Abstract
Abstract The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in S n {S^{n}} . In particular, we show that if the geodesic ball is a hemisphere, then all these continua are unbounded. It is also shown that the phenomenon of global symmetry-breaking bifurcation of such solutions occurs. Since the problem is variational and SO ( n ) {\operatorname{SO}(n)} -symmetric, we apply the techniques of equivariant bifurcation theory to prove the main results of this article. As the topological tool, we use the degree theory for SO ( n ) {\operatorname{SO}(n)} -invariant strongly indefinite functionals defined in [A. Gołȩbiewska and S. A. Rybicki, Global bifurcations of critical orbits of G-invariant strongly indefinite functionals, Nonlinear Anal. 74 2011, 5, 1823–1834].
Highlights
The aim of this paper is to study continua of solutions of boundary value problems for non-cooperative elliptic systems considered on geodesic balls in Sn, i.e. systems of the form
Our purpose is to study the phenomenon of global bifurcations of weak solutions of system (1.1)
In other words we have studied closed connected sets of weak solutions of this system, satisfying a Symmetric Rabinowitz alternative
Summary
The aim of this paper is to study continua of solutions of boundary value problems for non-cooperative elliptic systems considered on geodesic balls in Sn, i.e. systems of the form. We discuss only some results concerning symmetric nonlinear problems, where the authors have eliminated one of the Rabinowitz alternatives showing that some (or all) global solution branches are bounded (or unbounded). The author has proved the existence of an unbounded continuum of non-radially symmetric solutions of this problem bifurcating from the second eigenvalue of the Laplace operator. In this article we consider weak solutions of problem (1.1) as orbits of critical points of an SO(n)-invariant functional defined on a suitably chosen infinite-dimensional orthogonal representation of SO(n) This justifies an application of a special degree, i.e. the degree for equivariant gradient maps, see [10, 20]. Properties of the eigenvalues and eigenspaces of the Laplace-Beltrami operator considered on geodesic balls in Sn (with Dirichlet boundary conditions) are described in Remark 2.4, Theorem 2.5 and Corollary 2.6. In Lemmas 4.10, 4.11 we prove formulas for bifurcation indexes
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