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 the 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 the global symmetry-breaking phenomenon occurs.


Introduction
In this paper, we study bifurcations of weak solutions of elliptic systems of the form: In particular, we are interested in the equivariant case. Namely, we assume that on the space R m there is defined an action of the compact Lie group Γ and ∇F is the Γequivariant 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, containing (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 [6], [9], [14]. A similar method has been also used in [8] for the system with the Neumann boundary conditions with the infinity instead of the critical point. The phenomenon of symmetry breaking for elliptic systems with the Neumann boundary conditions has been considered by the third author in [22]. solutions emanating from orbits of radial ones. To obtain this result, we generalise the result of Dancer in Lemma 3. 9.
In Section 4 we illustrate our results with a few examples. Using the properties of the eigenspaces of the Laplace operator (with the Neumann boundary conditions) on the ball, we verify 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 or the properties of eigenspaces of the Laplace operator on a ball. However, it is not easy to find the full description 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 sub(G) the set of closed subgroups of G. For u from a given G-space X we denote by the G(u) the orbit through u and G u stands for the isotropy group of u.
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 CI 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. More precise description can be found in 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 Γ stands for a compact Lie group and R m is an orthogonal representation of the group Γ. 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 first Sobolev space on B N and consider a separable Hilbert space Denote by G the group Γ × SO(N), where SO(N) 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 Φ : Computing the gradient of Φ with respect to u we obtain: Moreover, from imbedding theorems and the assumption (B1) it follows that the operator ∇ u Φ is a completely continuous perturbation of the identity.

Linear equation.
In this subsection we consider the equation (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 Φ 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 the Neumann boundary conditions) on the ball. Write V −∆ (β k ) for the eigenspace of −∆ corresponding to β k ∈ σ(−∆; B N ). In Appendix we give a more precise description of these eigenspaces. By the spectral theorem it follows Let α 1 , . . . , α m denote the eigenvalues of A (not necessarily distinct) with corresponding eigenvectors f 1 , . . . , f m , which form an orthonormal basis of R m . Let In the lemma below we characterise the operator L λA , given by the formula (2.8).
The proof of this lemma is standard, see for example the proof of Lemma 3.2 in [8]. Let us denote by σ(L) the spectrum of a linear operator L : H → H. From the above lemma there 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
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 BIF (respectively GLOB). Note that directly from the above definitions it follows that GLOB ⊂ BIF.
2.3. Admissible pair. The notion of the admissible pair has been introduced in [16]. Fix a compact Lie group G and let H ∈ sub(G). Denote by (H) G the conjugacy class of H. Proof. Let us denote by H the group {e} × SO(N) and recall that G = Γ × SO(N). Moreover, letK 1 ,K 2 ∈ sub(H). By definition of H there are K 1 , K 2 ∈ sub(SO(N)) such Thus (K 1 ) H = (K 2 ) H and the proof is complete.

Main Results
Consider the nonlinear system (2.1) with a potential F satisfying (B1), (B2). Fix u 0 ∈ (∇F ) −1 (0) such that the orbit Γ(u 0 ) is non-degenerate. We put two additional assumptions: From the assumption (B3) we conclude that the gradient of the functional associated with the equation (2.1) has the following form: From the assumption (B4) it follows that Gũ 0 = {e} × SO(N).
3.1. Bifurcation from the critical orbit. Following the standard notation we denote the linear part of Proof. We first observe that for all λ ∈ R, since G(ũ 0 ) is a critical orbit of Φ(·, λ), we have dim ker From the definition of Λ such a choice is always possible.
Since λ 0 ± ε / ∈ Λ, Lemma 3.1 implies that λ 0 ± ε / ∈ BIF and therefore G(ũ 0 ) ⊂ H is an isolated critical orbit of the G-invariant functionals Φ(·, λ 0 ± ε) : H → R. 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 [12]) for the flows induced by the operators −∇ u Φ(·, λ 0 ± ε). Therefore, the indices λ 0 +ε)) are well-defined. In the following we study when they are not equal. Then
For n 1 put H n = n k=1 H k and Φ n = Φ |H n ×R : is a critical orbit of Φ n (·, λ 0 ±ε) for n 1. Note that, from the choice of ε and the definition of Φ n , it is a non-degenerate one.
Since ∇ u Φ(·, λ) is a completely continuous perturbation of the identity for all λ ∈ R, from the definition of the infinite dimensional equivariant Conley index, see [12], the assertion of the theorem is equivalent to for n sufficiently large.
It is known that the G-action on H given by (2.3) defines a Gũ 0 -action onH. Recall that Gũ 0 = {e} × SO(N). HenceH is an orthogonal SO(N)-representation. For Hence {ũ 0 } is an isolated invariant set (in the sense of the Conley index theory) of the flows generated by −∇Ψ n ± . Note that since Gũ 0 = {e} × SO(N), by Lemma 2.8 the pair (G, Gũ 0 ) is admissible. Therefore, using Fact 5.5 we obtain that the assertion reduces to 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

equivariant linear map and is given by the formula
From the homotopy invariance of the Conley index, see Theorem 5.3, we obtain 3) 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 βn−(λ 0 ±ε)α j 1+βn > 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, What is left is to show that . Denote by W(λ) the direct sum of the eigenspaces of Id − L λA (i.e. of ∇ξ n 0 λ ) corresponding to the negative eigenvalues and by V(λ) the eigenspace corresponding to the zero eigenvalue. Note that from Corollary 2.2, In both cases we have In the second equality we use the fact that S W(λ 0 −ε)⊕V(λ 0 ) is SO(N)-homeomorphic to S W(λ 0 −ε) ∧ S V(λ 0 ) and the formula for multiplication in U(SO(N)), see (5.1). Then we use invertibility of χ SO(N ) (S W(λ 0 −ε) ) in U(SO(N)), see [9].
It is known that, in general, the change of the Conley index along the family of trivial solutions, does not imply the global bifurcation. However, using the relation between the Conley index and the degree for strongly indefinite functionals, under some assumptions one can prove the existence of connected sets of bifurcating solutions. It occurs that (C1)-(C3) are this kind of assumptions. Proof. Let U ⊂ H be an open, bounded and G-invariant subset such that ∇ u Φ(·, λ 0 ± ε) −1 (0) ∩ U = G(ũ 0 ). Denote by ∇ G -deg(·, ·) the degree for equivariant gradient maps of the form completely continuous perturbation of the identity, defined in [17]. From the definition of this degree, for n 0 sufficiently large, where Φ n 0 is defined as in the proof of Theorem 3.3. The latter equality is the relation between the Conley index and the degree proved by Gęba in [7], see also Corollary 1 in [10]. From Theorem 3.3 and Fact 5.5 we have From the equivariant version of the Rabinowitz alternative, see for example Theorem 3.3 of [9], the change of the degree for G-equivariant gradient maps implies a global bifurcation, so we obtain the assertion.
In Theorem 3.5 we have proved that if the assumption (C3) is satisfied, then 0 ∈ GLOB. On the other hand, repeating the reasoning from the proof of this theorem it is easy to show that if the number α j ∈σ + (A) µ A (α j ) − α j ∈σ − (A) µ A (α j ) is even, then the Euler characteristics χ G (CI G (G(ũ 0 ), −∇ u Φ n 0 (·, ε)) and χ G (CI G (G(ũ 0 ), −∇ u Φ n 0 (·, −ε))) are equal. Therefore, we do not know whether 0 ∈ GLOB. However, under the assumption weaker than (C3) we can prove the result concerning the local bifurcation. Theorem 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 = 0 and α j ∈σ 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 CI SO(N ) ({ũ 0 }, −∇Ψ n 0 ± ) are SO(N)-homotopy types of S dim W(±ε) . Using information from [16] (namely Theorem 3.1 and the equality (2.11)) and from [13] (Lemma 1.88) we obtain that CI G (G(ũ 0 ), −∇ u Φ n 0 (·, ±ε)) are G-homotopy types of From Proposition 1.53 of [13], 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 G-homotopy 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 [24]. Analysis similar to that in the proof of Theorem 3.4 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 equation (2.1) at which the global symmetry-breaking phenomenon occurs. Here and thereafter we use the notation of Section 3.1. Recall that T denotes the set of trivial solutions.

Definition 3.7. 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 symmetrybreaking phenomenon 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 prove this theorem we first verify the following lemma: λ 0 ). Note that H = U 1 ⊕ U 2 and the spaces U 1 and U 2 are G-representations. For u ∈ H we put u = (u 1 , u 2 ) ∈ U 1 ⊕ U 2 . In particular, sinceũ 0 ∈ H 1 , we identify this element with (ũ 0 , 0) ∈ The equation 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 (3.5) 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 3.9 generalises the lemma due to Dancer from [3]. 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 3.8 in the case dim G(ũ 0 ) > 0, since ker ∇ 2 u Φ(ũ 0 , λ 0 ) contains constant (and therefore radially symmetric) functions from the space tangent to the orbit.
Proof of Theorem 3.8. Note that Theorem 3.5 implies that λ 0 ∈ GLOB. Moreover, from Corollary 2.3 we have Since α j 1 , . . . , α js = 0 are such that V −∆ (α j i ) SO(N ) = {0} for every i = 1, . . . , s, we conclude that Lemma 3.9 yields that there exists U ⊂ H×R such that if ∇ u Φ(u, λ) = 0 and (u, λ) ∈ U \T then there exists u ∈ ker ∇ 2 (N), to prove that G u = Gũ 0 it suffices to note that the isotropy group of u is not of the form Note that if the assumptions of Theorem 3.8 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 3.8 we obtain a connected family of orbits of non-radial solutions bifurcating from the set of radial ones.

Illustration
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 the Neumann boundary conditions) on the ball, we verify assumptions (C1)-(C3). More precisely we apply the information collected in Subsection 5.3.
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 5.7), the assumption (C1) is satisfied and from Theorem 3. 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 3.6 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.

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.
5.1. The equivariant implicit function theorem. Below we reformulate an equivariant version of the implicit function theorem in infinite dimensional spaces, due to Dancer (see [4], paragraph 3).
and v ∈ B δ (v 0 , H 2 ) and these are the only

Equivariant Conley index.
In this subsection we collect properties of the equivariant Conley index. For a fuller treatment we refer to [2], [7] in the finite dimensional case and to [12] for the infinite dimensional case.
Let G be a compact Lie group and suppose that Ω is a G-invariant subset of a finite dimensional G-representation V. The G-equivariant Conley index of an isolated invariant set of a (local) flow is defined as a G-homotopy type of a pointed G-space, see [2], [7]. If f : Ω → 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 . We denote by CI G (S, f ) the Conley index of an isolated invariant set S of the flow generated by f . Put S V = D 1 (0, V)/∂D 1 (0, V) and denote by [S V ] G a G-homotopy type of a pointed G-space S V . From the definition of the Conley index and the Hartman-Grobman theorem there follows (see also [21]): The following theorem is a direct consequence of the Continuation Property of the Conley index, see [2]: 1)).
The Conley index of a flow generated by a gradient map is homotopy type of a pointed finite G-CW-complex, see Proposition 5.6 of [7] 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 . The full description of this theory one can find for example in [23], [24].
Suppose now that U is a G-invariant subset of an infinite dimensional Hilbert space, which is an orthogonal G-representation H. The G-equivariant Conley index of an isolated invariant set of a (local) G-LS-flow is defined as a G-homotopy type of a G-equivariant spectrum, see [12]. As before, if F : U → H is a G-equivariant map of class C 1 and it is a completely continuous perturbation of the identity, then it generates a local G-LSflow. We denote by CI G (S, F ) the Conley index of an isolated invariant set S of the flow generated by F .

Eigenspaces of the Laplace operator.
In this subsection we introduce basic properties of the eigenspaces of the Laplace operator (with the 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], [15]. 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 .