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

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 their existence is straightforward to determine. In contrast, non-trivial solutions may only exist for a particular range of parameter values and often exhibit exotic properties that 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. Phenomena which admit the symmetries of a certain group $G$ (represented by the $G$-equivariance of the associated equations) may be studied as equivariant bifurcation problems, with additional consideration placed on 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 M. Poincare [13]. More recently, the Local/Equivariant Brouwer Degrees and their infinite dimensional generalizations – the Local/Equivariant Leray-Schauder Degrees – have proven prodigiously effective in obtaining existence results for solutions to a wide class of nonlinear differential equations. Somewhat remarkably, these degree theories are also instrumental in solving classical and equivariant bifurcation problems. The first use of the Leray-Schauder degree for the detection of (local) bifurcation in 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 the unboundedness of a branch of non-trivial solutions. In this paper, we employ equivariant analogues of Krasnosel’skii and Rabinowitz type results 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 not applicable for systems which lack variational structure. The degree theoretic methods used 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.

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] or, alternatively, the older text [5]. 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.

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)

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

In this section we present a concise exposition, following [4] and [5], 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.

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

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

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

Before approaching the question of bifurcation in the equation (1), we must derive a workable formula for the computation of the degree $G\text{\rm-deg}(\mathscr{A}(\alpha),B(\mathscr{H}))$, where $B(\mathscr{H}):=\{u\in\mathscr{H}:\|u\|_{\mathscr{H}}<1\}$ is the open unit ball in $\mathscr{H}$ and $\alpha\in\mathbb{R}$ is any parameter value for which $\mathscr{A}(\alpha):\mathscr{H}\rightarrow\mathscr{H}$ is an isomorphism. 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], [5]). As a $\Gamma$-representation, $V=\mathbb{R}^{k}$ is also a natural $\Gamma\times\mathbb{Z}_{2}$-representation with the $\Gamma\times\mathbb{Z}_{2}$-isotypic decomposition

$\displaystyle V=V_{1}\oplus V_{2}\oplus\cdots\oplus V_{r},$

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 our 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

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

On the other hand, for every $m\in{\mathbb{N}}$ we denote by $\mathcal{W}_{m}\simeq{\mathbb{C}}$ the irreducible $O(2)$-representation equipped with the $O(2)$-action

$$(\theta,z):=e^{im\theta}\cdot z,\;(\kappa,z):=\overline{z},\;z\in\mathcal{W}_{m},$$

and by $\mathcal{W}_{0}\simeq\mathbb{R}$ the irreducible $O(2)$-representation with the trivial $O(2)$-action. 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

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

Clearly, one has

$$\mathscr{E}_{nm}\simeq\mathcal{W}_{m}\otimes V,$$

for each $(n,m)\in{\mathbb{N}}\times{\mathbb{N}}\cup\{0\}$ such that $\mathscr{H}$ 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

(16)

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

and

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

one has

$$\mathscr{E}_{nm}^{j}\simeq\mathcal{W}_{m}\otimes\mathcal{V}_{j}^{-},\;(n,m)\in{\mathbb{N}}\times{\mathbb{N}}\cup\{0\},\;j\in\{1,2,\ldots,r\}.$$

Hence, $\mathscr{H}$ admits the $G$-isotypic

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

To be clear, each $G$-isotypic component $\mathscr{H}_{m,j}$ is modeled on the irreducible $G$-representation $\mathcal{V}_{m,j}:=\mathcal{W}_{m}\otimes\mathcal{V}_{j}^{-}$.