Global symmetry-breaking bifurcations of critical orbits of invariant functionals

In this article we present a method of study of a global symmetry-breaking bifurcation of critical orbits of invariant functionals. As a topological tool we use the degree for equivariant gradient maps. We underline that many known results on bifurcations of non-radial solutions of elliptic PDE's from the families of radial ones are consequences of our theory.


1.
Introduction. Let us consider the following equation where Ω ⊂ R n is a ball or annulus and f : R × R → R is a continuous function. The phenomenon of symmetry-breaking of solutions of problem (1.1) has been studied by many authors under various assumptions on f, see for instance [4]- [7], [15]- [20], [22]- [24], [26,29], [27]- [35]. Of course this list is far from being complete. At the beginning the authors prove the existence of radially non-degenerate families of radial solutions of problem (1.1). The radial non-degeneracy excludes a bifurcation of radial solutions of problem (1.1) from these families. Next they show a change of the Morse index computed along these families. After restriction to the subspace fixed by the group O(n − 1) they obtain a change of the Morse index by an odd number, which implies a change of the Leray-Schauder degree. Finally a change of the Leray-Schauder degree implies a global bifurcation of non-radial solutions of problem (1.1). Another possible approach is to apply the Crandall-Rabinowitz bifurcation theorem or reduce the problem to a finite-dimensional one and apply ∇ u Φ(u, λ) = 0, (1.2) from the line of trivial solutions {0} × R, whose isotropy group is different from G.
To prove our abstract results we use the degree for equivariant gradient maps, see [3]- [2], [11,30] i.e. a topological invariant which is suitable for study of homotopy classes of equivariant gradient maps.
To study bifurcations of non-radial solutions of problem (1.1) from the family of radial ones in our abstract approach we put H = H 1 0 (Ω) and G = SO(n). Problem (1.1) is SO(n)-invariant and possesses, under some growth conditions, variational structure i.e. its solutions one can consider as critical SO(n)-orbits of an SO(n)invariant functional Φ : H 1 0 (Ω) × R → R. Note that radial solutions of problem (1.1) correspond to the solutions of equation (1.2) whose isotropy group equal SO(n) i.e. solutions fixed by the group SO(n).
Our main contribution in this article are abstract symmetry-breaking bifurcation theorems for solutions of equation (1.2). We underline that we prove simultaneously the existence of symmetry-breaking and global bifurcation phenomenon of solutions of problem (1.2).
We claim that most of results on bifurcations of non-radial solutions of elliptic PDE's from families of radial ones proved in the cited articles follow from Theorem 3.3, see the discussion in the last section.
After this introduction our article is organized as follows.
In Section 2 we derive formulas for indices of isolated critical points of invariant functionals. Namely, we consider a G-invariant functional Φ : H → R of the class C 2 , where H is a separable Hilbert space which is an orthogonal representation of a compact Lie group G. We assume that 0 ∈ H is an isolated critical point of Φ (we allow also degenerate critical points!) and compute an index of this point in terms of the degree for G-equivariant gradient maps ∇ G -deg(·, ·) ∈ U (G), see [3,11,14,30] for a definition and properties of this degree. In other words we compute is an open ball in H of radius α centered at 0 ∈ H and (U (G), +, * ) is the Euler ring of G with the unit denoted by I, see [8,9] for a definition and properties of U (G). The main results of this section are Theorems 2.1, 2.3 and Corollaries 1, 2. In Theorem 2.1 we assume that the second derivative ∇ 2 Φ(0) is an isomorphism. Whereas in Theorem 2.3 and Corollaries 1, 2 we consider the degenerate case. We apply these theorems in the next section to prove sufficient conditions for the existence of a global symmetry-breaking bifurcation of solutions of problem (1.2).
The main result of Section 3 is Theorem 3.3. To prove this theorem we combine Theorem 3.2 with some results of Section 2. Theorem 3.3 is very important from the point of view of applications. We underline that assumptions of this theorem are relatively easy to verify i.e. this verification is reduced to some reasonings in representation theory of G. Moreover, since G is connected, we have reduced the computations in the Euler ring U (G) to equivalent but much simpler computations in the Euler ring U (T ), where T ⊂ G is a maximal torus, see Remark 2. These verifications are simple because the additive and multiplicative structures of the Euler ring U (T ) are completely described, see [21].
In Section 4 we have shown that well known results on bifurcations of non-radial solutions from the families of radial ones due to Ramaswamy and Srikanth [27,34], Cerami [4], Pacard [26] and Smoller and Wasserman [32,33] are consequences of Theorem 3.3.

2.
Index of an isolated critical point of an invariant functional. In this section we compute indices of isolated critical points of G-invariant functionals in terms of the degree for equivariant gradient maps. Let (H, ·, · ) be a separable Hilbert space, which is an orthogonal representation of a compact Lie group G. Fix u 0 ∈ H and define an isotropy group G u0 of u 0 by G u0 = {g ∈ G : , k ∈ N, the space of G-invariant functionals of the class C k i.e. functionals Φ : H → R of the class C k satisfying condition Φ(gu) = Φ(u) for every g ∈ G and u ∈ H. Moreover, let C k−1 G (H, H) denote the space of G-equivariant operators of the class C k−1 i.e. operators Ψ : H → H of the class C k−1 such that Ψ(gu) = gΨ(u) for every g ∈ G and u ∈ H. It is a known fact that if Φ ∈ C k G (H, R) then ∇Φ ∈ C k−1 G (H, H), where ∇Φ is the gradient of Φ. It is clear that if ∇Φ(u 0 ) = 0 then the gradient ∇Φ vanishes on the orbit G(u 0 ) i.e. G(u 0 ) ⊂ (∇Φ) −1 (0). In this article deg B , deg LS stand for the Brouwer and the Leray-Schauder degree, respectively. Let (U (G), +, * ) be the Euler ring of G with the unit denoted by I, see [8,9] for a definition and properties of the Euler ring of a compact Lie group G.
Denote by C k G (H, R) the set of functionals Φ ∈ C k G (H, R) whose gradient ∇Φ ∈ C k−1 G (H, H) satisfies the following assumptions: satisfying assumption (a1) and choose an open bounded and G- be an infinite-dimensional generalization of the degree for G-equivariant gradient maps due to Gȩba, see [11,14,21,30].
In this section we compute the index of an isolated critical point 0 ∈ H of a functional Φ ∈ C 2 G (H, R) using the degree for G-equivariant gradient maps. In other words we compute More precisely, we consider the following four cases Throughout this section we denote by H − the maximal subrepresentation of H on which the G-equivariant self-adjoint operator ) is the spectrum of the self-adjoint, compact and G-equivariant operator ∇ 2 η(0) and H(λ i ) is the eigenspace of ∇ 2 η(0) corresponding to the eigenvalue λ i ∈ σ(∇ 2 η(0)). We point out that H(λ i ) is a finite-dimensional orthogonal representation of G.
In the theorem below we consider the simplest case. Namely, we compute the index of a non-degenerate critical point of Φ ∈ C 2 G (H, R).
Consequently, taking into account the properties of the degree for equivariant gradient maps, see [11,14,21,30,31], we express the degree ∇ G -deg(∇Φ, B α (H)) ∈ U (G) as a degree of a finite-dimensional map. To be more precise, applying the homotopy invariance of the degree we obtain Next using the equality ∇ G -deg(Id, B α (H + )) = I ∈ U (G) and the product formula for the degree we obtain If additionally L is positively definite then H = H + , L = L + = ∇ 2 Φ(0) and consequently which completes the proof.
From now on we consider a degenerate isolated critical point 0 ∈ H of the func- . The principal significance of the lemma below is that it allows one to choose, in the homotopy class of equivariant gradient maps of ∇Φ, a product map whose degree is relatively easy to compute. We will use this lemma in the proofs of all the theorems of this subsection. A proof of this lemma one can find in [10].
In the theorem below we compute the index of a degenerate isolated critical point of a G-invariant functional Φ ∈ C 2 G (H, R), using the degree for G-equivariant gradient maps. This is the most general theorem of this section i.e. we do not assume any restrictions on the location of ker ∇ 2 Φ(0) with respect to the set of fixed points H G .
Proof. Combining the Splitting Lemma 2.2 with the product formula for the degree, see [30], we obtain the following Finally, by equality ∇ G -deg(Id, B α (H + )) = I ∈ U (G) we obtain If additionally L is positively defined then H 1 = H + and L = L + . Consequently, once more applying the equality ∇ G -deg(Id, B α (H + )) = I ∈ U (G) we obtain that which completes the proof.
From now on deg stands for the Brouwer degree if dim H G < ∞ and for the Leray-Schauder degree otherwise. In the corollary below we assume that ker ∇ 2 Φ(0) consists only of fixed points of the action of the group G. This assumption allows us to express the index of the critical point 0 ∈ H of the functional Φ in terms of the degrees of linear maps.
Proof. Taking into account that H 0 ⊂ H G , by Lemma 2.2 and the homotopy invariance of the degree, see [30], we obtain Consequently by the product formula for the degree, see [30], we obtain . Finally, by the properties of the degree we obtain which completes the proof.
If L is positively defined then H 1 = H + , L = L + and consequently In the following corollary, contrary to Corollary 1, we assume that ker ∇ 2 Φ(0) \ {0} is the fixed point free subset.

Corollary 2. Under the assumptions of theorem 2.3, if moreover ker
Proof. Since H 0 ∩ H G = {0}, applying Lemma 2.2 and the homotopy invariance of the degree, see [30], we obtain Consequently by the product formula for the degree, see [30], we obtain which completes the proof.
If additionally L is positively defined then H 1 = H + , L = L + and consequently 3. Symmetry-breaking bifurcation of critical orbits. In this section we have proved the main abstract symmetry-breaking theorems of our article. Namely, we have formulated sufficient conditions for the existence of a global symmetry-breaking bifurcation of critical orbits of invariant functionals. Let H be an orthogonal representation of a compact Lie group G and Ω ⊂ H be an open and G-invariant neighborhood of 0 ∈ H. Consider an open G-invariant subset Ω × R ⊂ H × R with an action defined by g(u, λ) = (gu, λ) for every (u, λ) ∈ Ω × R and g ∈ G. Fix λ ± ∈ R ∪ {±∞} and denote by C k G (Ω × (λ − , λ + ), R) the class of invariant functionals Φ such that the gradient of Φ with respect to the first coordinate , R) and consider the following equation In other words a point (0, λ 0 ) ∈ T is a local symmetry-breaking bifurcation point of solutions of equation (3.1), provided that in a sufficiently small neighborhood of (0, λ 0 ) ∈ Ω × (λ − , λ + ) the isotropy group of every non-trivial solution of equation (3.1) is different from G. Directly from the above definitions it follows that GLOB G ⊂ GLOB ⊂ BIF and GLOB G ⊂ BIF G ⊂ BIF.
Our aim is to study sufficient conditions for the existence of global symmetrybreaking bifurcation points of solutions of equation (3.1).
A change of any reasonable degree theory along the family of trivial solutions implies a global bifurcation of non-trivial solutions from this family. In the theorem below we have formulated a sufficient condition for the existence of a global symmetry-breaking bifurcation from the family T . Since the proof of this theorem is standard, we omit it. R). Assume that BIF G = ∅ and that there are λ − < λ < λ < λ + such that Note that conditions (A1), (A2) result from the following assumption Assumption (A) is stronger than assumptions (A1), (A2) but in real applications it is easier to verify this assumption than conditions (A1), (A2). In fact, since The theorem below provides a criterion for the existence of a phenomenon of a global symmetry-breaking bifurcation of critical orbits of invariant functionals. It will prove extremely useful in the study of symmetry breaking of non-radial solutions of elliptic differential equations considered on a ball or an annulus in R n with the group of symmetry SO(n). Proof. Let us first observe that since ∇ 2 u Φ(0, λ ), ∇ 2 u Φ(0, λ ) are isomorphisms, (0, λ ), (0, λ ) / ∈ BIF. Additionally, from assumption (A) it follows that BIF G = ∅. The basic idea of the proof is to apply Theorem 3.2. By Theorem 2.1 for ν ∈ { , } we obtain We claim that from assumption (A) it follows that dim(H ). The rest of the proof is a direct consequence of Theorem 3.2. (·, λ ), B α (H)). The rest of the proof is a direct consequence of Theorem 3.2.
Remark 1. Note that if in the above theorem dim H − > dim H − then applying a version of the Conley index defined in [12] one can prove that there is a local symmetry-breaking bifurcation of solutions of equation (3.1) from the interval {0}× (λ , λ ) i.e. BIF \ BIF G = ∅.
Remark 2. Let T ⊂ G be the maximal torus of a connected compact Lie group G. Computations in the ring U (T ) are much simpler than that in the ring U (G), see [21]. Note that we can treat G-representations H ν − , ν ∈ { , }, as orthogonal Trepresentations, with the induced T -action. Moreover, it is easier to verify condition [21]. To prove the above theorem we have shown that [11], where χ G is the G-equivariant Euler characteristic, see [8,9]. In other words, to prove the above theorem we have shown that 4. Remarks on symmetry-breaking of solutions of elliptic PDE's. In this section we have shown that some of well known results concerning symmetrybreaking of solutions of elliptic differential equations are consequences of Theorem 3.3. Below we show that some of well known results due to Ramaswamy and Srikanth follow from our results. We begin our discussion with a result due to Ramaswamy and Srikanth [27]. Consider a problem where 1 < p < (n + 2)/(n − 2). We treat the Sobolev space H 1 0 (B n ) as an orthogonal representation of SO(n) with an SO(n)-action defined by (gu)(x) = u(g −1 x) for every u ∈ H 1 0 (B n ) and g ∈ SO(n).
The following theorem is due to Ramaswamy and Srikanth [27]. We show that it is also a direct consequence of Theorem 3.3.
, R) be a functional defined by (4.2). The study of solutions of problem (4.1) is equivalent to the study of solutions of equation ∇ u Φ(u, λ) = 0. What is left is to show that (0, s 0 ) ∈ GLOB \ GLOB SO(n) . In order to prove this theorem it is enough to show that the functional Φ satisfies the assumptions of Theorem 3.3. First of all note that from assumption (a3) it follows that the potential Φ satisfies assumption (A). By assumption (a5) we obtain that ∇ 2 u Φ(0, s) is an isomorphism for every s ∈ (s 0 − , s 0 ) ∪ (s 0 , s 0 + ). From Lemma 4.1 it follows that a change of the Morse index of ∇ 2 u Φ(0, s) occurs when the parameter s crosses the value s 0 . Therefore for s ∈ (s 0 − , s 0 ), s ∈ (s 0 , s 0 + ) we obtain m − (∇ 2 u Φ(0, s )) = m − (∇ 2 u Φ(0, s )). The rest of the proof is a direct consequence of Theorem 3.3.
Remark 3. One can prove the symmetry breaking results due to Cerami [4] applying Theorem 3.3. The non-degeneracy condition considered by Cerami forces a change of the Morse index along the family of radial solutions. Using the techniques of equivariant degree theory the results due to Pacard [26] have been improved in [29]. Whereas results due to Smoller and Wassermann have been improved in [31]. Finally we would like to point out that the results due to Dancer [7] do not follow from our theory. Dancer has used a subtle reasoning on cones to prove his results.