Bifurcations of Nonconstant Solutions of the Ginzburg-Landau Equation

and Applied Analysis 3 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 Section 4 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 paper S1 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 e → [ cos θ − sin θ sin θ cos θ ] . Consider a real Hilbert space H, 〈·, ·〉 H which is an orthogonal S1-representation. The S1-action on the space H×Rwe 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 Su0 {g ∈ S1 : gu0 u0}. Assume that Su0 { S1 ifu0 0 {1} ifu0 / 0 . Hence, S 1 u0 is a manifold such that dimS1 u0 { 0 ifu0 0 1 ifu0 / 0 . A functional Φ : H × R → R is called S 1-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 C will be denoted by C S1 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 ∈ C S1 H×R,R then ∇uΦ ∈ Ck−1 S1 H×R,H , where ∇uΦ is the gradient ofΦwith respect to the first coordinate. Note that if ∇uΦ u0, λ0 0, then the gradient ∇uΦ vanishes on the orbit S1 u0 × {λ0}. Fix Φ ∈ C1 S1 H ×R,R . It is of our interest to study solutions of the following equation: