The aim of this paper is to study global bifurcations of non-constant solutions of some nonlinear elliptic systems, namely the system on a sphere and the Neumann problem on a ball. We study the bifurcation phenomenon from families of constant solutions given by critical points of the potentials. Considering this problem in the presence of additional symmetries of a compact Lie group, we study orbits of solutions and, in particular, we do not require the critical points to be isolated. Moreover, we allow the considered orbits of critical points to be degenerate. To prove the bifurcation, we compute the index of an isolated degenerate critical orbit in an abstract situation. This index is given in terms of the degree for equivariant gradient maps.

本文的目的是研究一些非线性椭圆系统的非常数解的全局分岔，即球面上的系统和球上的诺伊曼问题。我们研究了由势的临界点给出的常数解族的分岔现象。考虑到存在紧李群的额外对称性时的这个问题，我们研究解的轨道，特别是，我们不需要隔离关键点。此外，我们允许考虑的临界点轨道退化。为了证明分岔，我们计算了抽象情况下孤立退化临界轨道的指数。该指数是根据等变梯度图的程度给出的。