1 Introduction

In this paper, we study bifurcations of weak solutions of elliptic systems of the form:

$$\begin{aligned} \left\{ \begin{array}{ll} - \triangle u = \lambda \nabla F(u ) &{} \quad \text {in } B^N \\ \frac{\partial u}{\partial \nu } = 0 &{}\quad \text {on } S^{N-1}, \end{array}\right. \end{aligned}$$
(1.1)

where \(B^N\) is the open unit ball in \(\mathbb {R}^N\), \(S^{N-1}=\partial B^N\) and the function \(F:\mathbb {R}^m \rightarrow \mathbb {R}\) satisfies additional assumptions, see Sect. 2.

In particular, we are interested in the equivariant case. Namely, we assume that on the space \(\mathbb {R}^m\) there is defined an action of a compact Lie group \(\Gamma \) and \(\nabla F\) is a \(\Gamma \)-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 \(\nabla F^{-1}(0).\) For \(u_0 \in \nabla F^{-1}(0)\) the constant function \({\tilde{u}}_0 \equiv u_0\) is a solution of (1.1) for all \(\lambda \in \mathbb {R}\). Therefore, we obtain the family of trivial solutions \(\{{\tilde{u}}_0\} \times \mathbb {R}\). Investigating the change of the Conley index for different levels \(\lambda \in \mathbb {R}\), one can obtain a sequence of nontrivial weak solutions bifurcating from the point \(({\tilde{u}}_0, \lambda _0)\), for some values \(\lambda _0 \in \mathbb {R}.\) Investigating the change of the topological degree, one can prove the existence of the continuum, emanating from \(({\tilde{u}}_0, \lambda _0)\), 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 \(\nabla F\) is a \(\Gamma \)-equivariant mapping, we obtain that for \(u_0 \in \nabla F^{-1}(0)\) also \(\gamma u_0 \in \nabla F^{-1}(0)\) for all \(\gamma \in \Gamma \). 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 \(\Phi \) defined on a suitable Hilbert space \(\mathbb {H}\). 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 \(\mathcal {G}=\Gamma \times SO(N).\) Since \(\mathbb {R}^m\) is a \(\Gamma \)-representation (by assumption) and \(B^N\) is an SO(N)-invariant set, it follows that the space \(\mathbb {H}\) is a \(\mathcal {G}\)-representation. Moreover, for \(u_0 \in (\nabla F)^{-1}(0)\), \((g{\tilde{u}}_0, \lambda )\) is a critical point of \(\Phi \) for all \(g\in \mathcal {G}, \lambda \in \mathbb {R}.\)

Therefore we can consider the set of trivial solutions \(\mathcal {T}=\mathcal {G}({\tilde{u}}_0) \times \mathbb {R}\). We are going to investigate bifurcations of nontrivial solutions from the family \(\mathcal {T}\). 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 \(\mathcal {G}({\tilde{u}}_0) \times \{\lambda _0\}\).

We also consider global symmetry breaking phenomena at the orbit \(\mathcal {G}({\tilde{u}}_0) \times \{\lambda _0\}\). 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 \(\Gamma \) 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.

1.1 Notation

Suppose that G is a compact Lie group. We denote by \(\overline{{\text {sub}}}(G)\) 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 \(\chi _G(\cdot )\) to denote the G-equivariant Euler characteristic of a pointed finite G-CW-complex. Moreover, the symbols \(CI_{G}(S, f)\) and \({\mathcal {C}}{\mathcal {I}}_{G}(S, f)\) 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 \(\mathbb {H}\) and \(u_0 \in \mathbb {H}\) we denote by \(B_{\delta }(u_0,\mathbb {H})\) (respectively \(D_{\delta }(u_0,\mathbb {H})\)) the open (respectively closed) ball in \(\mathbb {H}\) centred at \(u_0\) and with radius \(\delta \). In particular, we use the symbol \(B^N\) for the open ball if \(\delta =1\), \(u_0=0\) and \(\mathbb {H}=\mathbb {R}^N\) and we write \(S^{N-1}\) for \(\partial B^N.\)

2 Preliminaries

Throughout this paper \(\Gamma \) stands for a compact Lie group and \(\mathbb {R}^m\) is an orthogonal representation of the group \(\Gamma \). Consider \(F:\mathbb {R}^m \rightarrow \mathbb {R}\) satisfying:

  1. (B1)

    \(F \in C^2(\mathbb {R}^m , \mathbb {R})\) is such that for every \(u \in \mathbb {R}^m\) we have \(|\nabla ^2 F(u) | \le a + b |u|^{q}\) where \(a,b \in \mathbb {R}\) and \(1<q < \frac{4}{N-2}\) for \(N \ge 3\) and \( 1< q< \infty \) for \(N=2,\)

  2. (B2)

    F is \(\Gamma \)-invariant, i.e. \(F(\gamma u) =F(u)\) for every \(\gamma \in \Gamma \), \(u\in \mathbb {R}^m\).

Our aim is to study bifurcations of weak solutions of the nonlinear Neumann problem, parameterised by \(\lambda \in \mathbb {R}\),

$$\begin{aligned} \left\{ \begin{array}{ll} - \triangle u = \lambda \nabla F(u ) &{} \quad \text {in } B^N \\ \frac{\partial u}{\partial \nu } = 0 &{}\quad \text {on } S^{N-1}. \end{array}\right. \end{aligned}$$
(2.1)

Denote by \(H^1(B^N)\) the usual Sobolev space on \(B^N\) and consider the separable Hilbert space \(\mathbb {H}= \bigoplus _{i=1}^{m} H^1(B^N)\) with the scalar product

$$\begin{aligned} \displaystyle \langle v, w \rangle _{\mathbb {H}}= & {} \displaystyle \sum _{i=1}^m \langle v_i, w_i \rangle _{H^1( B^N)} = \sum _{i=1}^m \int \limits _{B^N} (\nabla v_i(x), \nabla w_i(x)) +\, v_i(x)\cdot w_i(x) dx. \end{aligned}$$
(2.2)

Denote by \(\mathcal {G}\) the group \(\Gamma \times SO(N)\), where SO(N) is the special orthogonal group in dimension N. Note that the space \(\mathbb {H}\) with the scalar product given by (2.2) is an orthogonal \({\mathcal {G}}\)-representation with the \({\mathcal {G}}\)-action given by

$$\begin{aligned} (\gamma , \alpha ) (u)(x)= \gamma u({\alpha }^{-1}x)\ \text { for }\ (\gamma , \alpha ) \in {\mathcal {G}}, u \in \mathbb {H}, x\in B^N. \end{aligned}$$
(2.3)

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 \(\Phi :\mathbb {H}\times \mathbb {R}\rightarrow \mathbb {R}\) defined by

$$\begin{aligned} \Phi (u,\lambda ) = \frac{1}{2} \int \limits _{B^N} |\nabla u(x)|^2 dx- \lambda \int \limits _{B^N} F(u(x))dx. \end{aligned}$$
(2.4)

Computing the gradient of \(\Phi \) with respect to u we obtain:

$$\begin{aligned} \langle \nabla _{u} \Phi (u,\lambda ), v \rangle _{\mathbb {H}}= & {} \int \limits _{B^N} (\nabla u(x), \nabla v(x) )-\, (\lambda \nabla F (u(x)), v(x) ) dx, \ u,v \in \mathbb {H}. \end{aligned}$$
(2.5)

Moreover,

$$\begin{aligned} \begin{aligned} \left<\nabla ^2_u \Phi (u, \lambda )w, v\right>_{\mathbb {H}}&=\int \limits _{B^N} ( \nabla w(x),\nabla v(x))\\&\quad -(\lambda \nabla ^2 F(u(x))w(x), v(x)) dx, \ u, w, v \in \mathbb {H}. \end{aligned} \end{aligned}$$

Assumption (B2) implies that \(\nabla _u \Phi :\mathbb {H}\times \mathbb {R}\rightarrow \mathbb {H}\) is \({\mathcal {G}}\)-equivariant.

Moreover, from imbedding theorems and the assumption (B1) it follows that the operator \(\nabla _u \Phi \) is a completely continuous perturbation of the identity, see [21].

2.1 Linear equation

In this subsection we consider the Eq. (2.1) in the linear case, i.e. the system:

$$\begin{aligned} \left\{ \begin{array}{ll} - \triangle u = \lambda A u &{} \quad \text {in } B^N \\ \frac{\partial u}{\partial \nu } = 0 &{}\quad \text {on } S^{N-1}, \end{array} \right. \end{aligned}$$
(2.6)

where A is a real, symmetric \((m\times m)\)-matrix.

Using formula (2.4) we can associate with (2.6) the functional \(\Phi _A :\mathbb {H}\times \mathbb {R}\rightarrow \mathbb {R}\) given by

$$\begin{aligned} \Phi _A (u,\lambda ) = \frac{1}{2} \int \limits _{B^N} |\nabla u(x)|^2 dx - \frac{\lambda }{2} \int \limits _{B^N} (Au(x),u(x))dx. \end{aligned}$$
(2.7)

Note that from (2.5) for every \(v\in \mathbb {H}\) we have

$$\begin{aligned} \langle \nabla _u \Phi _A(u,\lambda ),v\rangle _{\mathbb {H}}= \langle u, v \rangle _{\mathbb {H}} - \langle L_{\lambda A} u, v \rangle _{\mathbb {H}}, \end{aligned}$$

where

$$\begin{aligned} \langle L_{\lambda A} u, v \rangle _{\mathbb {H}} = \int \limits _{B^N} (u(x), v(x)) + (\lambda A u(x), v(x)) dx. \end{aligned}$$
(2.8)

The existence and boundedness of the operator \(L_{\lambda A}:\mathbb {H}\rightarrow \mathbb {H}\) follow from the Riesz theorem. By definition \(L_{\lambda A}\) is self-adjoint.

Let us denote by \(\sigma (-\Delta ; B^N) = \{ 0= \beta _1< \beta _2< \ldots< \beta _k < \ldots \}\) the set of distinct eigenvalues of the Laplace operator (with Neumann boundary conditions) on the ball. Write \(\mathbb {V}_{-\Delta }(\beta _k)\) for the eigenspace of \(-\Delta \) corresponding to \(\beta _k \in \sigma (-\Delta ; B^N)\). In the “Appendix” we give a more precise description of these eigenspaces. By the spectral theorem it follows that \(\displaystyle H^1(B^N) = cl ( \bigoplus \nolimits _{k=1}^{\infty } \mathbb {V}_{-\Delta } (\beta _k)).\) Let us denote by \(\mathbb {H}_k\) the space \(\displaystyle \bigoplus \nolimits _{i=1}^m \mathbb {V}_{-\Delta }(\beta _k).\) In particular, \(u=\sum \nolimits _{k=1}^{\infty }u_k\) for every \(u\in \mathbb {H}\), where \(u_k\in \mathbb {H}_k\).

Let \(\alpha _1,\ldots ,\alpha _m\) denote the eigenvalues of A (not necessarily distinct) with corresponding eigenvectors \(f_1,\ldots ,f_m\), which form an orthonormal basis of \(\mathbb {R}^m\).

Let \(\pi _j:\mathbb {H}\rightarrow H^1(B^N)\) be the projection such that \(\pi _j(u)(x)=(u(x),f_j)\), \(j=1,\ldots , m\). Clearly, if \(u_k \in \mathbb {H}_k,\) then \(\pi _j(u_k) \in \mathbb {V}_{-\Delta }(\beta _k)\) for \(j=1, \ldots , m.\)

In the lemma below we characterise the operator \(L_{\lambda A}\), given by the formula (2.8).

Lemma 2.1

For every \(u\in \mathbb {H}\)

$$\begin{aligned} L_{\lambda A} u=\sum \limits _{k=1}^{\infty }\sum \limits _{j=1}^m \frac{1+\lambda \alpha _j}{1+\beta _k} \pi _j(u_k) \cdot f_j. \end{aligned}$$

The proof of this lemma is standard, see for example the proof of Lemma 3.2 in [9].

Let us denote by \(\sigma (L)\) the spectrum of a linear operator \(L:\mathbb {H}\rightarrow \mathbb {H}\). From the above lemma it immediately follows the corollary:

Corollary 2.2

Let \(L_{\lambda A}\) be defined by (2.8). Then:

$$\begin{aligned} \sigma (L_{\lambda A})=\left\{ \frac{1+\lambda \alpha _j}{1+\beta _k}:\alpha _j \in \sigma (A), \beta _k\in \sigma (-\Delta ; B^N) \right\} . \end{aligned}$$

Moreover,

$$\begin{aligned} \sigma (Id-L_{\lambda A})=\left\{ \frac{\beta _k-\lambda \alpha _j}{1+\beta _k}:\alpha _j \in \sigma (A), \beta _k\in \sigma (-\Delta ; B^N) \right\} . \end{aligned}$$

Fix eigenvalues \(\alpha _{j_0}\in \sigma (A)\) and \(\beta _{k_0}\in \sigma (-\Delta ; B^N)\). Let \(\mathbb {V}_A(\alpha _{j_0})\) be the eigenspace associated with the eigenvalue \(\alpha _{j_0}\) and \(\mu _{A}(\alpha _{j_0}) = \dim \mathbb {V}_A(\alpha _{j_0})\). Let \(\Pi _{j_0}:\mathbb {R}^m\rightarrow \mathbb {R}^m\) be the orthogonal projection such that \(\Pi _{j_0}(\mathbb {R}^m)=\mathbb {V}_A(\alpha _{j_0})\) and define \({\tilde{\Pi }}_{j_0}:\mathbb {H}\rightarrow \mathbb {H}\) by \(({\tilde{\Pi }}_{j_0}(u))(x)=\Pi _{j_0}(u(x))\). Denote

$$\begin{aligned} \mathbb {V}_{-\Delta }(\beta _{k_0})^{\mu _{A}(\alpha _{j_0})}={\tilde{\Pi }}_{j_0}\left( \bigoplus \limits _{j=1}^m\mathbb {V}_{-\Delta }(\beta _{k_0}) \right) . \end{aligned}$$

It follows that

$$\begin{aligned} \mathbb {V}_{-\Delta }(\beta _{k_0})^{\mu _{A}(\alpha _{j_0})}= \mathrm {span}\left\{ h\cdot f:h\in \mathbb {V}_{-\Delta }(\beta _{k_0}), f\in \mathbb {V}_A(\alpha _{j_0}) \right\} \subset \mathbb {H}. \end{aligned}$$

From Lemma 2.1 we obtain:

Corollary 2.3

If \(\sigma (\lambda A)\cap \sigma (-\Delta ; B^N)=\{\alpha _{j_1},\ldots , \alpha _{j_s}\}\), then

$$\begin{aligned} \ker ( Id - L_{\lambda A})=\mathbb {V}_{-\Delta }(\alpha _{j_1})^{\mu _{\lambda A}(\alpha _{j_1})}\oplus \cdots \oplus \mathbb {V}_{-\Delta }(\alpha _{j_s})^{\mu _{\lambda A}(\alpha _{j_s})}. \end{aligned}$$

2.2 The notion of bifurcation from the critical orbit

Fix \(u_0 \in (\nabla F)^{-1}(0)\). Since F is \(\Gamma \)-invariant, and therefore \(\nabla F\) is \(\Gamma \)-equivariant, \(\gamma u_0 \in (\nabla F)^{-1}(0)\) for all \(\gamma \in \Gamma \), i.e. \(\Gamma (u_0) \subset (\nabla F)^{-1}(0).\) We call such a set a critical orbit of F.

Note that \(T_{u_0} \Gamma (u_0) \subset \ker \nabla ^2 F (u_0)\) and therefore \(\dim \ker \nabla ^2 F (u_0) \ge \dim T_{u_0} \Gamma (u_0) = \dim \Gamma (u_0) .\) We assume that in this inequality there holds:

$$\begin{aligned} \dim \ker \nabla ^2 F (u_0) = \dim \Gamma (u_0).\end{aligned}$$
(2.9)

We call such an orbit non-degenerate.

By the equivariant Morse lemma, see [31], from (2.9) we conclude that \(\Gamma (u_0)\) is isolated in \((\nabla F)^{-1}(0).\)

Since \(u_0 \in (\nabla F)^{-1}(0)\), the constant function \({\tilde{u}}_0\equiv u_0\) is a solution of the problem (2.1) for all \(\lambda \in \mathbb {R}\). Therefore, \(({\tilde{u}}_0, \lambda )\), and consequently \((\gamma {\tilde{u}}_0, \lambda )\) for every \(\gamma \in \Gamma \), is a critical point of the functional \(\Phi \) given by (2.4). Since from (2.3) we have \(\mathcal {G}({\tilde{u}}_0)=\Gamma ({\tilde{u}}_0)\), we obtain a critical orbit of \(\Phi \) and therefore a \({\mathcal {G}}\)-orbit of weak solutions of (2.1) for all \(\lambda \in \mathbb {R}\). Hence we can consider a family of solutions \(\mathcal {T}= {\mathcal {G}}({\tilde{u}}_0) \times \mathbb {R}\subset \mathbb {H}\times \mathbb {R}\). We call the elements of \(\mathcal {T}\) the trivial solutions of (2.1). Put \(\mathcal {N}=\{(v, \lambda ) \in (\mathbb {H}\times \mathbb {R}) \setminus \mathcal {T}:\nabla _v\Phi (v, \lambda )=0\}.\)

Definition 2.4

A local bifurcation from the orbit \({\mathcal {G}}({\tilde{u}}_0) \times \{\lambda _0\} \subset \mathcal {T}\) of solutions of (2.1) occurs if the point \(({\tilde{u}}_0, \lambda _0)\) is an accumulation point of the set \(\mathcal {N}\).

Remark 2.5

Note that if \(({\tilde{u}}_0, \lambda _0)\) is an accumulation point of \(\mathcal {N}\) then for all \(g \in \mathcal {G}\), \((g{\tilde{u}}_0, \lambda _0)\) is also an accumulation point, since \(\mathbb {H}\) is an orthogonal representation of \(\mathcal {G}\). Therefore \(\mathcal {G}({\tilde{u}}_0)\subset cl(\mathcal {N})\).

Definition 2.6

A global bifurcation from the orbit \({\mathcal {G}}({\tilde{u}}_0) \times \{\lambda _0\} \subset \mathcal {T}\) of solutions of (2.1) occurs if there is a connected component \(\mathcal {C}(\lambda _0)\) of \(cl (\mathcal {N})\), containing \({\mathcal {G}}({\tilde{u}}_0) \times \{\lambda _0\}\), such that either \(\mathcal {C}(\lambda _0)\cap (\mathcal {T}\setminus ({\mathcal {G}}({\tilde{u}}_0)\times \{\lambda _0\}))\ne \emptyset \) or \(\mathcal {C}(\lambda _0)\) is unbounded.

The set of all \(\lambda _0 \in \mathbb {R}\) such that a local (respectively global) bifurcation from the orbit \({\mathcal {G}}({\tilde{u}}_0) \times \{\lambda _0\} \) occurs we denote by BIF (respectively GLOB). Note that directly from the above definitions it follows that \(GLOB \subset BIF.\)

2.3 Admissible pair

The notion of an admissible pair has been introduced in [18].

Fix a compact Lie group G and let \(H\in \overline{{\text {sub}}}(G)\). Denote by \((H)_G\) the conjugacy class of H.

Definition 2.7

A pair (G, H) is called admissible, if for any \(K_1,K_2\in \overline{{\text {sub}}}(H)\) the following condition is satisfied: if \((K_1)_H\ne (K_2)_H\), then \((K_1)_G\ne (K_2)_G\).

Lemma 2.8

The pair \((\Gamma \times SO(N), \{e\}\times SO(N))\) is admissible.

Proof

Let us denote by H the group \(\{e\} \times SO(N)\) and recall that \(\mathcal {G}=\Gamma \times SO(N).\) Moreover, let \({\tilde{K}}_1, {\tilde{K}}_2 \in \overline{{\text {sub}}}(H).\) By definition of H there are \(K_1, K_2 \in \overline{{\text {sub}}}(SO(N))\) such that \({\tilde{K}}_1 = \{e\} \times K_1\) and \({\tilde{K}}_2 = \{e\} \times K_2\). Suppose that \(({\tilde{K}}_1)_{\mathcal {G}}= ({\tilde{K}}_2)_{\mathcal {G}}\), i.e. \((\{e\} \times K_1)_{\mathcal {G}} =(\{e\} \times K_2)_{\mathcal {G}}.\) Therefore there exists \({(\gamma , \alpha )\in {\mathcal {G}}}\) such that \(\{e\} \times K_1 = (\gamma , \alpha )(\{e\} \times K_2)(\gamma , \alpha )^{-1}\) and hence

$$\begin{aligned} \{e\} \times K_1= & {} \{\gamma e \gamma ^{-1}\} \times \alpha K_2 \alpha ^{-1} = \{e\} \times \alpha K_2 \alpha ^{-1}\\= & {} (e,\alpha ) (\{e\} \times K_2)(e,\alpha )^{-1}. \end{aligned}$$

Thus \(({\tilde{K}}_1)_H = ({\tilde{K}}_2)_H\) and the proof is complete. \(\square \)

3 Main results

Consider the nonlinear system (2.1) with a potential F satisfying (B1), (B2). Fix \(u_0 \in (\nabla F)^{-1}(0)\) such that the orbit \(\Gamma (u_0)\) is non-degenerate. We make two additional assumptions:

  1. (B3)

    \(F(u) = \frac{1}{2} (Au, u)- (A u_0,u) + g(u-u_0)\), where A is a real symmetric \((m\times m)\)-matrix and \(\nabla g (u) = o(|u|)\) for \(|u| \rightarrow 0\),

  2. (B4)

    \(\Gamma _{u_0}=\{e\}.\)

From the assumption (B3) we conclude that the gradient of the functional associated with the Eq. (2.1) has the following form:

$$\begin{aligned} \nabla _u \Phi (u, \lambda )= u - {\tilde{u}}_0 - L_{\lambda A}(u-{\tilde{u}}_0)+\lambda \nabla \eta (u-{\tilde{u}}_0),\end{aligned}$$

where \(L_{\lambda A}:\mathbb {H}\rightarrow \mathbb {H}\) is an SO(N)-equivariant operator given by (2.8). Moreover, \(\nabla \eta :\mathbb {H}\rightarrow \mathbb {H}\) given by \(\langle \nabla \eta (u), v\rangle _{\mathbb {H}}= \int _{B^N}(\nabla g(u(x)),v(x)) dx\) is an SO(N)-equivariant operator such that \(\nabla \eta (u)=o(\Vert u\Vert _{\mathbb {H}})\) for \(\Vert u\Vert _{\mathbb {H}}\rightarrow 0.\)

From the assumption (B4) it follows that \(\mathcal {G}_{{\tilde{u}}_0}=\{e\} \times SO(N).\)

3.1 Bifurcation from the critical orbit

Following the standard notation we denote the linear part of \(\nabla _u \Phi (\cdot , \lambda )\) at \({\tilde{u}}_0\) by \(\nabla ^2_u \Phi ({\tilde{u}}_0, \lambda )u,\) thus \(\nabla ^2_u \Phi ({\tilde{u}}_0, \lambda )u=u - L_{\lambda A} u\).

Let us denote by \(\Lambda \) the set \(\bigcup _{\alpha _j \in \sigma (A) \setminus \{0\}} \bigcup _{\beta _k \in \sigma (-\Delta ;B^N)} \{\frac{\beta _k}{\alpha _j}\}.\)

Lemma 3.1

If \(\lambda _0 \in BIF,\) then \(\lambda _0 \in \Lambda .\)

Proof

We first observe that for all \(\lambda \in \mathbb {R}\), since \(\mathcal {G}({\tilde{u}}_0)\) is a critical orbit of \(\Phi (\cdot ,\lambda )\), we have \(\dim \ker \nabla ^2_u\Phi ({\tilde{u}}_0,\lambda )\ge \dim (\mathcal {G}({\tilde{u}}_0)\times \{\lambda \})\).

Moreover if \(\lambda _0 \in BIF\), this inequality is strict. Indeed, if \(\dim \ker \nabla ^2_u\Phi ({\tilde{u}}_0,\lambda _0)= \dim (\mathcal {G}({\tilde{u}}_0)\times \{\lambda _0\})\), then by the equivariant implicit function theorem (see Theorem 5.1) there exists \(\varepsilon >0\) such that the only solutions of the equation \(\nabla _u \Phi (u, \lambda ) = 0\) are elements of \({\mathcal {G}}({\tilde{u}}_0) \times \{\lambda \}\) for \(\lambda \in (\lambda _0 - \varepsilon , \lambda _0 + \varepsilon ).\) From this we obtain \(\lambda _0 \not \in BIF.\) Therefore, if \(\lambda _0 \in BIF,\)

$$\begin{aligned} \dim \ker \nabla ^2_u \Phi ({\tilde{u}}_0, \lambda _0) > \dim ({\mathcal {G}}({\tilde{u}}_0) \times \{\lambda _0\}).\end{aligned}$$
(3.1)

Since \(\mathcal {G}({\tilde{u}}_0)=\Gamma ({\tilde{u}}_0),\) we conclude from (2.9) and (3.1) that \(\displaystyle \dim \ker \nabla ^2_u \Phi ({\tilde{u}}_0, \lambda _0) > \dim ({\mathcal {G}}({\tilde{u}}_0) \times \{\lambda _0\})= \dim \ker \nabla ^2 F (u_0),\) i.e. \(\dim \ker \left( Id - L_{\lambda _0 A}\right) > \dim \ker A.\) Using Corollary 2.2 we obtain that this condition is satisfied if and only if \(\{(\alpha _j, \beta _k) \in \sigma (A) \times \sigma (-\Delta ; B^N):\beta _k = \lambda _0 \alpha _j\} \ne \{(0,0)\}.\) Therefore there are \((\alpha _j, \beta _k) \in \sigma (A)\setminus \{0\} \times \sigma (-\Delta ,B^N)\) such that \(\beta _k=\lambda _0\cdot \alpha _j\), i.e. \(\lambda _0 \in \Lambda .\) \(\square \)

Fix \(\lambda _0 \in \Lambda \) and choose \(\varepsilon >0\) such that \(\Lambda \cap [\lambda _0-\varepsilon ,\lambda _0+\varepsilon ]=\{\lambda _0\}\). From the definition of \(\Lambda \) such a choice is always possible.

Since \(\lambda _0 \pm \varepsilon \notin \Lambda ,\) Lemma 3.1 implies that \(\lambda _0\pm \varepsilon \notin BIF\) and therefore \(\mathcal {G}({\tilde{u}}_0) \subset \mathbb {H}\) is an isolated critical orbit of the \(\mathcal {G}\)-invariant functionals \(\Phi (\cdot , \lambda _0 \pm \varepsilon ) :\mathbb {H}\rightarrow \mathbb {R}.\) From this and the properties of flows induced by gradient operators, we conclude that \(\mathcal {G}({\tilde{u}}_0)\) is also an isolated invariant set (in the sense of the equivariant Conley index theory, see [13]) for the flows induced by the operators \(-\nabla _u \Phi (\cdot , \lambda _0 \pm \varepsilon ) \). Therefore, the indices \(\mathcal {C}\mathcal {I}_{{\mathcal {G}}}({\mathcal {G}}({\tilde{u}}_0),-\nabla _u\Phi (\cdot , \lambda _0-\varepsilon ))\), \(\mathcal {C}\mathcal {I}_{{\mathcal {G}}}({\mathcal {G}}({\tilde{u}}_0),-\nabla _u\Phi (\cdot , \lambda _0+\varepsilon ))\) are well-defined. In the following we study when they are not equal.

Assume that \(\sigma (\lambda _0 A)\cap \sigma (-\Delta ; B^N)=\{\alpha _{j_1},\ldots , \alpha _{j_s}\}.\) We consider the conditions:

  1. (C1)

    \(\lambda _0\ne 0\) and there is \(i\in \{1,\ldots ,s\}\) satisfying \(\dim \mathbb {V}_{-\Delta }(\alpha _{j_i})>1\),

  2. (C2)

    \(\lambda _0\ne 0\), \(\dim \mathbb {V}_{-\Delta }(\alpha _{j_i})=1\) for every \(i\in \{1,\ldots ,s\}\) and \(\dim \ker (Id-L_{\lambda _0A}) - \dim \ker A\) is an odd number,

  3. (C3)

    \(\lambda _0= 0\) and \(\sum _{\alpha \in \sigma _+(A)} \mu _A(\alpha )-\sum _{\alpha \in \sigma _-(A)} \mu _A(\alpha )\) is odd.

Remark 3.2

Note that we can reformulate conditions (C1)–(C3) in the following way:

  1. (C1’)

    \(\lambda _0\ne 0\) and there is \(i\in \{1,\ldots ,s\}\) such that \(\mathbb {V}_{-\Delta }(\alpha _{j_i})\) is a nontrivial SO(N)-representation,

  2. (C2’)

    \(\lambda _0\ne 0\), \(\dim \mathbb {V}_{-\Delta }(\alpha _{j_i})=1\) for every \(i\in \{1,\ldots ,s\}\) and \(\sum ^s_{i=1} \mu _{\lambda _0 A}(\alpha _{j_i})-\mu _A(0)\) is odd,

  3. (C3’)

    \(\lambda _0=0\) and \(m-\dim \ker A\) is odd.

Indeed,

  1. (1)

    \(\dim \mathbb {V}_{-\Delta }(\alpha _{j_i})>1\) if and only if \(\mathbb {V}_{-\Delta }(\alpha _{j_i})\) is a nontrivial SO(N)-representation, see Remark 5.11;

  2. (2)

    since \(\dim \mathbb {V}_{-\Delta }(\alpha _{j_i})=1\), from Corollary 2.3 we obtain \(\dim \ker (Id-L_{\lambda _0A})=\sum ^s_{i=1} \mu _{\lambda _0 A}(\alpha _{j_i})\);

  3. (3)

    since \(\sum _{\alpha \in \sigma _+(A)} \mu _A(\alpha )+\sum _{\alpha \in \sigma _-(A)} \mu _A(\alpha )+\mu _A(0)=m\), if \(m-\dim \ker A\) is odd, then so is \(\sum _{\alpha \in \sigma _+(A)} \mu _A(\alpha )-\sum _{\alpha \in \sigma _-(A)} \mu _A(\alpha )\).

Theorem 3.3

Assume that \(\lambda _0 \in \Lambda \) and one of the conditions (C1)–(C3) is satisfied. Then

$$\begin{aligned} {\mathcal {C}}{\mathcal {I}}_{{\mathcal {G}}}({\mathcal {G}}({\tilde{u}}_0),-\nabla _u\Phi (\cdot , \lambda _0-\varepsilon )) \ne {\mathcal {C}}{\mathcal {I}}_{{\mathcal {G}}}({\mathcal {G}}({\tilde{u}}_0),-\nabla _u\Phi (\cdot , \lambda _0+\varepsilon )). \end{aligned}$$

Proof

Denote by \({\tilde{\mathbb {H}}}\subset \mathbb {H}\) the linear subspace normal to \({\mathcal {G}}({\tilde{u}}_0)\) at \({\tilde{u}}_0\), i.e. \({\tilde{\mathbb {H}}}=T_{{\tilde{u}}_0}^{\perp } {\mathcal {G}}({\tilde{u}}_0)\subset \mathbb {H}\). We start the proof with showing that we can reduce comparing the Conley indices \(\mathcal {C}\mathcal {I}_{{\mathcal {G}}}({\mathcal {G}}({\tilde{u}}_0),-\nabla _u\Phi (\cdot , \lambda _0\pm \varepsilon ))\) to comparing Euler characteristics of some indices on the space \({\tilde{\mathbb {H}}}\).

For \(n\ge 1\) put \(\mathbb {H}^n=\bigoplus _{k=1}^{n}\mathbb {H}_k\) and

$$\begin{aligned} \Phi ^n=\Phi _{|\mathbb {H}^n\times \mathbb {R}}:\mathbb {H}^n\times \mathbb {R}\rightarrow \mathbb {R}. \end{aligned}$$
(3.2)

Note that \(\mathcal {G}({\tilde{u}}_0)=\Gamma ({\tilde{u}}_0) \subset T_{{\tilde{u}}_0} {\Gamma }({\tilde{u}}_0) \oplus T_{{\tilde{u}}_0}^{\perp }{\Gamma }({\tilde{u}}_0) \approx \mathbb {R}^m \approx \mathbb {H}_1\) (by \(T_{{\tilde{u}}_0}^{\perp }{\Gamma }({\tilde{u}}_0)\) we understand the complement of the space \(T_{{\tilde{u}}_0} {\Gamma }({\tilde{u}}_0)\) in \(\mathbb {H}_1\)). Therefore \(\mathcal {G}({\tilde{u}}_0)\) is a critical orbit of \(\Phi ^n(\cdot , \lambda _0 \pm \varepsilon )\) for \(n \ge 1\). Note that, from the choice of \(\varepsilon \) and the definition of \(\Phi ^n\), this orbit is non-degenerate.

Since \(\nabla _u \Phi (\cdot ,\lambda )\) is a completely continuous perturbation of the identity for all \(\lambda \in \mathbb {R}\), from the definition of the infinite dimensional equivariant Conley index, see Sect. 5.2, the assertion of the theorem is equivalent to

$$\begin{aligned} CI_{\mathcal {G}} (\mathcal {G}({\tilde{u}}_0), -\nabla _u\Phi ^n(\cdot , \lambda _0-\varepsilon )) \ne CI_{\mathcal {G}} (\mathcal {G}({\tilde{u}}_0), -\nabla _u\Phi ^n(\cdot , \lambda _0+\varepsilon )) \end{aligned}$$

for n sufficiently large. Obviously, the above inequality is implied by

$$\begin{aligned} \chi _{\mathcal {G}}(CI_{\mathcal {G}} (\mathcal {G}({\tilde{u}}_0), -\nabla _u\Phi ^n(\cdot , \lambda _0-\varepsilon ))) \ne \chi _{\mathcal {G}}(CI_{\mathcal {G}} (\mathcal {G}({\tilde{u}}_0), -\nabla _u\Phi ^n(\cdot , \lambda _0+\varepsilon ))). \end{aligned}$$
(3.3)

It is known that the \(\mathcal {G}\)-action on \(\mathbb {H}\) given by (2.3) defines a \(\mathcal {G}_{{\tilde{u}}_0}\)-action on \({\tilde{\mathbb {H}}}\). Recall that \(\mathcal {G}_{{\tilde{u}}_0}=\{e\} \times SO(N).\) Hence \({\tilde{\mathbb {H}}}\) is an orthogonal SO(N)-representation.

For \(n\ge 1\) put \({\tilde{\mathbb {H}}}^n=\mathbb {H}^n \cap {\tilde{\mathbb {H}}}=T_{{\tilde{u}}_0}^{\perp } \Gamma ({\tilde{u}}_0)\oplus \bigoplus _{k=2}^{n}\mathbb {H}_k \) and define \(\Psi ^n_{\pm }=\Phi ^n(\cdot , \lambda _0 \pm \varepsilon )_{|{\tilde{\mathbb {H}}}^n}:{\tilde{\mathbb {H}}}^n \rightarrow \mathbb {R}.\) From this definition the functionals \(\Psi ^n_{\pm }\) are SO(N)-invariant. Since \(\mathcal {G}({\tilde{u}}_0)\) is a non-degenerate critical orbit of \(\Phi ^n(\cdot , \lambda _0\pm \varepsilon )\), \({\tilde{u}}_0\in {\tilde{\mathbb {H}}}\) is a non-degenerate critical point of \(\Psi ^n_{\pm }\). Hence \(\{{\tilde{u}}_0\}\) is an isolated invariant set (in the sense of the Conley index theory) of the flows generated by \(-\nabla \Psi ^n _{\pm }\).

Note that since \(\mathcal {G}_{{\tilde{u}}_0}=\{e\} \times SO(N)\), by Lemma 2.8 the pair \(({\mathcal {G}}, {\mathcal {G}}_{{\tilde{u}}_0})\) is admissible. Therefore, using Fact 5.6 we obtain that the assertion reduces to

$$\begin{aligned} \chi _{\mathcal {G}_{{\tilde{u}}_0}}(CI_{{\mathcal {G}_{{\tilde{u}}_0}}}(\{{\tilde{u}}_0\},-\nabla \Psi ^n_-)) \ne \chi _{\mathcal {G}_{{\tilde{u}}_0}}( CI_{{\mathcal {G}_{{\tilde{u}}_0}}}(\{{\tilde{u}}_0\},-\nabla \Psi ^n_+)) \end{aligned}$$

for \(n\in \mathbb {N}\) sufficiently large. It is easy to see that this inequality is equivalent to

$$\begin{aligned} \chi _{SO(N)}(CI_{SO(N)}(\{{\tilde{u}}_0\},-\nabla \Psi ^n_-)) \ne \chi _{SO(N)}( CI_{SO(N)}(\{{\tilde{u}}_0\},-\nabla \Psi ^n_+)). \end{aligned}$$
(3.4)

We proceed to show that there exists \(n_0 \in \mathbb {N}\) such that for \(n \ge n_0\)

$$\begin{aligned} CI_{SO(N)}\left( \{{\tilde{u}}_0\},-\nabla \Psi ^n_{\pm }\right) =CI_{SO(N)}\left( \{{\tilde{u}}_0\},-\nabla \Psi ^{n_0}_{\pm }\right) . \end{aligned}$$
(3.5)

Let \({\nu } \in \mathbb {N}.\) For \(\delta >0\) sufficiently small and \(\lambda \in [\lambda _0 -\varepsilon , \lambda _0+\varepsilon ]\) we define SO(N)-equivariant gradient homotopy \(H_{\lambda }^{\nu }:(D_{\delta }({\tilde{u}}_0,{\tilde{\mathbb {H}}}^{\nu })\times [0,1], \partial D_{\delta }({\tilde{u}}_0,{\tilde{\mathbb {H}}}^{\nu }) \times [0,1]) \rightarrow ({\tilde{\mathbb {H}}}^{\nu },{\tilde{\mathbb {H}}}^{\nu }\setminus \{0\})\) by

$$\begin{aligned} H_{\lambda }^{\nu }(u,t)= u -{\tilde{u}}_0 - L_{\lambda A} (u -{\tilde{u}}_0)+t \lambda _0 P_{\nu } \circ \nabla \eta (u - {\tilde{u}}_0), \end{aligned}$$

where \(P_{\nu }:{\tilde{\mathbb {H}}} \rightarrow {\tilde{\mathbb {H}}}^{\nu }\) is the orthogonal SO(N)-equivariant projection onto \({\tilde{\mathbb {H}}}^{\nu }.\) Note that from Lemma 2.1 we have \(P_{\nu } \circ L_{\lambda A} = L_{\lambda A} \circ P_{\nu }\) and hence this homotopy is well-defined.

Let us denote by \(\xi _{\lambda }^{\nu }:{\tilde{\mathbb {H}}}^{\nu }\rightarrow \mathbb {R}\) the SO(N)-invariant potential of \(H^{\nu }_{\lambda }(\cdot ,0).\) It is clear that \(\nabla \xi _{\lambda }^{\nu }:{\tilde{\mathbb {H}}}^{\nu } \rightarrow {\tilde{\mathbb {H}}}^{\nu }\) is a self-adjoint SO(N)-equivariant linear map and is given by the formula \(\nabla \xi _{\lambda }^{\nu }=(Id-L_{\lambda A})_{|{\tilde{\mathbb {H}}}^{\nu }}.\) From the homotopy invariance of the Conley index, see Theorem 5.3, we obtain

$$\begin{aligned} CI_{SO(N)}(\{{\tilde{u}}_0\},-\nabla \Psi ^{\nu }_{\pm })= CI_{SO(N)}(\{{\tilde{u}}_0\},-\nabla \xi ^{{\nu }}_{\lambda _0 \pm \varepsilon }). \end{aligned}$$
(3.6)

Recall that \((\beta _k)\) denotes the sequence of the eigenvalues of the Neumann Laplacian and note that \(\beta _k\rightarrow +\infty \). Therefore, there exists \(n_0\in \mathbb {N}\) such that the inequalities \(\frac{\beta _n-(\lambda _0\pm \varepsilon )\alpha _j}{1+\beta _n}>0\) hold for every \(n\ge n_0\) and \(\alpha _j \in \sigma (A)\). Hence, by Corollary 2.2, there exists \(n_0\in \mathbb {N}\) such that \(m^-(\nabla \xi _{\lambda _0 \pm \varepsilon }^n)=m^-(\nabla \xi _{\lambda _0 \pm \varepsilon }^{n_0})\) for every \(n\ge n_0\), where \(m^-(\cdot )\) is the Morse index. Since \((\nabla \xi ^{n}_{\lambda _0 \pm \varepsilon })_{|{\tilde{\mathbb {H}}}^{n_0}}=\nabla \xi ^{n_0}_{\lambda _0 \pm \varepsilon }\), the eigenspaces corresponding to the negative eigenvalues of \(\nabla \xi ^{n}_{\lambda _0 \pm \varepsilon }\) and \(\nabla \xi ^{n_0}_{\lambda _0 \pm \varepsilon }\) are the same SO(N)-representations. Thus, from Theorem 5.2,

$$\begin{aligned} CI_{SO(N)}\left( \{{\tilde{u}}_0\},-\nabla \xi ^{n}_{\lambda _0 \pm \varepsilon }\right) = CI_{SO(N)}\left( \{{\tilde{u}}_0\},-\nabla \xi ^{n_0}_{\lambda _0 \pm \varepsilon }\right) , \end{aligned}$$

which implies (3.5).

To finish the proof of (3.4), and therefore also of the assertion, we will show that

$$\begin{aligned} \chi _{SO(N)}\left( CI_{SO(N)}(\{{\tilde{u}}_0\}, -\nabla \Psi ^{n_0}_+)\right) \ne \chi _{SO(N)}\left( CI_{SO(N)}(\{{\tilde{u}}_0\}, -\nabla \Psi ^{n_0}_-)\right) . \end{aligned}$$

Denote by \(\mathcal {W}(\lambda )\) the direct sum of the eigenspaces of \(Id-L_{\lambda A}\) (i.e. of \(\nabla \xi ^{n_0}_{\lambda }\)) corresponding to the negative eigenvalues and by \(\mathcal {V}(\lambda )\) the eigenspace corresponding to the zero eigenvalue. Note that from Corollary 2.2,

$$\begin{aligned}\mathcal {W}(\lambda )= \Bigg (\bigoplus _{\alpha _j\in \sigma ( A)} \ \bigoplus _{\begin{array}{c} \beta _k \in \sigma (-\Delta ;B^N)\\ \beta _k<\lambda \alpha _j \end{array}} \mathbb {V}_{-\Delta }(\beta _k)^{\mu _{ A}(\alpha _{j})}\Bigg )\cap {\tilde{\mathbb {H}}},\end{aligned}$$
$$\begin{aligned}\mathcal {V}(\lambda )=\Bigg (\bigoplus _{\alpha _j\in \sigma (A)} \ \bigoplus _{\begin{array}{c} \beta _k \in \sigma (-\Delta ;B^N)\\ \beta _k=\lambda \alpha _j \end{array}} \mathbb {V}_{-\Delta }(\beta _k)^{\mu _{ A}(\alpha _{j})}\Bigg )\cap {\tilde{\mathbb {H}}}.\end{aligned}$$

From Theorem 5.2, \(CI_{SO(N)}(\{{\tilde{u}}_0\},-\nabla \xi ^{n_0}_{\lambda _0 \pm \varepsilon })\) are SO(N)-homotopy types of \(S^{\mathcal {W}(\lambda _0\pm \varepsilon )}.\) Hence, from (3.6),

$$\begin{aligned} \chi _{SO(N)}\left( CI_{SO(N)}(\{{\tilde{u}}_0\}, -\nabla \Psi ^{n_0}_{\pm })\right) = \chi _{SO(N)}\left( S^{\mathcal {W}(\lambda _0\pm \varepsilon )}\right) . \end{aligned}$$
  1. (1)

    Suppose that \(\lambda _0>0\) and \(\varepsilon \) is such that \(\lambda _0 - \varepsilon >0\). Recall that \(\beta _k \ge 0\) for all \(\beta _k \in \sigma (-\Delta ;B^N).\) Then \(\mathcal {W}(\lambda _0+\varepsilon )=\mathcal {W}(\lambda _0-\varepsilon )\oplus \mathcal {V}(\lambda _0).\) If the assumption (C1) is satisfied, then, by Theorem 5.4 and Remark  5.11, we obtain \(\chi _{SO(N)}(S^{\mathcal {V}(\lambda _0)})\ne \mathbb {I}\in U(SO(N)).\) Similarly, if (C2) is fulfilled, then \(\mathcal {V}(\lambda _0)\) is a trivial SO(N)-representation and, from Corollary 2.2 and the definition of \({\tilde{\mathbb {H}}}\), \(\dim \mathcal {V}(\lambda _0)=\dim \ker (Id-L_{\lambda _0A}) - \dim \ker A\) is odd. Therefore:

    $$\begin{aligned} \chi _{SO(N)}(S^{\mathcal {V}(\lambda _0)})= & {} (-1)^{\dim \mathcal {V}(\lambda _0)}\chi _{SO(N)}\left( SO(N)/SO(N)^+\right) =-\mathbb {I}. \end{aligned}$$

    In both cases we have

    $$\begin{aligned}&\chi _{SO(N)}\left( CI_{SO(N)}(\{{\tilde{u}}_0\},-\nabla \Psi ^{n_0}_{+})\right) \\&\quad = \chi _{SO(N)}(S^{\mathcal {W}(\lambda _0-\varepsilon )\oplus \mathcal {V}(\lambda _0)})\\&\quad = \chi _{SO(N)}(S^{\mathcal {W}(\lambda _0-\varepsilon )})\star \chi _{SO(N)}(S^{\mathcal {V}(\lambda _0)}) \\&\quad \ne \chi _{SO(N)}(S^{\mathcal {W}(\lambda _0-\varepsilon )})\\&\quad =\chi _{SO(N)}\left( CI_{SO(N)}(\{{\tilde{u}}_0\},-\nabla \Psi ^{n_0}_{-})\right) . \end{aligned}$$

    In the second equality we use the fact that \(S^{\mathcal {W}(\lambda _0-\varepsilon )\oplus \mathcal {V}(\lambda _0)}\) is SO(N)-homeomorphic to \(S^{\mathcal {W}(\lambda _0-\varepsilon )}\wedge S^{\mathcal {V}(\lambda _0)}\) and the formula for multiplication in U(SO(N)), see (5.1). Then we use invertibility of \(\chi _{SO(N)}(S^{\mathcal {W}(\lambda _0-\varepsilon )})\) in U(SO(N)), see [10].

  2. (2)

    Suppose that \(\lambda _0<0\). Then \(\mathcal {W}(\lambda _0-\varepsilon )=\mathcal {W}(\lambda _0+\varepsilon )\oplus \mathcal {V}(\lambda _0) \) and hence

    $$\begin{aligned}&\chi _{SO(N)}\left( CI_{SO(N)}(\{{\tilde{u}}_0\},-\nabla \Psi ^{n_0}_{+})\right) \ne \chi _{SO(N)}\left( CI_{SO(N)}(\{{\tilde{u}}_0\},-\nabla \Psi ^{n_0}_{-})\right) , \end{aligned}$$

    as before.

  3. (3)

    Finally, suppose that \(\lambda _0=0\). Then, since

    $$\begin{aligned} \mathcal {W}(\pm \varepsilon )= \bigoplus \limits _{\alpha _j\in \sigma _{\pm }(A)}\mathbb {V}_{-\Delta }(0)^{\mu _{A}(\alpha _j)}, \end{aligned}$$

    and therefore \(\mathcal {W}(\pm \varepsilon )\) are trivial SO(N)-representations,

    $$\begin{aligned}&\chi _{SO(N)}\left( CI_{SO(N)}(\{{\tilde{u}}_0\},-\nabla \Psi ^{n_0}_{\pm })\right) \\&\quad =\chi _{SO(N)}(S^{\mathcal {W}(\pm \varepsilon )})\\&\quad =(-1)^{\dim \mathcal {W}(\pm \varepsilon )}\cdot \chi _{SO(N)}\left( SO(N)/SO(N)^+\right) \\&\quad =(-1)^{\dim \mathcal {W}(\pm \varepsilon )} \cdot \mathbb {I}. \end{aligned}$$

    Hence, because the assumption (C3) implies that \(\dim \mathcal {W}(\varepsilon )-\dim \mathcal {W}(-\varepsilon )\) is odd, we have

    $$\begin{aligned}&\chi _{SO(N)}\left( CI_{SO(N)}(\{{\tilde{u}}_0\},-\nabla \Psi ^{n_0}_{+})\right) \ne \chi _{SO(N)}\left( CI_{SO(N)}(\{{\tilde{u}}_0\},-\nabla \Psi ^{n_0}_{-})\right) , \end{aligned}$$

    which completes the proof.

\(\square \)

Remark 3.4

Note that in the proof of Theorem 3.3 we have obtained that there is in fact a change of the Euler characteristic of the Conley index at \({\mathcal {G}}({\tilde{u}}_0)\times \{\lambda _0\}\). More precisely, we have the inequality (3.3), i.e.

$$\begin{aligned}&\chi _{\mathcal {G}} (CI_{{\mathcal {G}}}({\mathcal {G}}({\tilde{u}}_0),-\nabla _u\Phi ^{n}(\cdot , \lambda _0-\varepsilon )))\ne \chi _{\mathcal {G}}(CI_{{\mathcal {G}}}({\mathcal {G}}({\tilde{u}}_0),-\nabla _u\Phi ^{n}(\cdot , \lambda _0+\varepsilon ))), \end{aligned}$$

where \(\Phi ^n\) is defined by (3.2) and n is sufficiently large.

Now we are in a position to prove one of the main results of our paper, namely the global bifurcation theorem.

Theorem 3.5

Consider the system (2.1) with the potential F and \(u_0 \in \nabla F^{-1}(0)\) satisfying assumptions (B1)–(B4). Assume that \(\lambda _0 \in \Lambda \) and one of the conditions (C1)–(C3) is satisfied. Then a global bifurcation of solutions of (2.1) occurs from the orbit \({\mathcal {G}}({\tilde{u}}_0) \times \{\lambda _0\}\).

Proof

Throughout the proof we follow the notation of the proof of Theorem 3.3. Let \(\mathcal {U}\subset \mathbb {H}\) be an open, bounded and \({\mathcal {G}}\)-invariant subset such that \(\nabla _u \Phi (\cdot , \lambda _0\pm \varepsilon )^{-1}(0)\cap \mathcal {U}={\mathcal {G}}({\tilde{u}}_0).\) From Theorem 5.7 it follows that to prove the assertion it is enough to show

$$\begin{aligned}&\nabla _{{\mathcal {G}}}\text {-}\mathrm {deg}(\nabla _u\Phi (\cdot , \lambda _0+\varepsilon ),\mathcal {U})\ne \nabla _{{\mathcal {G}}}\text {-}\mathrm {deg}(\nabla _u\Phi (\cdot , \lambda _0-\varepsilon ),\mathcal {U}), \end{aligned}$$

where \(\nabla _{\mathcal {G}}\text {-}\mathrm {deg}(\cdot , \cdot )\) is the degree for equivariant gradient maps, see Sect. 5.3. From the definition of the degree, see (5.2), it follows that

$$\begin{aligned}&\nabla _{{\mathcal {G}}}\text {-}\mathrm {deg}(\nabla _u\Phi (\cdot , \lambda _0\pm \varepsilon ),\mathcal {U}) =\nabla _{{\mathcal {G}}}\text {-}\mathrm {deg}(\nabla _u\Phi ^{n_0}(\cdot , \lambda _0\pm \varepsilon ),\mathcal {U}\cap \mathbb {H}^{n_0}) \end{aligned}$$

for \(n_0\) sufficiently large. Note that \({\mathcal {G}}({\tilde{u}}_0)\subset \mathcal {U}\cap \mathbb {H}^{n_0}\) and hence, by Theorem 5.8, we obtain

$$\begin{aligned}&\nabla _{{\mathcal {G}}}\text {-}\mathrm {deg}(\nabla _u\Phi ^{n_0}(\cdot , \lambda _0\pm \varepsilon ),\mathcal {U}\cap \mathbb {H}^{n_0})=\chi _{{\mathcal {G}}}\left( CI_{{\mathcal {G}}}({\mathcal {G}}({\tilde{u}}_0),-\nabla _u\Phi ^{n_0}(\cdot , \lambda _0\pm \varepsilon ))\right) . \end{aligned}$$

Therefore, the assertion follows from Remark 3.4.

\(\square \)

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 \(u_0 \in \nabla F^{-1}(0)\) satisfying assumptions (B1)–(B4). Assume that \(\lambda _0 \in \Lambda \) and one of the conditions (C1)–(C3) is satisfied. Then a local bifurcation of solutions of (2.1) occurs from the orbit \({\mathcal {G}}({\tilde{u}}_0) \times \{\lambda _0\}\).

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 \(\mathcal {C}\mathcal {I}_{{\mathcal {G}}}({\mathcal {G}}({\tilde{u}}_0),-\nabla _u\Phi (\cdot , \lambda _{0}-\varepsilon )) \ne \mathcal {C}\mathcal {I}_{{\mathcal {G}}}({\mathcal {G}}({\tilde{u}}_0),-\nabla _u\Phi (\cdot , \lambda _{0}+\varepsilon )),\) for sufficiently small \(\varepsilon >0.\) 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 \(0\in GLOB\). On the other hand, repeating the argument from the proof of this theorem it is easy to show that if the number \(\sum _{\alpha _j \in \sigma _+(A)} \mu _A(\alpha _j)-\sum _{\alpha _j \in \sigma _-(A)} \mu _A(\alpha _j)\) is even, then the Euler characteristics \(\chi _{\mathcal {G}}\left( CI_{\mathcal {G}}(\mathcal {G}({\tilde{u}}_0),-\nabla _u\Phi ^{n_0}(\cdot ,\varepsilon )\right) \) and \(\chi _{\mathcal {G}}\left( CI_{\mathcal {G}}(\mathcal {G}({\tilde{u}}_0),-\nabla _u\Phi ^{n_0}(\cdot ,-\varepsilon ))\right) \) are equal. Therefore, we do not know whether \(0 \in GLOB\). 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 \(u_0 \in \nabla F^{-1}(0)\) satisfying assumptions (B1)–(B4). Assume that \(\lambda _0 =0\) and \(\sum _{\alpha _j \in \sigma _+(A)} \mu _A(\alpha _j)\ne \sum _{\alpha _j \in \sigma _-(A)} \mu _A(\alpha _j)\). Then a local bifurcation of solutions of (2.1) occurs from the orbit \({\mathcal {G}}({\tilde{u}}_0) \times \{0\}\).

Proof

Using the notation of the proof of Theorem 3.3, we observe that \(\mathcal {W}(\pm \varepsilon )\) are trivial SO(N)-representations. Therefore \(CI_{SO(N)}(\{{\tilde{u}}_0\},-\nabla \Psi ^{n_0}_{\pm })\) are SO(N)-homotopy types of \(S^{\dim \mathcal {W}(\pm \varepsilon )}\). Using information from [18] (namely Theorem 3.1 and the equality (2.11)) and from [14] (Lemma 1.88) we obtain that \(CI_{\mathcal {G}}(\mathcal {G}({\tilde{u}}_0),-\nabla _u\Phi ^{n_0}(\cdot ,\pm \varepsilon ))\) are \(\mathcal {G}\)-homotopy types of

$$\begin{aligned}\left( \mathcal {G}/\mathcal {G}_{{\tilde{u}}_0}\times S^{\dim \mathcal {W}(\pm \varepsilon )}\right) /\left( \mathcal {G}/\mathcal {G}_{{\tilde{u}}_0}\times \{*\}\right) .\end{aligned}$$

From Proposition 1.53 of [14], we obtain that the above is \(\mathcal {G}\)-homotopy equivalent to

$$\begin{aligned}X_{\pm }=\left( \mathcal {G}({\tilde{u}}_0)\times S^{\dim \mathcal {W}(\pm \varepsilon )}\right) /\left( \mathcal {G}({\tilde{u}}_0)\times \{*\}\right) .\end{aligned}$$

But \(X_+\) and \(X_-\) are different \(\mathcal {G}\)-homotopy types. Indeed, if \(X_+\) and \(X_-\) are the same \(\mathcal {G}\)-homotopy types, then the orbit spaces \(X_{+}/\mathcal {G}\) and \(X_{-}/\mathcal {G}\) are the same homotopy types. This is impossible, since the spaces \(X_{\pm }/\mathcal {G}\) are homotopy types of \(S^{\dim \mathcal {W}(\pm \varepsilon )}\), 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. \(\square \)

3.2 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 \(\mathcal {T}\) denotes the set of trivial solutions.

Definition 3.9

We say that a global symmetry breaking phenomenon occurs at the orbit \(\mathcal {G}({\tilde{u}}_0)\times \{\lambda _{0}\}\) if \(\lambda _0\in GLOB\) and there exists \(U\subset \mathbb {H}\times \mathbb {R}\) such that \(\mathcal {G}({\tilde{u}}_0)\times \{\lambda _{0}\}\subset U\) and \(\mathcal {G}_{(u,\lambda )}\ne \mathcal {G}_{({\tilde{u}}_0, \lambda _0)}\) for all \((u,\lambda )\in (U\cap (\nabla _u\Phi )^{-1}(0))\setminus \mathcal {T}\).

Note that since the group \(\mathcal {G}\) acts trivially on the set of parameters \(\lambda \), the condition \(\mathcal {G}_{(u,\lambda )}\ne \mathcal {G}_{({\tilde{u}}_0, \lambda _0)}\) is equivalent to \(\mathcal {G}_u \ne \mathcal {G}_{{\tilde{u}}_0}\). 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):

Theorem 3.10

Consider the system (2.1) with the potential F and \(u_0 \in \nabla F^{-1}(0)\) satisfying assumptions (B1)–(B4). Fix \(\lambda _0\in \Lambda \) and suppose that \(\sigma (\lambda _0 A)\cap \sigma (-\Delta ; B^N)\setminus \{0\}=\{\alpha _{j_1},\ldots , \alpha _{j_s}\}\) and \(\mathbb {V}_{-\Delta }(\alpha _{j_i})^{SO(N)}=\{0\}\) for every \(i=1,\ldots ,s\). Then the global symmetry breaking phenomenon occurs at the orbit \(\mathcal {G}({\tilde{u}}_0)\times \{\lambda _{0}\}\).

Note that the assumption \(\mathbb {V}_{-\Delta }(\alpha _{j_i})^{SO(N)}=\{0\}\) means that there is no radially symmetric eigenfunction associated with \(\alpha _{j_i}\).

To prove this theorem we first verify the following lemma:

Lemma 3.11

Fix \(\lambda _0 \in \Lambda \). Then there exists \(U\subset \mathbb {H}\times \mathbb {R}\) such that \(\mathcal {G}({\tilde{u}}_0)\times \{\lambda _{0}\}\subset U\) and for all \((u,\lambda )\in (U\cap (\nabla _u\Phi )^{-1}(0))\setminus \mathcal {T}\) there exists \({\overline{u}}\in \ker \nabla _u^2\Phi _{|\mathbb {H}_1^{\perp }}({\tilde{u}}_0,\lambda _0)\setminus \{0\}\) such that \(\mathcal {G}_u\subset \mathcal {G}_{{\overline{u}}}\).

Proof

Consider \( \mathbb {U}_1=\mathrm { im \;}\nabla _u^2\Phi _{|\mathbb {H}_1^{\perp }}({\tilde{u}}_0,\lambda _0)\oplus \mathbb {H}_1\) and \(\mathbb {U}_2=\ker \nabla _u^2\Phi _{|\mathbb {H}_1^{\perp }}({\tilde{u}}_0,\lambda _0). \) Note that \(\mathbb {H}=\mathbb {U}_1\oplus \mathbb {U}_2\) and the spaces \(\mathbb {U}_1\) and \(\mathbb {U}_2\) are \({\mathcal {G}}\)-representations. For \(u\in \mathbb {H}\) we put \(u=(u_1,u_2)\in \mathbb {U}_1\oplus \mathbb {U}_2\). In particular, since \({\tilde{u}}_0\in \mathbb {H}_1\), we identify this element with \(({\tilde{u}}_0,0)\in \mathbb {U}_1\oplus \mathbb {U}_2\).

The equation

$$\begin{aligned} \nabla _u\Phi (u,\lambda )=0 \end{aligned}$$
(3.7)

is equivalent to the system

$$\begin{aligned} \pi _1(\nabla _u\Phi (u_1,u_2,\lambda ))=0, \end{aligned}$$
(3.8)
$$\begin{aligned} \pi _2(\nabla _u\Phi (u_1,u_2,\lambda ))=0, \end{aligned}$$
(3.9)

where \(\pi _1:\mathbb {H}\rightarrow \mathbb {U}_1\) and \(\pi _2:\mathbb {H}\rightarrow \mathbb {U}_2\) are \({\mathcal {G}}\)-equivariant projections. Moreover, since \(\mathcal {G}({\tilde{u}}_0)\subset \mathbb {H}_1\subset \mathbb {U}_1\),

$$\begin{aligned}\dim \ker \nabla ^2_u\Phi _{|\mathbb {U}_1}({\tilde{u}}_0,\lambda _{0})=\dim \mathcal {G}({\tilde{u}}_0), \end{aligned}$$

i.e. \(\mathcal {G}({\tilde{u}}_0)\) is a non-degenerate critical orbit of \(\Phi (\cdot ,\lambda _0)_{|\mathbb {U}_1}\). Therefore, by the equivariant implicit function theorem (see Theorem 5.1) applied to the functional \(\Phi :\mathbb {U}_1\oplus (\mathbb {U}_2 \times \mathbb {R}) \rightarrow \mathbb {R}\), the point \((0, \lambda _0)\) and the Eq. (3.8), there exist open sets \(\mathcal {O}_{0}\subset \mathbb {U}_2\), \(\mathcal {O}_{\lambda _0}\subset \mathbb {R}\) such that \(0\in \mathcal {O}_{0},\lambda _0 \in \mathcal {O}_{\lambda _0}\) and a \({\mathcal {G}}\)-equivariant map \(\tau :\mathcal {G}({\tilde{u}}_0)\times \mathcal {O}_{0}\times \mathcal {O}_{\lambda _0}\rightarrow \mathbb {U}_1 \) such that

  1. (i)

    \(\tau (u_1,0,\lambda _0)=u_1\) for \(u_1\in \mathcal {G}({\tilde{u}}_0)\),

  2. (ii)

    \(\pi _1(\nabla _u\Phi (\tau (u_1,u_2,\lambda ),u_2,\lambda ))=0\) if \(u_1 \in \mathcal {G}({\tilde{u}}_0), u_2\in \mathcal {O}_{0}\) and \(\lambda \in \mathcal {O}_{\lambda _0}\) and these are the only solutions of \(\pi _1(\nabla _u\Phi (u_1,u_2,\lambda ))=0\) near the orbit if \(u_2\in \mathcal {O}_{0}\) and \(\lambda \in \mathcal {O}_{\lambda _0}.\)

Hence all the solutions of the Eq. (3.8), and consequently the solutions of (3.9) and (3.7), can have (in the neighbourhood of the orbit) only the following isotropy groups:

$$\begin{aligned} {\mathcal {G}}_{(\tau (u_1,u_2,\lambda ),u_2,\lambda )}={\mathcal {G}}_{\tau (u_1,u_2,\lambda )}\cap {\mathcal {G}}_{u_2}\cap {\mathcal {G}}_{\lambda }= {\mathcal {G}}_{\tau (u_1,u_2,\lambda )}\cap {\mathcal {G}}_{u_2}\subset {\mathcal {G}}_{u_2}. \end{aligned}$$

Denote this neighbourhood by U. More precisely, \(U=int(\tau (\mathcal {G}({\tilde{u}}_0)\times \mathcal {O}_{0}\times \mathcal {O}_{\lambda _0}))\times \mathcal {O}_{0}\times \mathcal {O}_{\lambda _0}\). To finish the proof observe that in the case \(u_2=0\) we have \((\tau (u_1,0,\lambda ),0,\lambda )\in \mathbb {U}_1\times \{0\}\times \mathbb {R}\) for \(u_1\in \mathcal {G}({\tilde{u}}_0)\), \(\lambda \in \mathcal {O}_{\lambda _0}\). Considering only the solutions of (3.8) and observing that such solutions in \(\mathbb {U}_1\times \{0\}\times \mathbb {R}\) are the trivial ones, we obtain \((\tau (u_1,0,\lambda ),0,\lambda )\in \mathcal {T}\), which completes the proof.

\(\square \)

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 \(\dim \mathcal {G}({\tilde{u}}_0)>0\), since \(\ker \nabla _u^2\Phi ({\tilde{u}}_0,\lambda _0)\) 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 \(\lambda _0\in GLOB\). Moreover, from Corollary 2.3 we have

$$\begin{aligned} \ker \nabla _u^2\Phi _{|\mathbb {H}_1^{\perp }}({\tilde{u}}_0,\lambda _0)= & {} \ker ( Id - L_{\lambda _0 A})\cap \mathbb {H}_1^{\perp }\\= & {} \mathbb {V}_{-\Delta }(\alpha _{j_1})^{\mu _{\lambda _0A}(\alpha _{j_1})}\oplus \cdots \oplus \mathbb {V}_{-\Delta }(\alpha _{j_s})^{\mu _{\lambda _0A}(\alpha _{j_s})}. \end{aligned}$$

Since \(\alpha _{j_1},\ldots , \alpha _{j_s}\ne 0\) are such that \(\mathbb {V}_{-\Delta }(\alpha _{j_i})^{SO(N)}=\{0\}\) for every \(i=1,\ldots ,s\), we conclude that

$$\begin{aligned} \ker \nabla _u^2\Phi _{|\mathbb {H}_1^{\perp }}({\tilde{u}}_0,\lambda _0)^{SO(N)}=\{0\}. \end{aligned}$$
(3.10)

Lemma 3.11 yields that there exists \(U\subset \mathbb {H}\times \mathbb {R}\) such that if \(\nabla _u \Phi (u,\lambda )=0\) and \((u,\lambda )\in U\setminus \mathcal {T}\) then there exists \({\overline{u}}\in \ker \nabla _u^2\Phi _{|\mathbb {H}_1^{\perp }}({\tilde{u}}_0,\lambda _0)\setminus \{0\}\) such that \(\mathcal {G}_u\subset \mathcal {G}_{{\overline{u}}}\). Since \(\mathcal {G}_{{\tilde{u}}_0}=\{e\}\times SO(N)\), to prove that \(\mathcal {G}_u\ne \mathcal {G}_{{\tilde{u}}_0}\) it suffices to note that the isotropy group of \({\overline{u}}\) is not of the form \(H\times SO(N)\), where \(H\in \overline{{\text {sub}}}(\Gamma )\). Indeed, if \({\mathcal {G}}_{{\overline{u}}}= H\times SO(N)\), then \({\overline{u}}(\alpha ^{-1} x)= {\overline{u}}(x)\) for every \(\alpha \in SO(N)\), \(x\in B^N\), i.e. \(SO(N)_{{\overline{u}}}=SO(N)\) and therefore from (3.10) we obtain \( {\overline{u}} =0\), which contradicts \({\overline{u}}\in \ker \nabla _u^2\Phi _{|\mathbb {H}_1^{\perp }}({\tilde{u}}_0,\lambda _0)\setminus \{0\}\). \(\square \)

Note that if the assumptions of Theorem 3.10 are satisfied, i.e. \(\ker \nabla _u^2\Phi _{|\mathbb {H}_1^{\perp }}({\tilde{u}}_0,\lambda _0)^{SO(N)}=\{0\}\), 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 \(\lambda _0 \in BIF.\) By the proof of Lemma 3.11 we deduce that there is a neighbourhood of the orbit \(\mathcal {G}({\tilde{u}}_0)\times \{\lambda _0\}\) such that all nontrivial solutions of \(\nabla _u\Phi (u,\lambda )=0\) can have only isotropy groups of the form \({\mathcal {G}}_{\tau (u_1,u_2,\lambda )}\cap {\mathcal {G}}_{u_2}\). Note that \(u_1\in \mathcal {G}({\tilde{u}}_0)\) and hence \(\mathcal {G}_{u_1}=\{e\}\times SO(N)\).

Consider the additional assumption:

$$\begin{aligned}\ker \nabla _u^2\Phi _{|\mathbb {H}_1^{\perp }}({\tilde{u}}_0,\lambda _0)^{SO(N)}=\ker \nabla _u^2\Phi _{|\mathbb {H}_1^{\perp }}({\tilde{u}}_0,\lambda _0).\end{aligned}$$

Then \({\mathcal {G}}_{u_2}= \Gamma _{u_2}\times SO(N)\). Therefore by the proof of Lemma 3.11, and since a \({\mathcal {G}}\)-equivariant function \(\tau \) increases isotropy groups (i.e. \({\mathcal {G}}_{(u_1,u_2,\lambda )} \subset {\mathcal {G}}_{\tau (u_1,u_2,\lambda )}\)), we have

$$\begin{aligned} \mathcal {G}_{u_1}\cap \mathcal {G}_{u_2}= & {} (\{e\}\times SO(N))\cap (\Gamma _{u_2}\times SO(N))\\= & {} \{e\}\times SO(N)\subset {\mathcal {G}}_{\tau (u_1,u_2,\lambda )}\cap {\mathcal {G}}_{u_2}, \end{aligned}$$

i.e. solutions of \(\nabla _u\Phi (u,\lambda )=0\) in the neighbourhood of the orbit \( \mathcal {G}({\tilde{u}}_0) \times \{\lambda _0\}\) have isotropy groups of the form \(H\times SO(N)\), where \(H\in \overline{{\text {sub}}}(\Gamma )\). Hence all solutions from the neighbourhood of the orbit are radial.

Remark 3.13

Fix \(\lambda _0\in \Lambda \) and suppose that \(\sigma (\lambda _0 A)\cap \sigma (-\Delta ; B^N)\setminus \{0\}=\{\alpha _{j_1},\ldots , \alpha _{j_s}\}\) are such that \(\alpha _{j_1},\ldots , \alpha _{j_s}\notin \mathcal {A}_0\), where \(\mathcal {A}_0\) is defined in Sect. 5.4. Then from Remark 5.12 it follows that \(\mathbb {V}_{-\Delta }(\alpha _{j_i})^{SO(N)}=\{0\}\) and therefore the assumptions of Theorem 3.10 are satisfied. Hence the global symmetry breaking phenomenon occurs at the orbit \(\mathcal {G}({\tilde{u}}_0)\times \{\lambda _{0}\}\).

4 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.

Example 1

Consider the system (2.1) for \(N=2\) with a potential F and \(u_0 \in \nabla F^{-1}(0)\) satisfying assumptions (B1)–(B4). Assume that \(\lambda _0 \in \mathbb {R}\setminus \{0 \}\) and \(\sigma (\lambda _0 A) \cap \sigma (-\Delta ; B^2)\setminus \{0\}= \{\alpha \},\) where \(\sqrt{\alpha }\) is not a root of \(J_0'(x)=0\) for \(J_0\) being the Bessel function of order 0. Following the notation of Sect. 5.4 it means that \(\alpha \not \in \mathcal {A}_0\).

In this situation, from Theorem 5.9 and Fact 5.10 the assumption (C1) of Sect. 3 is satisfied. By Theorem 3.5 we obtain that a global bifurcation occurs from the orbit \(\mathcal {G}({\tilde{u}}_0) \times \{\lambda _0\}\).

Moreover, from Remark 5.12 it follows that \(\mathbb {V}_{-\Delta }(\alpha )^{SO(2)}=\{0\}.\) Then by Theorem 3.10 the global symmetry breaking occurs at the orbit \(\mathcal {G}({\tilde{u}}_0)\times \{\lambda _0\}.\)

Example 2

Consider the system (2.1) for \(N=2\) with the potential F and \(u_0 \in \nabla F^{-1}(0)\) satisfying assumptions (B1)–(B4). Assume that \(\lambda _0 \in \mathbb {R}\setminus \{0 \},\) \(\sigma (\lambda _0 A) \cap \sigma (-\Delta ; B^2)\setminus \{0\}= \{\alpha _1, \ldots ,\alpha _s \}\) and there exists \(i\in \{1, \ldots ,s\}\) such that \(\sqrt{\alpha _i}\) is not a root of \(J_0'(x)=0\).

As in Example 1, a global bifurcation occurs from the orbit \(\mathcal {G}({\tilde{u}}_0) \times \{\lambda _0\}.\) If moreover \(\alpha _i \not \in \mathcal {A}_0\) for all \(i\in \{1, \ldots ,s\}\), then the global symmetry breaking occurs at the orbit \(\mathcal {G}({\tilde{u}}_0)\times \{\lambda _0\}.\)

Example 3

Consider the system (2.1) for \(N=3\) with the potential F and \(u_0 \in \nabla F^{-1}(0)\) satisfying assumptions (B1)–(B4). Assume that \(\lambda _0 \in \mathbb {R}\setminus \{0 \}\), \(\sigma (\lambda _0 A) \cap \sigma (-\Delta ; B^3)\setminus \{0\}= \{\alpha _1, \ldots ,\alpha _s \}\) and there exists \(i\in \{1, \ldots ,s\}\) such that \(\sqrt{\alpha _i}\) is not a solution of the equation:

$$\begin{aligned}J_{\frac{1}{2}}'(x)-\frac{1}{2x}J_{\frac{1}{2}}(x)=0,\end{aligned}$$

where \(J_{\frac{1}{2}}\) is the Bessel function of order \(\frac{1}{2}\). Therefore \(\alpha _i \not \in \mathcal {A}_0.\)

In this situation, since \(\mathcal {H}_l^3 \subset \mathbb {V}_{-\Delta }(\alpha _i)\) 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 \(\mathcal {G}({\tilde{u}}_0) \times \{\lambda _0\}\).

Moreover, if \(\alpha _i \not \in \mathcal {A}_0\) for all \(i\in \{1, \ldots ,s\}\), then from Remark 5.12 we conclude that \(\mathbb {V}_{-\Delta }(\alpha _i)^{SO(3)}=\{0\}\) for all \(i\in \{1, \ldots ,s\}\). Therefore, by Theorem 3.10 it follows that the global symmetry breaking occurs at the orbit \(\mathcal {G}({\tilde{u}}_0)\times \{\lambda _0\}.\)

Example 4

Consider the system (2.1) with the potential F and \(u_0 \in \nabla F^{-1}(0)\) satisfying assumptions (B1)–(B4). Assume that \(\lambda _0 \in \mathbb {R}\setminus \{0 \}\) and that \(\sigma (\lambda _0 A) \cap \sigma (-\Delta ; B^N)\setminus \{0\}= \{\alpha _1, \ldots ,\alpha _s \},\) where \(\sqrt{\alpha _i}\) is a solution of the equation

$$\begin{aligned} J'_{\frac{N-2}{2}}(x)-\frac{N-2}{2x} J_{\frac{N-2}{2}}(x)=0 \end{aligned}$$

for every \(i \in \{1, \ldots , s\}\).

If there exists \(i \in \{1, \ldots ,s\}\) such that \(\dim \mathbb {V}_{-\Delta }(\alpha _i)>1\) then the assumption (C1) is satisfied and by Theorem 3.5 we obtain that a global bifurcation occurs from the orbit \(\mathcal {G}({\tilde{u}}_0) \times \{\lambda _0\}\).

If \(\dim \mathbb {V}_{-\Delta }(\alpha _{i})=1\) for all \(i\in \{1, \ldots ,s\}\), then we assume additionally that \(\sum _{i=1}^s \mu _{\lambda _0 A}(\alpha _i)-\mu _A(0)\) 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 \(\mathcal {G}({\tilde{u}}_0) \times \{\lambda _0\}\).

Note that, if \(\dim \mathbb {V}_{-\Delta }(\alpha _{i})=1\) for all \(i\in \{1, \ldots ,s\}\), then \(\ker ( Id - L_{\lambda _0 A})^{SO(N)}=\ker ( Id - L_{\lambda _0 A})\) (see Remark 5.11(2)). Therefore, from Remark 3.12, we conclude that all nontrivial solutions at a neighbourhood of \(\mathcal {G}({\tilde{u}}_0) \times \{\lambda _0\}\) (bifurcating from this orbit) are radial, i.e. there is no symmetry breaking at the orbit.

Example 5

Consider the system (2.1) with the potential F and \(u_0 \in \nabla F^{-1}(0)\) satisfying assumptions (B1)–(B4). Assume that \(\lambda _0=0\).

If \(m-\dim \ker A\) is odd, then the assumption (C3) is satisfied and we obtain a global bifurcation from the orbit \(\mathcal {G}({\tilde{u}}_0) \times \{0\}.\) If \(m-\dim \ker A > 0\), then Theorem 3.8 implies a local bifurcation from the orbit \(\mathcal {G}({\tilde{u}}_0) \times \{0\}.\)

As in Example 4, it is easy to see that all nontrivial solutions at a neighbourhood of the orbit are radial.