# Bifurcations of Nonconstant Solutions of the Ginzburg-Landau Equation

Authors: Norimichi Hirano; Sławomir Rybicki
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study local and global bifurcations of nonconstant solutions of the Ginzburg-Landau equation from the families of constant ones. As the topological tools we use the equivariant Conley index and the degree for equivariant gradient maps.


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## Body

## 1. Introduction

Let us consider the following Ginzburg-Landau problem:(1.1)-(∇-iA(x))2u(x)=λ(1-|u(x)|2)u(x)inΩ,u=0or∂u∂νon∂Ω,
where λ∈ℝ,Ω⊂ℝN(N=2,3) is an open, bounded domain with smooth boundary ∂Ω, ν(x) is an outward normal to Ω at x∈∂Ω,A∈C0(clΩ,ℝN) and u∈H1(Ω,ℂ).There is a vast literature on this problem. Bifurcations of solutions of the Ginzburg-Landau-type problems have been considered by many authors, see, for instance, [1–10] and references therein. Usually the authors study local bifurcations of nonzero solutions of problem (1.1) with Dirichlet boundary condition by using the Crandall-Rabinowitz bifurcation theorem, the Krasnosiel'ski bifurcation theorem for potential operators, the Lyapunov-Schmidt reduction, the center manifold theorem, the attractor bifurcation theorem, or the implicit function theorem. On the other hand, the global bifurcations of solutions of the one-dimensional Ginzburg-Landau model have been studied in [3]. Using the Brouwer degree the authors have proved the existence of a closed connected set of asymmetric solutions which connect the global curve of symmetric solutions to an asymmetric normal state solution.Our goal is to study the existence of nonconstant solutions of problem (1.1) with Neumann boundary condition. We apply the equivariant bifurcation theory technique. First of all, we study families of constant solutions of problem (1.1) and describe them assuming that the norm of magnetic field A is constant, that is, ∥A(x)∥=const for all x∈clΩ, where ∥·∥ is the usual Euclidean norm. We distinguish two cases ∥A∥=0 and ∥A∥=const≠0. Next we find necessary and sufficient conditions for the existence of local and global bifurcation points of nonconstant solutions from these families. Problem (1.1) is S1-symmetric, that is, if u∈H1(Ω,ℂ) is a solution of this problem, then eiθu is. Therefore, we consider solutions of (1.1) as critical orbits of S1-invariant functionals. The basic idea is to apply the S1-equivariant Conley index, see [11, 12], and the degree for S1-equivariant gradient maps, see [13–18], to obtain a local and global bifurcation of critical S1-orbits of these functionals. The choice of these invariants seems to be the best adapted to our theory. Since the Leray-Schauder degree is not applicable (see the remarks under Corollary 4.2) we have chosen the invariants which are suitable for the study of critical orbits of invariant functionals.After this introduction our paper is organized as follows.In Section2 we have summarized without proofs the relevant abstract material on the equivariant bifurcation theory. In the next sections we have applied these abstract results to the study of local and global bifurcation of nonconstant solutions of problem (1.1). Since the Ginzburg-Landau equation is S1-symmetric, we consider in this section only S1-symmetric variational bifurcation problems. The notion of a local and global bifurcation of critical S1-orbits of families of S1-invariant C1-functionals has been introduced in Definition 2.1. The necessary condition for the existence of bifurcation points of critical S1-orbits has been formulated in Lemma 2.3. The important point to note here is the form of the functional (2.2). Namely, we consider S1-invariant functionals whose gradients are of the form of compact perturbation of the identity. In Theorem 2.4 we have formulated sufficient conditions for the existence of global bifurcations  of S1-orbits of critical points of S1-invariant functionals. Sufficient condition for the existence of local bifurcation of critical S1-orbits has been presented in Theorem 2.6. In Remarks 2.5 and 2.7, we have reformulated assumptions of Theorems 2.4 and 2.6, respectively, to make them easier to understand.In Section3 we study bifurcations of nonconstant solutions of the Ginzburg-Landau bifurcating from the set of constant solutions. In the first part of this section we show that the functional (F) corresponding to the Ginzburg-Landau equation satisfies all the assumptions of the functional considered in Section 2. We consider two cases of the Ginzburg-Landau equation.In Section3.1 we assume that the magnetic field A vanishes. In Lemma 3.2 we have described the set of constant solutions of the Ginzburg-Landau system (3.7), which consists of two families. Moreover, we have proved the necessary condition for the existence of bifurcation of nonconstant solutions from these families. The sets of local and global bifurcation points of nonconstant solutions of system (3.7) have been described in Theorem 3.3.In Section3.2 we assume that the norm of the magnetic field A is constant and different from 0. Without loss of generality we assume that this norm is equal to 1. The structure of this subsection is similar to that of Section 3.1. In Lemma 3.4 we have described the set of constant solutions of system (3.2), which consists of three families. Moreover, we have proved sufficient conditions for the existence of local and global bifurcations of nonconstant solutions of system (3.2) from these families. The necessary conditions for the existence of local and global bifurcation of nonconstant solutions of system (3.2) from the families of constant solutions have been proved in Theorem 3.5.In Section4 we have shown that we cannot use the Leray-Schauder degree and the famous Rabinowitz alternative to study solutions of problem (1.1). Moreover, we have formulated an open question concerning bifurcations of nonconstant solutions of problem (1.1). This question is at present far from being solved. Finally, we have shown that for domains Ω with sufficiently small volume the first eigenvalue of the magnetic Laplace operator -ΔA equals 1. This property has allowed us to simplify the  formulation of Theorem 3.5, see Corollary 4.2.In the appendix we have recalled for the convenience of the reader some material on equivariant algebraic topology thus making our presentation self-contained.


## 2. Bifurcations of Critical Orbits

In this section we summarize without proofs the relevant material on the equivariant bifurcation theory. In the next section we will apply these abstract results to the study of nonconstant solutions of the Ginzburg-Landau equation.Throughout this paperS1 stands for the group of complex numbers of module 1. We identify this group with the group of special orthogonal two-dimensional matrices SO(2) as follows eiθ→[cosθ-sinθsinθcosθ]. Consider a real Hilbert space (ℍ,〈·,·〉ℍ) which is an orthogonal S1-representation. The S1-action on the space ℍ×ℝ we define by g(u,λ)=(gu,λ). For u0∈ℍ define the orbit of u0 by S1(u0)={gu0:g∈S1} and the isotropy group of u0 by Su01={g∈S1:gu0=u0}. Assume that Su01={S1ifu0=0{1}ifu0≠0. Hence, S1(u0) is a manifold such that dimS1(u0)={0ifu0=01ifu0≠0. A functional Φ:ℍ×ℝ→ℝ is called S1-invariant provided that Φ(gu,λ)=Φ(u,λ) for every g∈S1 and u∈ℍ. The space of S1-invariant functionals of the class Ck will be denoted by CS1k(ℍ×ℝ,ℝ). An operator Ψ:ℍ×ℝ→ℍ is said to be S1-equivariant if Ψ(gu,λ)=gΨ(u,λ) for every g∈S1 and u∈ℍ. It is a known fact that if Φ∈CS1k(ℍ×ℝ,ℝ) then ∇uΦ∈CS1k-1(ℍ×ℝ,ℍ), where ∇uΦ is the gradient of Φ with respect to the first coordinate. Note that if ∇uΦ(u0,λ0)=0, then the gradient ∇uΦ vanishes on the orbit S1(u0)×{λ0}.FixΦ∈CS11(ℍ×ℝ,ℝ). It is of our interest to study solutions of the following equation:
(2.1)∇uΦ(u,λ)=0.We are going to apply the bifurcation technique forS1-orbits of critical points of S1-invariant functionals. More precisely, we will apply the S1-equivariant Conley index [11, 12] and the degree for S1-equivariant gradient maps [16, 17] to prove a local and a global bifurcation of critical S1-orbits of problem (2.1).Fixk∈ℕ and {λ11,λ21,…,λ1k,λ2k}⊂ℝ∪{±∞} such that λ1i<λ2i for i=1,…,k. For i=1,…,k define a connected family ℱi={S1(uλ)×{λ}:λ∈(λ1i,λ2i)}, where uλ=ζi(λ) and ζi∈C0((λ1i,λ2i),ℍ). Finally define ℱ=ℱ1∪⋯∪ℱk and assume that ℱ⊂(∇uΦ)-1(0). The set ℱ is called the set of trivial solutions of problem (2.1). Define 𝒩={(u,λ)∈ℍ×ℝ:∇uΦ(u,λ)=0and(u,λ)∉ℱ}. Fix (uλ0,λ0)∈ℱ and denote by 𝒞(uλ0,λ0)⊂ℍ×ℝ, a connected component of cl(𝒩) such that (uλ0,λ0)∈𝒞(uλ0,λ0).Definition 2.1.
A point(uλ0,λ0)∈ℱ is said to be a local bifurcation point of solutions of (2.1), if (uλ0,λ0)∈cl(𝒩). The set of local bifurcation points will be denoted by ℬℐℱ. A point (uλ0,λ0)∈ℱ is said to be a global bifurcation point of solutions of (2.1), if either 𝒞(uλ0,λ0)∩(ℱ∖{(uλ0,λ0)})≠∅ or 𝒞(uλ0,λ0) is unbounded. The set of global bifurcation points will be denoted by 𝒢ℒ𝒪ℬ.In other words a point(uλ0,λ0)∈ℱ is a bifurcation point of nontrivial solutions of (2.1) provided that it is an accumulation point of nontrivial solutions of this equation. A bifurcation point (uλ0,λ0)∈ℱ is a global bifurcation point of nontrivial solutions of (2.1) provided that a connected set 𝒞(uλ0,λ0) of nontrivial solutions of (2.1) bifurcating from this point satisfies the Rabinowitz-type alternative, that is, either 𝒞(uλ0,λ0) is unbounded or meets the set ℱ at least at two times.Remark 2.2.
Directly from the above definition it follows that𝒢ℒ𝒪ℬ⊂ℬℐℱ. Moreover, if (u0,λ0)∈ℬℐℱ ((u0,λ0)∈𝒢ℒ𝒪ℬ), then (gu0,λ0)∈ℬℐℱ ((gu0,λ0)∈𝒢ℒ𝒪ℬ) for every g∈S1.From now on we assume that the functionalΦ∈CS12(ℍ×ℝ,ℝ) is of the following form:
(2.2)Φ(u,λ)=12〈u,u〉ℍ+η(u,λ),
where ∇uη:ℍ×ℝ→ℍ is a compact operator.The natural question is the following: what is the necessary condition for the existence of bifurcation points of solutions of (2.1)?In the lemma below we answer the above-stated question.Lemma 2.3.
If(uλ0,λ0)∈ℬℐℱ, then dimker∇u2Φ(uλ0,λ0)>dimS1(uλ0).Fixi0∈{1,…,k} and (uλ0,λ0)∈ℱi0 such that there is ϵ>0 satisfying the following conditions:(1)
[λ0-ϵ,λ0+ϵ]⊂(λ1i0,λ2i0),(2)
ifλ∈[λ0-ϵ,λ0+ϵ] and dimker∇u2Φ(uλ,λ)>dimS1(uλ), then λ=λ0.SinceS1(uλ0±ϵ) is a nondegenerate critical S1-orbit of the functional Φ(·,λ0±ϵ), there is an open bounded S1-invariant subset Ω⊂ℍ satisfying (∇uΦ(·,λ0±ϵ))-1∩clΩ=S1(uλ0±ϵ). Under these assumptions one can compute the index of an isolated critical orbit S1(uλ0±ϵ) in terms of the degree for S1-equivariant gradient maps, see [13, 15–17], that is, ∇S1-deg(∇uΦ(·,λ0±ϵ),Ω)∈U(S1), where U(S1) is the Euler ring of the group S1, see [19, 20]. For the convenience of the reader, one has reminded the definition of the Euler ring of the groups S1 in appendix.It is a known fact that change of any reasonable degree along the set of trivial solutions implies a global bifurcation of zeroes. In the theorem below we formulate sufficient condition for the existence of a global bifurcation of criticalS1-orbits of problem (2.1). The proof of the following theorem is standard and therefore we omit it.Theorem 2.4.
If∇S1-deg(∇uΦ(·,λ0+ϵ),Ω)≠∇S1-deg(∇uΦ(·,λ0-ϵ),Ω), then (uλ0,λ0)∈𝒢ℒ𝒪ℬ.Finite-dimensional equivariant Conley index has been considered in [12]. Infinite-dimensional generalisation of equivariant Conley index one can find in [11]. In this paper one considers the S1-equivariant Conley index 𝒞ℐS1(·,λ) defined by a flow induces by -∇uΦ(·,λ).Assume additionally thatSuλ1={e} for every (uλ,λ)∈ℱi0. Since S1(uλ0±ϵ) is isolated in ∇uΦ(·,λ0±ϵ)-1(0), it is an isolated invariant set in the sense of the S1-equivariant Conley index theory.It is a known fact, see Lemma 5.7 of [16], that 𝒞ℐS1(S1(uλ0±ϵ),λ0±ϵ) is an S1-CW-complex which consists of a base point and one S1-cell of dimension m-(∇u2Φ(uλ0±ϵ,λ0±ϵ)) and of isotropy group {e}, where m-(·) is the Morse index.Remark 2.5.
Note that(2.3)χS1𝒞ℐS1(S1(uλ0±ϵ),λ0±ϵ)=∇S1-deg(∇uΦ(·,λ0±ϵ),Ω)=(-1)m-(∇u2Φ(uλ0±ϵ,λ0±ϵ))χS1(S1/{e}+)∈U(S1),
where χS1 is the S1-equivariant Euler characteristic, see [19, 20]. The assumption of the above theorem is a little bit mysterious. Taking into account the above, it can be equivalently formulated in the following way: m-(∇u2Φ(uλ0+ϵ,λ0+ϵ))+m-(∇u2Φ(uλ0-ϵ,λ0-ϵ)) is odd. We underline that since ∇uΦ is of the form of compact perturbation of the identity, these Morse indices are finite.A finite-dimensional version of the following theorem has been proved in [12]. We can literally repeat this proof replacing the finite-dimensional S1-equivariant Conley index by its infinite-dimensional generalization.Theorem 2.6.
If𝒞ℐS1(S1(uλ0+ϵ),λ0+ϵ)≠𝒞ℐS1(S1(uλ0-ϵ),λ0-ϵ), then (uλ0,λ0)∈ℬℐℱ.Remark 2.7.
Similarly as in the case of Theorem2.4 one can reformulate the assumption of the above theorem. Equivalent but easier to understand formulation is the following:
(2.4)m-(∇u2Φ(uλ0+ϵ,λ0+ϵ))≠m-(∇u2Φ(uλ0-ϵ,λ0-ϵ)).


## 3. Results

In this section we prove the main results of our paper. Namely, we study bifurcations of nonconstant solutions of the following Ginzburg-Landau equation:(3.1)-(∇-iA(x))2u(x)=λ(1-|u(x)|2)u(x),inΩ,∂u∂ν=0,on∂Ω,
where λ∈ℝ,Ω⊂ℝN(N=2,3) is an open, bounded domain with smooth boundary ∂Ω, ν(x) is an outward normal to Ω at x∈∂Ω,A∈C0(clΩ,ℝN) and u∈H1(Ω,ℂ).After making in problem (3.1) a linear transformation u=v+iw for v,w∈H1(Ω,ℝ) we obtain an equivalent system of real equations
(3.2)-Δv(x)-2〈A(x),∇w(x)〉+∥A(x)∥2v(x)=λ(1-v(x)2-w(x)2)v(x),inΩ,-Δw(x)+2〈A(x),∇v(x)〉+∥A(x)∥2w(x)=λ(1-v(x)2-w(x)2)w(x),inΩ,∂v∂ν=∂w∂ν=0,on∂Ω.We are going to consider solutions of system (3.2) as critical points of S1-invariant functional of the class C2.Define scalar products〈·,·〉HA1(Ω,ℂ),〈·,·〉H1(Ω,ℂ):H1(Ω,ℂ)⊕H1(Ω,ℂ)→ℂ as follows:
(3.3)〈u1,u2〉HA1(Ω,ℂ)=∫Ω(∇u1(x)-iA(x)u1(x),∇u2(x)-iA(x)u2(x)¯)+u1(x)u2(x)¯dx,〈u1,u2〉H1(Ω,ℂ)=∫Ω(∇u1(x),∇u2(x)¯)+u1(x)u2(x)¯dx.We underline that norms∥·∥HA1(Ω,ℂ),∥·∥H1(Ω,ℂ) are equivalent, see [10, 21]. Now define scalar products 〈·,·〉ℍA,〈·,·〉ℍ:ℍ⊕ℍ→ℝ by
(3.4)〈(v1,w1),(v2,w2)〉ℍA=ℜ〈v1+iw1,v2+iw2〉HA1(Ω,ℂ),〈(v1,w1),(v2,w2)〉ℍ=ℜ〈v1+iw1,v2+iw2〉H1(Ω,ℂ),
where ℍ=H1(Ω,ℝ)⊕H1(Ω,ℝ). For simplicity of notation put 0=(0,0)∈ℍ. For v,w∈H1(Ω,ℝ)) put u=v+iw∈H1(Ω,ℂ). Since ∥u∥H1(Ω,ℂ)=∥(v,w)∥ℍ and ∥u∥HA1(Ω,ℂ)=∥(v,w)∥ℍA, norms ∥·∥ℍA,∥·∥ℍ are equivalent. From now on we consider ℍ as a Hilbert space with the scalar product 〈·,·〉ℍA. It is easy to check that ℍ is an orthogonal S1-representation with S1-action given by gu=(gut)t, where ut is the transposition of u. Define a map F∈C∞(ℝ2×ℝ,ℝ) by F((v,w),λ)=λ((1/2)(v2+w2)-(1/4)(v2+w2)2) and a functional Φ∈C2(ℍ×ℝ,ℝ) as follows:
(F)Φ((v,w),λ)=12∥(v,w)∥ℍA2-∫ΩF((v(x),w(x)),λ)+12(v(x)2+w(x)2)dx=12∫Ω|∇z(x)-iA(x)z(x)|2dx-∫Ωλ(12|z|2-14|z|4)dx=12∥z∥HA1(Ω,ℂ)2-∫Ωλ+12|z|2-λ4|z|4dx,
where z=v+iw.Remark 3.1.
It is easy to verify thatΦ∈CS12(ℍ×ℝ,ℝ). Indeed, this is a standard fact that the functional Φ is of the class C2. What is left is to show that the functional Φ is S1-invariant. For (v,w)∈ℍ put z=v+iw and note that Φ(eiθ(v,w),λ)=(1/2)∥eiθz∥HA1(Ω,ℂ)2-∫Ω((λ+1)/2)|eiθz|2-(λ/4)|eiθz|4dx=Φ((v,w),λ). Moreover, it is clear that for u=(v,w) we have Φ(u,λ)=(1/2)∥u∥ℍA2-((λ+1)/2)〈Lu,u〉ℍA-η(u,λ), where  (1)
L:ℍ→ℍ is a linear, compact, bounded, self-adjoint, positively definite, and S1-equivariant operator,(2)
η:ℍ×ℝ→ℝ is a S1-invariant functional of the class C2 such that
(a)
∇uη:ℍ×ℝ→ℍ is a compact, S1-equivariant operator,(b)
∇uη(0,λ)=0,(c)
∇u2η(u,λ)=o(∥u∥ℍ) at u=0, uniformly on bounded λ-intervals.
It is easy to verify that the gradient∇uΦ:ℍ×ℝ→ℍ is an S1-equivariant operator of the class C1 of the form ∇uΦ(u,λ)=u-(λ+1)Lu-∇uη(u,λ). Fix u0=(v0,w0)∈ℍ, λ'∈ℝ and note that the study of ker∇u2Φ(u0,λ′) is equivalent to the study of solutions of the following system:
(3.5)LA[ϕ1ϕ2]=λ'[1-3v02-w02-2v0w0-2v0w01-v02-3w02][ϕ1ϕ2],
where LA[ϕ1ϕ2]=[-Δϕ1(x)-2〈A(x),∇ϕ2(x)〉+∥A(x)∥2ϕ1-Δϕ2(x)+2〈A(x),∇ϕ1(x)〉+∥A(x)∥2ϕ2].
It is a known fact thatS1-orbits of solutions of system (3.2) are in one to one correspondence with the critical S1-orbits of the functional Φ, that is, with the S1-orbits of solutions of the following equation:
(3.6)∇uΦ(u,λ)=0.
From now on we study bifurcations of solutions of the above equation. For simplicity of notations we writeΔA instead of (∇-iA(x))2. Let σ(-ΔA)={0≦λ1<λ2<⋯<λk<⋯} denote the set of eigenvalues of the following eigenvalue problem {-ΔAϕ=λϕinΩ;∂ϕ/∂v=0on∂Ω. It is known that λ1=0 if and only if A=0, see [8]. It is clear that σ(-ΔA)=σ(LA).


### 3.1. Case∥A∥=0

In this section we study bifurcations of solutions of the simplified Ginzburg-Landau equation, that is, we assume that the magnetic fieldA vanishes. Such an equation has been considered in [22]. To underline that A=θ we write Δθ instead ΔA. Note that Δθϕ=Δϕ1+iΔϕ2 for ϕ=ϕ1+iϕ2∈H1(Ω,ℂ), where Δ is the usual Laplace operator on H1(Ω,ℝ). Note that σ(-Δθ)=σ(-Δ) (with Neumann boundary data). If λk∈σ(-Δ), then 𝕍-Δ(λk) denotes the eigenspace of -Δ corresponding to λk. Finally put σodd(-Δ)={λk∈σ(-Δ):dim𝕍-Δ(λk)isodd}.SinceA=θ, system (3.2) has the following form
(3.7)-Δv=λ(1-v2-w2)v,inΩ,-Δw=λ(1-v2-w2)w,inΩ,∂v∂ν=∂w∂ν=0,on∂Ω.The simplest solutions of system (3.7) are constant solutions. We call them trivial and denote ℱ. The set of trivial solutions consists of two families ℱ1={0}×ℝ and ℱ2=S1((1,0))×ℝ. In the lemma below we have described the set of trivial solutions of system (3.7) and have proved the necessary conditions for the existence of bifurcation points of solutions of this problem.Lemma 3.2.
Under the above assumptions,(3.8)(∇uΦ)-1(0)∩{((v,w),λ)∈ℍ×ℝ:v=constandw=const}=ℱ=ℱ1∪ℱ2,
where ℱ1={0}×ℝ and ℱ2=S1((1,0))×ℝ. Moreover,(1)
if(0,λ')∈ℬℐℱ, then λ'∈σ(-Δ) and ker∇u2Φ(0,λ′)=𝕍-Δ(λ′)⊕𝕍-Δ(λ′),(2)
if((1,0),λ′)∈ℬℐℱ, then -2λ'∈σ(-Δ).Proof.
It is easy to check that the set of constant solutions of problem (3.7) is equal ℱ.(1)
By Lemma2.3, if (0,λ′)∈ℬℐℱ, then dimker∇u2Φ(0,λ′)>0. Putting in (3.5) ((v0,w0),λ0)=(0,λ′) we obtain {-Δϕ1=λ'ϕ1-Δϕ2=λ'ϕ2, which completes the proof.(2)
By Lemma2.3, if ((1,0),λ′)∈ℬℐℱ, then dimker∇u2Φ((1,0),λ′)>1. Putting in (3.5) ((v0,w0),λ0)=((1,0),λ′) we obtain {-Δϕ1=-2λ'ϕ1-Δϕ2=0, which completes the proof.In the theorem below we study local and global bifurcations of nonconstant solutions of system (3.7) from the set of constant solutions. The idea of proof is natural. We compute the S1-equivariant Conley index and the degree for S1-equivariant gradient maps along the families of constant solutions and determine levels at which these invariants change.Theorem 3.3.
Under the above assumptions,(a)
(3.9)ℬℐℱ=(ℬℐℱ∩ℱ1)∪(ℬℐℱ∩ℱ2)=⋃λk∈σ(-Δ)‍{(0,λk)∈ℍ×ℝ}∪⋃λk∈σ(-Δ)‍{S1((1,0))×{-λk/2}⊂ℍ×ℝ},(b)
(3.10)𝒢ℒ𝒪ℬ=(𝒢ℒ𝒪ℬ∩ℱ1)∪(𝒢ℒ𝒪ℬ∩ℱ2)=⋃λk∈σ(-Δ)‍{(0,λk)∈ℍ×ℝ}∪⋃λk∈σodd(-Δ)‍{S1((1,0))×{-λk/2}⊂ℍ×ℝ}.
Moreover, ifλk0>0, then the continuum 𝒞(0,λk0) is either unbounded or 𝒞(0,λk0)∩(ℍ×{0})≠∅.Proof.
Fix an arbitraryλk0∈σ(-Δ). By Lemma 3.2 we have ker∇u2Φ(0,λk0)=𝕍-Δ(λk0)⊕𝕍-Δ(λk0). It is simple matter to check that ker∇u2Φ(0,λk0) is a nontrivial even-dimensional orthogonal S1-representation. Combining Theorems 4.5 and 4.7 of [20] we obtain (0,λk0)∈𝒢ℒ𝒪ℬ. If λk0>0 and 𝒞(0,λk0)∩ℍ×{0}=∅, then applying Theorem 4.7 of [20] we obtain that the continuum 𝒞(0,λk0)⊂ℍ×(0,+∞) is unbounded.
Chooseϵ>0 such that
(3.11)[-λk0-ϵ2,-λk0+ϵ2]∩{-λk2:λk∈σ(-Δ)}={-λk02}.
To shorten notation setλk0±=-((λk0±ϵ)/2), λ^k0=-(λk0/2), u0=(1,0), T±ϵ=∇u2Φ(u0,λk0±) and define 𝕍k0=𝕍-Δ(λ1)⊕⋯⊕𝕍-Δ(λk0),𝕍k0-1=𝕍-Δ(λ1)⊕⋯⊕𝕍-Δ(λk0-1). It is understood that 𝕍0={0}.
We claim that(c1)
dimkerT±ϵ=1,(c2)
T+ϵ is negatively defined on 𝕍k0⊕{0}⊂ℍ,(c3)
T-ϵ is negatively defined on 𝕍k0-1⊕{0}⊂ℍ.
Indeed, like in the proof of Lemma3.2 the study of kerT±ϵ is equivalent to the study of solutions of the following system:
(3.12)-Δϕ1=(λk0±ϵ)ϕ1,-Δϕ2=0.
From condition (3.11) it follows that the linear space of solutions of (3.12) is spanned by (ϕ1,ϕ2)=(0,1). Hence, dimkerT±ϵ=1. Examining (3.12) we obtain that(1)
the operatorT+ϵ is negatively defined on the space 𝕍k0⊕{0}⊂ℍ,(2)
the operatorT-ϵ is negatively defined on the space 𝕍k0-1⊕{0}⊂ℍ, which completes the proof of (c1), (c2), and (c3).
SincedimkerT±ϵ=1, there is an open, bounded, and S1-invariant subset Ω⊂ℍ such that ∇uΦ(·,λk0±)-1(0)∩Ω=S1((1,0)). Moreover, since the isotropy group of every (v0,w0)∈ℍ∖{0} is trivial, we obtain(1)
∇S1-deg(∇uΦ(·,λk0+),Ω)=(-1)dim𝕍k0·χS1(S1/{e}+)∈U(S1),(2)
∇S1-deg(∇uΦ(·,λk0-),Ω)=(-1)dim𝕍k0-1·χS1(S1/{e}+)∈U(S1).
Taking into account thatdim𝕍k0=μ-Δ(λk0)+dim𝕍k0-1 and the above we obtain
(3.13)∇S1-deg(∇uΦ(·,λk0+),Ω)=(-1)μ-Δ(λk0)∇S1-deg(∇uΦ(·,λk0-),Ω).
From the above it follows that(3.14)λk0∈σodd(-Δ)iff∇S1-deg(∇uΦ(·,λk0+),Ω)≠∇S1-deg(∇uΦ(·,λk0-),Ω).
Hence, by Theorem2.4 we obtain (u0,λ^k0)∈𝒢ℒ𝒪ℬ provided that λk0∈σodd(-Δ), which completes the proof of (b). Fix λk0∈σ(-Δ) and note that(i)
the Conley index𝒞ℐS1(S1(u0),λk0+) is an S1-CW-complex with S1-CW-decomposition {(0,S1)},{(dim𝕍k0,{e})},(ii)
the Conley index𝒞ℐS1(S1(u0),λk0-) is an S1-CW-complex with S1-CW-decomposition {(0,S1)}, {(dim𝕍k0-1,{e})}.
Note thatχS1(𝒞ℐS1(S1u0,λk0±)=∇S1-deg(∇uΦ(·,λk0±)),Ω), where χS1 is the S1-equivariant Euler characteristic, see [19, 20].
Since the Conley indices𝒞ℐS1(S1(u0),λk0+),𝒞ℐS1(S1u0,λk0-) are not S1-homotopically equivalent, applying Theorem 2.6 we obtain (u0,λ^k0)∈ℬℐℱ, which completes the proof of (a).


### 3.2. Case∥A∥=const≠0

In this section we study bifurcations of solutions of the Ginzburg-Landau equation (3.1) assuming that ∥A(x)∥=1 for every x∈clΩ.In the lemma below we have described the set of constant solutions of system (3.2) and have proved the necessary conditions for the existence of bifurcation points of solutions of this problem. To simplify notation we put uλ=(((λ-1)/λ),0).Lemma 3.4.
Under the above assumptions,(3.15)(∇uΦ)-1(0)∩{((v,w),λ)∈ℍ×ℝ:v=constandw=const}=ℱ=ℱ1∪ℱ2∪ℱ3,
where ℱ1={0}×ℝ,ℱ2={S1(uλ)×{λ}:λ>1},ℱ3={S1(uλ)×{λ}:λ<0}.
Moreover,(1)
if(0,λ′)∈ℬℐℱ∩ℱ1, then λ'∈σ(-ΔA),(2)
ifλ'∈(1,+∞) and (uλ′,λ′)∈ℬℐℱ∩ℱ2, then λ'∈(1,3/2) and 3-2λ′∈σ(-ΔA)∩(0,1),(3)
ifλ'∈(-∞,0) and (uλ′,λ′)∈ℬℐℱ∩ℱ3, then 3-2λ'∈σ(-ΔA).Proof.
First of all we are looking for constant solutions of system (3.2). Put (v,w)≡(c1,c2)∈ℝ2 in (3.2). Taking into account that ∥A(x)∥=1 we obtain system (3.2) in the following form {c1=λ(1-c12-c22)c1;c2=λ(1-c12-c22)c2. Solving this system we obtain the following set of solutions:
(3.16){0}×ℝ∪⋃λ∈(-∞,0)∪(1,+∞)‍S1(uλ)×{λ}⊂(∇uΦ)-1(0)⊂ℍ×ℝ,
which completes the proof.(1)
Applying Lemma2.3 we obtain that if (0,λ′)∈ℬℐℱ∩ℱ1, then dimker∇u2Φ(0,λ′)>0. Putting in (3.5) ((v0,w0),λ0)=(0,λ′) we obtain the following system LA[ϕ1ϕ2]=λ'[ϕ1ϕ2]. Since σ(LA)=σ(-ΔA), the proof is completed.(2)
Fixλ'∈(1,+∞). Putting in (3.5) ((v0,w0),λ0)=(uλ′,λ′) we obtain the following system LA[ϕ1ϕ2]=[(3-2λ')ϕ1ϕ2]. Note that the vector (ϕ1,ϕ2)=(0,1) solves this system. Hence, dimker∇u2Φ(uλ′,λ′)≧1. By Lemma 2.3, if (uλ′,λ′)∈ℬℐℱ∩ℱ2, then dimker∇u2Φ(uλ′,λ′)>1. Since σ(-ΔA)∩(-∞,0]=∅, dimker∇u2Φ(uλ′,λ′)>1 if and only if 3-2λ'∈σ(-ΔA)∩(0,1) and λ'∈(1,3/2) which completes the proof.(3)
Fixλ'∈(-∞,0). Putting in (3.5) ((v0,w0),λ0)=(uλ′,λ′) we obtain the following system LA[ϕ1ϕ2]=[(3-2λ')ϕ1ϕ2]. Note that the vector (ϕ1,ϕ2)=(0,1) solves this system, that is, dimker∇u2Φ(uλ′,λ′)≧1. By Lemma 2.3, if (uλ′,λ′)∈ℬℐℱ∩ℱ3, then dimker∇u2Φ(uλ′,λ′)>1. Since λ'<0, dimker∇u2Φ(uλ′,λ′)>1 if and only if 3-2λ'∈σ(-ΔA), which completes the proof.Putσodd(-ΔA)={λk∈σ(-ΔA):dimℂ𝕍-ΔA(λk)isodd}. In the theorem below we study local and global bifurcations of nonconstant solutions of Ginzburg-Landau equation (3.2). The proof is similar in spirit to that of Theorem 3.3. For simplicity of notation put uk=((2-λk)/(3-λk),0) and λ^k=((3-λk)/2).Theorem 3.5.
Under the above assumptions,(a)
(3.17)ℬℐℱ=(ℬℐℱ∩ℱ1)∪(ℬℐℱ∩ℱ2)∪(ℬℐℱ∩ℱ3)=⋃λk∈σ(-ΔA)‍{(0,λk)∈ℍ×ℝ}∪⋃λk∈σ(-ΔA)∩(0,1)‍S1(uk)×{λ^k}∪⋃λk∈σ(-ΔA)∩(3,+∞)‍S1(uk)×{λ^k},(b)
(3.18)𝒢ℒ𝒪ℬ=(𝒢ℒ𝒪ℬ∩ℱ1)∪(𝒢ℒ𝒪ℬ∩ℱ2)∪(𝒢ℒ𝒪ℬ∩ℱ3)=⋃λk∈σ(-ΔA)‍{(0,λk)∈ℍ×ℝ}∪⋃λk∈σodd(-ΔA)∩(0,1)‍S1(uk)×{λ^k}∪⋃λk∈σodd(-ΔA)∩(3,+∞)‍S1(uk)×{λ^k}.
Moreover, for everyλk∈σ(-ΔA)(1)
𝒞(0,λk)∩ℍ×{0}=∅,(2)
either the continuum𝒞(0,λk)⊂ℍ×(0,+∞) is unbounded or 𝒞(0,λk)∩𝒢ℒ𝒪ℬ∩ℱ2≠∅.Proof.
Bifurcations from the family
ℱ
1. Note that ker∇u2Φ(0,λk) is a nontrivial even-dimensional orthogonal S1-representation for every λk∈σ(-ΔA). Combining Theorems 4.5 and 4.7 of [20] we obtain (0,λk)∈𝒢ℒ𝒪ℬ∩ℱ1 and for every λk∈σ(-ΔA) the continuum 𝒞(0,λk)⊂ℍ×ℝ is unbounded or C(0,λk)∩ℱ2≠∅. Taking into account that ∇u2Φ(0,0) is an isomorphism we obtain 𝒞(0,λk)⊂ℍ×(0,+∞).
Bifurcations from the family
ℱ
3. Fix an arbitrary λk0∈σ(-ΔA)∩(3,+∞) and choose ϵ>0 such that
(3.19)[3-λk0-ϵ2,3-λk0+ϵ2]∩{3-λk2:λk∈σ(-ΔA)}={3-λk02}.
For brevity setλk0±=(3-λk0±ϵ)/2, λ^k0=(3-λk0)/2, uλk0±=((2-λk0±ϵ)/(3-λk0±ϵ),0), u^λk0=((2-λk0)/(3-λk0),0), T±ϵ=∇u2Φ(uλk0±,λk0±), 𝕍k=𝕍-ΔA(λ1)⊕⋯⊕𝕍-ΔA(λk) and 𝕎=⊕λk<1𝕍-ΔA(λk). It is understood that 𝕍0={0} and 𝕎={0} provided that σ(-ΔA)∩(-∞,0)=∅. We claim that(c1)
dimkerT±ϵ=1,(c2)
T+ϵ is negatively defined on the space of dimension dimℂ𝕍k0-1+dimℂ𝕎,(c3)
T-ϵ is negatively defined on the space of dimension dimℂ𝕍k0+dimℂ𝕎.
Indeed, like in the proof of Lemma3.4 the study of kerT±ε is equivalent to the study of solutions of the following system:
(3.20)LA[ϕ1ϕ2]=[(λk0∓ϵ)ϕ1ϕ2].
From condition (3.19) it follows that the linear space of solutions of (3.20) is spanned by (ϕ1,ϕ2)=(0,1). Hence, dimkerT±=1. Fix λk∈σ(-ΔA) and ϕ1+iϕ2∈𝕍-ΔA(λk). Then (3.20) is equivalent to the following system {(λk-λk0±ϵ)ϕ1=0;(λk-1)ϕ2=0. That is why(1)
the operatorT+(ϕ1,ϕ2) is negatively definite if and only if ϕ1∈𝕍k0-1 and ϕ2∈𝕎,(2)
the operatorT-(ϕ1,ϕ2) is negatively definite if and only if ϕ1∈𝕍k0 and ϕ2∈𝕎,which completes the proof of (c1), (c2), and (c3).
SincedimkerT±ϵ=1, there is an open, bounded, and S1-invariant subset Ω⊂ℍ such that ∇uΦ(·,λk0±)-1(0)∩Ω=S1(uλk0±)×{λk0±}. Moreover, since the isotropy group of every (v0,w0)∈ℍ∖{0} is trivial, we obtain(1)
∇S1-deg(∇uΦ(·,λk0+),Ω)=(-1)dimℂ(𝕍k0-1⊕𝕎)·χS1(S1/{e}+)∈U(S1),(2)
∇S1-deg(∇uΦ(·,λk0-),Ω)=(-1)dimℂ(𝕍k0⊕𝕎)·χS1(S1/{e}+)∈U(S1).
Taking into account thatdimℂ𝕍k0=μ-ΔAℂ(λk0)+dimℂ𝕍k0-1 and the above we obtain
(3.21)∇S1-deg(∇uΦ(·,λk0-),Ω)=(-1)μ-ΔAℂ(λk0)∇S1-deg(∇uΦ(·,λk0+),Ω).
From the above it follows that(3.22)μ-ΔAℂ(λk0)isoddif and only if∇S1-deg(∇uΦ(·,λk0-),Ω)≠∇S1-deg(∇uΦ(·,λk0+),Ω).
Hence, by Theorem2.4 we obtain (u^λk0,λ^k0)∈𝒢ℒ𝒪ℬ∩ℱ3, provided that λk0∈σodd(-ΔA). Fix λk0∈σ(-ΔA) and note that (i)
the Conley index𝒞ℐS1(S1(uλk0+),λk0+ϵ) is an S1-CW-complex with S1-CW-decomposition {(0,S1)},{(dimℂ(𝕍k0-1⊕𝕎),{e})},(ii)
the Conley index𝒞ℐS1(S1(uλk0-),λk0-ϵ) is an S1-CW-complex with the following S1-CW-decomposition {(0,S1)},{(dimℂ(𝕍k0⊕𝕎),{e})}.
Since the Conley indices𝒞ℐS1(S1(uλk0+),λk0+ϵ),𝒞ℐS1(S1(uλk0-),λk0-ϵ) are not S1-homotopically equivalent, applying Theorem 2.6 we obtain (u^λk0,λ^k0)∈ℬℐℱ∩ℱ3.
Bifurcations from the family
ℱ
2. This case can be handled in much the same way as bifurcations from the family ℱ3.


## 3.1. Case∥A∥=0

In this section we study bifurcations of solutions of the simplified Ginzburg-Landau equation, that is, we assume that the magnetic fieldA vanishes. Such an equation has been considered in [22]. To underline that A=θ we write Δθ instead ΔA. Note that Δθϕ=Δϕ1+iΔϕ2 for ϕ=ϕ1+iϕ2∈H1(Ω,ℂ), where Δ is the usual Laplace operator on H1(Ω,ℝ). Note that σ(-Δθ)=σ(-Δ) (with Neumann boundary data). If λk∈σ(-Δ), then 𝕍-Δ(λk) denotes the eigenspace of -Δ corresponding to λk. Finally put σodd(-Δ)={λk∈σ(-Δ):dim𝕍-Δ(λk)isodd}.SinceA=θ, system (3.2) has the following form
(3.7)-Δv=λ(1-v2-w2)v,inΩ,-Δw=λ(1-v2-w2)w,inΩ,∂v∂ν=∂w∂ν=0,on∂Ω.The simplest solutions of system (3.7) are constant solutions. We call them trivial and denote ℱ. The set of trivial solutions consists of two families ℱ1={0}×ℝ and ℱ2=S1((1,0))×ℝ. In the lemma below we have described the set of trivial solutions of system (3.7) and have proved the necessary conditions for the existence of bifurcation points of solutions of this problem.Lemma 3.2.
Under the above assumptions,(3.8)(∇uΦ)-1(0)∩{((v,w),λ)∈ℍ×ℝ:v=constandw=const}=ℱ=ℱ1∪ℱ2,
where ℱ1={0}×ℝ and ℱ2=S1((1,0))×ℝ. Moreover,(1)
if(0,λ')∈ℬℐℱ, then λ'∈σ(-Δ) and ker∇u2Φ(0,λ′)=𝕍-Δ(λ′)⊕𝕍-Δ(λ′),(2)
if((1,0),λ′)∈ℬℐℱ, then -2λ'∈σ(-Δ).Proof.
It is easy to check that the set of constant solutions of problem (3.7) is equal ℱ.(1)
By Lemma2.3, if (0,λ′)∈ℬℐℱ, then dimker∇u2Φ(0,λ′)>0. Putting in (3.5) ((v0,w0),λ0)=(0,λ′) we obtain {-Δϕ1=λ'ϕ1-Δϕ2=λ'ϕ2, which completes the proof.(2)
By Lemma2.3, if ((1,0),λ′)∈ℬℐℱ, then dimker∇u2Φ((1,0),λ′)>1. Putting in (3.5) ((v0,w0),λ0)=((1,0),λ′) we obtain {-Δϕ1=-2λ'ϕ1-Δϕ2=0, which completes the proof.In the theorem below we study local and global bifurcations of nonconstant solutions of system (3.7) from the set of constant solutions. The idea of proof is natural. We compute the S1-equivariant Conley index and the degree for S1-equivariant gradient maps along the families of constant solutions and determine levels at which these invariants change.Theorem 3.3.
Under the above assumptions,(a)
(3.9)ℬℐℱ=(ℬℐℱ∩ℱ1)∪(ℬℐℱ∩ℱ2)=⋃λk∈σ(-Δ)‍{(0,λk)∈ℍ×ℝ}∪⋃λk∈σ(-Δ)‍{S1((1,0))×{-λk/2}⊂ℍ×ℝ},(b)
(3.10)𝒢ℒ𝒪ℬ=(𝒢ℒ𝒪ℬ∩ℱ1)∪(𝒢ℒ𝒪ℬ∩ℱ2)=⋃λk∈σ(-Δ)‍{(0,λk)∈ℍ×ℝ}∪⋃λk∈σodd(-Δ)‍{S1((1,0))×{-λk/2}⊂ℍ×ℝ}.
Moreover, ifλk0>0, then the continuum 𝒞(0,λk0) is either unbounded or 𝒞(0,λk0)∩(ℍ×{0})≠∅.Proof.
Fix an arbitraryλk0∈σ(-Δ). By Lemma 3.2 we have ker∇u2Φ(0,λk0)=𝕍-Δ(λk0)⊕𝕍-Δ(λk0). It is simple matter to check that ker∇u2Φ(0,λk0) is a nontrivial even-dimensional orthogonal S1-representation. Combining Theorems 4.5 and 4.7 of [20] we obtain (0,λk0)∈𝒢ℒ𝒪ℬ. If λk0>0 and 𝒞(0,λk0)∩ℍ×{0}=∅, then applying Theorem 4.7 of [20] we obtain that the continuum 𝒞(0,λk0)⊂ℍ×(0,+∞) is unbounded.
Chooseϵ>0 such that
(3.11)[-λk0-ϵ2,-λk0+ϵ2]∩{-λk2:λk∈σ(-Δ)}={-λk02}.
To shorten notation setλk0±=-((λk0±ϵ)/2), λ^k0=-(λk0/2), u0=(1,0), T±ϵ=∇u2Φ(u0,λk0±) and define 𝕍k0=𝕍-Δ(λ1)⊕⋯⊕𝕍-Δ(λk0),𝕍k0-1=𝕍-Δ(λ1)⊕⋯⊕𝕍-Δ(λk0-1). It is understood that 𝕍0={0}.
We claim that(c1)
dimkerT±ϵ=1,(c2)
T+ϵ is negatively defined on 𝕍k0⊕{0}⊂ℍ,(c3)
T-ϵ is negatively defined on 𝕍k0-1⊕{0}⊂ℍ.
Indeed, like in the proof of Lemma3.2 the study of kerT±ϵ is equivalent to the study of solutions of the following system:
(3.12)-Δϕ1=(λk0±ϵ)ϕ1,-Δϕ2=0.
From condition (3.11) it follows that the linear space of solutions of (3.12) is spanned by (ϕ1,ϕ2)=(0,1). Hence, dimkerT±ϵ=1. Examining (3.12) we obtain that(1)
the operatorT+ϵ is negatively defined on the space 𝕍k0⊕{0}⊂ℍ,(2)
the operatorT-ϵ is negatively defined on the space 𝕍k0-1⊕{0}⊂ℍ, which completes the proof of (c1), (c2), and (c3).
SincedimkerT±ϵ=1, there is an open, bounded, and S1-invariant subset Ω⊂ℍ such that ∇uΦ(·,λk0±)-1(0)∩Ω=S1((1,0)). Moreover, since the isotropy group of every (v0,w0)∈ℍ∖{0} is trivial, we obtain(1)
∇S1-deg(∇uΦ(·,λk0+),Ω)=(-1)dim𝕍k0·χS1(S1/{e}+)∈U(S1),(2)
∇S1-deg(∇uΦ(·,λk0-),Ω)=(-1)dim𝕍k0-1·χS1(S1/{e}+)∈U(S1).
Taking into account thatdim𝕍k0=μ-Δ(λk0)+dim𝕍k0-1 and the above we obtain
(3.13)∇S1-deg(∇uΦ(·,λk0+),Ω)=(-1)μ-Δ(λk0)∇S1-deg(∇uΦ(·,λk0-),Ω).
From the above it follows that(3.14)λk0∈σodd(-Δ)iff∇S1-deg(∇uΦ(·,λk0+),Ω)≠∇S1-deg(∇uΦ(·,λk0-),Ω).
Hence, by Theorem2.4 we obtain (u0,λ^k0)∈𝒢ℒ𝒪ℬ provided that λk0∈σodd(-Δ), which completes the proof of (b). Fix λk0∈σ(-Δ) and note that(i)
the Conley index𝒞ℐS1(S1(u0),λk0+) is an S1-CW-complex with S1-CW-decomposition {(0,S1)},{(dim𝕍k0,{e})},(ii)
the Conley index𝒞ℐS1(S1(u0),λk0-) is an S1-CW-complex with S1-CW-decomposition {(0,S1)}, {(dim𝕍k0-1,{e})}.
Note thatχS1(𝒞ℐS1(S1u0,λk0±)=∇S1-deg(∇uΦ(·,λk0±)),Ω), where χS1 is the S1-equivariant Euler characteristic, see [19, 20].
Since the Conley indices𝒞ℐS1(S1(u0),λk0+),𝒞ℐS1(S1u0,λk0-) are not S1-homotopically equivalent, applying Theorem 2.6 we obtain (u0,λ^k0)∈ℬℐℱ, which completes the proof of (a).


## 3.2. Case∥A∥=const≠0

In this section we study bifurcations of solutions of the Ginzburg-Landau equation (3.1) assuming that ∥A(x)∥=1 for every x∈clΩ.In the lemma below we have described the set of constant solutions of system (3.2) and have proved the necessary conditions for the existence of bifurcation points of solutions of this problem. To simplify notation we put uλ=(((λ-1)/λ),0).Lemma 3.4.
Under the above assumptions,(3.15)(∇uΦ)-1(0)∩{((v,w),λ)∈ℍ×ℝ:v=constandw=const}=ℱ=ℱ1∪ℱ2∪ℱ3,
where ℱ1={0}×ℝ,ℱ2={S1(uλ)×{λ}:λ>1},ℱ3={S1(uλ)×{λ}:λ<0}.
Moreover,(1)
if(0,λ′)∈ℬℐℱ∩ℱ1, then λ'∈σ(-ΔA),(2)
ifλ'∈(1,+∞) and (uλ′,λ′)∈ℬℐℱ∩ℱ2, then λ'∈(1,3/2) and 3-2λ′∈σ(-ΔA)∩(0,1),(3)
ifλ'∈(-∞,0) and (uλ′,λ′)∈ℬℐℱ∩ℱ3, then 3-2λ'∈σ(-ΔA).Proof.
First of all we are looking for constant solutions of system (3.2). Put (v,w)≡(c1,c2)∈ℝ2 in (3.2). Taking into account that ∥A(x)∥=1 we obtain system (3.2) in the following form {c1=λ(1-c12-c22)c1;c2=λ(1-c12-c22)c2. Solving this system we obtain the following set of solutions:
(3.16){0}×ℝ∪⋃λ∈(-∞,0)∪(1,+∞)‍S1(uλ)×{λ}⊂(∇uΦ)-1(0)⊂ℍ×ℝ,
which completes the proof.(1)
Applying Lemma2.3 we obtain that if (0,λ′)∈ℬℐℱ∩ℱ1, then dimker∇u2Φ(0,λ′)>0. Putting in (3.5) ((v0,w0),λ0)=(0,λ′) we obtain the following system LA[ϕ1ϕ2]=λ'[ϕ1ϕ2]. Since σ(LA)=σ(-ΔA), the proof is completed.(2)
Fixλ'∈(1,+∞). Putting in (3.5) ((v0,w0),λ0)=(uλ′,λ′) we obtain the following system LA[ϕ1ϕ2]=[(3-2λ')ϕ1ϕ2]. Note that the vector (ϕ1,ϕ2)=(0,1) solves this system. Hence, dimker∇u2Φ(uλ′,λ′)≧1. By Lemma 2.3, if (uλ′,λ′)∈ℬℐℱ∩ℱ2, then dimker∇u2Φ(uλ′,λ′)>1. Since σ(-ΔA)∩(-∞,0]=∅, dimker∇u2Φ(uλ′,λ′)>1 if and only if 3-2λ'∈σ(-ΔA)∩(0,1) and λ'∈(1,3/2) which completes the proof.(3)
Fixλ'∈(-∞,0). Putting in (3.5) ((v0,w0),λ0)=(uλ′,λ′) we obtain the following system LA[ϕ1ϕ2]=[(3-2λ')ϕ1ϕ2]. Note that the vector (ϕ1,ϕ2)=(0,1) solves this system, that is, dimker∇u2Φ(uλ′,λ′)≧1. By Lemma 2.3, if (uλ′,λ′)∈ℬℐℱ∩ℱ3, then dimker∇u2Φ(uλ′,λ′)>1. Since λ'<0, dimker∇u2Φ(uλ′,λ′)>1 if and only if 3-2λ'∈σ(-ΔA), which completes the proof.Putσodd(-ΔA)={λk∈σ(-ΔA):dimℂ𝕍-ΔA(λk)isodd}. In the theorem below we study local and global bifurcations of nonconstant solutions of Ginzburg-Landau equation (3.2). The proof is similar in spirit to that of Theorem 3.3. For simplicity of notation put uk=((2-λk)/(3-λk),0) and λ^k=((3-λk)/2).Theorem 3.5.
Under the above assumptions,(a)
(3.17)ℬℐℱ=(ℬℐℱ∩ℱ1)∪(ℬℐℱ∩ℱ2)∪(ℬℐℱ∩ℱ3)=⋃λk∈σ(-ΔA)‍{(0,λk)∈ℍ×ℝ}∪⋃λk∈σ(-ΔA)∩(0,1)‍S1(uk)×{λ^k}∪⋃λk∈σ(-ΔA)∩(3,+∞)‍S1(uk)×{λ^k},(b)
(3.18)𝒢ℒ𝒪ℬ=(𝒢ℒ𝒪ℬ∩ℱ1)∪(𝒢ℒ𝒪ℬ∩ℱ2)∪(𝒢ℒ𝒪ℬ∩ℱ3)=⋃λk∈σ(-ΔA)‍{(0,λk)∈ℍ×ℝ}∪⋃λk∈σodd(-ΔA)∩(0,1)‍S1(uk)×{λ^k}∪⋃λk∈σodd(-ΔA)∩(3,+∞)‍S1(uk)×{λ^k}.
Moreover, for everyλk∈σ(-ΔA)(1)
𝒞(0,λk)∩ℍ×{0}=∅,(2)
either the continuum𝒞(0,λk)⊂ℍ×(0,+∞) is unbounded or 𝒞(0,λk)∩𝒢ℒ𝒪ℬ∩ℱ2≠∅.Proof.
Bifurcations from the family
ℱ
1. Note that ker∇u2Φ(0,λk) is a nontrivial even-dimensional orthogonal S1-representation for every λk∈σ(-ΔA). Combining Theorems 4.5 and 4.7 of [20] we obtain (0,λk)∈𝒢ℒ𝒪ℬ∩ℱ1 and for every λk∈σ(-ΔA) the continuum 𝒞(0,λk)⊂ℍ×ℝ is unbounded or C(0,λk)∩ℱ2≠∅. Taking into account that ∇u2Φ(0,0) is an isomorphism we obtain 𝒞(0,λk)⊂ℍ×(0,+∞).
Bifurcations from the family
ℱ
3. Fix an arbitrary λk0∈σ(-ΔA)∩(3,+∞) and choose ϵ>0 such that
(3.19)[3-λk0-ϵ2,3-λk0+ϵ2]∩{3-λk2:λk∈σ(-ΔA)}={3-λk02}.
For brevity setλk0±=(3-λk0±ϵ)/2, λ^k0=(3-λk0)/2, uλk0±=((2-λk0±ϵ)/(3-λk0±ϵ),0), u^λk0=((2-λk0)/(3-λk0),0), T±ϵ=∇u2Φ(uλk0±,λk0±), 𝕍k=𝕍-ΔA(λ1)⊕⋯⊕𝕍-ΔA(λk) and 𝕎=⊕λk<1𝕍-ΔA(λk). It is understood that 𝕍0={0} and 𝕎={0} provided that σ(-ΔA)∩(-∞,0)=∅. We claim that(c1)
dimkerT±ϵ=1,(c2)
T+ϵ is negatively defined on the space of dimension dimℂ𝕍k0-1+dimℂ𝕎,(c3)
T-ϵ is negatively defined on the space of dimension dimℂ𝕍k0+dimℂ𝕎.
Indeed, like in the proof of Lemma3.4 the study of kerT±ε is equivalent to the study of solutions of the following system:
(3.20)LA[ϕ1ϕ2]=[(λk0∓ϵ)ϕ1ϕ2].
From condition (3.19) it follows that the linear space of solutions of (3.20) is spanned by (ϕ1,ϕ2)=(0,1). Hence, dimkerT±=1. Fix λk∈σ(-ΔA) and ϕ1+iϕ2∈𝕍-ΔA(λk). Then (3.20) is equivalent to the following system {(λk-λk0±ϵ)ϕ1=0;(λk-1)ϕ2=0. That is why(1)
the operatorT+(ϕ1,ϕ2) is negatively definite if and only if ϕ1∈𝕍k0-1 and ϕ2∈𝕎,(2)
the operatorT-(ϕ1,ϕ2) is negatively definite if and only if ϕ1∈𝕍k0 and ϕ2∈𝕎,which completes the proof of (c1), (c2), and (c3).
SincedimkerT±ϵ=1, there is an open, bounded, and S1-invariant subset Ω⊂ℍ such that ∇uΦ(·,λk0±)-1(0)∩Ω=S1(uλk0±)×{λk0±}. Moreover, since the isotropy group of every (v0,w0)∈ℍ∖{0} is trivial, we obtain(1)
∇S1-deg(∇uΦ(·,λk0+),Ω)=(-1)dimℂ(𝕍k0-1⊕𝕎)·χS1(S1/{e}+)∈U(S1),(2)
∇S1-deg(∇uΦ(·,λk0-),Ω)=(-1)dimℂ(𝕍k0⊕𝕎)·χS1(S1/{e}+)∈U(S1).
Taking into account thatdimℂ𝕍k0=μ-ΔAℂ(λk0)+dimℂ𝕍k0-1 and the above we obtain
(3.21)∇S1-deg(∇uΦ(·,λk0-),Ω)=(-1)μ-ΔAℂ(λk0)∇S1-deg(∇uΦ(·,λk0+),Ω).
From the above it follows that(3.22)μ-ΔAℂ(λk0)isoddif and only if∇S1-deg(∇uΦ(·,λk0-),Ω)≠∇S1-deg(∇uΦ(·,λk0+),Ω).
Hence, by Theorem2.4 we obtain (u^λk0,λ^k0)∈𝒢ℒ𝒪ℬ∩ℱ3, provided that λk0∈σodd(-ΔA). Fix λk0∈σ(-ΔA) and note that (i)
the Conley index𝒞ℐS1(S1(uλk0+),λk0+ϵ) is an S1-CW-complex with S1-CW-decomposition {(0,S1)},{(dimℂ(𝕍k0-1⊕𝕎),{e})},(ii)
the Conley index𝒞ℐS1(S1(uλk0-),λk0-ϵ) is an S1-CW-complex with the following S1-CW-decomposition {(0,S1)},{(dimℂ(𝕍k0⊕𝕎),{e})}.
Since the Conley indices𝒞ℐS1(S1(uλk0+),λk0+ϵ),𝒞ℐS1(S1(uλk0-),λk0-ϵ) are not S1-homotopically equivalent, applying Theorem 2.6 we obtain (u^λk0,λ^k0)∈ℬℐℱ∩ℱ3.
Bifurcations from the family
ℱ
2. This case can be handled in much the same way as bifurcations from the family ℱ3.


## 4. Remarks and Open Questions

We can substantially simplify Theorem3.5 assuming that the first eigenvalue of the magnetic Laplace operator -ΔA is equal to 1. In the lemma below we show that for a domain Ω of sufficiently small volume this assumption is satisfied.Lemma 4.1.
Assume that∥A(x)∥=1forallx∈Ω. Then there is ϵ>0 such that if volΩ<ϵ, then the first eigenvalue of the magnetic Laplace operator -ΔA is equal to 1, that is, σ(-ΔA)={λ1=1<λ2<⋯}.Proof.
Let us consider an eigenvalue problem-ΔAϕ=λϕ. Putting ϕ=ϕ1+iϕ2 we obtain equivalent system of the form
(4.1)-Δϕ1(x)-2〈A(x),∇ϕ2(x)〉+∥A(x)∥2ϕ1=λϕ1,-Δϕ2(x)+2〈A(x),∇ϕ1(x)〉+∥A(x)∥2ϕ2=λϕ2.
Multiplying equations byϕ1,ϕ2, respectively, and integrating by parts we obtain the following equality:
(4.2)∫Ω∥∇ϕ1(x)∥2+∥∇ϕ2(x)∥-2〈A(x),ϕ1(x)∇ϕ2(x)-ϕ2(x)∇ϕ1(x)〉dx=(λ-1)∫Ωϕ12(x)+ϕ22(x)dx.
We remind that1∈σ(-ΔA). To complete the proof it is enough to show that for every ϕ1+iϕ2∈𝕍-ΔA(1)⊥⊂ℍ the following inequality holds true:
(4.3)∫Ω∥∇ϕ1(x)∥2+∥∇ϕ2(x)∥-2〈A(x),ϕ1(x)∇ϕ2(x)-ϕ2(x)∇ϕ1(x)〉dx>0.
Letμ2(Ω) be the first positive eigenvalue of the Laplace operator -Δ under the Neumann boundary condition. Taking into account that ∥A(x)∥=1 we obtain the following estimation:
(4.4)∫Ω∥∇ϕ1(x)∥2+∥∇ϕ2(x)∥2-2〈A(x),ϕ1(x)∇ϕ2(x)-ϕ2(x)∇ϕ1(x)〉dx≧∫Ω∥∇ϕ1(x)∥2+∥∇ϕ2(x)∥2-2∥ϕ1(x)∇ϕ2(x)-ϕ2(x)∇ϕ1(x)∥dx≧∫Ω∥∇ϕ1(x)∥2+∥∇ϕ2(x)∥2-2(|ϕ1(x)|∥∇ϕ2(x)∥+|ϕ2(x)|∥∇ϕ1(x)∥)dx≧|∇ϕ1|22+|∇ϕ2|22-2(|ϕ1|2|∇ϕ2|2+|ϕ2|2|∇ϕ1|2)≧|∇ϕ1|22+|∇ϕ2|22-4μ2(Ω)|∇ϕ1|2|∇ϕ2|2=(|∇ϕ1|2-|∇ϕ2|2)2+(2-4μ2(Ω))|∇ϕ1|2|∇ϕ2|2,
where |·|2 is the L2(Ω) norm. By the Cheeger's inequality we have μ2(Ω)→∞ as volΩ→0, see [23], that is, for any M>0 exists ϵ>0 such that if volΩ<ϵ, then μ2(Ω)>M. To complete the proof it is enough to choose ϵ>0 for M=4.Combining Lemmas3.4 and 4.1 with Theorem 3.5 we obtain the following corollary. In this corollary we use the notation of Theorem 3.5. Since σ(-ΔA)∩(0,1)=∅, the sets ℬℐℱ,𝒢ℒ𝒪ℬ are less complicated. What is important is that we obtain unbounded continua 𝒞(0,λk), that is, we have excluded one possibility of behavior of bifurcating continua in the famous Rabinowitz alternative.Corollary 4.2.
Choose a domainΩ such that volΩ<ϵ, where ϵ is given by Lemma 4.1. Then λ1=1 and σ(-ΔA)∩(0,1)=∅. By Lemma 3.4 one obtains ℬℐℱ∩ℱ2=∅. Applying Theorem 3.5 one obtains(a)
(4.5)ℬℐℱ=⋃λk∈σ(-ΔA)‍{(0,λk)∈ℍ×ℝ}∪⋃λk∈σ(-ΔA)∩(3,+∞)‍S1(uk)×{λ^k},(b)
(4.6)𝒢ℒ𝒪ℬ=⋃λk∈σ(-ΔA){(0,λk)∈ℍ×ℝ}∪⋃λk∈σodd(-ΔA)∩(3,+∞)S1(uk)×{λ^k}.
Moreover, for everyλk∈σ(-ΔA) the continuum 𝒞(0,λk)⊂ℍ×(0,+∞) is unbounded.Note that in the proofs of Theorems3.3 and 3.5 one cannot replace the degree for S1-equivariant gradient maps by the Leray-Schauder degree. Indeed, since ΩS1=∅, degLS(∇uΦ(·,λk0±),Ω)=0∈ℤ, see [24, 25]. The famous Rabinowitz alternative, see [26], is not applicable in our paper because ker∇u2Φ(0,λ^k0) is even. Moreover, if the multiplicity μ-Δ(λk0)(μ-ΔAℂ(λk0)) is even, then one cannot apply the degree for S1-equivariant gradient maps to prove the existence of a global bifurcation, see assumption (3.14) (see assumption (3.22)). On the other hand, to prove the existence of a local bifurcation one applies the S1-equivariant Conley index.It is a known fact that change of the Conley index implies only a local bifurcation of critical points, see [27–30] for examples and discussion. Therefore, it is natural to rise the following question. Fix λk0∈σ(-Δ)∖σodd(-Δ) and λ'k0∈(σ(-Δ)∖σodd(-Δ))∩((1,3/2)∪(3,+∞)).Is it true that((1,0),-(λk0/2)),(((2-λ'k0)/(3-λk'0),0),(3-λk'0)/2)∈𝒢ℒ𝒪ℬ?This question is at present far from being solved.From now on one replaces the groupS1 with an arbitrary compact connected Lie group G and assume that Φ∈CG2(ℍ×ℝ,ℝ) is of the form Φ(u,λ)=(1/2)∥u∥ℍ2-(λ/2)〈Lu,u〉ℍ-η(u,λ), where(1)
L:ℍ→ℍ is a linear, compact, bounded, self-adjoint, positively definite, and G-equivariant operator,(2)
η:ℍ×ℝ→ℝ is a G-invariant functional of the class C2 such that
(a)
∇uη:ℍ×ℝ→ℍ is a compact, G-equivariant operator,(b)
∇uη(0,λ)=0,(c)
∇u2η(u,λ)=o(∥u∥ℍ) at u=0, uniformly on bounded λ-intervals.The gradient∇uΦ:ℍ×ℝ→ℍ is a G-equivariant operator of the class C1 of the form ∇uΦ(u,λ)=u-λLu-∇uη(u,λ). It is clear that ∇uΦ(0,λ)=0 for any λ∈ℝ. Define ℒ(λ)=Id-λL. Following [31] one can introduce a notion of nonlinear eigenvalue of ℒ(λ).Definition 4.3.
λ
0
>
0 is a nonlinear eigenvalue of ℒ(λ), if (0,λ0)∈ℍ×ℝ is a bifurcation point of solutions of equation ∇uΦ(u,λ)=0 from the curve {0}×ℝ⊂ℍ×ℝ for any η∈CG2(ℍ×ℝ,ℝ) satisfying the above conditions.
Forgetting about variational structure andG-symmetries one can study bifurcations of solutions of the equation ∇uΦ(u,λ)=0 applying the Leray-Schauder degree. It is clear that λ0 is a nonlinear eigenvalue of ℒ(λ) if and only if λ0-1 is an eigenvalue of L of odd multiplicity, see, for instance, Theorem 1.2.1 of [31].
Computing the bifurcation index in term of the degree forG-equivariant gradient maps we obtain the following theorem, see Theorem 4.5 of [20].Theorem 4.4.
λ
0
>
0 is a nonlinear eigenvalue of ℒ(λ)if and only if  λ0-1 is an eigenvalue of odd multiplicity or the eigenspace 𝕍L(λ0-1) of L corresponding to the eigenvalue λ0-1 is a nontrivial G-representation.In this paperG=S1. All the eigenvalues of L are of even multiplicity and corresponding eigenspaces of L are nontrivial S1-representations. Summing up, inverse of any eigenvalue of L is a nonlinear eigenvalue of ℒ(λ). Therefore, one obtains global bifurcations of critical S1-orbits from the family ℱ1 in Theorems 3.3 and 3.5. Note that if Ω is of sufficiently small volume, then continua bifurcating from family ℱ1 are unbounded, see Corollary 4.2.Finally we underline that we have proved local and global bifurcations of criticalS1-orbits from connected sets of trivial solutions which are not of the form {0}×(α,β)⊂ℍ×ℝ, see local and global bifurcations from families ℱ2,ℱ3 in Theorems 3.3 and 3.5.



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