Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls

Sławomir Rybicki; Naoki Shioji; Piotr Stefaniak

doi:10.1515/ans-2018-0012

Article Open Access

Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls

Sławomir Rybicki , Naoki Shioji and Piotr Stefaniak

Published/Copyright: March 29, 2018

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From the journal Advanced Nonlinear Studies Volume 18 Issue 4

Abstract

The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in Sn. In particular, we show that if the geodesic ball is a hemisphere, then all these continua are unbounded. It is also shown that the phenomenon of global symmetry-breaking bifurcation of such solutions occurs. Since the problem is variational and SO⁡(n)-symmetric, we apply the techniques of equivariant bifurcation theory to prove the main results of this article. As the topological tool, we use the degree theory for SO⁡(n)-invariant strongly indefinite functionals defined in [A. Gołȩbiewska and S. A. Rybicki, Global bifurcations of critical orbits of G-invariant strongly indefinite functionals, Nonlinear Anal. 74 2011, 5, 1823–1834].

Keywords: Symmetric Rabinowitz Alternative; Global Symmetry-Breaking Bifurcations; Non-cooperative Elliptic Systems; Equivariant Degree

MSC 2010: 35B32; 35J20

1 Introduction

The aim of this paper is to study closed connected sets (continua) of solutions of boundary value problems for non-cooperative elliptic systems considered on geodesic balls in Sn, i.e., systems of the form

(1.1) { Λ ⁢ Δ S n ⁢ u = ∇ u ⁡ F ⁢ ( u , λ ) in ⁢ B ⁢ ( γ ) , u = 0 on ⁢ ∂ ⁡ B ⁢ ( γ ) ,

where Λ=diag⁡(α1,…,αp), αi=±1, ΔSn is the Laplace–Beltrami operator on the sphere Sn, B⁢(γ)⊂Sn is the geodesic ball of radius γ>0 centered at the north pole, F∈C2⁢(ℝp×ℝ,ℝ) fulfils ∇u⁡F⁢(u,λ)=λ⁢u+∇u⁡η⁢(u,λ), where ∇u is the gradient with respect to u, η∈C2⁢(ℝp×ℝ,ℝ), ∇u⁡η⁢(0,λ)=0 and ∇u2⁡η⁢(0,λ)=0 for every λ∈ℝ.

More precisely speaking, our purpose is to investigate the phenomenon of global bifurcations of weak solutions of system (1.1). In other words, we study closed connected sets of weak solutions of this system satisfying the symmetric Rabinowitz alternative; see Theorem A.3. For the classical Rabinowitz alternative, we refer the reader to [5, 14, 17, 18, 19].

Global bifurcations of solutions of nonlinear problems have been studied under various conditions by many authors. Some references and a discussion of these results can be found in [22]. Below we discuss only some results concerning symmetric nonlinear problems where the authors have eliminated one of the Rabinowitz alternatives showing that some (or all) global solution branches are either bounded or unbounded.

An elliptic boundary value problem on a two-dimensional annulus has been considered by Dancer [7]. He has studied the bifurcation of non-radially symmetric solutions from radially symmetric positive ones. The existence of many distinct global branches of non-symmetric solutions which do not intersect has been shown in this article.

Healey and Kielhöfer [13] have studied global symmetry-breaking equilibria of the van der Waals–Cahn–Hilliard phase-field model on the sphere S2. What is interesting is that they have proved that all the continua of nontrivial solutions are bounded! Thanks to the classical Rabinowitz alternative (because the unboundedness of continua has been eliminated), these continua meet the set of trivial solutions in at least two points. A general class of quasi-linear elliptic systems has been considered in [12]. It has been proved that, under additional assumptions on nodal sets of the eigenvalues of the linearized problem, some of the continua of nontrivial solutions are separated and that is why they are unbounded.

The Neumann problem on a two-dimensional ball has been considered in [15]. It has been proved that there are unbounded continua consisting of non-radially symmetric solutions emanating from the second and third eigenvalues of the Laplace operator. The Neumann problem on a ball of any dimensions has been studied by Miyamoto [16]. He has proved the existence of an unbounded continuum of non-radially symmetric solutions of this problem bifurcating from the second eigenvalue of the Laplace operator. Under additional assumptions, this continuum is unbounded in λ-direction.

A nonlinear eigenvalue problem on the sphere Sn-1 has been considered in [20]. It has been proved that any continuum of nontrivial solutions bifurcating from the trivial ones is unbounded. Similar results for the non-cooperative systems of elliptic differential equations have been obtained in [22].

In this article, we consider weak solutions of problem (1.1) as SO⁡(n)-orbits of critical points of an SO⁡(n)-invariant functional defined on a suitably chosen infinite-dimensional orthogonal representation of SO⁡(n). This justifies an application of a special degree theory, i.e., the degree for equivariant gradient maps, see [9, 10, 21] for the definition and properties of this invariant. It is worth to point out that this degree is an element of the Euler ring U⁢(SO⁡(n)) of SO⁡(n); see [25, 24] for the definition and properties of this ring. We emphasize that the advantage of using the degree for equivariant gradient maps lies in the fact that it allows us to distinguish homotopy classes of equivariant gradient maps which are not distinguished by the Leray–Schauder degree, i.e., for the class of equivariant gradient maps the degree for equivariant gradient maps is more sensitive than the Leray–Schauder degree.

We have proved that if the geodesic ball is a hemisphere B⁢(π2), then any continuum of weak solutions of system (1.1) which bifurcates from the set of trivial ones is unbounded; see Theorems 3.1 and 3.2. In other words, one of the Rabinowitz alternatives is eliminated by showing that all global solution branches are unbounded.

How did we prove it? Applying the degree for strongly indefinite SO⁡(n)-invariant functionals, we associate a bifurcation index defined by formula (A.4) to each point at which the necessary condition for bifurcation is satisfied. Next we show that for any choice of a finite number of these indices, their sum is nontrivial in the Euler ring U⁢(SO⁡(n)). Hence the symmetric Rabinowitz alternative (see Theorem A.3) implies unboundedness of the bifurcating continua. We have received it through a careful analysis of the eigenspaces of the Laplace–Beltrami operator ΔSn as representations of SO⁡(n); see Remark 2.6, Theorem 2.7 and Corollary 2.8. We emphasize that in the classical Rabinowitz alternative the sum of bifurcation indices is computed in the ring of integers ℤ because they are defined in terms of the Leray–Schauder degrees. Since such indices equal ±2 or 0, it is impossible to show that for any choice of a finite number of bifurcation indices their sum is nontrivial in ℤ.

It would be desirable to prove unboundedness of continua for a geodesic ball B⁢(γ) of any radius 0<γ<π, but at this moment this problem is far from being solved.

For a geodesic ball of an arbitrary radius 0<γ<π, we have characterized bifurcation points at which the global symmetry-breaking phenomenon occurs; see Theorem 3.5 and Corollary 3.6. Finally, a necessary condition for the existence of bounded continua is presented in Theorem 3.3.

After this introduction, our article is organized as follows: In Section 2, we recall basic properties of non-cooperative elliptic systems considered on geodesic balls and eigenvalues of the Laplace–Beltrami operator on these balls. The main problem is given by formula (2.1). The associated functional is defined by formula (2.2). Its properties are described in Lemma 2.2. We introduce a notion of a local bifurcation of solutions of nonlinear problems in Definition 2.4. The set of parameters at which the bifurcation of solutions of problem (2.1) can occur is described in Lemma 2.3 and Theorem 2.5. Finally, properties of the eigenvalues and eigenspaces of the Laplace–Beltrami operator considered on geodesic balls in Sn (with Dirichlet boundary conditions) are described in Remark 2.6, Theorem 2.7 and Corollary 2.8.

In Section 3, the main results of this article are stated and proved. The unboundedness of continua of solutions of system (1.1) on a hemisphere, i.e., for γ=π2, are proved in Theorems 3.1 and 3.2. A characterization of bifurcation points of this system at which the global symmetry-breaking of solutions phenomenon occurs on a geodesic ball of an arbitrary radius is given by Theorem 3.5 and Corollary 3.6. A necessary condition for the existence of bounded continua of solutions of problem (1.1) are formulated in Theorem 3.3.

In Section A, for the convenience of the reader, we have repeated the relevant material on equivariant bifurcation theory, thus making our exposition self-contained. For the definition of the Euler ring U⁢(G) of a compact Lie group G we refer the reader to [25, 24]. Since most of the computations in this article are done in the Euler ring U⁢(SO⁡(2)), we have reminded the definition of this ring; see Definition A.1. Next we have reminded the classification of orthogonal representations of SO⁡(2); see (A.2). A definition of the bifurcation index, which is an element of the Euler ring U⁢(SO⁡(n)), is given by formula (A.4). Remark A.2 allows us to reduce difficult computations in the Euler ring U⁢(SO⁡(n)) to much simpler computations in the Euler ring of SO⁡(2). The essential role in the proofs of the main results of this article plays the symmetric Rabinowitz alternative; see Theorem A.3. In the technical Lemmas A.13 and A.14, we have proved formulas for bifurcation indices which play crucial role in the proofs of our results.

2 Preliminaries

In this section, we remind some definitions from the equivariant topology. Moreover, we present a variational setting of our problem and study properties of the associated functional.

Throughout this section, G stands for a real compact Lie group. Let X be a topological space. An action of G on X is a continuous map ρ:G×X→X such that

ρ ⁢ ( g , ρ ⁢ ( h , x ) ) = ρ ⁢ ( g ⁢ h , x ) for g,h∈G,x∈X;
ρ ⁢ ( e , x ) = x for x∈X and the unit e∈G.

A G-space is a pair (X,ρ) consisting of a space together with an action of G on X. Usually the G-space (X,ρ) is denoted just by the underlying topological space X, and ρ⁢(g,x) is denoted by gx. An action of G on X is called trivial if g⁢x=x for all x∈X,g∈G. For each x∈X, the set G⁢(x)={g⁢x:g∈G} is called the orbit through x and Gx={g∈G:g⁢x=x} is called the isotropy group of x. A subset A of a G-space X is said to be G-invariant if for all x∈A and g∈G we have g⁢x∈A, i.e., G⁢(x)⊂A for any x∈A. By XG we denote the space of fixed points of the action of the group G on X, i.e., XG={x∈X:g⁢x=x⁢ for all ⁢g∈G}.

Suppose that X and Y are G-spaces. A continuous map f:X→Y is called G-equivariant if for all g∈G and x∈X the equality f⁢(g⁢x)=g⁢f⁢(x) holds true. Moreover, if Y=ℝ with the trivial action of G, then a map f:X→ℝ is said to be G-invariant.

Definition 2.1.

Let 𝕍, 𝕍′ be representations of G. We say that 𝕍 and 𝕍′ are equivalent if there is a G-equivariant linear isomorphism L:𝕍→𝕍′. For the sake of simplicity, we denote this relation briefly by 𝕍≈G𝕍′.

Throughout this article, SO⁡(n) stands for the real special orthogonal group.

Consider the sphere Sn={x∈ℝn+1:∥x∥=1} and the metric between two points p,q∈Sn defined by d⁢(p,q)=infω⁡∫ab|ω′⁢(t)|⁢𝑑t, where ω ranges over all continuous, piecewise C1 paths ω:[a,b]→Sn for which ω⁢(a)=p and ω⁢(b)=q. Define the geodesic ball in Sn centered at N=(0,…,0,1) and with radius γ∈(0,π) by B⁢(γ)={q∈Sn:d⁢(N,q)<γ}. The geodesic ball B⁢(γ) is an SO⁡(n)-invariant subset of the representation ℝn+1 of the group SO⁡(n) with the action SO⁡(n)×ℝn+1→ℝn+1 given by

( g , x ) = ( g , ( x 1 , … , x n , x n + 1 ) ) ↦ ( g ⁢ ( x 1 , x 2 , … , x n ) , x n + 1 ) .

Consider the following system of equations:

(2.1) { Λ ⁢ Δ S n ⁢ u = ∇ u ⁡ F ⁢ ( u , λ ) in ⁢ B ⁢ ( γ ) , u = 0 on ⁢ ∂ ⁡ B ⁢ ( γ ) ,

with the following assumptions:

F ∈ C 2 ⁢ ( ℝ p × ℝ , ℝ ) , ∇u⁡F⁢(u,λ)=(∇u1⁡F⁢(u,λ),…,∇up⁡F⁢(u,λ)).
∇ u ⁡ F ⁢ ( u , λ ) = λ ⁢ u + ∇ u ⁡ η ⁢ ( u , λ ) , where η∈C2⁢(ℝp×ℝ,ℝ), and for every λ∈ℝ follows ∇u⁡η⁢(0,λ)=0 and ∇u2⁡η⁢(0,λ)=0.
There exist C>0 and 1≤s<(n+2)⁢(n-2)-1 such that |∇u2⁡F⁢(u,λ)|≤C⁢(1+|u|s-1) (if n=2, we assume that s∈[1,+∞)).
Λ = diag ⁡ ( α 1 , … , α p ) , α i ∈ { - 1 , 1 } .

If the coefficients αi are not of the same sign, we call system (2.1) non-cooperative. From now on, p- (resp. p+) stands for the number of negative (resp. positive) αi, i=1,…,p.

Let H01⁢(B⁢(γ)) denote the Sobolev space with the inner product

〈 u , v 〉 H 0 1 ⁢ ( B ⁢ ( γ ) ) = ∫ B ⁢ ( γ ) 〈 ∇ ⁡ u ⁢ ( x ) , ∇ ⁡ v ⁢ ( x ) 〉 ⁢ 𝑑 σ

for all u,v∈H01⁢(B⁢(γ)). The space H01⁢(B⁢(γ)) is an orthogonal representation of SO⁡(n) with the action given by SO⁡(n)×H01⁢(B⁢(γ))∋(g,u)↦g⁢u∈H01⁢(B⁢(γ)), where (g⁢u)⁢(x)=u⁢(g-1⁢x). Let ℍ be the direct sum of p copies of the representation H01⁢(B⁢(γ)), i.e., ℍ=⊕i=1pH01⁢(B⁢(γ)). We consider ℍ×ℝ as a representation of SO⁡(n) with the action given by SO⁡(n)×(ℍ×ℝ)∋(g,(u,λ))↦(g⁢u,λ)∈ℍ×ℝ.

Define a family of SO⁡(n)-invariant functionals Φ:ℍ×ℝ→ℝ of the class C2 by

(2.2) Φ ⁢ ( u , λ ) = 1 2 ⁢ ∫ B ⁢ ( γ ) ∑ i = 1 p ( - α i ⁢ | ∇ ⁡ u i ⁢ ( x ) | 2 ) ⁢ d ⁢ σ - ∫ B ⁢ ( γ ) F ⁢ ( u ⁢ ( x ) , λ ) ⁢ 𝑑 σ ,

and note that

Φ ⁢ ( u , λ ) = ∑ i = 1 p 1 2 ⁢ ∫ B ⁢ ( γ ) ( - α i ⁢ | ∇ ⁡ u i ⁢ ( x ) | 2 ) ⁢ 𝑑 σ - λ 2 ⁢ ∫ B ⁢ ( γ ) | u ⁢ ( x ) | 2 ⁢ 𝑑 σ - ∫ B ⁢ ( γ ) η ⁢ ( u ⁢ ( x ) , λ ) ⁢ 𝑑 σ

= - 1 2 ⁢ ∑ i = 1 p α i ⁢ ∥ u i ∥ H 0 1 ⁢ ( B ⁢ ( γ ) ) 2 - λ 2 ⁢ ∫ B ⁢ ( γ ) | u i ⁢ ( x ) | 2 ⁢ 𝑑 σ - ∫ B ⁢ ( γ ) η ⁢ ( u ⁢ ( x ) , λ ) ⁢ 𝑑 σ .

Define T:H01⁢(B⁢(γ))→H01⁢(B⁢(γ)) and η0:ℍ×ℝ→ℝ by

〈 T ⁢ u , v 〉 H 0 1 ⁢ ( B ⁢ ( γ ) ) = ∫ B ⁢ ( γ ) u ⁢ ( x ) ⁢ v ⁢ ( x ) ⁢ 𝑑 σ , η 0 ⁢ ( u , λ ) = ∫ B ⁢ ( γ ) η ⁢ ( u ⁢ ( x ) , λ ) ⁢ 𝑑 σ

for all v∈H01⁢(B⁢(γ)).

Since the functional Φ⁢(⋅,λ) is SO⁡(n)-invariant, its gradient ∇u⁡Φ⁢(⋅,λ) is SO⁡(n)-equivariant. The following lemma is a direct consequence of the above computations.

Lemma 2.2.

Under the above assumptions, we have

∇ u ⁡ Φ ⁢ ( u , λ ) = L ⁢ u - λ ⁢ K ⁢ ( u ) - ∇ u ⁡ η 0 ⁢ ( u , λ ) = ( - α 1 ⁢ u 1 - λ ⁢ T ⁢ u 1 , … , - α p ⁢ u p - λ ⁢ T ⁢ u p ) - ∇ u ⁡ η 0 ⁢ ( u , λ ) ,

where the following assertions hold:

L : ℍ → ℍ given by L ⁢ ( u 1 , … , u p ) = ( - α 1 ⁢ u 1 , … , - α p ⁢ u p ) is a self-adjoint, bounded SO ⁡ ( n ) -equivariant Fredholm operator.
K = ( T , … , T ) : ℍ → ℍ is a self-adjoint, bounded, completely continuous SO ⁡ ( n ) -equivariant operator.
∇ u ⁡ η 0 ⁢ ( u , λ ) : ℍ × ℝ → ℍ is a completely continuous, SO ⁡ ( n ) -equivariant operator such that ∇ u ⁡ η 0 ⁢ ( 0 , λ ) = 0 and ∇ u 2 ⁡ η 0 ⁢ ( 0 , λ ) = 0 for every λ ∈ ℝ .
u = ( u 1 , … , u p ) ∈ ℍ is a weak solution of system ( 2.1 ) if and only if ∇ u ⁡ Φ ⁢ ( u , λ ) = 0 , i.e., u is a critical point of Φ.

Denote by σ(-ΔSn;B(γ))={0<λ1<λ2<⋯} the set of all eigenvalues of the problem

(2.3) { - Δ S n ⁢ u ⁢ ( x ) = λ ⁢ u ⁢ ( x ) in ⁢ B ⁢ ( γ ) , u ⁢ ( x ) = 0 on ⁢ ∂ ⁡ B ⁢ ( γ ) ,

and by V-ΔSnγ⁢(λm0) the eigenspace of -ΔSn associated with the eigenvalue λm0∈σ⁢(-ΔSn;B⁢(γ)). Set

σ - ⁢ ( - Δ S n ; B ⁢ ( γ ) ) = { - λ m : λ m ∈ σ ⁢ ( - Δ S n ; B ⁢ ( γ ) ) }

and

𝒫 γ ⁢ ( Φ ) = { λ 0 ∈ ℝ : ∇ u 2 ⁡ Φ ⁢ ( 0 , λ 0 ) = L - λ 0 ⁢ K ⁢ is not an isomorphism } .

In the next lemma, we formulate basic properties of eigenvalues and eigenspaces of the Laplace–Beltrami operator on the geodesic ball B⁢(γ). We omit an easy proof of this lemma.

Lemma 2.3.

Under the above assumptions, the following assertions hold:

We have

𝒫 γ ⁢ ( Φ ) = { σ ⁢ ( - Δ S n ; B ⁢ ( γ ) ) when ⁢ p - = p , σ - ⁢ ( - Δ S n ; B ⁢ ( γ ) ) when ⁢ p + = p , σ ⁢ ( - Δ S n ; B ⁢ ( γ ) ) ∪ σ - ⁢ ( - Δ S n ; B ⁢ ( γ ) ) when ⁢ p - ⁢ p + > 0 .
σ ⁢ ( K ) = { 1 λ m : λ m ∈ σ ⁢ ( - Δ S n ; B ⁢ ( γ ) ) } .
We have

V K ⁢ ( 1 λ m ) = ⊕ i = 1 p V - Δ S n γ ⁢ ( λ m ) .
For every λ m ∈ σ ⁢ ( - Δ S n ; B ⁢ ( γ ) ) , the eigenspace V - Δ S n γ ⁢ ( λ m ) ⊂ H 0 1 ⁢ ( B ⁢ ( γ ) ) is a finite dimensional orthogonal representation of SO ⁡ ( n ) .
We have

H 0 1 ⁢ ( B ⁢ ( γ ) ) = cl ⁢ ( ⊕ m = 1 ∞ V - Δ S n γ ⁢ ( λ m ) ) .

Moreover, for λ0∈Pγ⁢(Φ), the following assertions hold:

If λ 0 > 0 , then

p - > 0 , λ 0 ∈ σ ⁢ ( - Δ S n ; B ⁢ ( γ ) ) , ker ⁢ ∇ u 2 ⁡ Φ ⁢ ( 0 , λ 0 ) = ⊕ i = 1 p - V - Δ S n γ ⁢ ( λ 0 ) .
If λ 0 < 0 , then

p + > 0 , - λ 0 ∈ σ ⁢ ( - Δ S n ; B ⁢ ( γ ) ) , ker ⁢ ∇ u 2 ⁡ Φ ⁢ ( 0 , - λ 0 ) = ⊕ i = 1 p + V - Δ S n γ ⁢ ( - λ 0 ) .

Let us remind that ∇u⁡Φ⁢(0,λ)=0 for every λ∈ℝ.

Definition 2.4.

A solution (0,λ) of the equation ∇u⁡Φ⁢(u,λ)=0 is said to be trivial. A point (0,λ0) is said to be a bifurcation point of solutions of the equation ∇u⁡Φ⁢(u,λ)=0 if

( 0 , λ 0 ) ∈ cl ⁢ { ( u , λ ) ∈ ℍ × ℝ : ∇ u ⁡ Φ ⁢ ( u , λ ) = 0 , u ≠ 0 } .

In the following theorem, we formulate a necessary condition for the existence of bifurcation points of solutions of the equation ∇u⁡Φ⁢(u,λ)=0. This theorem is a direct consequence of Lemmas 2.2 and 2.3 and the implicit function theorem.

Theorem 2.5.

If (0,λ0) is a bifurcation point of solutions of the equation ∇u⁡Φ⁢(u,λ)=0, then λ0∈Pγ⁢(Φ).

Let (t,θ) be the geodesic spherical coordinate on Sn. The eigenvectors of problem (2.3) are of the form u⁢(t,θ)=Tm⁢(λ,t)⁢vm⁢(θ) (see [3, 4]), where m≥0 and vm⁢(θ) is a spherical harmonic of n variables and degree m, i.e., vm is a solution of the equation

- Δ S n - 1 ⁢ v ⁢ ( θ ) = β m ⁢ v ⁢ ( θ ) , where ⁢ β m = m ⁢ ( m + n - 2 ) ,

and Tm⁢(λ,t) is a solution of the equation

T ′′ ⁢ ( t ) + ( n - 1 ) ⁢ ( cot ⁡ t ) ⁢ T ′ ⁢ ( t ) + ( λ - β m sin 2 ⁡ t ) ⁢ T ⁢ ( t ) = 0 .

The explicit formula for Tm⁢(λ,t) is given in [3].

For m≥0, define Amγ={λ>0:Tm⁢(λ,γ)=0}. From the general theory of the eigenvalue problem, the set Amγ is countable and σ⁢(-ΔSn;B⁢(γ))=⋃m=0∞Amγ; see [3]. Moreover, if λ0∈Amγ and λ0∉Am′γ for all m′≠m, then V-ΔSnγ⁢(λ0) is equivalent as a representation of SO⁡(n) to ℋmn, where ℋmn denotes the linear space of harmonic, homogeneous polynomials of n independent variables, of degree m, restricted to the sphere Sn-1; see Section A.

Remark 2.6.

Fix λ0∈σ⁢(-ΔSn;B⁢(γ)) and define Γγ⁢(λ0)={m≥0:λ0∈Amγ}. Since the multiplicity of λ0 is finite, we have card⁢(Γγ⁢(λ0))<∞. Without loss of generality, one can assume that Γγ⁢(λ0)={m1,…,mq} and that 0≤m1<⋯<mq. Finally, we obtain that

V - Δ S n γ ⁢ ( λ 0 ) ≈ SO ⁡ ( n ) ℋ m 1 n ⊕ ⋯ ⊕ ℋ m q n ,

i.e., the representations V-ΔSnγ⁢(λ0) and ℋm1n⊕⋯⊕ℋmqn are SO⁡(n)-equivalent.

In the theorem below, we discuss the special case of a hemisphere, i.e., γ=π2.

Theorem 2.7 ([3, Theorem 6]).

Suppose that γ=π2. Then

σ ⁢ ( - Δ S n ; B ⁢ ( π 2 ) ) = { λ m = m ⁢ ( n + m - 1 ) : m ∈ ℕ }

and

V - Δ S n π / 2 ⁢ ( λ m ) ≈ SO ⁡ ( n ) ⊕ l : ∃ p ∈ ℕ ∪ { 0 } m = 2 ⁢ p + l + 1 ℋ l n .

Moreover, the multiplicity of λm is (n+m-2n-1) for every m∈N.

The following corollary is a direct consequence of the above theorem.

Corollary 2.8.

Fix λm∈σ⁢(-ΔSn;B⁢(π2)).

If m is even, then

V - Δ S n π / 2 ⁢ ( λ m ) ≈ SO ⁡ ( n ) ℋ 1 n ⊕ ℋ 3 n ⊕ ⋯ ⊕ ℋ m - 1 n .
If m is odd, then

V - Δ S n π / 2 ⁢ ( λ m ) ≈ SO ⁡ ( n ) ℋ 0 n ⊕ ℋ 2 n ⊕ ⋯ ⊕ ℋ m - 1 n .

Consequently,

ℋ m - 1 n ⊂ V - Δ S n π / 2 ⁢ ( λ m ) 𝑎𝑛𝑑 ℋ m - 1 n ⊄ V - Δ S n π / 2 ⁢ ( λ m ^ )

for every 0<m^<m.

3 Main Results

In this section, we study continua of weak solutions of non-cooperative elliptic systems considered on a geodesic ball B⁢(γ) where γ∈(0,π). Consider system (2.1) with F and Λ satisfying assumptions (A1)–(A4) of Section 2.

Recall that u∈ℍ=⊕i=1pH01⁢(B⁢(γ)) is a weak solution of the above system if and only if u is a critical point of the functional Φ:ℍ×ℝ→ℝ given by formula (2.2). That is why we study in this section solutions of the equation ∇u⁡Φ⁢(u,λ)=0.

Denote by C⁢(λ0)⊂ℍ×ℝ the continuum of cl⁢{(u,λ)∈ℍ×ℝ:∇u⁡Φ⁢(u,λ)=0,u≠0} containing (0,λ0).

Our first aim is to prove unboundedness of continua of weak solutions of system (2.1) with γ=π2, bifurcating from the set of trivial ones.

To prove the famous global bifurcation theorem, Rabinowitz has applied the Leray–Schauder degree, being an element of the ring of integers ℤ. Since the system of elliptic differential equations considered in this article is SO⁡(n)-symmetric and has a variational structure, to study the phenomenon of global bifurcations of weak solutions of this system we apply a more subtle invariant defined for the class of equivariant gradient operators, i.e., we apply the degree for SO⁡(n)-invariant strongly indefinite functionals, defined in [10], being an element of the Euler ring U⁢(SO⁡(n)).

Our idea to prove unboundedness of the bifurcating continua of weak solutions is to eliminate one of the possibilities in the symmetric Rabinowitz alternative; see Theorem A.3. More precisely speaking, first we attach to every λ∈𝒫π/2⁢(Φ) the bifurcation index ℬ⁢ℐ⁢ℱSO⁡(n)⁢(λ)∈U⁢(SO⁡(n)) defined in terms of the degree for SO⁡(n)-invariant strongly indefinite functionals. Next we prove that for any choice of λ1,…,λs∈𝒫π/2⁢(Φ), the sum of bifurcation indices ℬ⁢ℐ⁢ℱSO⁡(n)⁢(λ1)+⋯+ℬ⁢ℐ⁢ℱSO⁡(n)⁢(λs)∈U⁢(SO⁡(n)) is nontrivial in U⁢(SO⁡(n)). Note that the nontriviality of this sum implies the unboundedness of every bifurcating continua.

Since computations in the Euler ring U⁢(SO⁡(n)) are difficult, we reduce them to simpler computations in the better-known ring U⁢(SO⁡(2)), i.e., we show that the sum ℬ⁢ℐ⁢ℱSO⁡(2)⁢(λ1)+⋯+ℬ⁢ℐ⁢ℱSO⁡(2)⁢(λs) is nontrivial in U⁢(SO⁡(2)), which implies that the sum ℬ⁢ℐ⁢ℱSO⁡(n)⁢(λ1)+⋯+ℬ⁢ℐ⁢ℱSO⁡(n)⁢(λs) is nontrivial in U⁢(SO⁡(n)); see Remark A.2. To do this, we carefully study the structure of the eigenspaces of the Laplace–Beltrami operator as representations of the group SO⁡(2) given in Corollaries 2.8 and A.8. More precisely speaking, we apply the fact that for any m there is an element in V-ΔSnπ/2⁢(λm) whose orbit type does not appear as an orbit type of elements of V-ΔSnπ/2⁢(λm^) for 0<m^<m. We follow this idea to prove unboundedness of the bifurcating continua in Theorems 3.1, 3.2 and 3.3.

Let μm=dim⁡V-ΔSnπ/2⁢(λm) and νm=μ1+⋯+μm for m∈ℕ.

Theorem 3.1.

Fix λm0∈σ⁢(-ΔSn;B⁢(π2))∖{λ1}. Then the continuum C⁢(±λm0)⊂H×R of weak solutions of system (2.1) is unbounded.

Proof.

We prove the unboundedness of the continuum C⁢(λm0). The proof for the continuum C⁢(-λm0) is literally the same and left to the reader. Since λm0≠λ1, from Theorem A.6 it follows that V-ΔSn⁢(λm0) is a nontrivial representation of SO⁡(n). Suppose, contrary to our claim, that the continuum C⁢(λm0)⊂ℍ×ℝ is bounded. Then from the symmetric Rabinowitz alternative (see Theorem A.3) it follows that

C ⁢ ( λ m 0 ) ∩ ( { 0 } × ℝ ) = { 0 } × { λ ^ 1 , … , λ ^ s } ⊂ { 0 } × 𝒫 π / 2 ⁢ ( Φ ) ,

(3.1) ∑ j = 1 s ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ ^ j ) = Θ ∈ U ⁢ ( SO ⁡ ( n ) ) .

Without loss of generality, one can assume that

λ ^ 1 < ⋯ < λ ^ s ′ < 0 < λ ^ s ′ + 1 < ⋯ < λ ^ s .

Since {λ^1,…,λ^s}⊂𝒫π/2⁢(Φ), there are λm1,…,λms′,λms′+1,…,λms∈σ⁢(-ΔSn;B⁢(π2)) such that λ^j=-λmj for j=1,…,s′ and λ^j=λmj for j=s′+1,…,s, i.e.,

- λ m 1 < ⋯ < - λ m s ′ < 0 < λ m s ′ + 1 < ⋯ < λ m s

and m1>⋯>ms′, ms>⋯>ms′+1.

It is difficult to verify formula (3.1) because the ring U⁢(SO⁡(n)) has a complicated structure. That is why we apply the homomorphism i⋆:U⁢(SO⁡(n))→U⁢(SO⁡(2)) defined in Remark A.2, to reduce the computations to the much easier but a little bit technical computations in the ring U⁢(SO⁡(2)).

Namely, since i⋆ is a homomorphism, to finish the proof we will investigate the following equality:

(3.2) ∑ j = 1 s ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ ^ j ) = i ⋆ ⁢ ( ∑ j = 1 s ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ ^ j ) ) = Θ ∈ U ⁢ ( SO ⁡ ( 2 ) ) .

To complete the proof it is enough to show that the above equality does not hold true. From equality (3.2) we obtain

(3.3) ∑ j = 1 s ′ ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( - λ m j ) + ∑ j = s ′ + 1 s ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ m j ) = Θ ∈ U ⁢ ( SO ⁡ ( 2 ) ) .

What is left is to show that equality (3.3) is never satisfied. In the rest of the proof, we consider four cases.

Case: p-,p+∈2⁢N. Since p-,p+ are even, from Lemmas A.13 (ii) and A.14 (ii) it follows that

ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ m s ′ + 1 ) , … , ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ m s ) ∈ U - ⁢ ( SO ⁡ ( 2 ) )

and that

ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( - λ m 1 ) , … , ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( - λ m s ′ ) ∈ U - ⁢ ( SO ⁡ ( 2 ) ) .

To complete the proof it is enough to note that ℬ⁢ℐ⁢ℱSO⁡(2)⁢(λms)∈U-⁢(SO⁡(2))∖{Θ}. Indeed, by Lemma A.13 (i) it follows that ℬ⁢ℐ⁢ℱSO⁡(2)⁢(λms)≠Θ, which contradicts equality (3.3).

Case: p-∈2⁢N,p+∈2⁢N+1. Since p- is even, from Lemmas A.13 (i) and A.13 (ii) we obtain that

ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ m s ′ + 1 ) , … , ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ m s ) ∈ U - ⁢ ( SO ⁡ ( 2 ) ) ∖ { Θ } .

From Lemma A.13 (i) we obtain

ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ m s ) = ( 0 , α 1 , … , α m s - 2 , - p - , 0 , … ) ∈ U - ⁢ ( SO ⁡ ( 2 ) ) .

Moreover, for j=s′+1,…,s-1, if ℬ⁢ℐ⁢ℱSO⁡(2)⁢(λmj)=(α0,α1,…,αk,…), then αk=0 for k≥ms-1. From Lemma A.14 (i) we obtain

ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( - λ m 1 ) = ( α 0 , α 1 , … , α m 1 - 2 , ( - 1 ) 1 + ν m 1 ⁢ p + ⁢ p + , 0 , … ) ∈ U ⁢ ( SO ⁡ ( 2 ) ) .

Moreover, for j=2,…,s′, if

ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( - λ m j ) = ( α 0 , α 1 , … , α k , … ) ,

then αk=0 for k≥m1-1. From the above reasoning and formula (3.3) it follows that m1=ms and that the ms-th coordinate of formula (3.3) equals -p-+(-1)1+νm1⁢p+⁢p+=0. Since p-≠p+, formula (3.3) is not fulfilled, a contradiction.

Case: p-∈2⁢N+1,p+∈2⁢N. The proof is in fact the same as the proof of the previous case.

Case: p-,p+∈2⁢N+1. In the first case, we have considered the numbers p-, p+ of the same even parity. Now the numbers p-, p+ are of the same but odd parity. In this case, the bifurcation indices are not elements of U-⁢(SO⁡(2)). Taking into account Lemmas A.13 (i) and A.14 (i) and formula (3.3), we obtain m1=ms. Moreover, the ms-th coordinate of formula (3.3) has the following form:

( - 1 ) 1 + ν m 1 ⁢ p - ⁢ p - + ( - 1 ) 1 + ν m 1 ⁢ p + ⁢ p + = 0 .

Thus we obtain p-=-p+, a contradiction. ∎

In the theorem below, we describe continua C⁢(±λ1)⊂ℍ×ℝ of weak solutions of system (2.1), i.e., continua bifurcating from the first eigenvalue ±λ1.

Theorem 3.2.

If p∓ is odd and λ1∈σ⁢(-ΔSn;B⁢(π2)), then the continuum C⁢(±λ1)⊂H×R of weak solutions of system (2.1) is unbounded.

Proof.

We prove this theorem for p->0. The proof for p+>0 is literally the same and left to the reader. From Lemma A.13 (i) it follows that ℬ⁢ℐ⁢ℱSO⁡(2)⁢(λ1)=(-2,0,…,0,…)∈U⁢(SO⁡(2)). Suppose contrary to our claim that the continuum C⁢(λ1) is bounded. Then by the symmetric Rabinowitz alternative (see Theorem A.3) the continuum C⁢(λ1) meets the set of trivial solutions {0}×ℝ⊂ℍ×ℝ at a finite number of points. By Theorem 3.1, the continua C⁢(±λm), m>1, are unbounded. Therefore, C⁢(λ1)∩({0}×ℝ)={0}×{-λ1,λ1}. Moreover, ℬ⁢ℐ⁢ℱSO⁡(n)⁢(λ1)+ℬ⁢ℐ⁢ℱSO⁡(n)⁢(-λ1)=Θ∈U⁢(SO⁡(n)), and consequently

(3.4) ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ 1 ) + ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( - λ 1 ) = i ⋆ ⁢ ( ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ 1 ) + ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( - λ 1 ) ) = Θ ∈ U ⁢ ( SO ⁡ ( 2 ) ) .

By Lemma A.14 (i), we have ℬ⁢ℐ⁢ℱSO⁡(2)⁢(-λ1)=((-1)p+-1,0,…,0,…)∈U⁢(SO⁡(2)). Therefore,

ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ 1 ) SO ⁡ ( 2 ) + ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( - λ 1 ) SO ⁡ ( 2 ) = - 2 + ( - 1 ) p + - 1 = - 3 + ( - 1 ) p + ≠ 0 ,

which contradicts equality (3.4). ∎

From now on, we consider system (2.1) on a geodesic ball B⁢(γ)⊂Sn with γ∈(0,π). Since in this case the structure of eigenspaces as representations of SO⁡(2) is not known explicitly, the reasoning from the proofs of Theorems 3.1 and 3.2 cannot be repeated.

In the theorem below, we formulate necessary conditions for boundedness of continua of weak solutions of system (2.1).

Theorem 3.3.

Fix λm0∈σ⁢(-ΔSn;B⁢(γ)) such that dim⁡V-ΔSnγ⁢(λm0)>1. Then if p∓>0 is even and the continuum C⁢(±λm0)⊂H×R is bounded, then p±>0 is odd and

C ⁢ ( ± λ m 0 ) ∩ ( { 0 } × σ ∓ ⁢ ( - Δ S n ; B ⁢ ( γ ) ) ) ≠ ∅ .

Proof.

We prove this theorem for even p->0. The proof for even p+>0 is literally the same and left to the reader. Suppose, contrary to our claim, that p->0 is even, the continuum C⁢(λm0)⊂ℍ×ℝ is bounded and p+>0 is even or C⁢(λm0)∩({0}×σ-⁢(-ΔSn;B⁢(γ)))=∅.

Note that λm0∈Alγ for l>0 because dim⁡V-ΔSnγ⁢(λm0)>1 and dim⁡ℋn0=1; see Theorem A.5. Therefore, combining Lemma 2.3 and Remark 2.6 with Theorems A.5 and A.6, we obtain that V-ΔSnγ⁢(λm0) is a nontrivial representation of SO⁡(n). Since the continuum C⁢(λm0)⊂ℍ×ℝ is bounded, from the symmetric Rabinowitz alternative (Theorem A.3) it follows that

C ⁢ ( λ m 0 ) ∩ ( { 0 } × ℝ ) = { 0 } × { λ ^ 1 , … , λ ^ s } ⊂ { 0 } × 𝒫 γ ⁢ ( Φ ) ,

∑ j = 1 s ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ ^ j ) = Θ ∈ U ⁢ ( SO ⁡ ( n ) ) .

Without loss of generality, one can assume that

λ ^ 1 < ⋯ < λ ^ s ′ < 0 < λ ^ s ′ + 1 < ⋯ < λ ^ s .

Since {λ^1,…,λ^s}⊂𝒫γ⁢(Φ), there are

λ m 1 , … , λ m s ′ , λ m s ′ + 1 , … , λ m s ∈ σ ⁢ ( - Δ S n ; B ⁢ ( γ ) )

such that λ^j=-λmj for j=1,…,s′ and λ^j=λmj for j=s′+1,…,s, i.e.,

- λ m 1 < ⋯ < - λ m s ′ < 0 < λ m s ′ + 1 < ⋯ < λ m s

and m1>⋯>ms′,ms>⋯>ms′+1.

By Remark A.2, we obtain

∑ j = 1 s ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ ^ j ) = i ⋆ ⁢ ( ∑ j = 1 s ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ ^ j ) ) = Θ ∈ U ⁢ ( SO ⁡ ( 2 ) ) .

That is why we obtain the following equality:

(3.5) ∑ j = 1 s ′ ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( - λ m j ) + ∑ j = s ′ + 1 s ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ m j ) = Θ ∈ U ⁢ ( SO ⁡ ( 2 ) ) .

Since p- is even, taking into account Lemmas A.13 (i) and A.13 (ii), we obtain

(3.6) ∑ j = s ′ + 1 s ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ m j ) ∈ U - ⁢ ( SO ⁡ ( 2 ) ) ∖ { Θ } .

Comparing formulas (3.5) and (3.6), we obtain that

C ⁢ ( λ m 0 ) ∩ ( { 0 } × σ - ⁢ ( - Δ S n ; B ⁢ ( γ ) ) ) ≠ ∅ and p + > 0 ⁢ is odd.

Indeed, if p+ is even, then by Lemma A.14 (ii) we obtain

(3.7) ∑ j = 1 s ′ ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( - λ m j ) ∈ U - ⁢ ( SO ⁡ ( 2 ) ) .

But formulas (3.6) and (3.7) contradict formula (3.5), which implies that p+ is odd, a contradiction. ∎

Definition 3.4.

We say that (0,λm0)∈ℍ×ℝ is a global symmetry-breaking bifurcation point of solutions of system (2.1) if there exists an open SO⁡(n)-invariant neighborhood U⊂ℍ×ℝ of (0,λm0) such that the isotropy group SO(n)(u,λ) of every element (u,λ)∈(U∩C⁢(λm0))∖({0}×ℝ) is different from SO⁡(n).

In the theorem below, we characterize global symmetry-breaking points of weak solutions of system (2.1).

Theorem 3.5.

Assume that λm0∈σ⁢(-ΔSn,B⁢(γ))∖A0γ. If p∓>0, then (0,±λm0)∈H×R is a global symmetry-breaking bifurcation point of solutions of system (2.1).

Proof.

We prove this theorem for p->0. The proof for p+>0 is literally the same and left to the reader. Since λm0∉A0γ, from Remark 2.6 we obtain Γγ⁢(λm0)={m1,…,mq} and 0<m1<⋯<mq. Moreover,

V - Δ S n γ ⁢ ( λ m 0 ) ≈ SO ⁡ ( n ) ℋ m 1 n ⊕ ⋯ ⊕ ℋ m q n .

From Lemma 2.3 we obtain

ker ⁢ ∇ 2 ⁡ Φ ⁢ ( 0 , λ m 0 ) = ⊕ i = 1 p - V - Δ S n γ ⁢ ( λ m 0 ) .

Summing up, we obtain

ker ⁢ ∇ 2 ⁡ Φ ⁢ ( 0 , λ m 0 ) = ⊕ i = 1 p - ( ℋ m 1 n ⊕ ⋯ ⊕ ℋ m q n ) .

Since (ℋmn)SO⁡(n)={0} for every m>0, we have

( ker ⁢ ∇ 2 ⁡ Φ ⁢ ( 0 , λ m 0 ) ) SO ⁡ ( n ) = ⊕ i = 1 p - ( ( ℋ m 1 n ) SO ⁡ ( n ) ⊕ ⋯ ⊕ ( ℋ m q n ) SO ⁡ ( n ) ) = { 0 } .

The rest of the proof is a consequence of Theorem A.4. ∎

From Remark 2.6 and Corollary 2.8 follows that if λm0∈σ⁢(-ΔSn,B⁢(π2)) is such that m0 is even, then λm0∉A0π/2. Therefore, in the case γ=π2 we obtain the following corollary.

Corollary 3.6.

Fix λm0∈Pπ/2⁢(Φ). If m0∈Z is even, then the point (0,λm0)∈H×R is a global symmetry-breaking point of weak solutions of system (2.1).

Communicated by Paul Rabinowitz

Funding source: Narodowe Centrum Nauki

Award Identifier / Grant number: DEC-2012/05/B/ST1/02165

Funding statement: This work was partially supported by the National Science Center, Poland, under grant number DEC-2012/05/B/ST1/02165.

A Appendix

In this section, to make this article self-contained, we present all the material concerning equivariant bifurcation theory which we need in the proofs of the results of this paper.

Definition A.1.

The Euler ring of SO⁡(2) is defined by U⁢(SO⁡(2))=ℤ⊕⊕i=1∞ℤ and for

a = ( a 0 , a 1 , a 2 , … ) , b = ( b 0 , b 1 , b 2 , … ) ∈ ℤ ⊕ ⊕ i = 1 ∞ ℤ

we put

a + b = ( a 0 + b 0 , a 1 + b 1 , a 2 + b 2 , … ) ,

(A.1) a * b = ( a 0 ⁢ b 0 , a 1 ⁢ b 0 + a 0 ⁢ b 1 , a 2 ⁢ b 0 + a 0 ⁢ b 2 , … ) .

The element Θ=(0,0,0,…)∈U⁢(SO⁡(2)) is the neutral element and 𝕀=(1,0,0,…)∈U⁢(SO⁡(2)) is the unit.

The definition above agrees with that of [25, 24], where one can find further information, in particular the definition of the Euler ring U⁢(G) where G is a compact Lie group.

For a=(a0,a1,a2,…)∈U⁢(SO⁡(2)), the term a0 corresponds to the isotropy group SO⁡(2) and ai to the cyclic subgroup of SO⁡(2) which is isomorphic to the cyclic subgroup ℤi of S1 for i∈ℕ.

Put

U + ⁢ ( SO ⁡ ( 2 ) ) = { ( a 0 , a 1 , a 2 , … ) ∈ U ⁢ ( SO ⁡ ( 2 ) ) : a i ≥ 0 ⁢ for all ⁢ i ∈ ℕ ∪ { 0 } } ,

U - ⁢ ( SO ⁡ ( 2 ) ) = { ( a 0 , a 1 , a 2 , … ) ∈ U ⁢ ( SO ⁡ ( 2 ) ) : a i ≤ 0 ⁢ for all ⁢ i ∈ ℕ ∪ { 0 } } .

The degree for G-invariant strongly indefinite functionals ∇G⁡-deg⁢(⋅,⋅) is an element of the Euler ring U⁢(G); see [10] for the definition. For the general theory of the equivariant degree, we refer the reader to [2].

Let m∈ℕ and denote by ℝ⁢[1,m] the two-dimensional representation of SO⁡(2) with a linear SO⁡(2)-action defined by

( [ cos ⁡ φ - sin ⁡ φ sin ⁡ φ cos ⁡ φ ] , [ x y ] ) ↦ [ cos ⁡ m ⁢ φ - sin ⁡ m ⁢ φ sin ⁡ m ⁢ φ cos ⁡ m ⁢ φ ] ⁢ [ x y ] .

Note that if v∈ℝ⁢[1,m]∖{0}, then the isotropy group SO(2)v={g∈SO(2):gv=v} is isomorphic to ℤm. For k,m∈ℕ, we will denote by ℝ⁢[k,m] the direct sum of k copies of the representation ℝ⁢[1,m]. For k∈ℕ, we denote by ℝ⁢[k,0] the trivial k-dimensional representation of SO⁡(2).

It is known that any finite-dimensional, orthogonal representation 𝕍 of SO⁡(2) is equivalent to the representation of the form ℝ⁢[k0,0]⊕ℝ⁢[k1,m1]⊕⋯⊕ℝ⁢[kr,mr]; see [1]. Therefore, without loss of generality one can assume that

(A.2) 𝕍 = ℝ ⁢ [ k 0 , 0 ] ⊕ ℝ ⁢ [ k 1 , m 1 ] ⊕ ⋯ ⊕ ℝ ⁢ [ k r , m r ] .

Below, we present the formula for the degree of SO⁡(2)-equivariant gradient maps of the map

- Id : ( B ⁢ ( 𝕍 ) , S ⁢ ( 𝕍 ) ) → ( 𝕍 , 𝕍 ∖ { 0 } ) ,

where B⁢(𝕍) is an open ball in 𝕍 of radius 1 centered at the origin. Namely, it is known that

∇ SO ⁡ ( 2 ) ⁡ - ⁢ deg ⁡ ( - Id , B ⁢ ( 𝕍 ) ) = ( α 0 , α 1 , … , α i , … ) ∈ U ⁢ ( SO ⁡ ( 2 ) ) ,

where (see [9])

(A.3) α i = { ( - 1 ) k 0 if ⁢ i = 0 , ( - 1 ) k 0 + 1 ⁢ k p if ⁢ i = m p , p = 1 , … , r , 0 for ⁢ i ∉ { 0 , m 1 , … , m r } .

Now, with the functional Φ given by (2.2) we assign a bifurcation index in terms of the degree for SO⁡(n)-invariant strongly indefinite functionals. Fix λ0∈𝒫γ⁢(Φ) and define the SO⁡(n)-bifurcation index

ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ 0 ) ∈ U ⁢ ( SO ⁡ ( n ) )

(A.4) ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ 0 ) = ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( ∇ ⁡ Φ ⁢ ( ⋅ , λ 0 + ϵ ) , B δ ⁢ ( ℍ ) ) - ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( ∇ ⁡ Φ ⁢ ( ⋅ , λ 0 - ϵ ) , B δ ⁢ ( ℍ ) ) ,

where δ,ϵ>0 are sufficiently small.

Remark A.2.

The natural inclusion i:SO⁡(2)→SO⁡(n) defined by

i ⁢ ( g ) = [ g 0 0 Id n - 2 ]

induces a ring homomorphism i⋆:U⁢(SO⁡(n))→U⁢(SO⁡(2)). We define the SO⁡(2)-bifurcation index

ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ 0 ) ∈ U ⁢ ( SO ⁡ ( 2 ) )

by ℬ⁢ℐ⁢ℱSO⁡(2)⁢(λ0)=i⋆⁢(ℬ⁢ℐ⁢ℱSO⁡(n)⁢(λ0)). It is easy to see that

i ⋆ ⁢ ( ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ 0 ) ) = ∇ SO ⁡ ( 2 ) ⁡ - ⁢ deg ⁡ ( ∇ ⁡ Φ ⁢ ( ⋅ , λ 0 + ϵ ) , B δ ⁢ ( ℍ ) ) - ∇ SO ⁡ ( 2 ) ⁡ - ⁢ deg ⁡ ( ∇ ⁡ Φ ⁢ ( ⋅ , λ 0 - ϵ ) , B δ ⁢ ( ℍ ) ) .

The following theorem is a symmetric version of the famous Rabinowitz alternative; see [18, 19]. The Rabinowitz theorem says that a change of the Leray–Schauder degree (nontriviality of the bifurcation index) along the line of trivial solutions implies a global bifurcation of solutions of a nonlinear eigenvalue problem. The proof of this theorem is standard; see for instance [5, 8, 14, 17, 18, 19].

Since ∇u⁡Φ⁢(⋅,λ) is a family of strongly-indefinite SO⁡(n)-equivariant operators, it is enough to replace in the classical proof the Leray–Schauder degree by the degree for SO⁡(n)-invariant strongly indefinite functionals.

Finally, note that under the assumptions of the following theorem for λm0∈σ⁢(-ΔSn;B⁢(γ)) the bifurcation indices ℬ⁢ℐ⁢ℱSO⁡(n)⁢(λm0),ℬ⁢ℐ⁢ℱSO⁡(n)⁢(-λm0)∈U⁢(SO⁡(n)) are nontrivial. This is a consequence of Lemmas A.13 and A.14.

Theorem A.3 (Symmetric Rabinowitz alternative).

Fix λm0∈σ⁢(-ΔSn;B⁢(γ)).

If V - Δ S n γ ⁢ ( λ m 0 ) is a nontrivial representation of SO ⁡ ( n ) or p - ⋅ dim ⁡ V - Δ S n γ ⁢ ( λ m 0 ) is odd, then either C ⁢ ( λ m 0 ) is unbounded in ℍ × ℝ or the following assertions hold:
1. C ⁢ ( λ m 0 ) ⊂ ℍ × ℝ is bounded.
2. C ⁢ ( λ m 0 ) ∩ ( { 0 } × ℝ ) = { 0 } × { λ ^ 1 , … , λ ^ s } ⊂ { 0 } × 𝒫 γ ⁢ ( Φ ) and
  
  ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ ^ 1 ) + ⋯ + ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ ^ s ) = Θ ∈ U ⁢ ( SO ⁡ ( n ) ) .
If V - Δ S n γ ⁢ ( λ m 0 ) is a nontrivial representation of SO ⁡ ( n ) or p + ⋅ dim ⁡ V - Δ S n γ ⁢ ( λ m 0 ) is odd, then either C ⁢ ( - λ m 0 ) is unbounded in ℍ × ℝ or the following assertions hold:
1. C ⁢ ( - λ m 0 ) ⊂ ℍ × ℝ is bounded.
2. C ⁢ ( - λ m 0 ) ∩ ( { 0 } × ℝ ) = { 0 } × { λ ^ 1 , … , λ ^ s } ⊂ { 0 } × 𝒫 γ ⁢ ( Φ ) and
  
  ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ ^ 1 ) + ⋯ + ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ ^ s ) = Θ ∈ U ⁢ ( SO ⁡ ( n ) ) .

To characterize bifurcation points of system (2.1) at which the symmetry-breaking phenomenon occurs, we use the following theorem. Here we locally control the isotropy groups of the bifurcating solutions by the isotropy groups of elements of ker⁢∇u2⁡Φ⁢(0,λm). The proof of this theorem is a natural application of the Lyapunov–Schmidt reduction; it can be found for instance in [6].

Theorem A.4.

Suppose λm0∈Pγ⁢(Φ). Then there exists an open SO⁡(n)-invariant neighborhood U⊂H×R of (0,λm0) such that for all (u^,λ)∈(U∩(∇u⁡Φ)-1⁢(0))∖({0}×R) there exists u¯∈ker⁢∇u2⁡Φ⁢(0,λm0)∖{0} such that SO(n)u^=SO(n)u¯. Moreover, if ker⁢∇u2⁡Φ⁢(0,λm0)SO⁡(n)={0}, then for all (u^,λ)∈(U∩(∇u⁡Ψ)-1⁢(0))∖({0}×R) we have SO(n)u^≠SO(n).

In the theorem below, we formulate the basic properties of ΔSn-1. Recall that ℋmn denotes the linear space of harmonic, homogeneous polynomials of n independent variables, of degree m, restricted to the sphere Sn-1.

Theorem A.5 ([23, Theorem 4.1]).

The eigenvalues of ΔSn-1 are

λ m = m ⁢ ( m + n - 2 ) , m = 0 , 1 , 2 , … .

If V-ΔSn-1⁢(λm) is the eigenspace of -ΔSn-1 belonging to λm, then

V - Δ S n - 1 ⁢ ( λ m ) = ℋ m n .
We have

dim ⁡ V - Δ S n - 1 ⁢ ( λ m ) = { 1 if ⁢ n = 2 , m = 0 , 2 if ⁢ n = 2 , m ≥ 1 , ( 2 ⁢ m + n - 2 ) ⁢ ( n - 3 + m ) ! m ! ⁢ ( n - 2 ) ! if ⁢ n ≥ 3 , m ≥ 0 .
L 2 ⁢ ( S n - 1 ) = cl ⁢ ( ⊕ m = 0 ∞ V - Δ S n - 1 ⁢ ( λ m ) ) .

One can consider the space ℋmn as a representation of SO⁡(n) with the action given by the formula

SO ⁡ ( n ) × ℋ m n ∋ ( g , u ⁢ ( x ) ) ↦ u ⁢ ( g - 1 ⁢ x ) ∈ ℋ m n .

Theorem A.6 ([11, Theorem 5.1]).

For every m≥1, the space Hmn is a nontrivial, irreducible representation of SO⁡(n). Moreover, the space H0n is a trivial representation.

One can find a proof of the above theorem in [26].

One can consider the space ℋmn as a representation of SO⁡(2). Recall that

SO ⁡ ( 2 ) = { g ⁢ ( φ ) = [ cos ⁡ φ sin ⁡ φ - sin ⁡ φ cos ⁡ φ ] : φ ∈ [ 0 , 2 ⁢ π ) }

and define the action of SO⁡(2) on ℋmn by

SO ⁡ ( 2 ) × ℋ m n ∋ ( g ⁢ ( φ ) , u ⁢ ( x ) ) ↦ u ⁢ ( i ⁢ ( g ⁢ ( φ ) ) - 1 ⁢ x ) = u ⁢ ( i ⁢ ( g ⁢ ( - φ ) ) ⁢ x ) ∈ ℋ m n .

In other words, if u⁢(x1,…,xn)∈ℋmn, then the action of SO⁡(2) is given by

( g ⁢ ( φ ) , u ) ⁢ ( x 1 , … , x n ) ↦ u ⁢ ( x 1 ⁢ cos ⁡ φ + x 2 ⁢ sin ⁡ φ , - x 1 ⁢ sin ⁡ φ + x 2 ⁢ cos ⁡ φ , x 3 , … , x n ) .

To compute equivariant bifurcation indices we will use some properties of ℋmn as representations of SO⁡(2). Let us remind that spherical coordinates have the following form:

x 1 = sin ⁡ θ n - 1 ⁢ ⋯ ⁢ sin ⁡ θ 2 ⁢ sin ⁡ θ 1 ,

x 2 = sin ⁡ θ n - 1 ⁢ ⋯ ⁢ sin ⁡ θ 2 ⁢ cos ⁡ θ 1 ,

⁢ ⋮

x n - 1 = sin ⁡ θ n - 1 ⁢ cos ⁡ θ n - 2 ,

x n = cos ⁡ θ n - 1 ,

where 0≤θ1<2⁢π and 0≤θk<π, k≠1.

Lemma A.7 ([26, Chapter IX]).

An orthonormal basis of Hmn, n≥3, m>0, is given by functions of the form

C M ⁢ ( θ 2 , … , θ n - 1 ) ⁢ cos ⁡ ( m n - 2 ⁢ θ 1 ) , C M ⁢ ( θ 2 , … , θ n - 1 ) ⁢ sin ⁡ ( m n - 2 ⁢ θ 1 ) ,

where M=(m0,…,mn-3,mn-2), m=m0≥m1≥⋯≥mn-2≥0 and CM are the Gegenbauer polynomials.

As a consequence of the above lemma, we obtain a description of ℋmn, n≥3, m>0, as representations of SO⁡(2).

Corollary A.8.

Note that since the actions of SO⁡(2) on Hmn in the polar coordinates are given by

( g ⁢ ( φ ) , C M ⁢ ( θ 2 , … , θ n - 1 ) ⁢ cos ⁡ ( m n - 2 ⁢ θ 1 ) ) ↦ C M ⁢ ( θ 2 , … , θ n - 1 ) ⁢ cos ⁡ ( m n - 2 ⁢ ( θ 1 - φ ) ) ,

( g ⁢ ( φ ) , C M ⁢ ( θ 2 , … , θ n - 1 ) ⁢ sin ⁡ ( m n - 2 ⁢ θ 1 ) ) ↦ C M ⁢ ( θ 2 , … , θ n - 1 ) ⁢ sin ⁡ ( m n - 2 ⁢ ( θ 1 - φ ) ) ,

the space

span ℝ ⁡ { C M ⁢ ( θ 2 , … , θ n - 1 ) ⁢ cos ⁡ ( m n - 2 ⁢ θ 1 ) , C M ⁢ ( θ 2 , … , θ n - 1 ) ⁢ sin ⁡ ( m n - 2 ⁢ θ 1 ) }

is a two-dimensional representation of SO⁡(2) equivalent to the representation R⁢[1,mn-2] with 0<mn-2≤m. If mn-2=0, then

span ℝ ⁡ { C M ⁢ ( θ 2 , … , θ n - 1 ) ⁢ cos ⁡ ( m n - 2 ⁢ θ 1 ) , C M ⁢ ( θ 2 , … , θ n - 1 ) ⁢ sin ⁡ ( m n - 2 ⁢ θ 1 ) } = ℝ ⁢ [ 1 , 0 ]

is a one-dimensional trivial representation of SO⁡(2).

Moreover, there are numbers k0,…,km-1≥0 such that

ℋ m n ≈ SO ⁡ ( 2 ) ℝ ⁢ [ k 0 , 0 ] ⊕ ℝ ⁢ [ k 1 , 1 ] ⊕ ⋯ ⊕ ℝ ⁢ [ k m - 1 , m - 1 ] ⊕ ℝ ⁢ [ k m , m ] .

Note also that km=1 since there is only one possible M=(m0,…,mn-3,mn-2) such that m=mn-2, i.e., M=(m,m,…,m).

Additionally, combining the formula for the dimension of Hm2 (see Theorem A.5) and the above reasoning, we obtain Hm2≈SO⁡(2)R⁢[1,m], m≥0.

Combining the above corollary with Corollary 2.8, we obtain that

ℝ ⁢ [ 1 , m - 1 ] ⊂ V - Δ S n π / 2 ⁢ ( λ m ) ,

and if

ℝ ⁢ [ 1 , m ^ ] ⊂ V - Δ S n π / 2 ⁢ ( λ m ) ,

then 0<m^<m, i.e., there is an element of V-ΔSnπ/2⁢(λm) with the isotropy group isomorphic to the cyclic subgroup ℤm-1⊂S1, while no element of V-ΔSnπ/2⁢(λm^), 0<m^<m, has an isotropy group isomorphic to ℤm-1.

Taking into consideration the above, we obtain a formula for the degree for SO⁡(2)-equivariant gradient maps of -Id computed on the unit ball B⁢(ℋmn) in ℋmn.

Corollary A.9.

Combining formula (A.3) with Corollary A.8, for n≥3 and m>0 we obtain

∇ SO ⁡ ( 2 ) ⁡ -deg ⁢ ( - Id , B ⁢ ( ℋ m n ) ) = ( α 0 , α 1 , … , α i , … ) ∈ U ⁢ ( SO ⁡ ( 2 ) ) ,

where

α i = { ( - 1 ) k 0 if ⁢ i = 0 , ( - 1 ) k 0 + 1 if ⁢ i = m , ( - 1 ) k 0 + 1 ⁢ k i if ⁢ i = 1 , … , m - 1 , 0 if ⁢ i > m .

Moreover, for m>0,

∇ SO ⁡ ( 2 ) ⁡ -deg ⁢ ( - Id , B ⁢ ( ℋ m 2 ) ) = ( α 0 , α 1 , … , α i , … ) ∈ U ⁢ ( SO ⁡ ( 2 ) ) ,

where

α i = { 1 if ⁢ i = 0 , - 1 if ⁢ i = m , 0 if ⁢ i ≠ 0 , m .

By using Corollaries A.8 and A.9 above, one can determine the last nontrivial coordinate of the degree ∇SO⁡(2)⁡-deg⁢(-Id,B⁢(ℋmn)); see the remark below.

Remark A.10.

Suppose that n≥2 and m≥0. Then from the above corollary it follows that if

∇ SO ⁡ ( 2 ) ⁡ -deg ⁢ ( - Id , B ⁢ ( ℋ m n ) ) = ( α 0 , α 1 , … , α i , … ) ∈ U ⁢ ( SO ⁡ ( 2 ) ) ,

then αm=(-1)dim⁡ℋmn+1 and αi=0 for every i>m.

To illustrate the above lemma we consider the following examples.

Example A.11.

Suppose that n=2 and m≥0. Then ℋm2=spanℝ⁡{cos⁡m⁢φ,sin⁡m⁢φ} and ℋm2≈ℝ⁢[1,m], where the action of SO⁡(2) (≈S1) is given by shift in time.

Example A.12.

Suppose that n=3 and m≥0. Then ℋm3 is equivalent to a representation of SO⁡(2) of the form ℝ⁢[1,0]⊕ℝ⁢[1,1]⊕⋯⊕ℝ⁢[1,m].

Define

V m 0 - ⊕ V m 0 0 ⊕ V m 0 + := ⊕ i = 1 m 0 - 1 V - Δ S n γ ⁢ ( λ i ) ⊕ V - Δ S n γ ⁢ ( λ m 0 ) ⊕ ⊕ i = m 0 + 1 ∞ V - Δ S n γ ⁢ ( λ i ) .

Set μm0=dim⁡V-ΔSnγ⁢(λm0) and νm0=μ1+⋯+μm0 for m0∈ℕ.

In the two lemmas below, we present formulas for bifurcation indices and their properties. We prove only Lemma A.13. The proof of Lemma A.14 is in spirit the same as the proof of Lemma A.13.

Lemma A.13.

Assume that p->0 and fix λm0∈σ⁢(-ΔSn;B⁢(γ)). Then

(A.5) ℬ ℐ ℱ SO ⁡ ( n ) ( λ m 0 ) = ∇ SO ⁡ ( n ) - deg ( - Id , B ( V m 0 - ) ) p - ∗ ( ( ∇ SO ⁡ ( n ) - deg ( - Id , B ( V - Δ S n γ ( λ m 0 ) ) ) ) p - - 𝕀 ) ∈ U ( SO ( n ) ) .

Moreover, we have the following assertions:

If ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ m 0 ) = i ⋆ ⁢ ( ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ m 0 ) ) = ( α 0 , α 1 , … , α k , … ) , then

α k = { ( - 1 ) ν m 0 ⁢ p - - ( - 1 ) ν m 0 - 1 ⁢ p - if ⁢ k = 0 , ( - 1 ) 1 + ν m 0 ⁢ p - ⁢ p - if ⁢ k = m 0 - 1 , 0 if ⁢ k ≥ m 0 .
If p - is even, then

ℬ ℐ ℱ SO ⁡ ( 2 ) ( λ m 0 ) = ∇ SO ⁡ ( 2 ) - deg ( - Id , B ( V - Δ S n γ ( λ m 0 ) ) ) p - - 𝕀 ∈ U - ( SO ( 2 ) ) .
If dim ⁡ V - Δ S n γ ⁢ ( λ m 0 ) and p - ⋅ dim ⁡ V m 0 - are even, then

ℬ ℐ ℱ SO ⁡ ( 2 ) ( λ m 0 ) = ∇ SO ⁡ ( 2 ) - deg ( - Id , B ( V - Δ S n γ ( λ m 0 ) ) ) p - - 𝕀 ∈ U - ( SO ( 2 ) ) .
If dim ⁡ V - Δ S n γ ⁢ ( λ m 0 ) is even and p - ⋅ dim ⁡ V m 0 - is odd, then

ℬ ℐ ℱ SO ⁡ ( 2 ) ( λ m 0 ) = 𝕀 - ∇ SO ⁡ ( 2 ) - deg ( - Id , B ( V - Δ S n γ ( λ m 0 ) ) ) p - ∈ U + ( SO ( 2 ) ) .

Proof.

Since ϵ>0 is sufficiently small,

∇ 2 ⁡ Φ ⁢ ( 0 , λ m 0 ± ϵ ) = ( - α 1 ⁢ u 1 - ( λ m 0 ± ϵ ) ⁢ T ⁢ u 1 , … , - α p ⁢ u p - ( λ m 0 ± ϵ ) ⁢ T ⁢ u p )

is an isomorphism (a product of isomorphisms) and that is why by the Cartesian product formula for the degree for invariant strongly indefinite functionals (see [10, Theorem 3.2]) we obtain

ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ m 0 ) = ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( ∇ u 2 ⁡ Φ ⁢ ( 0 , λ m 0 + ϵ ) , B ⁢ ( ℍ ) ) - ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( ∇ u 2 ⁡ Φ ⁢ ( 0 , λ m 0 - ϵ ) , B ⁢ ( ℍ ) )

= ∏ i = 1 p ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( - α i ⁢ Id - ( λ m 0 + ϵ ) ⁢ T , B ⁢ ( H 0 1 ⁢ ( B ⁢ ( γ ) ) ) )

- ∏ i = 1 p ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( - α i ⁢ Id - ( λ m 0 - ϵ ) ⁢ T , B ⁢ ( H 0 1 ⁢ ( B ⁢ ( γ ) ) ) ) .

Since ∇SO⁡(n)⁡-⁢deg⁡(-Id,B⁢(ℍ01⁢(B⁢(γ))))=𝕀∈U⁢(SO⁡(n)) (see [10]), we obtain

ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ m 0 ) = ∏ α i = - 1 ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( Id - ( λ m 0 + ϵ ) ⁢ T , B ⁢ ( H 0 1 ⁢ ( B ⁢ ( γ ) ) ) )

- ∏ α i = - 1 p ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( Id - ( λ m 0 - ϵ ) ⁢ T , B ⁢ ( H 0 1 ⁢ ( B ⁢ ( γ ) ) ) )

= ( ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( Id - ( λ m 0 + ϵ ) ⁢ T , B ⁢ ( H 0 1 ⁢ ( B ⁢ ( γ ) ) ) ) ) p -

- ( ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( Id - ( λ m 0 - ϵ ) ⁢ T , B ⁢ ( H 0 1 ⁢ ( B ⁢ ( γ ) ) ) ) ) p - .

Note that

∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( Id - ( λ m 0 + ϵ ) ⁢ T , B ⁢ ( H 0 1 ⁢ ( B ⁢ ( γ ) ) ) )

= ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( - Id , B ⁢ ( V m - ) ) ∗ ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( - Id , B ⁢ ( V m 0 ) ) ∗ ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( Id , B ⁢ ( V m + ) )

and

∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( Id - ( λ m 0 - ϵ ) ⁢ T , B ⁢ ( H 0 1 ⁢ ( B ⁢ ( γ ) ) ) )

= ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( - Id , B ⁢ ( V m - ) ) ∗ ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( Id , B ⁢ ( V m 0 ) ) ∗ ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( Id , B ⁢ ( V m + ) ) .

Since for any representation W of SO⁡(n) we have ∇SO⁡(n)⁡-⁢deg⁡(Id,B⁢(W))=𝕀∈U⁢(SO⁡(n)), we obtain

ℬ ℐ ℱ SO ⁡ ( n ) ( λ m 0 ) = ∇ SO ⁡ ( n ) - deg ( - Id , B ( V m - ) ) p - ∗ ∇ SO ⁡ ( n ) - deg ( - Id , B ( V m 0 ) ) p - - ∇ SO ⁡ ( n ) - deg ( - Id , B ( V m - ) ) p -

= ∇ SO ⁡ ( n ) - deg ( - Id , B ( V m - ) ) p - ∗ ( ∇ SO ⁡ ( n ) - deg ( - Id , B ( V m 0 ) ) p - - 𝕀 ) ,

which completes the proof of the first part of this lemma.

(i) Taking into account formula (A.5), we obtain

ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ m 0 ) = i ⋆ ⁢ ( ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( λ m 0 ) )

(A.6) = ∇ SO ⁡ ( 2 ) - deg ( - Id , B ( V m 0 - ) ) p - ∗ ( ( ∇ SO ⁡ ( 2 ) - deg ( - Id , B ( V - Δ S n γ ( λ m 0 ) ) ) ) p - - 𝕀 ) ∈ U ( SO ( 2 ) ) .

By the Cartesian product formula for the degree for SO⁡(2)-equivariant gradient maps, we have

∇ SO ⁡ ( 2 ) - deg ( - Id , B ( V m 0 - ) ) p - = ∇ SO ⁡ ( 2 ) - deg ( - Id , B ( V m 0 - × ⋯ × V m 0 - ) ) = β = ( β 0 , β 1 , … , β k , … ) .

Combining Corollaries 2.8 and A.9 with Remark A.10, we obtain

β k = { ( - 1 ) ν m 0 - 1 ⁢ p - if ⁢ k = 0 , 0 if ⁢ k ≥ m 0 - 1 ,

and

∇ SO ⁡ ( 2 ) - deg ( - Id , B ( V - Δ S n γ ( λ m 0 ) ) ) p - = ∇ SO ⁡ ( 2 ) - deg ( - Id , B ( V - Δ S n γ ( λ m 0 ) × ⋯ × V - Δ S n γ ( λ m 0 ) ) )

= δ = ( δ 0 , δ 1 , … , δ k , … ) .

Once more applying Corollaries 2.8 and A.9 together with Remark A.10, we obtain

δ k = { ( - 1 ) μ m 0 ⁢ p - if ⁢ k = 0 , ( - 1 ) 1 + μ m 0 ⁢ p - ⁢ p - if ⁢ k = m 0 - 1 , 0 if ⁢ k ≥ m 0 .

Now formula (A.6) has the form

ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( λ m 0 ) = β ∗ ( δ - 𝕀 ) = α = ( α 0 , α 1 , … , α k , … ) .

By (A.1), we obtain

α k = { ( - 1 ) 1 + ν m 0 ⁢ p - ⁢ p - if ⁢ k = m 0 - 1 , 0 if ⁢ k ≥ m 0 ,

which completes the proof.

(ii) Set

∇ SO ⁡ ( n ) - deg ( - Id , B ( V m 0 - ) ) p - = ( α 0 , α 1 , … , α k , … ) ,

∇ SO ⁡ ( n ) - deg ( - Id , B ( V - Δ S n γ ( λ m 0 ) ) ) p - - 𝕀 = ( β 0 , β 1 , … , β k , … ) .

Since p->0 is even, combining formulas (A.1) and (A.3), we obtain that α0=1,β0=0 and βk≤0 for k≥1. Finally, by (A.1) we obtain

( 1 , α 1 , … , α k , … ) ∗ ( 0 , β 1 , … , β k , … ) = ( 0 , β 1 , … , β k , … ) ∈ U - ⁢ ( SO ⁡ ( 2 ) ) ,

which completes the proof.

(iii) Set

∇ SO ⁡ ( n ) - deg ( - Id , B ( V m 0 - ) ) p - = ( α 0 , α 1 , … , α k , … ) ,

∇ SO ⁡ ( n ) - deg ( - Id , B ( V - Δ S n γ ( λ m 0 ) ) ) p - - 𝕀 = ( β 0 , β 1 , … , β k , … ) .

Since p-⋅dim⁡Vm0- and dim⁡V-ΔSnγ⁢(λm0) are even, combining formulas (A.1) and (A.3), we obtain that α0=1, β0=0 and βk≤0 for k≥1. Finally, by formula (A.1) we obtain

( 1 , α 1 , … , α k , … ) ∗ ( 0 , β 1 , … , β k , … ) = ( 0 , β 1 , … , β k , … ) ∈ U - ⁢ ( SO ⁡ ( 2 ) ) ,

which completes the proof.

(iv) Set

∇ SO ⁡ ( n ) - deg ( - Id , B ( V m 0 - ) ) p - = ( α 0 , α 1 , … , α k , … ) ,

∇ SO ⁡ ( n ) - deg ( - Id , B ( V - Δ S n γ ( λ m 0 ) ) ) p - - 𝕀 = ( β 0 , β 1 , … , β k , … ) .

Since p-⋅dim⁡Vm0- is odd and dim⁡V-ΔSnγ⁢(λm0) is even, combining formulas (A.1) and (A.3), we obtain that α0=-1, β0=0 and βk≤0 for k≥1. Finally, by formula (A.1) we obtain

( - 1 , α 1 , … , α k , … ) ∗ ( 0 , β 1 , … , β k , … ) = ( 0 , - β 1 , … , - β k , … ) ∈ U + ⁢ ( SO ⁡ ( 2 ) ) ,

which completes the proof. ∎

In the following lemma, we consider the case p+>0. Since the proof of the following lemma is literally the same as that of the above lemma, it is left to the reader.

Lemma A.14.

Assume that p+>0 and fix λm0∈σ⁢(-ΔSn;B⁢(γ)). Then

ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( - λ m 0 ) = ( ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( - Id , B ⁢ ( V m 0 - ) ) ) - p + ∗ ( ( ∇ SO ⁡ ( n ) ⁡ - ⁢ deg ⁡ ( - Id , B ⁢ ( V - Δ S n γ ⁢ ( λ m 0 ) ) ) ) p + - 𝕀 ) .

Moreover, we have the following assertions:

If ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( 2 ) ⁢ ( - λ m 0 ) = i ⋆ ⁢ ( ℬ ⁢ ℐ ⁢ ℱ SO ⁡ ( n ) ⁢ ( - λ m 0 ) ) = ( α 0 , α 1 , … , α k , … ) , then

α k = { ( - 1 ) ν m 0 ⁢ p + - ( - 1 ) ν m 0 - 1 ⁢ p + if ⁢ k = 0 , ( - 1 ) 1 + ν m 0 ⁢ p + ⁢ p + if ⁢ k = m 0 - 1 , 0 if ⁢ k ≥ m 0 .
If p + is even, then

ℬ ℐ ℱ SO ⁡ ( 2 ) ( λ m 0 ) = ∇ SO ⁡ ( 2 ) - deg ( - Id , B ( V - Δ S n γ ( λ m 0 ) ) ) p + - 𝕀 ∈ U - ( SO ( 2 ) ) .
If dim ⁡ V - Δ S n ⁢ ( λ m 0 ) and p + ⋅ dim ⁡ V m 0 - are even, then

ℬ ℐ ℱ SO ⁡ ( 2 ) ( λ m 0 ) = ∇ SO ⁡ ( 2 ) - deg ( - Id , B ( V - Δ S n γ ( λ m 0 ) ) ) p + - 𝕀 ∈ U - ( SO ( 2 ) ) .
If dim ⁡ V - Δ S n ⁢ ( λ m 0 ) is even and p + ⋅ dim ⁡ V m 0 - is odd, then

ℬ ℐ ℱ SO ⁡ ( 2 ) ( λ m 0 ) = 𝕀 - ∇ SO ⁡ ( 2 ) - deg ( - Id , B ( V - Δ S n γ ( λ m 0 ) ) ) p + ∈ U + ( SO ( 2 ) ) .

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Received: 2018-01-08

Accepted: 2018-03-13

Published Online: 2018-03-29

Published in Print: 2018-11-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2018-0012

Keywords for this article

Symmetric Rabinowitz Alternative; Global Symmetry-Breaking Bifurcations; Non-cooperative Elliptic Systems; Equivariant Degree

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