Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unbounded sets of solutions of non-cooperative elliptic systems on symmetric spaces

Piotr Stefaniak

Abstract

The aim of this paper is to show that, for a class of non-cooperative elliptic systems on compact symmetric spaces, any continuum of nontrivial solutions bifurcating from the set of trivial solutions is unbounded. The main tool is the degree for invariant strongly indefinite functionals. The analysis relies on the torus-equivariant structure of the Laplace--Beltrami eigenspaces. The result is obtained by ruling out return to the trivial branch in an equivariant version of the Rabinowitz alternative.

Unbounded sets of solutions of non-cooperative elliptic systems on symmetric spaces

Abstract

The aim of this paper is to show that, for a class of non-cooperative elliptic systems on compact symmetric spaces, any continuum of nontrivial solutions bifurcating from the set of trivial solutions is unbounded. The main tool is the degree for invariant strongly indefinite functionals. The analysis relies on the torus-equivariant structure of the Laplace--Beltrami eigenspaces. The result is obtained by ruling out return to the trivial branch in an equivariant version of the Rabinowitz alternative.

Paper Structure

This paper contains 10 sections, 12 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representations and Torus Decomposition
  4. Laplace--Beltrami Spectrum
  5. The $\nabla_\mathbb{T}$-degree for strongly indefinite functionals
  6. Main Results
  7. System and Variational Structure
  8. Global bifurcations
  9. Proof of Theorem \ref{['thm:unbound']}
  10. Further remarks and examples

Key Result

Theorem 2.1

Let $G$ be a compact Lie group. Then: Moreover, if $\mu$ and $\nu$ are highest weights of two irreducible representations, then $\mu$ appears as a weight in the representation with highest weight $\nu$ only if $\mu \preceq \nu$. In particular, if $\mu$ and $\nu$ are incomparable (i.e., neither $\mu \preceq \nu$ nor $\nu \preceq \mu$), t

Theorems & Definitions (23)

  • Theorem 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • Theorem 2.8
  • ...and 13 more