Bifurcations of Solutions of SO(2)-Symmetric Nonlinear Problems with Variational Structure

Abstract

One of the strongest and the most popular tools in topological nonlinear analysis are the Brouwer topological degree and its infinite-dimensional generalization the Leray–Schauder topological degree, see [De], [Ll]. Usually the Leray–Schauder degree is used to prove the existence, local bifurcations, global bifurcations and continuation of solutions of nonlinear problems. The Krasnosel’skii local bifurcation theorem, see for example [Bro], [De], [Iz1], [Kras], [Ni], ensures the existence of bifurcation points of nontrivial solutions of parameterized nonlinear problems. By the Rabinowitz global bifurcation theorem, see for example [Bro], [De], [Iz1], [Ni], [Ra2], [Ra3], [Ra4], we obtain the existence of branching points of nontrivial solutions of parameterized nonlinear problems. Finally the Leray–Schauder continuation theorem, see [De], ensures the existence of continua of nontrivial solutions of parameterized nonlinear problems. All the theorems mentioned above play the crucial role in topological nonlinear analysis. The Brouwer degree is defined for continuous, admissible maps. On the other hand, the Leray–Schauder degree is defined for the class of admissible operators in the form of a compact perturbation of the identity. Therefore the natural question is the following. Is it possible to define, for a restricted class of operators, degree theory stronger than the Brouwer or the Leray–Schauder degree? Facing the problem of the existence of solutions of many nonlinear differential equations (Hamiltonian systems, wave equations, elliptic differential equations) one can study these solutions as critical points of functionals defined on suitably