\mathcal{V}_{1,j}^{H}}.$

We need one more result, this concerning the fate of orbit types belonging to
$\mathfrak{M}_{m}$ in the Burnside Ring product of basic degrees such as (20).
In particular, we find that for $m>0$ and $i,l\in\\{1,2,\ldots,r\\}$, the
coefficient of $(H)\in\mathfrak{M}_{m,i}\cap\mathfrak{M}_{m,l}$ is
$2$-nilpotent with respect to the Burnside Ring product
$\deg_{\mathcal{V}_{m,i}}\cdot\deg_{\mathcal{V}_{m,l}}\in A(G)$ in the case
that both $\dim\mathcal{V}_{m,i}^{H}$ and $\dim\mathcal{V}_{m,l}^{H}$ are odd.

######  Lemma 4.3.

Take $m>0$ and $i,l\in\\{1,2,\ldots,r\\}$. For
$(H)\in\mathfrak{M}_{1,i}\cap\mathfrak{M}_{1,l}$, one has

$\displaystyle\operatorname{coeff}^{{}^{m}H}(\deg_{\mathcal{V}_{m,i}}\cdot\deg_{\mathcal{V}_{m,l}})=\begin{cases}0\quad&\text{if
}\dim\mathcal{V}_{1,i}^{H}\text{ and }\dim\mathcal{V}_{1,l}^{H}\text{ are of
the same parity;}\\\ -x_{0}\quad&\text{else.}\end{cases}$

equivalently

$\displaystyle\operatorname{coeff}^{{}^{m}H}(\deg_{\mathcal{V}_{m,i}}\cdot\deg_{\mathcal{V}_{m,l}})=\frac{-x_{0}}{2}(1-(-1)^{\dim\mathcal{V}_{1,i}^{H}+\dim\mathcal{V}_{1,l}^{H}}).$

###### Proof.

Consider the Burnside Ring product of the relevant basic degrees

$\displaystyle\deg_{\mathcal{V}_{m,i}}\cdot\deg_{\mathcal{V}_{m,l}}$
$\displaystyle=\left((G)-y_{i}({}^{m}H)+a_{i}\right)\cdot\left((G)-y_{l}({}^{m}H)+a_{l}\right)$
$\displaystyle=(G)-(y_{i}+y_{l}-y_{i}y_{l}|W(H)|)(H)+a,$

where $a\in A(G)$ is such that $\operatorname{coeff}^{{}^{s}H}(a)=0$ for all
$s\in\\{1,2,\ldots\\}$, and put $y_{0}:=y_{i}+y_{l}-y_{i}y_{l}|W(H)|$. Now, if
$\dim\mathcal{V}_{1,i}^{H}$ and $\dim\mathcal{V}_{1,l}^{H}$ are both even,
then one has $y_{i}=y_{l}=0$ and the result follows. On the other hand, if
$\dim\mathcal{V}_{1,i}^{H}$ and $\dim\mathcal{V}_{1,l}^{H}$ are both odd, then
one has $y_{i}=y_{l}=x_{0}$ such that

$\displaystyle y_{0}=x_{0}(2-x_{0}|W(H)|),$

where in either of the cases, $x_{0}=2$ and $|W(H)|=1$ or $x_{0}=1$ and
$|W(H)|=2$, one has $2-x_{0}|W(H)|=0$. If instead one supposes that
$\dim\mathcal{V}_{1,i}^{H}$ and $\dim\mathcal{V}_{1,l}^{H}$ are of different
parities, then one has the two corresponding cases, $y_{i}=x_{0}$ and
$y_{l}=0$ or $y_{i}=0$ and $y_{l}=x_{0}$, both of which imply $y_{0}=x_{0}$.
$\square$ Naturally, the Lemma (4.3) can be generalized to an orbit type
$(H)\in\bigcap_{k=1}^{N}\mathfrak{M}_{1,j_{k}}$ and the Burnside Ring product
of the basic degrees $\\{\deg_{\mathcal{V}_{m,j_{k}}}\\}_{k=1}^{N}$ with
$\dim\mathcal{V}_{1,j_{k}}^{H}$ odd for an even number of
$j_{k}\in\\{1,2,\ldots,r\\}$.

######  Corollary 4.1.

Take $m>0,\;N\in\mathbb{N}$ and $j_{1},\ldots,j_{N}\in\\{1,2,\ldots,r\\}$. For
$(H)\in\bigcap_{k=1}^{N}\mathfrak{M}_{1,j_{k}}$, one has

$\displaystyle\operatorname{coeff}^{{}^{m}H}\left(\prod\limits_{k=1}^{N}\deg_{\mathcal{V}_{m,j_{k}}}\right)=\begin{cases}0\quad&\text{if
}|\\{j_{k}:2\nmid\dim\mathcal{V}_{1,j_{k}}^{H}\\}|\text{ is even};\\\
-x_{0}\quad&\text{otherwise}\end{cases}$

equivalently

$\displaystyle\operatorname{coeff}^{{}^{m}H}\left(\prod\limits_{k=1}^{N}\deg_{\mathcal{V}_{m,j_{k}}}\right)=\frac{-x_{0}}{2}\left(1-(-1)^{\sum\limits_{k=1}^{N}\dim\mathcal{V}_{1,j_{k}}^{H}}\right).$

Remarks 4.2 and 4.3 permit us to further refine our index set (19) as follows.
Let $(\alpha_{0},0)\in\Lambda$ be an isolated critical point with a deleted
regular neighborhood $\alpha_{0}^{-}<\alpha_{0}<\alpha_{0}^{+}$ on which
$\mathscr{A}(\alpha):\mathscr{H}\rightarrow\mathscr{H}$ is an isomorphism and
for a given $s\in\mathbb{N}$, $(H)\in\mathfrak{M}_{1}$ put

(25)
$\displaystyle\Sigma^{s}(\alpha_{0}^{\pm},H):=\\{(n,m,j)\in\Sigma(\alpha_{0}^{\pm}):\operatorname{coeff}^{{}^{s}H}(\deg_{\mathcal{V}_{m,j}})\neq
0\\},$ (26)
$\displaystyle\mathfrak{n}^{s}(\alpha_{0}^{\pm},H):=|\Sigma^{s}(\alpha_{0}^{\pm},H)|,$

and

(27)
$\displaystyle\mathfrak{m}^{s}(\alpha_{0}^{\pm},H):=|\\{(n,m,j)\in\Sigma(\alpha_{0}^{\pm}):({}^{s}H)<({}^{m}H)\text{
and }2\nmid\mathfrak{n}^{m}(\alpha_{0}^{\pm},H)\\}|.$

######  Remark 4.4.

Take $(H)\in\mathfrak{M}_{1}$, $(n,m,j)\in\Sigma$ and $s\in{\mathbb{N}}$. If
$\operatorname{coeff}^{{}^{s}H}(\deg_{\mathcal{V}_{m,j}})\neq 0$, then $m=s$.
On the other hand, if $({}^{s}H)\leq({}^{m}H)$, then $s\mid m$.

In particular, we are interested in keeping track of the numbers
$s\in{\mathbb{N}}$ for which the parities of
$\mathfrak{n}^{s}(\alpha_{0}^{\pm},H)$ disagree. With this in mind, put

(28) $\mathfrak{i}^{s}(\alpha_{0},H):=\begin{cases}-1\;&\text{ if
}\mathfrak{n}^{s}(\alpha_{0}^{-},H)\;\text{ is odd and
}\mathfrak{n}^{s}(\alpha_{0}^{+},H)\;\;\text{ is even}\\\ 1\;&\text{ if
}\mathfrak{n}^{s}(\alpha_{0}^{-},H)\;\text{ is even and
}\mathfrak{n}^{s}(\alpha_{0}^{+},H)\;\;\text{ is odd},\\\ 0\;&\;\text{
otherwise}.\end{cases}$

and also

(29)
$\displaystyle\mathfrak{s}(\alpha_{0},H):=\max\\{s\in{\mathbb{N}}:\mathfrak{i}^{s}(\alpha_{0},H)\neq
0\\}.$

######  Remark 4.5.

Notice that $\mathfrak{s}(\alpha_{0},H)$ is well defined and that the numbers
$\mathfrak{m}^{\mathfrak{s}(\alpha_{0},H)}(\alpha_{0}^{\pm},H)$ are of the
same parity for any isolated critical point $(\alpha_{0},0)\in\Lambda$ and
orbit type $(H)\in\mathfrak{M}_{1}$.

We are finally in a position to formulate our main local bifurcation result.

######  Theorem 4.1.

If $(\alpha_{0},0)\in\Lambda$ is an isolated critical point with a deleted
regular neighborhood $\alpha^{-}_{0}\leq\alpha_{0}\leq\alpha^{+}_{0}$ on which
the linear operator $\mathscr{A}(\alpha):\mathscr{H}\rightarrow\mathscr{H}$ is
an isomorphism and if there is an orbit type $(H)\in\mathfrak{M}_{1}$ and a
number $\mathfrak{s}:=\mathfrak{s}(\alpha_{0},H)\in{\mathbb{N}}$ such that
$\mathfrak{i}^{\mathfrak{s}}(\alpha_{0},H)\not=0$ and
$\mathfrak{i}^{m}(\alpha_{0},H)=0$ for all $m>\mathfrak{s}$, then

$\displaystyle\operatorname{coeff}^{{}^{s}H}(\omega_{G}(\alpha_{0}))=\begin{cases}(-1)^{{}^{\mathfrak{m}^{\mathfrak{s}}(\alpha^{-}_{0},H)}}\mathfrak{i}^{\mathfrak{s}}(\alpha_{0},H)x_{0}\quad&\text{
if }s=s_{0};\\\ 0\quad&\text{ if }s>s_{0}.\end{cases}$

###### Proof.

For convenience, take $\alpha^{*}_{0}\in\\{\alpha^{\pm}_{0}\\}$ and introduce
the notation

(30)
$\displaystyle\rho_{G}(\alpha^{*}_{0}):=\prod\limits_{(n,m,j)\in\Sigma(\alpha^{*}_{0})}\deg_{\mathcal{V}_{m,j}},$

such that the local bifurcation invariant (cf. 3.1) becomes

$\displaystyle\omega_{G}(\alpha_{0})=\rho_{G}(\alpha^{-}_{0})-\rho_{G}(\alpha^{+}_{0}).$

Also, put

$\displaystyle\Sigma^{0}(\alpha_{0}^{*},H):=\\{(n,m,j)\in\Sigma(\alpha_{0}^{*}):\forall_{s^{\prime}\in\mathbb{N}}\;\operatorname{coeff}^{{}^{s^{\prime}}H}(\deg_{\mathcal{V}_{m,j}})=0\\},$

and, for each $s=0,1,\ldots$ define

(31)
$\displaystyle\rho^{s}_{G}(\alpha^{*}_{0},H):=\prod\limits_{(n,m,j)\in\Sigma^{s}(\alpha^{*}_{0},H)}\deg_{\mathcal{V}_{m,j}}.$

Clearly the index set (19) has the partition

$\displaystyle\Sigma(\alpha_{0}^{*})=\bigcup\limits_{s\in\mathbb{N}\cup\\{0\\}}\Sigma^{s}(\alpha_{0}^{*},H),$

such that the product of basic degrees (30) becomes

$\displaystyle\rho_{G}(\alpha^{*}_{0})=\prod\limits_{s\in\mathbb{N}\cup\\{0\\}}\rho^{s}_{G}(\alpha^{*}_{0},H).$

It follows, from Lemma (4.3) and its Corollary (4.1), that each of (31) is of
the form

$\displaystyle\rho^{s}_{G}(\alpha^{*}_{0},H)=\begin{cases}(G)+b_{0}\quad&\text{
if }s=0;\\\ (G)-y_{s}(\alpha^{*}_{0})({}^{s}H)+b_{s}\quad&\text{ if }s\geq
1,\end{cases}$

where $b_{0},b_{s}\in A(G)$ are such that
$\operatorname{coeff}^{{}^{s^{\prime}}H}(b_{0})=\operatorname{coeff}^{{}^{s^{\prime}}H}(b_{s})=0$
for all $s^{\prime}\in\mathbb{N}$ and where the coefficients
$y_{s}(\alpha^{*}_{0})\in\mathbb{Z}$ are determined by the rule

$\displaystyle y_{s}(\alpha^{*}_{0})=\begin{cases}0\quad&\text{ if
}\mathfrak{n}^{s}(\alpha^{*}_{0},H)\text{ is even};\\\ x_{0}\quad&\text{ if
}\mathfrak{n}^{s}(\alpha^{*}_{0},H)\text{ is odd},\end{cases}$

equivalently

$\displaystyle
y_{s}(\alpha^{*}_{0})=\frac{x_{0}}{2}(1-(-1)^{\mathfrak{n}^{s}(\alpha^{*}_{0},H)}).$

Now, since $\mathfrak{n}^{s}(\alpha^{\pm}_{0},H)$ are of the same parity for
all $s>\mathfrak{s}$, one has

$\displaystyle\rho_{G}^{s}(\alpha^{-}_{0})\cdot\rho_{G}^{s}(\alpha^{+}_{0})=(G)+c_{s},$

where $c_{s}\in A(G)$ is such that
$\operatorname{coeff}^{{}^{s^{\prime}}H}(c_{s})=0$ for all
$s^{\prime}\in\mathbb{N}$. Therefore, the local bifurcation invariant can be
expressed in terms of the quantities (31) as follows

$\displaystyle\omega_{G}(\alpha_{0})=\prod\limits_{s>\mathfrak{s}}\rho_{G}^{s}(\alpha^{-}_{0},H)\cdot\left(\rho_{G}^{\mathfrak{s}}(\alpha^{-}_{0},H)-\rho_{G}^{\mathfrak{s}}(\alpha^{+}_{0},H)+\bm{\beta}\right)$

where $\bm{\beta}\in A(G)$ is such that
$\operatorname{coeff}^{{}^{s^{\prime}}H}(\bm{\beta})=0$ for all
$s^{\prime}\geq\mathfrak{s}$ . To complete the proof of Theorem 4.1, we will
need the following Lemma.

######  Lemma 4.4.

Take $s,s_{0}\in\mathbb{N}$. Using the notation (31), one has

$\displaystyle\operatorname{coeff}^{{}^{s_{0}}H}(\rho_{G}^{s}(\alpha^{*}_{0},H)\cdot\pm
x_{0}({}^{s_{0}}H))=\begin{cases}\mp x_{0}\quad&\text{ if
}({}^{s_{0}}H)\leq({}^{s}H)\text{ and
}\mathfrak{n}^{s}(\alpha^{*}_{0},H)\text{ is odd};\\\ \pm x_{0}\quad&\text{
otherwise}.\end{cases}$

###### Proof.

Indeed, consider the relevant Burnside Ring product

$\displaystyle\rho_{G}^{s}(\alpha^{*}_{0},H)\cdot\pm x_{0}({}^{s_{0}}H)$
$\displaystyle=\left((G)-y_{s}(\alpha^{*}_{0})({}^{s}H)+b_{s}\right)\cdot\left(\pm
x_{0}({}^{s_{0}}H)\right)$ (32) $\displaystyle=\pm x_{0}({}^{s_{0}}H)\mp
y_{s}(\alpha^{*})x_{0}({}^{s}H)\cdot({}^{s_{0}}H)+\bm{\alpha}$

where $\bm{\alpha}\in A(G)$ is such that
$\operatorname{coeff}^{{}^{s^{\prime}}H}(\bm{\alpha})=0$ for all
$s^{\prime}\in\mathbb{N}$. Now, if $\mathfrak{n}^{s}(\alpha^{*}_{0},H)$ is
even, then $y_{s}(\alpha^{*}_{0})=0$ and the result follows. Suppose instead
that $2\nmid\mathfrak{n}^{s}(\alpha^{*}_{0},H)$. Then (4.1) becomes

$\displaystyle\rho_{G}^{s}(\alpha^{*}_{0},H)\cdot\pm x_{0}({}^{s_{0}}H)=\pm
x_{0}(1-x_{0}d_{0})({}^{s_{0}}H)+\bm{\gamma}$

where $\bm{\gamma}\in A(G)$ is such that
$\operatorname{coeff}^{{}^{s^{\prime}}H}(\bm{\gamma})=0$ for all
$s^{\prime}\in\mathbb{N}$ and
$d_{0}:=\operatorname{coeff}^{{}^{s_{0}}H}(({}^{s}H)\cdot({}^{s_{0}}H))$ is
given by the recursive formula (cf. Appendix C) as follows

$\displaystyle
d_{0}:=\frac{n({}^{s_{0}}H,{}^{s}H)n({}^{s_{0}}H,{}^{s_{0}}H)|W(H)|^{2}}{|W(H)|}.$

Now, if $({}^{s_{0}}H)\nleq({}^{s}H)$, then $n({}^{s_{0}}H,{}^{s}H)=0$ and the
result follows. In the case that $({}^{s_{0}}H)\leq({}^{s}H)$, one has
$n({}^{s_{0}}H,{}^{s}H)=n({}^{s_{0}}H,{}^{s_{0}}H)=1$ such that $d_{0}=|W(H)|$
and the result follows from the fact that $x_{0}|W(H)|=2$. $\square$
Completion of the proof of Theorem 4.1 The result follows from Lemma 4.4
together with the observation that

$\displaystyle\operatorname{coeff}^{{}^{s}H}\left(\rho_{G}^{\mathfrak{s}}(\alpha^{-}_{0},H)-\rho_{G}^{\mathfrak{s}}(\alpha^{+}_{0},H)+\bm{\beta}\right)=\begin{cases}\mathfrak{i}^{\mathfrak{s}}(\alpha_{0},H)x_{0}\quad&\text{
if }s=\mathfrak{s};\\\ 0\quad&\text{ if }s>\mathfrak{s}.\end{cases}$

$\square$

######  Corollary 4.2.

Under the assumptions ($A_{1}$)-($A_{5}$), if $(\alpha_{0},0)\in\Lambda$ is an
isolated critical point with a deleted neighborhood
$[\alpha^{-},\alpha^{+}]\setminus\\{\alpha_{0}\\}$ on which the linear
operator $\mathscr{A}(\alpha):\mathscr{H}\rightarrow\mathscr{H}$ is an
isomorphism and if there is an orbit type $(H)\in\mathfrak{M}_{1}$ and a
number $\mathfrak{s}:=\mathfrak{s}(\alpha_{0},H)\in{\mathbb{N}}$ such that
$\mathfrak{i}^{\mathfrak{s}}(\alpha_{0},H)\not=0$ and
$\mathfrak{i}^{m}(\alpha_{0},H)=0$ for all $m>\mathfrak{s}$, then system (8)
admits a branch of non-radial solutions $\mathscr{C}$ with branching point
$(\alpha_{0},0)$ and with $(G_{u})\geq(^{\mathfrak{s}}H)$ for all
$u\in\mathscr{C}$.

###  4.2. Resolution of the Rabinowitz Alternative

Without an appropriate fixed point reduction of the bifurcation problem (8),
one is unable to guarantee that a branch of non-trivial solutions
$\mathcal{C}$ to (1) bifurcating from a given isolated critical point
$(\alpha_{0},0)\in\mathbb{R}\times\mathscr{H}$ whose existence has been
established using a Krasnosel’skii type result (e.g. by Theorem 4.1) is not
comprised of radial solutions. With this in mind, consider the subgroup
$\bm{K}:=\\{(-1,e,-1),(1,e,1)\\}\leq O(2)\times\Gamma\times{\mathbb{Z}}_{2}$
and denote by
$\mathscr{F}^{\bm{K}}:\mathbb{R}\times\mathscr{H}^{\bm{K}}\to\mathscr{H}^{\bm{K}}$
the restriction

(33)
$\mathscr{F}^{\bm{K}}:=\mathscr{F}|_{\mathbb{R}\times\mathscr{H}^{\bm{K}}},$

of the operator (7) to the $\bm{K}$-fixed point space $\mathscr{H}^{\bm{K}}$.
Clearly, any solution $(\alpha,u)\in\mathbb{R}\times\mathscr{H}^{\bm{K}}$ to
the equation

(34) $\mathscr{F}^{\bm{K}}(\alpha,u)=0,$

is also solution to (8). Notice also that any radial solution to (34) belongs
to the set of trivial solutions

$M^{\bm{K}}:=\\{(\alpha,0):\alpha\in\mathbb{R}\times\mathscr{H}^{\bm{K}}\\}.$

Therefore, any branch of of non-trivial solutions to the bifurcation problem
(34) consists solely of non-radial solutions. In this $\bm{K}$-fixed point
setting, we adapt each of the notions introduced in Section 3.2 used to
describe the Rabinowitz alternative for the equation (8) to the bifurcation
map (33). To begin, notice that the $\bm{K}$ fixed point space
$\mathscr{H}^{\bm{K}}$ is an isometric Hilbert representation of the group
$\mathcal{G}:=N(\bm{K})/\bm{K}=O(2)\times\Gamma$ with the
$\mathcal{G}$-isotypic decomposition

$\displaystyle\mathscr{H}^{\bm{K}}=\bigoplus_{j=1}^{r}\overline{\bigoplus\limits_{m=1}^{\infty}\mathscr{H}_{2m-1,j}},\quad\mathscr{H}_{2m-1,j}:=\overline{\bigoplus\limits_{n=1}^{\infty}\mathscr{E}_{n,2m-1}^{j}}.$

We denote by $\mathscr{S}^{\bm{K}}$ the set of non-trivial solutions to (34),
i.e.

$\mathscr{S}^{\bm{K}}:=\\{(\alpha,u)\in\mathbb{R}\times\mathscr{H}^{\bm{K}}:\mathscr{F}^{\bm{K}}(\alpha,0)=0\;\;\text{
and }\;\;u\not=0\\},$

and by
$\mathscr{A}^{\bm{K}}:\mathbb{R}\times\mathscr{H}^{\bm{K}}\to\mathscr{H}^{\bm{K}}$
the restriction of the operator
$\mathscr{A}:\mathbb{R}\times\mathscr{H}\rightarrow\mathscr{H}$ to
$\mathscr{H}^{\bm{K}}$, i.e. for each $\alpha\in\mathbb{R}$

(35)
$\mathscr{A}^{\bm{K}}(\alpha):=\mathscr{A}(\alpha)|_{\mathscr{H}^{\bm{K}}}:\mathscr{H}^{\bm{K}}\to\mathscr{H}^{\bm{K}}.$

As before, the critical set of (34), now denoted $\Lambda^{\bm{K}}$, is the
set of trivial solutions $(\alpha_{0},0)\in M^{\bm{K}}$ for which
$\mathscr{A}^{\bm{K}}(\alpha_{0})$ is not an isomorphism

$\Lambda^{\bm{K}}:=\\{(\alpha,0)\in\mathbb{R}\times\mathscr{H}^{\bm{K}}:\mathscr{A}^{\bm{K}}(\alpha):\mathscr{H}^{\bm{K}}\to\mathscr{H}^{\bm{K}}\;\;\text{
is not an isomorphism}\\}.$

Next, we describe the spectrum of (35) in terms of the spectra
$\sigma(\mathscr{A}^{j}_{n,m}(\alpha))$ (cf. Lemma 4.1) as follows

$\sigma(\mathscr{A}^{\bm{K}}(\alpha))=\bigcup_{m=1}^{\infty}\bigcup_{n=1}^{\infty}\bigcup_{j=1}^{r}\sigma(\mathscr{A}_{n,2m-1}^{j}(\alpha)).$

Assume that for a given $\alpha\in\mathbb{R}$, the operator
$\mathscr{A}^{\bm{K}}(\alpha):\mathscr{H}^{\bm{K}}\rightarrow\mathscr{H}^{\bm{K}}$
is an isomorphism. If we refine the index set (19) to include only those
indices $(n,m,j)\in\Sigma(\alpha)$ relevant to the $\bm{K}$-fixed point
setting with the notation

$\Sigma^{\bm{K}}(\alpha):=\\{(n,m,j)\in\Sigma(\alpha):2\nmid m\\},$

then the $\mathcal{G}$-equivariant degree
$\mathcal{G}\text{-deg}(\mathscr{A}^{\bm{K}}(\alpha),B(\mathscr{H}^{\bm{K}}))$
can be computed as follows

$\displaystyle\mathcal{G}\text{-deg}(\mathscr{A}^{\bm{K}}(\alpha),B(\mathscr{H}^{\bm{K}}))=\prod\limits_{(n,m,j)\in\Sigma^{\bm{K}}(\alpha)}\widetilde{\deg}_{\mathcal{V}_{j,m}},$

where, to distinguish between $G$-basic degrees and $\mathcal{G}$-basic
degrees, we have introduced the notation

$\widetilde{\deg}_{\mathcal{V}_{j,m}}:=\mathcal{G}\text{-deg}(-\operatorname{Id},B(\mathcal{V}_{j,m})).$

At this point, Lemma 4.2 can be reformulated for the map
$\mathscr{F}^{\bm{K}}$ as follows:

######  Lemma 4.5.

Under the assumptions ($A_{0}$)–($A_{5}$) and for any isolated critical point
$(\alpha_{0},0)\in\Lambda^{\bm{K}}$ with deleted regular neighborhood
$\alpha_{0}^{-}<\alpha_{0}<\alpha_{0}^{+}$, the local bifurcation invariant
$\omega_{\mathcal{G}}(\alpha_{0}):=\mathcal{G}\text{-deg}(\mathcal{A}^{\bm{K}}(\alpha_{0}^{-}),B(\mathscr{H}^{\bm{K}}))-\mathcal{G}\text{-deg}(\mathcal{A}^{\bm{K}}(\alpha_{0}^{+}),B(\mathscr{H}^{\bm{K}}))$
is given by

(36)
$\displaystyle\omega_{\mathcal{G}}(\alpha_{0})=\prod\limits_{(n,m,j)\in\Sigma^{\bm{K}}(\alpha^{-})}\widetilde{\deg}_{\mathcal{V}_{m,j}}-\prod\limits_{(n,m,j)\in\Sigma^{\bm{K}}(\alpha^{+})}\widetilde{\deg}_{\mathcal{V}_{m,j}}.$

Likewise, Theorem 3.2 becomes:

######  Theorem 4.2.

(Rabinowitz’ Alternative) Under the assumptions ($A_{0}$)–($A_{5}$), let
$\mathcal{U}\subset\mathbb{R}\times\mathscr{H}^{\bm{K}}$ be an open bounded
$\mathcal{G}$-invariant set with
$\partial\mathcal{U}\cap\Lambda^{\bm{K}}=\emptyset$ and
$(\alpha_{0},0)\in\mathcal{U}\cap\Lambda^{\bm{K}}$. If $\mathcal{C}$ is a
branch of nontrivial solutions to (34) bifurcating from the critical point
$(\alpha_{0},0)$, then one has the following alternative:

  1. $(a)$

either $\mathcal{C}\cap\partial\mathcal{U}\neq\emptyset$;

  2. $(b)$

or there exists a finite set

$\displaystyle\mathcal{C}\cap\Lambda^{\bm{K}}=\\{(\alpha_{0},0),(\alpha_{1},0),\ldots,(\alpha_{n},0)\\},$

satisfying the following relation

$\displaystyle\sum\limits_{k=1}^{n}\omega_{\mathcal{G}}(\alpha_{k})=0.$

To further simplify our exposition, we replace assumption ($A_{0}$) with:

  1. ($\tilde{A}_{0}$)

For each $j\in\\{1,2,\ldots,r\\}$ there exists a continuous and bounded map
$\mu_{j}:\mathbb{R}\rightarrow\mathbb{R}$ with

$A_{j}(\alpha):=A(\alpha)|_{V_{j}}=\mu_{j}(\alpha)\operatorname{Id}|_{V_{j}}.$

Take $(H)\in\mathfrak{M}_{1}$ and $(\alpha_{0},0)\in\Lambda^{\bm{K}}$. Under
assumption ($\tilde{A}_{0}$), there is some finite $m^{\prime}=0,1,\ldots$ for
which $m>m^{\prime}$ implies $(n,m,j)\notin\Sigma^{\bm{K}}(\alpha_{0})$.
Clearly, the quantity

(37)
$\displaystyle\bar{\mathfrak{s}}(H):=\max\limits_{(\alpha,0)\in\Lambda^{\bm{K}}}\\{\mathfrak{s}(\alpha,H)\\}$

and the set

(38)
$\displaystyle\mathfrak{J}(H):=\\{(\alpha,0)\in\Lambda^{\bm{K}}:\mathfrak{s}(\alpha,H)=\bar{\mathfrak{s}}(H)\\}.$

are well-defined. The elements of $\mathfrak{J}(H)$ can always be indexed
$(\alpha_{1},0),(\alpha_{2},0),\ldots\in\mathfrak{J}(H)$ in such a way that,
if $i<j$, then $\alpha_{i}<\alpha_{j}$. We are now in a position to formulate
our main global bifurcation result.

######  Theorem 4.3.

If there is an orbit type $(H)\in\mathfrak{M}_{1}$ with
$\bar{\mathfrak{s}}:=\bar{\mathfrak{s}}(H)$ and for which
$2\nmid|\mathfrak{J}(H)|$, then the system (1) admits an unbounded branch
$\mathscr{C}$ of non-radial solutions with symmetries at least
$({}^{\bar{\mathfrak{s}}}H)$. Moreover, we have the following alternative:
there exists $M>0$ such that, either

  * (a)

for all $\alpha>M$ with $(\alpha,0)\notin\Lambda^{\bm{K}}$ one has
$\mathscr{C}\cap\\{\alpha\\}\times\mathscr{H}\not=\emptyset$, or

  * (b)

for all $\alpha>-M$ with $(\alpha,0)\notin\Lambda^{\bm{K}}$ one has
$\mathscr{C}\cap\\{\alpha\\}\times\mathscr{H}\not=\emptyset$.

######  Remark 4.6.

Conditions (a) and (b) in Theorem 4.3 guarantee that the branch $\mathscr{C}$
of non-radial solutions extends indefinitely either into the direction of
increasing $\alpha\to\infty$ or into the decreasing $\alpha\to-\infty$.

###### Proof.

Notice that, for any critical point $(\alpha_{i},0)\in\mathfrak{J}(H)$, the
numbers $\mathfrak{m}^{\bar{\mathfrak{s}}}(\alpha_{i}^{\pm},H)$ have the same
parity (cf. Remark 4.5). Consider any other critical point
$(\alpha_{j},0)\in\mathfrak{J}(H)$ with $\alpha_{i}<\alpha_{j}$. We will show
that $\mathfrak{m}^{\bar{\mathfrak{s}}}(\alpha_{i}^{+},H)$ and
$\mathfrak{m}^{\bar{\mathfrak{s}}}(\alpha_{j}^{-},H)$ are also of the same
parity. Indeed, suppose for contradiction that
$\mathfrak{m}^{\bar{\mathfrak{s}}}(\alpha_{i}^{+},H)$ and
$\mathfrak{m}^{\bar{\mathfrak{s}}}(\alpha_{j}^{-},H)$ have different parities.
Then there is a folding $m>\bar{\mathfrak{s}}$ such that the sets
$\Sigma^{m}(\alpha_{i}^{+},H)$, $\Sigma^{m}(\alpha_{j}^{-},H)$ disagree for
some odd numbers of indices (equivalently, the numbers
$\mathfrak{n}^{m}(\alpha_{i}^{+},H)$, $\mathfrak{n}^{m}(\alpha_{j}^{-},H)$
have different parity). Consequently, there must be an intermediate critical
point $(\alpha_{k},0)\in\mathfrak{J}(H)$ with
$\alpha_{i}<\alpha_{k}<\alpha_{j}$ and $\mathfrak{i}^{m}(\alpha_{k},H)\neq 0$.
This is in contradiction with the assumption of maximality for
$\bar{\mathfrak{s}}$. It follows that the quantity
$(-1)^{\mathfrak{m}^{\bar{\mathfrak{s}}}(\alpha_{*}^{-},H)}$ is constant for
all critical points $(\alpha_{*},0)\in\mathfrak{J}(H)$. On the other hand, let
$(\alpha_{k},0),(\alpha_{k+1},0)\in\mathfrak{J}(H)$ be any two consecutive
critical points. From the definition of (38), the numbers
$\mathfrak{i}^{\bar{\mathfrak{s}}}(\alpha_{k},H),\mathfrak{i}^{\bar{\mathfrak{s}}}(\alpha_{k+1},H)$
must be non-zero. With an argument similar to that the one used above, it can
be shown that the numbers
$\mathfrak{n}^{\bar{\mathfrak{s}}}(\alpha_{k}^{+},H)$ and
$\mathfrak{n}^{\bar{\mathfrak{s}}}(\alpha_{k+1}^{-},H)$ have the same parity,
from which it follows that

$\displaystyle\mathfrak{i}^{\bar{\mathfrak{s}}}(\alpha_{k},H)\mathfrak{i}^{\bar{\mathfrak{s}}}(\alpha_{k+1},H)=-1.$

It follows then, from Theorem 4.1, that
$\operatorname{\operatorname{coeff}^{{}^{\bar{\mathfrak{s}}}H}(\omega_{G}(\alpha_{k}))}=-\operatorname{\operatorname{coeff}^{{}^{\bar{\mathfrak{s}}}H}(\omega_{G}(\alpha_{k+1}))}$
and also, for any critical point
$(\alpha_{0},0)\in\Lambda^{\bm{K}}\setminus\mathfrak{J}(H)$, that
$\operatorname{\operatorname{coeff}^{{}^{\bar{\mathfrak{s}}}H}(\omega_{G}(\alpha_{0}))}=0$.
Therefore, if $2\nmid|\mathfrak{J}(H)|$, there exists an unbounded branch
$\mathscr{C}$ of non-radial solutions bifurcating from each critical point in
$\mathfrak{J}(H)$ with symmetries at least $(^{\bar{\mathfrak{s}}}H)$.
Moreover, by Lemma 2.2, the branch $\mathscr{C}$ cannot extend to infinity
with respect to the magnitude of vectors $u$ belonging to $\mathscr{C}$ (for
any particular $\alpha$), the only option is that the branch $\mathscr{C}$
extends to infinity with respect to $\alpha$. $\square$

##  5\. Motivating Example: Vibrating Membranes with Symmetric Coupling

Equations involving the Laplacian operator are sometimes used to describe
time-invariant wave or diffusion processes, also called steady-state
phenomena. As a preliminary model for steady-state phenomena, consider the
Helmholtz equation

(39) $\begin{cases}-\Delta u=\lambda^{2}u,\\\ u|_{\partial D}=0,\end{cases}$

defined on the planar unit disc $D:=\\{z\in{\mathbb{C}}:|z|<1\\}$. Solutions
of (39) are called normal modes and take the form

$\displaystyle
u_{nm}(r,\theta)=J_{m}(\sqrt{s_{nm}}r)(A\cos(m\theta)+B\sin(m\theta)),\quad
A,B\in\mathbb{R},\;n\in{\mathbb{N}},\;m=0,1,\ldots$

where each $\sqrt{s_{nm}}$ can be calculated as the $n$-th root of the $m$-th
Bessel function of the first kind $J_{m}$. The classical Helmholtz equation,
as is well-known, has limited applicability to real-world problems, which are
often inherently nonlinear. For example, in order to study vibrations of a
membrane, it is standard to perturb the system (39) with a nonlinearity
$r:\overline{D}\times\mathbb{R}\to\mathbb{R}$ as follows

(40) $\begin{cases}-\Delta u=\lambda^{2}u+r(z,u),\\\ u|_{\partial
D}=0.\end{cases}$

Figure 1. An octahedral configuration of six coupled membranes.

Of course, the single membrane model (40) is still too simplistic to model
many physically integrated systems. A more realistic problem might instead
involve a configuration of some number of coupled oscillating membranes.
Consider, for example, a collection of six membranes arranged on corresponding
faces of a cube, each coupled to its adjacent neighbors and modeled by a
system of identical non-linear Helmholtz equations

(41) $\begin{cases}-\Delta u=f(\alpha,z,u),\quad u\in V\\\ u|_{\partial
D}=0,\end{cases}$

where $V=\mathbb{R}^{6}$ and $f:\mathbb{R}\times\overline{D}\times
V\rightarrow V$ is the vector-valued function

(42)
$f(\alpha,z,u)=\left[\begin{array}[]{c}f_{1}(\alpha,z,u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})\\\
f_{2}(\alpha,z,u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})\\\
f_{3}(\alpha,z,u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})\\\
f_{4}(\alpha,z,u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})\\\
f_{5}(\alpha,z,u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})\\\
f_{6}(\alpha,z,u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})\end{array}\right],\quad
u=\left[\begin{array}[]{c}u_{1}\\\ u_{2}\\\ u_{3}\\\ u_{4}\\\ u_{5}\\\
u_{6}\end{array}\right].$

satisfying the following assumptions:

  1. ($E_{1}$)

For all $\alpha\in\mathbb{R}$, $z\in D$,
$u=(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})^{T}\in\mathbb{R}^{6}$ and
$\tau\in\bm{O}$

$\displaystyle\tau
f(\alpha,z,u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})=f(\alpha,z,u_{\tau(1)},u_{\tau(2)},u_{\tau(3)},u_{\tau(4)},u_{\tau(5)},u_{\tau(6)});$

  2. ($E_{2}$)

For all $\alpha\in\mathbb{R}$, $z\in D$,
$u=(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})^{T}\in\mathbb{R}^{6}$ and
$e^{i\theta}\in O(2)$

$\displaystyle
f(\alpha,e^{i\theta}z,u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})=f(\alpha,z,u_{1},u_{2},u_{3},u_{4},u_{5},u_{6}).$

  3. ($E_{3}$)

For all $\alpha\in\mathbb{R}$, $z\in D$,
$u=(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})^{T}\in\mathbb{R}^{6}$,

$\displaystyle
f(\alpha,z,-u_{1},-u_{2},-u_{3},-u_{4},-u_{5},-u_{6})=-f(\alpha,z,u_{1},u_{2},u_{3},u_{4},u_{5},u_{6}).$

Here, $\bm{O}\leq S_{6}$ is used to denote the chiral symmetry group of the
cube, whose action on $V$ is described by the set of orientation preserving
permutations of the corresponding membranes. If we identify $S_{4}$ with
$\bm{O}$ according to the rule

$\displaystyle(1,2)\;\;\;\leftrightarrow\;\;(1,4)(2,3)(5,6),\quad(1,2,3)\;\;\leftrightarrow\;\;(1,4,6)(3,5,2),$
$\displaystyle(1,2)(3,4)\;\;\leftrightarrow\;\;(1,4)(2,3),\quad(1,2,3,4)\;\;\leftrightarrow\;\;(1,2,3,4),$

then the faces of the cube (see Figure LABEL:fig:6membranes) are naturally
permuted as follows

(43) $\displaystyle\forall_{\sigma\in S_{4}\leq
S_{6}}\;\;\sigma(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})^{T}=(u_{\sigma(1)},u_{\sigma(2)},u_{\sigma(3)},u_{\sigma(4)},u_{\sigma(5)},u_{\sigma(6)})^{T},\quad
u\in V.$

In this way, $V$ is an orthogonal $S_{4}$-representation with respect to the
action (43). The table of characters for $S_{4}$

con. classes | $(1)$ | $(1,2)$ | $(1,2)(3,4)$ | $(1,2,3)$ | $(1,2,3,4)$
---|---|---|---|---|---
$\chi_{0}$ | $1$ | $1$ | $1$ | $1$ | $1$
$\chi_{1}$ | $1$ | $-1$ | $1$ | $1$ | $-1$
$\chi_{2}$ | $2$ | $0$ | $2$ | $-1$ | $0$
$\chi_{3}$ | $3$ | $-1$ | $-1$ | $0$ | $1$
$\chi_{4}$ | $3$ | $1$ | $-1$ | $0$ | $-1$
$\chi_{V}$ | $6$ | $0$ | $2$ | $0$ | $2$
Table 1. Character Table for $S_{4}$.

reveals the relation

$\chi_{V}=\chi_{0}+\chi_{2}+\chi_{3},$

which can be used to obtain the following
$S_{4}\times{\mathbb{Z}}_{2}$-isotypic decomposition of $V$

$\displaystyle
V=\mathcal{V}_{0}^{-}\oplus\mathcal{V}^{-}_{2}\oplus\mathcal{V}^{-}_{3}.$

Under the assumptions ($E_{1}$)-($E_{3}$), the function $f$ clearly satisfies
conditions ($A_{1}$)-($A_{3}$). For simplicity, let us also assume that $f$
satisfies the conditions ($A_{4}$) and ($A_{5}$) with the matrix
$A(\alpha):=D_{u}f(0):V\rightarrow V$ given by

(44) $A(\alpha)=aI+\zeta(\alpha)C,$

where $I:V\rightarrow V$ is the identity matrix, $C:V\rightarrow V$ is the
weighted adjacency matrix

(45) $C=\left[\begin{array}[c]{cccccc}c&d&0&d&d&d\\\ d&c&d&0&d&d\\\
0&d&c&d&d&d\\\ d&0&d&c&d&d\\\ d&d&d&d&c&0\\\ d&d&d&d&0&c\\\
\end{array}\right],$

defined for some fixed $c,d\in\mathbb{R}$ satisfying $c>0$, $d<0$, $4d+c\geq
0$ and where $\zeta:\mathbb{R}\rightarrow\mathbb{R}$ is the sigmoid function

(46) $\displaystyle\zeta(\alpha)=\frac{1}{1+e^{-\alpha}}.$

$\alpha$$\zeta(\alpha)=\frac{1}{1+e^{-\alpha}}$$0.5$$0$$1.0$$0$ Figure 2.
Graph of the sigmoid function $\zeta(\alpha)$.

######  Remark 5.1.

The sigmoid function introduces a nonlinear dependence of the coupling in (41)
on the bifurcation parameter $\alpha\in\mathbb{R}$. Moreover, the asymptotic
behaviour of (46) imposes coupling strength saturation limits which might
reflect physical constraints such as, for example, bounded ranges for certain
material densities.

With the linearization (44), system (41) clearly satisfies the condition
($\tilde{A}_{0}$). Indeed, it can be shown that $A(\alpha):V\rightarrow V$
admits three eigenvalues with eigenspaces corresponding to the
$S_{4}$-isotypic components as follows

(47) $\displaystyle\mu_{0}(\alpha)$ $\displaystyle:=a+\zeta(\alpha)(c+4d),$
$\displaystyle E(\mu_{0}(\alpha))=\mathcal{V}_{0}^{-},$
$\displaystyle\mu_{2}(\alpha)$ $\displaystyle:=a+\zeta(\alpha)(c-2d),$
$\displaystyle E(\mu_{2}(\alpha))=\mathcal{V}_{2}^{-},$
$\displaystyle\mu_{3}(\alpha)$ $\displaystyle:=a+\zeta(\alpha)c,$
$\displaystyle E(\mu_{3}(\alpha))=\mathcal{V}_{3}^{-}.$

If $c,d\in\mathbb{R}$ are such that $c+4d,c-2d,c\neq 0$, then (42) satisfies
condition ($B_{3}$). Moreover, together with the strict monotonicity of the
eigenvalues (47) and the distinctness of the numbers $s_{nm}$, it follows that
each critical point of (41) can be specified by a unique triple
$(n,m,j)\in\Sigma$ with the notation

(48) $\displaystyle\alpha_{n,m,j}:=\mu_{j}^{-1}(s_{nm}).$

Let us make the system (41) more specific by choosing the following values for
the constants $a,c,d\in\mathbb{R}$

(49) $\displaystyle a=32,\;c=5,\;d=-1.$

For convenience, we include graphs of the three eigenvalues (47) with the
assignments (49) in Figure 3 and the numbers $s_{nm}$ for $m=0,\ldots,10$ and
$n=1,\ldots,9$ in Table 2.

|  $n=1$ |  $n=2$ |  $n=3$ |  $n=4$ |  $n=5$ |  $n=6$ |  $n=7$ |  $n=8$ |  $n=9$
---|---|---|---|---|---|---|---|---|---
m=0 |  5.783 |  30.471 |  74.887 |  139.04 |  222.932 |  326.563 |  449.934 |  593.043 |  755.891
m=1 |  14.682 |  49.218 |  103.499 |  177.521 |  271.282 |  384.782 |  518.021 |  671 |  843.718
m=2 |  26.374 |  70.85 |  135.021 |  218.92 |  322.555 |  445.928 |  589.038 |  751.888 |  934.476
m=3 |  40.706 |  95.278 |  169.395 |  263.201 |  376.625 |  509.98 |  662.968 |  835.693 |  1028.15
m=4 |  57.583 |  122.428 |  206.57 |  310.322 |  433.761 |  576.913 |  739.79 |  922.398 |  1124.74
m=5 |  76.939 |  152.241 |  246.495 |  360.245 |  493.631 |  646.702 |  819.483 |  1011.99 |  1224.21
m=6 |  98.726 |  184.67 |  289.13 |  412.934 |  556.303 |  719.321 |  902.024 |  1104.44 |  1326.56
m=7 |  122.908 |  219.67 |  334.436 |  468.356 |  621.751 |  794.743 |  987.392 |  1199.73 |  1431.77
m=8 |  149.453 |  257.21 |  382.38 |  526.481 |  689.946 |  872.946 |  1075.56 |  1297.84 |  1539.81
m=9 |  178.337 |  297.26 |  432.933 |  587.281 |  760.863 |  953.907 |  1166.52 |  1398.77 |  1650.68
m=10 |  209.54 |  339.793 |  486.07 |  650.732 |  834.48 |  1037.6 |  1260.24 |  1502.48 |  1764.35

Table 2. Squared Zeros of Bessel functions $s_{nm}$ for $0\leq m\leq 10$,
$1\leq n\leq 9$.
$\mu_{0}(\alpha)$$\mu_{2}(\alpha)$$\mu_{3}(\alpha)$$s_{14}=57.583$$s_{13}=40.706$$s_{21}=49.218$$s_{20}=30.471$
Figure 3. Graph of the eigenvalues $\mu_{0}(\alpha),\mu_{2}(\alpha)$ and
$\mu_{3}(\alpha)$ and the consecutive numbers $s_{20}<s_{13}<s_{21}<s_{14}$.

Notice that the critical set for (41) consists of exactly five critical points
that can be expressed, using the notation (48), as follows

(50)
$\displaystyle\Lambda=\\{(\alpha_{1,3,0},0),(\alpha_{1,3,2},0),(\alpha_{1,3,3},0),(\alpha_{2,1,2},0),(\alpha_{2,1,3},0)\\}.$

In order to employ the results presented in previous sections to the
bifurcation problem (41), we must first identify the maximal orbit types in
$\Phi_{0}(G;\mathscr{H}_{1}\setminus\\{0\\})$. The following G.A.P code can be
used to generate a complete list of the sets $\mathfrak{M}_{1,j}$ and the
corresponding Basic Degrees $\deg_{\mathcal{V}_{1,j}}$ for $j=1,2,3$.

⬇

1# A G.A.P Program for the computation of Maximal Orbit Types and Basic
Degrees associated with the G-isotypic decomposition V = V_0 \times V_2 \times
V_3

2

3LoadPackage ("EquiDeg");

4# create groups O(2)xS4xZ2

5o2:=OrthogonalGroupOverReal(2);

6s4:=SymmetricGroup(4);

7z2:=pCyclicGroup(2);

8# generate S4xZ2

9g1:=DirectProduct(s4,z2);

10# set names for subgroup conjugacy classes in S4xZ2

11SetCCSsAbbrv(g1, [ "Z1", "Z2", "D1z","D1","Z2m", "Z1p",

12"Z3", "V4", "D2z", "Z4", "D2", "D1p","D2d",

13"V4m", "D2p", "Z4d", "D3", "D3z", "Z3p",

14"V4p", "D4z", "D4d", "Z4p", "D4", "D4z",

15 "D4hd", "D3p", "A4", "D4p", "A4p", "S4", "S4m", "S4p"]);

16# generate O(2)xS4xZ2

17G:=DirectProduct(o2,g1);

18ccs:=ConjugacyClassesSubgroups(G);

19# find Maximal Orbit Types and Basic Degrees in first O(2)-isotypic
component

20irrs := Irr(G);

21# The G-istoypic components 0,2,3 are indexed in G.A.P as 3,5,7

22for i in [ 3,5,7 ] do

23 max_orbtyps := MaximalOrbitTypes( irrs[1,i] );

24 basic_deg := BasicDegree( irrs[1,i] );

25 PrintFormatted( "\n Basic Degree associated with irrep V_{1,j} where j=
\n", i );

26 View(basic_deg);

27 PrintFormatted( "\n Maximal Orbit Types in M_1,{} \n", i );

28 View(max_orbtyps);

29od;

30Print( "Done!\n" );

The output of the above program can be expressed using amalgamated notation
and an arbitrary $m>0$ folding (cf. (23) and (24)) as follows:

$\displaystyle\mathfrak{M}_{m,0}=$
$\displaystyle\Big{\\{}(D_{2m}^{D_{m}}\times^{S_{4}}S_{4}^{p})\Big{\\}},$
$\displaystyle\mathfrak{M}_{m,2}=$
$\displaystyle\Big{\\{}(D_{6m}^{{\mathbb{Z}}_{m}}\times^{V_{4}}S_{4}^{p}),(D_{2m}^{D_{m}}\times^{D_{4}}D_{4}^{p}),(D_{2m}^{D_{m}}\times^{D_{4}^{\hat{d}}}D_{4}^{p})\Big{\\}},$
$\displaystyle\mathfrak{M}_{m,3}=$
$\displaystyle\Big{\\{}(D_{2m}^{D_{m}}\times^{D_{4}^{z}}D_{4}^{p}),(D_{2m}^{D_{m}}\times^{D_{3}^{z}}D_{3}^{p}),(D_{2m}^{D_{m}}\times^{D_{2}^{d}}D_{4}^{z}),(D_{4m}^{{\mathbb{Z}}_{m}}\times^{{\mathbb{Z}}_{2}^{-}}D_{4}^{p}),(D_{6m}^{{\mathbb{Z}}_{m}}\times
D_{3}^{p})\Big{\\}},$ $\displaystyle\deg_{\mathcal{V}_{m,0}}$
$\displaystyle=(G)-(D_{2m}^{D_{m}}\times^{S_{4}}S_{4}^{p}),$
$\displaystyle\deg_{\mathcal{V}_{m,2}}$
$\displaystyle=(G)-2(D_{6m}^{{\mathbb{Z}}_{m}}\times^{V_{4}}S_{4}^{p})-(D_{2m}^{D_{m}}\times^{D_{4}}D_{4}^{p})-(D_{2m}^{D_{m}}\times^{D_{4}^{\hat{d}}}D_{4}^{p})$
$\displaystyle+(D_{2m}^{D_{m}}\times^{V_{4}}V_{4}^{p})+4(D_{2m}^{{\mathbb{Z}}_{m}}\times^{V_{4}}D_{4}^{p}),$
$\displaystyle\deg_{\mathcal{V}_{m,3}}$
$\displaystyle=(G)-(D_{2m}^{D_{m}}\times^{D_{4}^{z}}D_{4}^{p})-(D_{2m}^{D_{m}}\times^{D_{3}^{z}}D_{3}^{p})-(D_{2m}^{D_{m}}\times^{D_{2}^{d}}D_{4}^{z})$
$\displaystyle-2(D_{4m}^{{\mathbb{Z}}_{m}}\times^{{\mathbb{Z}}_{2}^{-}}D_{4}^{p})-2(D_{6m}^{{\mathbb{Z}}_{m}}\times
D_{3}^{p})+(D_{2m}^{{\mathbb{Z}}_{m}}\times^{{\mathbb{Z}}_{2}^{-}}D_{2}^{p})$
$\displaystyle+2(D_{2m}^{{\mathbb{Z}}_{m}}\times^{{\mathbb{Z}}_{2}^{-}}V_{4}^{p})+2(D_{2m}^{{\mathbb{Z}}_{m}}\times^{{\mathbb{Z}}_{2}^{-}}D_{4}^{z})+2(D_{2m}^{{\mathbb{Z}}_{m}}\times^{D_{1}^{z}}D_{4}^{z})-(D_{2m}^{{\mathbb{Z}}_{m}}\times{\mathbb{Z}}_{1}^{p})-2(D_{2m}^{{\mathbb{Z}}_{m}}\times
D_{2}^{p}).$

Take $j\in\\{0,2,3\\}$ and let $(H)$ be any orbit type in
$\mathfrak{M}_{1,j}$. From the coefficients of the Basic Degrees
$\deg_{\mathcal{V}_{1,j}}\in A(G)$, it is clear that the fixed point space
$\mathcal{V}_{1,j}^{H}$ has odd dimension. Since the critical set (50) is
manageably small, we can compute the cardinalities (26) for each of the
critical points $(\alpha_{n,m,j},0)\in\Lambda$ to obtain the quantities

$\displaystyle\mathfrak{s}(\alpha_{1,3,2},H)$
$\displaystyle=\mathfrak{s}(\alpha_{1,3,3},H)=\mathfrak{s}(\alpha_{1,3,0},H)=3,$
$\displaystyle\mathfrak{s}(\alpha_{2,1,2},H)$
$\displaystyle=\mathfrak{s}(\alpha_{2,1,3},H)=1.$

and

$\displaystyle\mathfrak{i}^{3}(\alpha_{1,3,2},H)$
$\displaystyle=1,\quad\mathfrak{i}^{3}(\alpha_{1,3,3},H)=-1,\quad\mathfrak{i}^{3}(\alpha_{1,3,0},H)=1,$
$\displaystyle\mathfrak{i}^{1}(\alpha_{2,1,2},H)$
$\displaystyle=1,\quad\mathfrak{i}^{1}(\alpha_{2,1,3},H)=-1.$

Therefore, from Theorem 4.1, it follows that every critical point
$(\alpha_{n,m,j},0)\in\Lambda$ is a branching point for branches of non-
trivial solutions to the problem (41) with symmetries at least $({}^{3}H)$ in
the case of
$(\alpha_{1,3,2},0),(\alpha_{1,3,3},0),(\alpha_{1,3,0},0)\in\Lambda$ and with
symmetries at least $(H)$ in the case of
$(\alpha_{2,1,2},0),(\alpha_{2,1,3},0)\in\Lambda$. The following G.A.P code
can be used to computationally verify the non-triviality of the local
bifurcation invariant (specifically, the non-triviality of the relevant
coefficients) at each of the critical points in terms of the basic degrees

$\deg_{\mathcal{V}_{3,2}},\deg_{\mathcal{V}_{3,3}},\deg_{\mathcal{V}_{3,0}},\deg_{\mathcal{V}_{1,2}},\deg_{\mathcal{V}_{1,3}}\in
A(G),$

and the unit Burnside Ring element $(G)$ according to the rule derived in
Lemma (4.2)

$\displaystyle\omega_{G}(\alpha_{1,3,2})$
$\displaystyle=(G)-\deg_{\mathcal{V}_{3,2}},$
$\displaystyle\omega_{G}(\alpha_{1,3,3})$
$\displaystyle=\deg_{\mathcal{V}_{3,2}}-\deg_{\mathcal{V}_{3,2}}\cdot\deg_{\mathcal{V}_{3,3}},$
$\displaystyle\omega_{G}(\alpha_{1,3,0})$
$\displaystyle=\deg_{\mathcal{V}_{3,2}}\cdot\deg_{\mathcal{V}_{3,3}}-\deg_{\mathcal{V}_{3,2}}\cdot\deg_{\mathcal{V}_{3,3}}\cdot\deg_{\mathcal{V}_{3,0}},$
$\displaystyle\omega_{G}(\alpha_{2,1,2})$
$\displaystyle=\deg_{\mathcal{V}_{3,2}}\cdot\deg_{\mathcal{V}_{3,3}}\cdot\deg_{\mathcal{V}_{3,0}}-\deg_{\mathcal{V}_{3,2}}\cdot\deg_{\mathcal{V}_{3,3}}\cdot\deg_{\mathcal{V}_{3,0}}\cdot\deg_{\mathcal{V}_{1,2}},$
$\displaystyle\omega_{G}(\alpha_{2,1,3})$
$\displaystyle=\deg_{\mathcal{V}_{3,2}}\cdot\deg_{\mathcal{V}_{3,3}}\cdot\deg_{\mathcal{V}_{3,0}}\cdot\deg_{\mathcal{V}_{1,2}}-\deg_{\mathcal{V}_{3,2}}\cdot\deg_{\mathcal{V}_{3,3}}\cdot\deg_{\mathcal{V}_{3,0}}\cdot\deg_{\mathcal{V}_{1,2}}\cdot\deg_{\mathcal{V}_{1,3}}.$

⬇

1# A G.A.P Program for the Burnside Ring product of Basic Degrees to be used
for the verification of non-triviality of local bifurcation invariants

2# Initialize the Burnside Ring A(G)

3AG:=BurnsideRing(G); H:=Basis(AG);

4# Initialize the unit element (G) \in A(G)

5U:=H[0,131];

6# Initialize the relevant Basic Degrees

7d_30 := BasicDegree(Irr(G)[3,3]);

8d_32 := BasicDegree(Irr(G)[3,5]);

9d_33 := BasicDegree(Irr(G)[3,7]);

10d_12 := BasicDegree(Irr(G)[1,5]);

11d_13 := BasicDegree(Irr(G)[1,7]);

12# Compute Products of Basic Degrees Cumulatively

13p_1 := d_32;

14p_2 := p_1*d_33;

15p_3 := p_2*d_30;

16p_4 := p_3*d_12;

17p_5 := p_4*d_13;

18# Compute the local bifurcation invariants w_{n,m,j} at the corresponding
critical points \alpha_{n,m,j}

19w_132 := U - p_1;

20w_133 := p_1 - p_2;

21w_130 := p_2 - p_3;

22w_212 := p_3 - p_4;

23w_213 := p_4 - p_5;

24# Sum the local bifurcation invariants to determine the Rabinowitz
Alternative

25sum := w_132 + w_133 + w_130 + w_212 + w_213

The output of the above program can be expressed using amalgamated notation as
follows:

$\displaystyle\omega_{G}(\alpha_{1,3,2})$
$\displaystyle=-4(D_{6}^{{\mathbb{Z}}_{3}}\times^{V_{4}}D_{4}^{p})-(D_{6}^{D_{3}}\times^{V_{4}}V_{4}^{p})+(D_{6}^{D_{3}}\times^{D_{4}}D_{4}^{p})+(D_{6}^{D_{3}}\times^{D_{4}^{\hat{d}}}D_{4}^{p})+2(D_{18}^{{\mathbb{Z}}_{3}}\times^{V_{4}}S_{4}^{p}),$
$\displaystyle\omega_{G}(\alpha_{1,3,3})$
$\displaystyle=4(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{1}}D_{1}^{p})+2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{1}}D_{2}^{p})+(D_{6}^{D_{3}}\times^{{\mathbb{Z}}_{1}}{\mathbb{Z}}_{1}^{p})-2(D_{6}^{{\mathbb{Z}}_{3}}\times^{D_{1}}D_{4}^{z})-2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{2}^{-}}D_{4}^{z})$
$\displaystyle\quad-2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{2}^{-}}V_{4}^{p})+2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{2}}{\mathbb{Z}}_{4}^{p})+2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{2}}D_{4}^{z})+2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{2}}V_{4}^{p})-(D_{6}^{D_{3}}\times^{D_{1}}D_{1}^{p})$
$\displaystyle\quad-(D_{6}^{D_{3}}\times^{D_{1}^{z}}D_{1}^{p})-(D_{6}^{D_{3}}\times^{{\mathbb{Z}}_{2}^{-}}D_{2}^{p})+(D_{6}^{D_{3}}\times^{{\mathbb{Z}}_{2}}D_{2}^{p})-2(D_{18}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{1}}D_{3}^{p})+2(D_{12}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{2}^{-}}D_{4}^{p})$
$\displaystyle\quad-2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{4}}D_{4}^{p})-2(D_{6}^{{\mathbb{Z}}_{3}}\times^{D_{2}^{z}}D_{4}^{p})+(D_{6}^{D_{3}}\times^{D_{2}^{d}}D_{4}^{z})-(D_{6}^{D_{3}}\times^{{\mathbb{Z}}_{4}}{\mathbb{Z}}_{4}^{p})-(D_{6}^{D_{3}}\times^{D_{2}^{z}}D_{4}^{z})$
$\displaystyle\quad+(D_{6}^{D_{3}}\times^{D_{3}^{z}}D_{3}^{p})+(D_{6}^{D_{3}}\times^{D_{4}^{z}}D_{4}^{p}),$
$\displaystyle\omega_{G}(\alpha_{1,3,0})$
$\displaystyle=2(D_{6}^{{\mathbb{Z}}_{3}}\times^{D_{1}}D_{4}^{z})-2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{2}}{\mathbb{Z}}_{4}^{p})-2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{2}}D_{4}^{z})-2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{2}}V_{4}^{p})+(D_{6}^{D_{3}}\times^{D_{1}}D_{1}^{p})$
$\displaystyle\quad-(D_{6}^{D_{3}}\times^{{\mathbb{Z}}_{2}}D_{2}^{p})-2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{3}}D_{3}^{p})-(D_{6}^{D_{3}}\times^{{\mathbb{Z}}_{3}}{\mathbb{Z}}_{3}^{p})+2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{4}}D_{4}^{p})+3(D_{6}^{{\mathbb{Z}}_{3}}\times^{V_{4}}D_{4}^{p})$
$\displaystyle\quad+(D_{6}^{D_{3}}\times^{{\mathbb{Z}}_{4}}{\mathbb{Z}}_{4}^{p})+(D_{6}^{D_{3}}\times^{V_{4}}V_{4}^{p})-2(D_{6}^{D_{3}}\times^{D_{4}}D_{4}^{p})+(D_{6}^{D_{3}}\times^{S_{4}}S_{4}^{p})$
$\displaystyle\omega_{G}(\alpha_{2,1,2})$
$\displaystyle=2(D_{2}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{1}}D_{2}^{p})+(D_{2}^{D_{1}}\times^{{\mathbb{Z}}_{1}}{\mathbb{Z}}_{1}^{p})-2(D_{2}^{{\mathbb{Z}}_{1}}\times^{D_{1}}D_{4}^{z})-2(D_{2}^{{\mathbb{Z}}_{1}}\times^{D_{1}^{z}}D_{4}^{z})-(D_{2}^{D_{1}}\times^{D_{1}}D_{1}^{p})$
$\displaystyle\quad-(D_{2}^{D_{1}}\times^{D_{1}^{z}}D_{1}^{p})+4(D_{6}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{1}}D_{3}^{p})-2(D_{2}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{4}}D_{4}^{p})+2(D_{2}^{{\mathbb{Z}}_{1}}\times^{D_{2}^{z}}D_{4}^{p})+2(D_{2}^{{\mathbb{Z}}_{1}}\times^{V_{4}}D_{4}^{p})$
$\displaystyle\quad-(D_{2}^{D_{1}}\times^{{\mathbb{Z}}_{4}}{\mathbb{Z}}_{4}^{p})+(D_{2}^{D_{1}}\times^{D_{2}^{z}}D_{4}^{z})+(D_{2}^{D_{1}}\times^{D_{4}}D_{4}^{p})-(D_{2}^{D_{1}}\times^{D_{4}^{\hat{d}}}D_{4}^{p})-2(D_{6}^{{\mathbb{Z}}_{1}}\times^{V_{4}}S_{4}^{p})$
$\displaystyle\omega_{G}(\alpha_{2,1,3})$
$\displaystyle=-4(D_{2}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{1}}D_{1}^{p})-4(D_{2}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{1}}D_{2}^{p})-2(D_{2}^{D_{1}}\times^{{\mathbb{Z}}_{1}}{\mathbb{Z}}_{1}^{p})+2(D_{2}^{{\mathbb{Z}}_{1}}\times^{D_{1}}D_{4}^{z})+2(D_{2}^{{\mathbb{Z}}_{1}}\times^{D_{1}^{z}}D_{4}^{z})$
$\displaystyle\quad+2(D_{2}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{2}^{-}}D_{4}^{z})+2(D_{2}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{2}^{-}}V_{4}^{p})+(D_{2}^{D_{1}}\times^{D_{1}}D_{1}^{p})+2(D_{2}^{D_{1}}\times^{D_{1}^{z}}D_{1}^{p})+(D_{2}^{D_{1}}\times^{{\mathbb{Z}}_{2}^{-}}D_{2}^{p})$
$\displaystyle\quad-2(D_{6}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{1}}D_{3}^{p})+2(D_{2}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{3}}D_{3}^{p})+(D_{2}^{D_{1}}\times^{{\mathbb{Z}}_{3}}{\mathbb{Z}}_{3}^{p})-2(D_{4}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{2}^{-}}D_{4}^{p})+2(D_{2}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{4}}D_{4}^{p})$
$\displaystyle\quad-(D_{2}^{D_{1}}\times^{D_{2}^{d}}D_{4}^{z})+(D_{2}^{D_{1}}\times^{{\mathbb{Z}}_{4}}{\mathbb{Z}}_{4}^{p})+(D_{2}^{D_{1}}\times^{D_{3}^{z}}D_{3}^{p})-(D_{2}^{D_{1}}\times^{D_{4}^{z}}D_{4}^{p}).$

Next, we can determine the global behaviour of some of the branches predicted
using Theorem (4.1) by establishing the parity of the set (38). Notice that
$\max\limits_{(\alpha,0)\in\Lambda}\\{\mathfrak{s}(\alpha,H)\\}\nmid 2$ is
obtained for exactly three critical points
$(\alpha_{1,3,0},0),(\alpha_{1,3,2},0),(\alpha_{1,3,3},0)$, i.e.

$\displaystyle\bar{\mathfrak{s}}(H)=3\;\text{ and
}\;\mathfrak{J}(H)=\\{(\alpha_{1,3,0},0),(\alpha_{1,3,2},0),(\alpha_{1,3,3},0)\\}.$

Therefore, any branch of non-trivial solutions with symmetries at least
$({}^{3}H)$ bifurcating from
$(\alpha_{1,3,0},0),(\alpha_{1,3,2},0),(\alpha_{1,3,3},0)$ consists only of
non-radial solutions and is unbounded. Of course, since the critical set (50)
is finite, one needs only sum the local bifurcation invariants (LABEL:) to
arrive at the same conclusion directly from the Rabinowitz alternative (cf.
Theorems 4.2 and 4.3).

$\displaystyle\sum_{(\alpha_{0},0)\in\Lambda}\omega_{G}(\alpha_{0})$
$\displaystyle=\omega_{G}(\alpha_{1,3,2})+\omega_{G}(\alpha_{1,3,3})+\omega_{G}(\alpha_{1,3,0})+\omega_{G}(\alpha_{2,1,2})+\omega_{G}(\alpha_{2,1,3})$
$\displaystyle=-4(D_{2}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{1}}D_{1}^{p})-2(D_{2}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{1}}D_{2}^{p})-(D_{2}^{D_{1}}\times^{{\mathbb{Z}}_{1}}{\mathbb{Z}}_{1}^{p})+2(D_{2}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{2}^{-}}D_{4}^{z})$
$\displaystyle\quad+2(D_{2}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{2}^{-}}V_{4}^{p})+(D_{2}^{D_{1}}\times^{D_{1}^{z}}D_{1}^{p})+(D_{2}^{D_{1}}\times^{{\mathbb{Z}}_{2}^{-}}D_{2}^{p})+2(D_{6}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{1}}D_{3}^{p})$
$\displaystyle\quad+2(D_{2}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{3}}D_{3}^{p})+(D_{2}^{D_{1}}\times^{{\mathbb{Z}}_{3}}{\mathbb{Z}}_{3}^{p})-2(D_{4}^{{\mathbb{Z}}_{1}}\times^{{\mathbb{Z}}_{2}^{-}}D_{4}^{p})+2(D_{2}^{{\mathbb{Z}}_{1}}\times^{D_{2}^{z}}D_{4}^{p})$
$\displaystyle\quad+2(D_{2}^{{\mathbb{Z}}_{1}}\times^{V_{4}}D_{4}^{p})-(D_{2}^{D_{1}}\times^{D_{2}^{d}}D_{4}^{z})+(D_{2}^{D_{1}}\times^{D_{2}^{z}}D_{4}^{z})-(D_{2}^{D_{1}}\times^{D_{3}^{z}}D_{3}^{p})$
$\displaystyle\quad-(D_{2}^{D_{1}}\times^{D_{4}^{z}}D_{4}^{p})+(D_{2}^{D_{1}}\times^{D_{4}}D_{4}^{p})-(D_{2}^{D_{1}}\times^{D_{4}^{\hat{d}}}D_{4}^{p})-2(D_{6}^{{\mathbb{Z}}_{1}}\times^{V_{4}}S_{4}^{p})$
$\displaystyle\quad+4(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{1}}D_{1}^{p})+2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{1}}D_{2}^{p})+(D_{6}^{D_{3}}\times^{{\mathbb{Z}}_{1}}{\mathbb{Z}}_{1}^{p})-2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{2}^{-}}D_{4}^{z})$
$\displaystyle\quad-2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{2}^{-}}V_{4}^{p})-(D_{6}^{D_{3}}\times^{D_{1}^{z}}D_{1}^{p})-(D_{6}^{D_{3}}\times^{{\mathbb{Z}}_{2}^{-}}D_{2}^{p})-2(D_{1}8^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{1}}D_{3}^{p})$
$\displaystyle\quad-2(D_{6}^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{3}}D_{3}^{p})-(D_{6}^{D_{3}}\times^{{\mathbb{Z}}_{3}}{\mathbb{Z}}_{3}^{p})+2(D_{1}2^{{\mathbb{Z}}_{3}}\times^{{\mathbb{Z}}_{2}^{-}}D_{4}^{p})-2(D_{6}^{{\mathbb{Z}}_{3}}\times^{D_{2}^{z}}D_{4}^{p})$
$\displaystyle\quad-2(D_{6}^{{\mathbb{Z}}_{3}}\times^{V_{4}}D_{4}^{p})+(D_{6}^{D_{3}}\times^{D_{2}^{d}}D_{4}^{z})-(D_{6}^{D_{3}}\times^{D_{2}^{z}}D_{4}^{z})+(D_{6}^{D_{3}}\times^{D_{3}^{z}}D_{3}^{p})$
$\displaystyle\quad+(D_{6}^{D_{3}}\times^{D_{4}^{z}}D_{4}^{p})-(D_{6}^{D_{3}}\times^{D_{4}}D_{4}^{p})+(D_{6}^{D_{3}}\times^{D_{4}^{\hat{d}}}D_{4}^{p})+2(D_{18}^{{\mathbb{Z}}_{3}}\times^{V_{4}}S_{4}^{p})$
$\displaystyle\quad+(D_{6}^{D_{3}}\times^{S_{4}}S_{4}^{p}).$

In conclusion, let’s consider an interpretation of the above results. Take the
maximal orbit type
$(H)=(D_{2}^{D_{1}}\times^{D_{4}}D_{4}^{p})\in\mathscr{M}_{1,2}$ and let
$\mathcal{C}$ be a branch of non-radial solutions with symmetries at least
$({}^{3}H)=(D_{6}^{D_{3}}\times^{D_{4}}D_{4}^{p})\in\mathscr{M}_{3,2}$
emerging from the first critical point $(\alpha_{1,3,2},0)\in\Lambda$. The
kernel of $\mathscr{A}(\alpha):\mathscr{H}\rightarrow\mathscr{H}$ at any
critical point $(\alpha_{n,m,j},0)\in\Lambda$ is given by the corresponding
irreducible $G$-representation $\mathscr{E}_{n,m}^{j}$ (cf. (14)), i.e.

$\text{\rm Ker\,}\mathscr{A}(\alpha_{1,3,2})=\mathscr{E}_{1,3}^{2}.$

In particular, any vector $\hat{u}\in\mathscr{H}$ with
$\mathscr{A}(\alpha_{1,3,2})\hat{u}=0$ must be of the form

$\hat{u}(r,\theta)\in\left\\{J_{3}(\sqrt{s_{13}}r)\Big{(}\cos(3\theta)\vec{a}+\sin(3\theta)\vec{b}\Big{)}:\vec{a},\,\vec{b}\in\text{span}\\{\vec{w}_{1},\vec{w}_{2}\\}\right\\},$

where $\vec{w}_{1},\vec{w}_{2}\in\mathcal{V}_{2}^{-}$ are the eigenvectors
$\vec{w}_{1}=(-1,1,-1,1,0,0)^{T}$ and $\vec{w}_{2}=(1,0,1,0,-1,-1)^{T}$.
Computing the generators of $(H)$ and examining their action on
$\mathscr{E}_{1,3}^{2}$, it can be shown that $\hat{u}\in\text{\rm
Ker\,}\mathscr{A}(\alpha_{0})^{H}$ if and only if $\vec{b}=0$ and
$\vec{a}=\vec{w}_{1}+2\vec{w}_{2}$. In particular, one has $\dim\text{\rm
Ker\,}\mathscr{A}(\alpha_{0})^{H}=1$, which implies, from the celebrated
theorem of Crandall-Rabinowitz (cf. [29]), that the branch $\mathcal{C}$ is
tangent to the solution
$\hat{u}(r,\theta)=J_{3}(\sqrt{s_{13}}r)\cos(3\theta)(\vec{w}_{1}+2\vec{w}_{2})$
at the point $(\alpha_{1,3,2},0)$. Therefore, $\hat{u}\in\mathscr{H}$ can
reasonably be used to estimate the behaviour of the branch $\mathcal{C}$ for
parameter values near $\alpha_{1,3,2}\in\mathbb{R}$.

Figure 4. Graph of the map
$\hat{u}(r,\theta)=J_{3}(\sqrt{s_{13}}r)\cos(3\theta)(\vec{w}_{1}+2\vec{w}_{2})$.

##  Appendix A Amalgamated Notation of Subgroups

The convention of amalgamated notation is a shorthand for the specification of
subgroups and their conjugacy classes in a product group $G_{1}\times G_{2}$.
In order to introduce this convention, we recall a well-known consequence of
Goursat’s Lemma (cf. [15]). Namely that, for any closed subgroup in $H\leq
G_{1}\times G_{2}$, there are two subgroups $K_{1}\leq G_{1}$, $K_{2}\leq
G_{2}$, a group $L$ and a pair of epimorphisms $\varphi_{2}:K_{1}\rightarrow
L$, $\varphi_{2}:K_{2}\rightarrow L$ such that

(51) $\displaystyle H=\\{(x,y)\in K_{1}\times
K_{2}\;:\;\varphi_{1}(x)=\varphi_{2}(y)\\}.$

Using amalgamated notation, the subgroup (51) is identified, up to its
conjugacy class in $\Phi_{0}(G_{1}\times G_{2})$, by the quintuple
$(K_{1},K_{2},L,Z_{1},Z_{2})$, where $Z_{1}:=\text{\rm Ker\,}\varphi_{1}$ and
$Z_{2}:=\text{\rm Ker\,}\varphi_{2}$, as follows

$\displaystyle(H)=(K_{1}{}^{Z_{1}}\times^{Z_{2}}_{L}K_{2}).$

In particular, one can always choose the group $L$ such that $L\simeq
K_{1}/Z_{1}$, in which case the conjugacy class of (51) is identified with the
quadruple $(K_{1},K_{1},Z_{1},Z_{2})$ as follows

(52) $\displaystyle(H)=(K_{1}{}^{Z_{1}}\times^{Z_{2}}K_{2}).$

This compact amalgamated decomposition (52) is the form with which subgroup
conjugacy classes are identified in this paper. In addition to the notation
defined above, the following list provides some ancillary shorthand used in
this paper for the identification of subgroups in
$S_{4}\times{\mathbb{Z}}_{2}$

$\displaystyle{\mathbb{Z}}_{2}^{-}$
$\displaystyle=\\{((1),1),((12)(34),-1)\\},$
$\displaystyle{\mathbb{Z}}_{4}^{-}$
$\displaystyle=\\{((1),1),((1324),-1),((12)(34),1),((1423),-1)\\},$
$\displaystyle D_{1}^{z}$ $\displaystyle=\\{((1),1),((12),-1)\\},$
$\displaystyle V_{4}^{-}$
$\displaystyle=\\{((1),1),((12)(34),1),((13)(24),-1),((14)(23),-1)\\},$
$\displaystyle D_{2}^{d}$
$\displaystyle=\\{((1),1),((12)(34),-1),((12),1),((34),-1)\\},$ $\displaystyle
D_{2}^{z}$ $\displaystyle=\\{((1),1),((12)(34),1),((12),-1),((34),-1)\\},$
$\displaystyle D_{3}^{z}$
$\displaystyle=\\{((1),1),((123),1),((132),1),((12),-1),((23),-1),((13),-1)\\},$
$\displaystyle D_{4}^{d}$
$\displaystyle=\\{((1),1),((1324),-1),((12)(34),1),((1423),-1),((34),1),$
$\displaystyle~{}~{}~{}~{}~{}((14)(23),-1),((12),1),((13)(24),-1)\\},$
$\displaystyle D_{4}^{\hat{d}}$
$\displaystyle=\\{((1),1),((1324),-1),((12)(34),1),((1423),-1),((34),-1),$
$\displaystyle~{}~{}~{}~{}~{}((14)(23),1),((12),-1),((13)(24),1)\\},$
$\displaystyle D_{4}^{z}$
$\displaystyle=\\{((1),1),((1324),1),((12)(34),1),((1423),1),((34),-1),$
$\displaystyle~{}~{}~{}~{}~{}((14)(23),-1),((12),-1),((13)(24),-1)\\},$
$\displaystyle S_{4}^{-}$
$\displaystyle=\\{((1),1),((12),-1),((12)(34),1),((123),1),((1234),-1),((13),-1),$
$\displaystyle~{}~{}~{}~{}~{}((13)(24),1),((132),1),((1342),-1),((14),-1),((14)(23),1),((142),1),$
$\displaystyle~{}~{}~{}~{}~{}((1324),-1),((23),-1),((124),1),((1243),-1),((24),-1),((134),1),$
$\displaystyle~{}~{}~{}~{}~{}((1423),-1),((34),-1),((143),1),((1432),-1),((243),1),((234),1)\\}.$

##  Appendix B Spectral Properties of Operator $\mathscr{A}$

In this section, we leverage the spectral properties of the Laplace operator
$\mathscr{L}:\mathscr{H}\rightarrow\mathscr{H}$ to describe the spectrum of

$\displaystyle\mathscr{A}(\alpha):=\operatorname{Id}-\mathscr{L}^{-1}A(\alpha):\mathscr{H}\rightarrow\mathscr{H}.$

To begin, consider the following eigenvalue problem defined on the planar unit
disc $D:=\\{z\in{\mathbb{C}}:|z|<1\\}$ with Dirichlet boundary conditions

$\displaystyle\begin{cases}-\Delta u=\lambda u,\quad
u\in\mathscr{H},\lambda\in\mathbb{R}\\\ u|_{\partial D}=0\end{cases}$

In polar coordinates, the Laplacian $\Delta$ takes the form

$\displaystyle\Delta u=\frac{\partial^{2}u}{\partial
r^{2}}+\frac{1}{r}\frac{\partial u}{\partial
r}+\frac{1}{r^{2}}\frac{\partial^{2}u}{\partial\theta^{2}},$

Assuming that $u\in\mathscr{H}$ can be separated $u(r,\theta)=R(r)\cdot
T(\theta)$, the eigenvalue problem $-\Delta u=\lambda u$ becomes

$\displaystyle\left(-R^{\prime\prime}-\frac{1}{r}R^{\prime}\right)T-\frac{1}{r^{2}}T^{\prime\prime}R$
$\displaystyle=\lambda RT.$

Rearranging terms, we obtain

(53) $\displaystyle\frac{r^{2}R^{\prime\prime}+rR^{\prime}}{R}+\lambda
r^{2}=-\frac{T^{\prime\prime}}{T}.$

Since the left and right-hand sides of equation (53) depend on different
variables, there must exist a constant $c$ such that

(54) $\displaystyle r^{2}R^{\prime\prime}+rR^{\prime}+(\lambda r^{2}-c)R=0,$
(55) $\displaystyle-T^{\prime\prime}=cT.$

To ensure that $u(r,\theta)$ is single-valued and continuous on $D$, we impose
the periodicity condition $T(\theta+2\pi)=T(\theta)$. Then, the angular part
(55) becomes the ordinary differential equation

$\displaystyle\begin{cases}T^{\prime\prime}(\theta)=-cT(\theta)\\\
T(\theta+2\pi)=T(\theta).\end{cases}$

which has eigenvalues $c=m^{2}$ for $m=0,1,2,\ldots$ with corresponding
eigenfunctions $T(\theta)=cos(m\theta)$ and $T(\theta)=\sin(m\theta)$. Notice
that, as a consequence of solving for $T(\theta)$, we have determined the
values of constant $c$:

$\displaystyle c=m^{2},m=0,1,2,\dots$

Then, equation (54) becomes

(56) $r^{2}R^{\prime\prime}+rR^{\prime}+(\lambda r^{2}-m^{2})R=0.$

Introducing the change of variables $\rho=\sqrt{\lambda}r$ and
$\widetilde{R}(\rho)=R(\sqrt{\lambda}r)$ and substituting into (56), we obtain
the classical Bessel equation

$\rho^{2}\widetilde{R}^{\prime\prime}(\rho)+\rho\widetilde{R}^{\prime}(\rho)+(\rho^{2}-m^{2})\widetilde{R}(\rho)=0.$

The bounded at zero solution is given by $\widetilde{R}(\rho)=J_{m}(\rho)$,
where $J_{m}$ stands for the $m$-th Bessel function of the first kind. It
follows that solutions to (54) are constant multiples of
$R(r)=J_{m}(\sqrt{\lambda}r)$ (notice that $R(r)$ is finite at zero).
Consequently, $u(r,\theta)=R(r)\Theta(\theta)$ satisfies the Dirichlet
condition if $R(1)=0$, i.e. $J_{m}(\sqrt{\lambda})=0$. In this way, we obtain
that the spectrum of the Laplace operator is given by

$\sigma(\mathscr{L})=\\{s_{nm}:\text{$\sqrt{s_{nm}}$ is the $n$-th positive
zero of $J_{m},\;n\in{\mathbb{N}}\;m=0,1,\ldots\\}$}.$

Moreover, corresponding to each eigenvalue $s_{nm}\in\sigma(\mathscr{L})$,
there is the eigenfunction

$u_{nm}(r,\theta)=J_{m}(\sqrt{s_{nm}}r)\left(A\cos(m\theta)+B\sin(m\theta)\right),$

where $A,B\in\mathbb{R}$, and also the eigenspace

$\displaystyle\mathscr{E}(s_{nm})=\text{span\,}\Big{\\{}J_{m}(\sqrt{s_{nm}}r)\Big{(}A\cos(m\theta)+B\sin(m\theta)\Big{)}:\vec{a},\,\vec{b}\in\mathbb{R}^{k},\;0\leq
n\leq k\Big{\\}}$

In this way, we have determined the spectrum of our operator $\mathscr{A}$

$\sigma(\mathscr{A})=\left\\{\xi_{n,m,j}:=1-\frac{\mu_{j}(\alpha)}{s_{nm}}:j=1,2,\dots,k,\;n\in{\mathbb{N}},\;m=0,1,2,\dots\right\\}.$

##  Appendix C Equivariant Brouwer Degree Background

Equivariant notation. Let $\mathcal{G}$ be a compact Lie Group. For any
subgroup $H\leq\mathcal{G}$, denote by $(H)$ its conjugacy class, by by $N(H)$
its normalizer by $W(H):=N(H)/H$ its Weyl group in $\mathcal{G}$. The set of
all subgroup conjugacy classes in $\mathcal{G}$, denoted
$\Phi(\mathcal{G}):=\\{(H):H\leq\mathcal{G}\\}$, has a natural partial order
defined as follows

$(H)\leq(K)\iff\exists_{g\in\mathcal{G}}\;\;gHg^{-1}\leq K.$

In particular, we put
$\Phi_{0}(\mathcal{G}):=\\{(H)\in\Phi(\mathcal{G})\;:\;\text{$W(H)$ is
finite}\\}$. For a $\mathcal{G}$-space $X$ and $x\in X$, denote by
$\mathcal{G}_{x}:=\\{g\in\mathcal{G}:gx=x\\}$ the isotropy group of $x$ and
call $(\mathcal{G}_{x})\in\Phi(\mathcal{G})$ the orbit type of $x\in X$. Put
$\Phi(\mathcal{G},X):=\\{(H)\in\Phi_{0}(\mathcal{G})\;:\;(H)=(\mathcal{G}_{x})\;\text{for
some $x\in X$}\\}$ and
$\Phi_{0}(\mathcal{G},X):=\Phi(\mathcal{G},X)\cap\Phi_{0}(\mathcal{G})$. For a
subgroup $H\leq\mathcal{G}$, the subspace $X^{H}:=\\{x\in
X:\mathcal{G}_{x}\geq H\\}$ is called the $H$-fixed-point subspace of $X$. If
$Y$ is another $\mathcal{G}$-space, then a continuous map $f:X\to Y$ is said
to be $\mathcal{G}$-equivariant if $f(gx)=gf(x)$ for each $x\in X$ and
$g\in\mathcal{G}$. The Burnside Ring and Axioms of Equivariant Brouwer Degree.
Denote by $\mathcal{M}^{G}$ the set of all admissible $G$-pairs by $A(G)$ the
Burnside ring of $G$. The following statement is the standard axiomatic
definition of the $G$-equivariant Brouwer degree (cf. [5]).

######  Theorem C.1.

There exists a unique map $G\text{\rm-deg}:\mathcal{M}^{G}\to A(G)$, that
assigns to every admissible $G$-pair $(f,\Omega)$ the Burnside Ring element

(57) $G\text{\rm-deg}(f,\Omega)=\sum_{(H)\in\Phi_{0}(G)}{n_{H}(H)},$

satisfying the following properties:

  * (Existence) If $n_{H}\neq 0$ for some $(H)\in\Phi_{0}(G)$ in (57), then there exists $x\in\Omega$ such that $f(x)=0$ and $(G_{x})\geq(H)$.

  * (Additivity) For any two disjoint open $G$-invariant subsets $\Omega_{1}$ and $\Omega_{2}$ with $f^{-1}(0)\cap\Omega\subset\Omega_{1}\cup\Omega_{2}$, one has

$\displaystyle G\text{\rm-deg}(f,\Omega)=G\text{\rm-
deg}(f,\Omega_{1})+G\text{\rm-deg}(f,\Omega_{2}).$

  * (Homotopy) For any $\Omega$-admissible $G$-homotopy, $h:[0,1]\times V\to V$, one has

$\displaystyle G\text{\rm-deg}(h_{t},\Omega)=\mathrm{constant}.$

  * (Normalization) For any open bounded neighborhood of the origin in an orthogonal $G$-representation $V$ with the identity operator $\operatorname{Id}:V\rightarrow V$, one has

$\displaystyle G\text{\rm-deg}(\operatorname{Id},\Omega)=(G).$

The following are additional properties of the map $G\text{\rm-deg}$ which can
be derived from the four axiomatic properties defined above:

  * (Multiplicativity) For any $(f_{1},\Omega_{1}),(f_{2},\Omega_{2})\in\mathcal{M}^{G}$,

$\displaystyle G\text{\rm-deg}(f_{1}\times
f_{2},\Omega_{1}\times\Omega_{2})=G\text{\rm-deg}(f_{1},\Omega_{1})\cdot
G\text{\rm-deg}(f_{2},\Omega_{2}),$

where the multiplication ‘$\cdot$’ is taken in the Burnside ring $A(G)$.

  * (Recurrence Formula) For an admissible $G$-pair $(f,\Omega)$, the $G$-degree (57) can be computed using the following Recurrence Formula:

(58)
$n_{H}=\frac{\deg(f^{H},\Omega^{H})-\sum_{(K)>(H)}{n_{K}\,n(H,K)\,\left|W(K)\right|}}{\left|W(H)\right|},$

where $\left|X\right|$ stands for the number of elements in the set $X$ and
$\deg(f^{H},\Omega^{H})$ is the Brouwer degree of the map $f^{H}:=f|_{V^{H}}$
on the set $\Omega^{H}\subset V^{H}$.

Computation of Brouwer equivariant degree. For an orthogonal
$G$-representation $V$ with the open unit ball $B(V):=\left\\{x\in
V:\left|x\right|<1\right\\}$, denote by
$\\{\mathcal{V}_{i}\\}_{i\in{\mathbb{N}}}$ the set of its irreducible
$G$-subrepresentations. In particular, define the _basic degree associated
with $\mathcal{V}_{i}$_ as follows

$\displaystyle\deg_{\mathcal{V}_{i}}:=G\text{\rm-
deg}(-\operatorname{Id},B(\mathcal{V}_{i})),$

Now, consider a $G$-equivariant linear isomorphism $T:V\to V$ and assume that
$V$ has a $G$-isotypic decomposition

$V=\bigoplus_{i\in{\mathbb{N}}}V_{i},$

where each isotypic component $V_{i}$ is equivalent to $m_{i}\in{\mathbb{N}}$
copies of the irreducible $G$-representation $\mathcal{V}_{i}$. From the
Multiplicativity property of the $G$-equivariant Brouwer Degree, one has

$\displaystyle G\text{\rm-deg}(T,B(V))=\prod_{i\in{\mathbb{N}}}G\text{\rm-
deg}(T_{i},B(V_{i}))=\prod_{i\in{\mathbb{N}}}\prod_{\mu\in\sigma_{-}(T)}\left(\deg_{\mathcal{V}_{i}}\right)^{m_{i}(\mu)}$

where $T_{i}=T|_{V_{i}}$ and $\sigma_{-}(T)$ denotes the real negative
spectrum of $T$.

Notice that the basic degrees can be effectively computed from (58):

$\displaystyle\deg_{\mathcal{V}_{i}}=\sum_{(H)}n_{H}(H),$

where

$\displaystyle
n_{H}=\frac{(-1)^{\dim\mathcal{V}_{i}^{H}}-\sum_{H<K}{n_{K}\,n(H,K)\,\left|W(K)\right|}}{\left|W(H)\right|}.$

The following fact is well-known (see for example [3]).

######  Lemma C.1.

For any irreducible $G$-representation $\mathcal{V}$, the basic degree
$\deg_{\mathcal{V}}\in A(G)$ is an involutive element of the Burnside Ring,
i.e.

$(\deg_{\mathcal{V}_{j}})^{2}=\deg_{\mathcal{V}_{j}}\cdot\deg_{\mathcal{V}_{j}}=(G).$

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