Bifurcations from the orbit of solutions of the Neumann problem

The purpose of this paper is to study weak solutions of a nonlinear Neumann problem considered on a ball. Assuming that the potential is invariant, we consider an orbit of critical points, i.e. we do not assume that critical points are isolated. We apply techniques of equivariant analysis to examine bifurcations from the orbits of trivial solutions. We formulate sufficient conditions for local and global bifurcations, in terms of the right-hand side of the system and eigenvalues of the Laplace operator. Moreover, we characterise orbits at which global symmetry breaking phenomena occur.


Introduction
In this paper, we study bifurcations of weak solutions of elliptic systems of the form: Communicated by P. Rabinowitz  In particular, we are interested in the equivariant case. Namely, we assume that on the space R m there is defined an action of a compact Lie group and ∇ F is a -equivariant mapping. Moreover, it is known that B N is SO(N )-invariant, where SO(N ) stands for the special orthogonal group in dimension N .
Consider the set ∇ F −1 (0). For u 0 ∈ ∇F −1 (0) the constant functionũ 0 ≡ u 0 is a solution of (1.1) for all λ ∈ R. Therefore, we obtain the family of trivial solutions {ũ 0 } × R. Investigating the change of the Conley index for different levels λ ∈ R, one can obtain a sequence of nontrivial weak solutions bifurcating from the point (ũ 0 , λ 0 ), for some values λ 0 ∈ R. Investigating the change of the topological degree, one can prove the existence of the continuum, emanating from (ũ 0 , λ 0 ), of nontrivial weak solutions of the system (i.e. the global bifurcation of weak solutions).
For a system of elliptic differential equations with Dirichlet boundary conditions such methods have been used in many papers, among others by the first and the second author in [7,10,15]. A similar method has been also used in [9] for the system with Neumann boundary conditions for bifurcation from infinity instead from critical points. The phenomenon of symmetry breaking for elliptic systems with Neumann boundary conditions has been considered by the third author in [27].
The results described above are obtained with the assumption that u 0 is an isolated critical point of the potential F.
Assuming that ∇ F is a -equivariant mapping, we obtain that for u 0 ∈ ∇F −1 (0) also γ u 0 ∈ ∇ F −1 (0) for all γ ∈ . It is therefore clear, that the assumption that the critical point u 0 is an isolated one, does not have to be satisfied in this case.
The method, that can be used in this situation, is an investigation of the index of the isolated orbit. Under some additional assumptions, this method has been recently proposed by Pérez-Chavela et al. [18]. In that paper it has been proved that the computation of the Conley index of the orbit can be in some cases reduced to computation of the index of a point from the space normal to the orbit.
To study weak solutions of the system (1.1) we apply variational methods, i.e. we associate with the system a functional defined on a suitable Hilbert space H. Its critical points are in one-to-one correspondence with weak solutions of (1.1). The tools we use are the finite and infinite dimensional equivariant Conley index (see [2,8] for the definition in the finite dimensional case and [13] for the infinite dimensional case) and the degree for equivariant gradient maps, defined in [22].
Consider the group G = × SO(N ). Since R m is a -representation (by assumption) and B N is an SO(N )-invariant set, it follows that the space H is a G-representation. Moreover, for u 0 ∈ (∇ F) −1 (0), (gũ 0 , λ) is a critical point of for all g ∈ G, λ ∈ R.
Therefore we can consider the set of trivial solutions T = G(ũ 0 ) × R. We are going to investigate bifurcations of nontrivial solutions from the family T . Our aim is to formulate necessary and sufficient conditions, in terms of the right-hand side of the system and of the eigenvalues of the Laplace operator, for a bifurcation from the orbit G(ũ 0 ) × {λ 0 }.
We also consider global symmetry breaking phenomena at the orbit G(ũ 0 ) × {λ 0 }. More precisely, knowing that the trivial solutions are radial, we study when the bifurcating solutions are non-radial. The analogous problem has been studied by the third author in [23,24] on the sphere and on the geodesic ball, with the use of a lemma due to Dancer (see [4]), characterising isotropy groups of bifurcating solutions. In our situation, if the group is not a discrete one, we cannot use this result. Therefore we generalise it.
After this introduction the paper is organised in the following way: In Sect. 2 we introduce the problem and recall some definitions. With an elliptic system on a ball we associate a functional. Next we study the properties of the linear system. We end this section with the definitions of local and global bifurcations from an orbit and of the admissible pair.
In Sect. 3 we formulate and prove the main results of this article, namely Theorem 3.5 concerning the global bifurcation of solutions, and Theorem 3.10, concerning the symmetry breaking problem. First we consider the phenomenon of bifurcation from the critical orbit. We start with some auxiliary results. In Lemma 3.1 we describe the set of parameters at which the bifurcation of solutions can occur. In Theorem 3.3 we investigate the change of the Conley index at the levels obtained in Lemma 3.1. This result is applied to prove Theorem 3.5. Since the global bifurcation implies the local one, we obtain the result concerning local bifurcations, see Corollary 3.6. This phenomenon can be proved also directly from Theorem 3.3, what is described in Remark 3.7. The local bifurcation of solutions, under weaker assumptions, is considered also in Theorem 3.8. Next we study the symmetry breaking problem. In Theorem 3.10 we prove the bifurcation of orbits of non-radial solutions emanating from orbits of radial ones. To obtain this result, we generalise the result of Dancer in Lemma 3.11.
In Sect. 4 we illustrate our results with a few examples. Using the properties of the eigenspaces of the Laplace operator (with Neumann boundary conditions) on the ball, we verify the assumptions of our main results. Section 5 is the "Appendix". In the main part of our paper we assume that the reader is familiar with some classical definitions and facts, concerning for example the equivariant Conley index, the degree for equivariant gradient maps or the properties of eigenspaces of the Laplace operator on a ball. However, it is not easy to find a detailed discussion of these properties. Therefore, for the completeness of the paper we collect in this section the information which we use to prove our main results. In this section we present also an equivariant version of the implicit function theorem in infinite dimensional spaces, due to Dancer.

Notation
Suppose that G is a compact Lie group. We denote by sub(G) the set of closed subgroups of G. For u from a given G-space X we denote by G(u) the orbit through u and G u stands for the isotropy group of u. By X G we denote the space of all fixed points of the action of the group G on X .
Further, by U (G) we denote the Euler ring of G and we use the symbol χ G (·) to denote the G-equivariant Euler characteristic of a pointed finite G-CW-complex. Moreover, the symbols C I G (S, f ) and CI G (S, f ) stand for the Conley indices of an isolated invariant set S of the flow generated by f , considered respectively in finite and infinite dimensional cases. A more precise description can be found in the "Appendix".
Finally, for a Hilbert space H and u 0 ∈ H we denote by B δ (u 0 , H) (respectively D δ (u 0 , H)) the open (respectively closed) ball in H centred at u 0 and with radius δ. In particular, we use the symbol B N for the open ball if δ = 1, u 0 = 0 and H = R N and we write S N −1 for ∂ B N .

Preliminaries
Throughout this paper stands for a compact Lie group and R m is an orthogonal representation of the group . Consider F : R m → R satisfying: Our aim is to study bifurcations of weak solutions of the nonlinear Neumann problem, parameterised by λ ∈ R, Denote by H 1 (B N ) the usual Sobolev space on B N and consider the separable Hilbert space is the special orthogonal group in dimension N . Note that the space H with the scalar product given by (2.2) is an orthogonal G-representation with the G-action given by It is well known that weak solutions of the problem (2.1) are in one-to-one correspondence with critical points (with respect to u) of the functional : H × R → R defined by Computing the gradient of with respect to u we obtain: Moreover, Moreover, from imbedding theorems and the assumption (B1) it follows that the operator ∇ u is a completely continuous perturbation of the identity, see [21].

Linear equation
In this subsection we consider the Eq. (2.1) in the linear case, i.e. the system: where A is a real, symmetric (m × m)-matrix.
Using formula (2.4) we can associate with (2.6) the functional A : H × R → R given by Note that from (2.5) for every v ∈ H we have The existence and boundedness of the operator L λA : H → H follow from the Riesz theorem. By definition L λA is self-adjoint. Let us denote by σ (− ; B N ) = {0 = β 1 < β 2 < . . . < β k < . . .} the set of distinct eigenvalues of the Laplace operator (with Neumann boundary conditions) on the ball. Write V − (β k ) for the eigenspace of − corresponding to β k ∈ σ (− ; B N ). In the "Appendix" we give a more precise description of these eigenspaces. By the spectral theorem it follows that In the lemma below we characterise the operator L λA , given by the formula (2.8).

Lemma 2.1 For every u ∈ H
The proof of this lemma is standard, see for example the proof of Lemma 3.2 in [9]. Let us denote by σ (L) the spectrum of a linear operator L : H → H. From the above lemma it immediately follows the corollary: Corollary 2.2 Let L λA be defined by (2.8). Then: Moreover, Fix eigenvalues α j 0 ∈ σ (A) and β k 0 ∈ σ (− ; B N ). Let V A (α j 0 ) be the eigenspace associated with the eigenvalue α j 0 and μ A ( It follows that From Lemma 2.1 we obtain:

The notion of bifurcation from the critical orbit
. We call such a set a critical orbit of F.
Note that T u 0 (u 0 ) ⊂ ker ∇ 2 F(u 0 ) and therefore dim ker . We assume that in this inequality there holds: We call such an orbit non-degenerate. By the equivariant Morse lemma, see [31], from (2.9) we conclude that (u 0 ) is isolated in (∇ F) −1 (0).
Since u 0 ∈ (∇ F) −1 (0), the constant functionũ 0 ≡ u 0 is a solution of the problem (2.1) for all λ ∈ R. Therefore, (ũ 0 , λ), and consequently (γũ 0 , λ) for every γ ∈ , is a critical point of the functional given by (2.4). Since from (2.3) we have G(ũ 0 ) = (ũ 0 ), we obtain a critical orbit of and therefore a G-orbit of weak solutions of (2.1) for all λ ∈ R. Hence we can consider a family of solutions T = G(ũ 0 ) × R ⊂ H × R. We call the elements of T the trivial solutions of (2.1). Remark 2.5 Note that if (ũ 0 , λ 0 ) is an accumulation point of N then for all g ∈ G, (gũ 0 , λ 0 ) is also an accumulation point, since H is an orthogonal representation of G. Therefore G(ũ 0 ) ⊂ cl(N ).

Definition 2.6 A global bifurcation from the orbit
The set of all λ 0 ∈ R such that a local (respectively global) bifurcation from the orbit G(ũ 0 ) × {λ 0 } occurs we denote by B I F (respectively G L O B). Note that directly from the above definitions it follows that G L O B ⊂ B I F.

Admissible pair
The notion of an admissible pair has been introduced in [18].
Fix a compact Lie group G and let H ∈ sub(G). Denote by (H ) G the conjugacy class of H.
Thus (K 1 ) H = (K 2 ) H and the proof is complete.

Main results
We make two additional assumptions: From the assumption (B3) we conclude that the gradient of the functional associated with the Eq. (2.1) has the following form: where L λA : H → H is an SO(N )-equivariant operator given by (2.8). Moreover, ∇η : From the assumption (B4) it follows that Gũ 0 = {e} × SO(N ).

Bifurcation from the critical orbit
Following the standard notation we denote the linear part of ∇ u (·, λ) atũ 0 by ∇ 2 Proof We first observe that for all λ ∈ R, since G(ũ 0 ) is a critical orbit of (·, λ), we have dim ker From the definition of such a choice is always possible. Since From this and the properties of flows induced by gradient operators, we conclude that G(ũ 0 ) is also an isolated invariant set (in the sense of the equivariant Conley index theory, see [13]) for the flows induced by the operators −∇ u (·, λ 0 ± ε). Therefore, the indices In the following we study when they are not equal. Assume We consider the conditions: Remark 3.2 Note that we can reformulate conditions (C1)-(C3) in the following way:

Theorem 3.3 Assume that λ 0 ∈ and one of the conditions (C1)-(C3) is satisfied. Then
We start the proof with showing that we can reduce comparing the Conley indices CI G (G(ũ 0 ), −∇ u (·, λ 0 ± ε)) to comparing Euler characteristics of some indices on the spaceH.
For n ≥ 1 put H n = n k=1 H k and n = |H n ×R : we understand the complement of the space Tũ 0 (ũ 0 ) in H 1 ). Therefore G(ũ 0 ) is a critical orbit of n (·, λ 0 ± ε) for n ≥ 1. Note that, from the choice of ε and the definition of n , this orbit is nondegenerate.
Since ∇ u (·, λ) is a completely continuous perturbation of the identity for all λ ∈ R, from the definition of the infinite dimensional equivariant Conley index, see Sect. 5.2, the assertion of the theorem is equivalent to for n sufficiently large. Obviously, the above inequality is implied by It is known that the G-action on H given by (2.3) defines a Gũ 0 -action onH. Recall that Hence {ũ 0 } is an isolated invariant set (in the sense of the Conley index theory) of the flows generated by −∇ n ± . Note that since Gũ 0 = {e} × SO(N ), by Lemma 2.8 the pair (G, Gũ 0 ) is admissible. Therefore, using Fact 5.6 we obtain that the assertion reduces to for n ∈ N sufficiently large. It is easy to see that this inequality is equivalent to We proceed to show that there exists n 0 ∈ N such that for n ≥ n 0 where P ν :H →H ν is the orthogonal SO(N )-equivariant projection ontoH ν . Note that from Lemma 2.1 we have P ν • L λA = L λA • P ν and hence this homotopy is well-defined. Let us denote by ξ ν λ :H ν → R the SO(N )-invariant potential of H ν λ (·, 0). It is clear that ∇ξ ν λ :H ν →H ν is a self-adjoint SO(N )-equivariant linear map and is given by the formula From the homotopy invariance of the Conley index, see Theorem 5.3, we obtain Recall that (β k ) denotes the sequence of the eigenvalues of the Neumann Laplacian and note that β k → +∞. Therefore, there exists n 0 ∈ N such that the inequalities β n −(λ 0 ±ε)α j 1+β n > 0 hold for every n ≥ n 0 and α j ∈ σ (A). Hence, by Corollary 2.2, there exists n 0 ∈ N such that m − (∇ξ n λ 0 ±ε ) = m − (∇ξ n 0 λ 0 ±ε ) for every n ≥ n 0 , where m − (·) is the Morse index. Since (∇ξ n λ 0 ±ε ) |H n 0 = ∇ξ n 0 λ 0 ±ε , the eigenspaces corresponding to the negative eigenvalues of ∇ξ n λ 0 ±ε and ∇ξ n 0 λ 0 ±ε are the same SO(N )-representations. Thus, from Theorem 5.2, To finish the proof of (3.4), and therefore also of the assertion, we will show that λ ) corresponding to the negative eigenvalues and by V(λ) the eigenspace corresponding to the zero eigenvalue. Note that from Corollary 2.2, (1) Suppose that λ 0 > 0 and ε is such that λ 0 − ε > 0. Recall that β k ≥ 0 for all β k ∈ σ (− ; B N ). Then W(λ 0 +ε) = W(λ 0 −ε)⊕V(λ 0 ). If the assumption (C1) is satisfied, then, by Theorem 5.4 and Remark 5.11, we obtain χ SO(N ) (S V(λ 0 ) ) = I ∈ U (SO(N )).
Now we are in a position to prove one of the main results of our paper, namely the global bifurcation theorem. Proof Throughout the proof we follow the notation of the proof of Theorem 3.3. Let U ⊂ H be an open, bounded and G-invariant subset such that ∇ u (·, λ 0 ± ε) −1 (0) ∩ U = G(ũ 0 ). From Theorem 5.7 it follows that to prove the assertion it is enough to show where ∇ G -deg(·, ·) is the degree for equivariant gradient maps, see Sect. 5.3. From the definition of the degree, see (5.2), it follows that for n 0 sufficiently large. Note that G(ũ 0 ) ⊂ U ∩ H n 0 and hence, by Theorem 5.8, we obtain Therefore, the assertion follows from Remark 3.4.
Since the global bifurcation implies the local one, as an immediate corollary of the above theorem we obtain the following: Corollary 3.6 Consider the system (2.1) with the potential F and u 0 ∈ ∇ F −1 (0) satisfying assumptions (B1)-(B4). Assume that λ 0 ∈ and one of the conditions (C1)-(C3) is satisfied. Then a local bifurcation of solutions of (2.1) occurs from the orbit G(ũ 0 ) × {λ 0 }.

Remark 3.7
The above corollary can be obtained directly from Theorem 3.3. Indeed, from this theorem it follows that if one of the conditions (C1)-(C3) is satisfied then , for sufficiently small ε > 0. Following for example the idea of the proof of Theorem 2.1 of [26], using the continuation property of the Conley index, one can prove that the change of the Conley index implies a local bifurcation of critical orbits.
In Theorem 3.5 we have proved that if the assumption (C3) is satisfied, then 0 ∈ G L O B. On the other hand, repeating the argument from the proof of this theorem it is easy to show that if the number ε)) and χ G (C I G (G(ũ 0 ), −∇ u n 0 (·, −ε))) are equal. Therefore, we do not know whether 0 ∈ G L O B. However, under the assumption weaker than (C3) we can prove the result concerning the local bifurcation.

Theorem 3.8 Consider the system (2.1) with the potential F and u
Then a local bifurcation of solutions of (2.1) occurs from the orbit G(ũ 0 ) × {0}.
Proof Using the notation of the proof of Theorem 3.3, we observe that W(±ε) are trivial SO(N )-representations. Therefore C I SO(N ) ({ũ 0 }, −∇ n 0 ± ) are SO(N )-homotopy types of S dim W(±ε) . Using information from [18] (namely Theorem 3.1 and the equality (2.11)) and from [14] (Lemma 1.88) we obtain that C I G (G(ũ 0 ), −∇ u n 0 (·, ±ε)) are G-homotopy types of From Proposition 1.53 of [14], we obtain that the above is G-homotopy equivalent to But X + and X − are different G-homotopy types. Indeed, if X + and X − are the same Ghomotopy types, then the orbit spaces X + /G and X − /G are the same homotopy types. This is impossible, since the spaces X ± /G are homotopy types of S dim W(±ε) , see [29]. Similarly as in Remark 3.7 we use the fact, that the change of the Conley indices implies the local bifurcation. This shows the assertion.

Symmetry breaking
In this section we consider the symmetry breaking problem, i.e. the change of the isotropy groups of solutions of (2.1) along connected sets. More precisely, we characterise bifurcation orbits of the Eq. (2.1) at which global symmetry breaking phenomena occur. Here and thereafter we use the notation of Sect. 3.1. Recall that T denotes the set of trivial solutions. Definition 3. 9 We say that a global symmetry breaking phenomenon occurs at the orbit Note that since the group G acts trivially on the set of parameters λ, the condition G (u,λ) = G (ũ 0 ,λ 0 ) is equivalent to G u = Gũ 0 . In particular we are interested in studying SO(N )symmetries of solutions. We say that the function u satisfying SO(N ) u = SO(N ) is radially symmetric.
Our aim in this section is to prove the following characterisation of global symmetry breaking phenomena of solutions of (2.1): Note that the assumption V − (α j i ) SO(N ) = {0} means that there is no radially symmetric eigenfunction associated with α j i .
To finish the proof observe that in the case u 2 = 0 we have (τ (u 1 , 0, λ), 0, λ) ∈ U 1 ×{0}×R for u 1 ∈ G(ũ 0 ), λ ∈ O λ 0 . Considering only the solutions of (3.8) and observing that such solutions in U 1 × {0} × R are the trivial ones, we obtain (τ (u 1 , 0, λ), 0, λ) ∈ T , which completes the proof. Lemma 3.11 generalises a lemma due to Dancer from [4]. Dancer's result states that if the kernel of the second derivative of the functional at a bifurcation point does not contain nonzero radially-symmetric elements, then at a neighbourhood of this point all nontrivial solutions are not radial. This lemma cannot be applied to prove Theorem 3.10 in the case dim G(ũ 0 ) > 0, since ker ∇ 2 u (ũ 0 , λ 0 ) contains constant (and therefore radially symmetric) functions from the space tangent to the orbit.
Proof of Theorem 3.10 Note that by the assumptions of the theorem, from Remark 5.11 it follows that the assumption (C1) is satisfied. Therefore Theorem 3.5 implies that λ 0 ∈ G L O B. Moreover, from Corollary 2.3 we have Note that if the assumptions of Theorem 3.10 are satisfied, i.e. ker ∇ 2 , then there is a neighbourhood U of the bifurcation orbit such that all nontrivial solutions from U are non-radial. In other words, in Theorem 3.10 we obtain a connected family of orbits of non-radial solutions bifurcating from the set of radial ones.

Remark 3.12
Let λ 0 ∈ B I F. By the proof of Lemma 3.11 we deduce that there is a neighbourhood of the orbit G(ũ 0 ) × {λ 0 } such that all nontrivial solutions of ∇ u (u, λ) = 0 can have only isotropy groups of the form G τ (u 1 ,u 2 ,λ) ∩ G u 2 . Note that u 1 ∈ G(ũ 0 ) and hence Consider the additional assumption: λ 0 ).

Examples
In this section we discuss a few examples in order to illustrate the abstract results proved in the previous section. Using the properties of the eigenspaces of the Laplace operator (with Neumann boundary conditions) on the ball, we verify assumptions (C1)-(C3). More precisely we apply the material collected in Sect. 5.4.
is the Bessel function of order 1 2 . Therefore α i / ∈ A 0 .
In this situation, since H 3 l ⊂ V − (α i ) for some l > 0 (by Fact 5.10), the assumption (C1) is satisfied and from Theorem 3.5 we obtain that a global bifurcation occurs from the orbit Moreover, if α i / ∈ A 0 for all i ∈ {1, . . . , s}, then from Remark 5.12 we conclude that V − (α i ) SO(3) = {0} for all i ∈ {1, . . . , s}. Therefore, by Theorem 3.10 it follows that the global symmetry breaking occurs at the orbit G(ũ 0 ) × {λ 0 }. Example 4 Consider the system (2.1) with the potential F and u 0 ∈ ∇F −1 (0) satisfying assumptions (B1)-(B4). Assume that λ 0 ∈ R \ {0} and that σ ( √ α i is a solution of the equation If there exists i ∈ {1, . . . , s} such that dim V − (α i ) > 1 then the assumption (C1) is satisfied and by Theorem 3.5 we obtain that a global bifurcation occurs from the orbit . . , s}, then we assume additionally that is an odd number. In this situation the assumption (C2) is satisfied and by Theorem 3.5 we obtain that a global bifurcation occurs from the orbit Remark 5.11(2)). Therefore, from Remark 3.12, we conclude that all nontrivial solutions at a neighbourhood of G(ũ 0 ) × {λ 0 } (bifurcating from this orbit) are radial, i.e. there is no symmetry breaking at the orbit.
If m − dim ker A is odd, then the assumption (C3) is satisfied and we obtain a global bifurcation from the orbit G(ũ 0 ) × {0}. If m − dim ker A > 0, then Theorem 3.8 implies a local bifurcation from the orbit G(ũ 0 ) × {0}.
As in Example 4, it is easy to see that all nontrivial solutions at a neighbourhood of the orbit are radial.
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Appendix
In the following section, to make the paper self-contained, we collect some classical definitions and facts which we use to prove our main results.

The equivariant implicit function theorem
Below we reformulate an equivariant version of the implicit function theorem in infinite dimensional spaces, due to Dancer (see [5], paragraph 3).

Theorem 5.1 Let G be a compact Lie group and suppose that
(i) H 1 , H 2 are Hilbert spaces, which are orthogonal G-representations, is Fredholm for every u ∈ H 1 , there is u 0 ∈ H 1 such that ∇ u (u 0 , v 0 ) = 0 and G(u 0 ) is a non-degenerate critical orbit of (·, v 0 ). τ (u, v), v) = 0 if u ∈ G(u 0 ) and v ∈ B δ (v 0 , H 2 ) and these are the only solutions H 2 ), the map u → τ (u, v) is one-to-one.

Equivariant Conley index
In this subsection we collect properties of the equivariant Conley index. For a more detailed exposition we refer to [2,8] in the finite dimensional case and to [13] for the infinite dimensional case. Note that the Conley index (in the finite dimensional as well as in the infinite dimensional case) is defined for an arbitrary compact Lie group G. In our paper we use it in special cases G = G = × SO (N ) and G = SO(N ).
Let G be a compact Lie group and suppose that is a G-invariant subset of a finite dimensional G-representation V. The G-equivariant Conley index of an isolating neighbourhood of a (local) flow is defined as the G-homotopy type of a pointed G-space, see [2,8]. If f : → V is a G-equivariant map of class C 1 , then it generates a local G-flow η, such that η(x 0 , ·) is the local solution of the problem y (t) = f (y(t)), y(0) = x 0 . Moreover, if S is an isolated invariant set of the flow, then there exists an isolating neighbourhood for this set. To simplify the notation in the main part of our paper, we denote by C I G (S, f ) the Conley index of an isolating neighbourhood of the isolated invariant set S of the flow generated by f .
From the definition of the Conley index and the Hartman-Grobman theorem it follows (see also [26]): The following theorem is a direct consequence of the Continuation Property of the Conley index, see [2]: The Conley index of a flow generated by a gradient map is the homotopy type of a pointed finite G-CW-complex, see Proposition 5.6 of [8] for the proof. With a G-homotopy type of a pointed finite G-CW-complex X one can associate a G-equivariant Euler characteristic χ G (X ), which is an element of the Euler ring U (G) with the unit I = χ G (G/G + ). The actions in U (G) are defined by where X ∨ Y is the wedge sum and X ∧ Y is the smash product of pointed finite G-CW-complexes X, Y . It is known that (U (G), +) is a free abelian group with basis we denote the set of conjugacy classes (H ) G of closed subgroups of the group G). A more detailed exposition of this theory can be found for example in [28,29].
The following theorem is an immediate consequence of Lemma 3.4 of [6]:

Theorem 5.4 If the group G is connected and V is a nontrivial G-representation, then
Consider the potential ϕ : V × R → R and assume that for λ − , λ + ∈ R the critical orbit G(ũ 0 ) of ϕ(·, λ ± ) is non-degenerate. In Sect. 3 we compare equivariant Conley indices C I G (G(ũ 0 ), −∇ϕ(·, λ ± )). Following the method introduced in [18], we want to reduce this problem to comparing the Euler characteristics of the Conley indices of potentials restricted to the space orthogonal to the orbit. This method bases on the relation between Conley indices obtained with the use of the smash product over Gũ 0 . To make our paper self-contained, we recall the relevant material.
Let H ∈ sub(G) and X be a pointed H -space with a base point * . Denote by G + the group G with disjoint base point added. The smash product of G + and X is defined by The orbit space of this action is called the smash product over H and denoted by G + ∧ H X, see [29]. Note that formula (g , [g, x]) → [g g, x] induces a G-action on G + ∧ H X and therefore this is a pointed G-space.

Theorem 5.5 (Theorem 3.1 of [18]) Let be an open, G-invariant subset of
In the following we briefly describe the infinite dimensional version of the G-equivariant Conley index of an isolated invariant set of a (local) G-LS-flow. This index is defined as a G-homotopy type of a G-equivariant spectrum, see [13]. Below we recall this definition in a special case. Namely, since in our computation we use the G-equivariant gradient maps of the form of a completely continuous perturbation of the identity, we consider only flows generated by such maps.
We start with a definition of a G-spectrum. Fix a sequence ξ = (V n ) ∞ n=0 of finitedimensional orthogonal G-representations. A pair of sequences: of finite pointed G-CW-complexes E n and morphisms n : S V n ∧ E n → E n+1 is called a G-spectrum if there exists n 0 > n(E) such that n is a G-homotopy equivalence for all n ≥ n 0 .
Let H be an infinite-dimensional, separable Hilbert space, which is an orthogonal Grepresentation. Assume that H = cl( ∞ n=0 H n ), where all subspaces H n are disjoint finitedimensional G-representations and put H n = n k=0 H k . Let ξ ∈ C 1 (H, R) be a G-invariant functional such that ∇ξ : H → H is a completely continuous, G-equivariant operator. Put Denote by τ n : H → H n the G-equivariant orthogonal projection. Define f n : H n → H n by setting f n (u) = u − τ n (∇ξ(u)) for any u ∈ H n . It is known (see [13]) that if X is an isolating neighbourhood for the flow generated by f , then X n = X ∩ H n is an isolating neighbourhood for the flow generated by f n , for n sufficiently large. It follows that X n admits an index pair (Y n , Z n ) with respect to the flow generated by f n , and consequently, From the continuation property of the Conley index, the sequence (E n ) ∞ n=n 0 = (Y n /Z n ) ∞ n=n 0 uniquely determines the G-homotopy type of a spectrum. The Gequivariant Conley index of X with respect to the flow generated by f , denoted by CI G (X, f ), is defined as this G-homotopy type.
As in the finite dimensional case, we use the simplified notation CI G (S, f ) for the Conley index of an isolating neighbourhood of the isolated invariant set S of the flow generated by f .

The degree for equivariant gradient maps
In this subsection we recall the definition and basic properties of the degree for G-equivariant gradient maps of the form of a completely continuous perturbation of the identity, denoted by ∇ G -deg(·, ·), defined by Rybicki [22]. This degree is an element of the Euler ring U (G). Let H be an infinite-dimensional, separable Hilbert space which is an orthogonal G-representation. We keep the notation of the previous section, namely we consider finite dimensional subrepresentations H n of H and the projections τ n : H → H n .
Assume that ⊂ H is an open, bounded and G-invariant subset and ξ ∈ C 1 (H, R) is a G-invariant map such that (1) ∇ξ : H → H is a completely continuous, G-equivariant operator, The degree of Rybicki is defined with the use of the degree for finite dimensional equivariant gradient maps introduced by Gęba [8], also being an element of the Euler ring U (G). For simplicity of notation, we use the same symbol ∇ G -deg(·, ·) for both degrees.
Restricting I d − ∇ξ to H n , one can study Gęba's degree of these restrictions. It appears (see [22]) that it stabilises for n sufficiently large, i.e. there is n 0 such that for every n ≥ n 0 .
Using this equality one can define the degree for infinite dimensional equivariant gradient maps. More precisely, fix n 0 as in the above lemma and define the degree for G-equivariant gradient maps of I d − ∇ξ on by A possible application of this degree is studying phenomena of global bifurcations and, in particular, an equivariant version of the Rabinowitz alternative. Let us make this more precise. Consider a family of G-invariant functionals ∈ C 2 (H × R, R) and suppose that there is u 0 ∈ H such that ∇ u (u 0 , λ) = 0 for every λ ∈ H. The invariance of implies that G(u 0 ) ⊂ (∇ u (·, λ)) −1 (0) for every λ ∈ H. We call the elements of G(u 0 ) × R the trivial solutions of ∇ u (u, λ) = 0.
Assume additionally that The proof of this theorem is standard. It is enough to replace in the classical proof (see [3,19,20]) the Leray-Schauder degree by the degree for equivariant gradient operators. Equivariant versions have been formulated in [22] (Theorem 4.9) and [10] (Theorem 3.3). A version for bifurcations from an orbit in finite dimensional representations has been proved in [16] (Theorem 3.4). Now we turn our attention to relations between the degree theory and the theory of the Conley index. For the Gęba's degree and the Conley index for equivariant maps defined on finite dimensional representations such a relation has been obtained by Gęba [8] (see also Corollary 1 in [11]). We recall it in the following theorem: Theorem 5.8 Let V be a finite dimensional G-representation and ϕ ∈ C 2 (V, R) a Ginvariant map such that G(u 0 ) ⊂ (∇ϕ) −1 (0) for some u 0 ∈ V. Assume that the orbit G(u 0 ) is non-degenerate and fix ⊂ V such that (∇ϕ) −1 (0) ∩ = G(u 0 ). Then ∇ G -deg(∇ϕ, ) = χ G (C I G (G(u 0 ), −∇ϕ)) .

Eigenspaces of the Laplace operator
In this subsection we introduce basic properties of the eigenspaces of the Laplace operator (with Neumann boundary conditions) on the ball. More precisely, we study the problem: These properties are known, but it is difficult to find a reference in the literature, except for the case N = 2, 3, see for example [1,17]. To make our article self-contained, we sketch here the general case. Let H N l denote the linear space of harmonic, homogeneous polynomials of N independent variables, of degree l, restricted to the sphere S N −1 . To illustrate the above description of the eigenspaces, we will look more closely at the cases N = 2, 3.