Applications of degree for \(S^1\)-equivariant gradient maps to variational nonlinear problems with \(S^1\)-symmetries
The author [Nonlinear Anal., Theory Methods Appl. 23, No. 1, 83-102 (1994; Zbl 0815.58027)] established a degree theory for \(S^1\)-equivariant orthogonal maps. If \(V\) is a finite-dimensional orthogonal representation of \(S^1\), call \(D_\alpha(V):=\{x\in V||x|<\alpha\}\) for \(\alpha>0\). Let \(f:V\to\mathbb{R}\) be a \(C^2\)-function such that \(0\in V\) is an isolated critical point and choose \(\alpha>0\) such that \((\text{grad}f)^{-1}(\{0\})\cap D_\alpha(V)=\{0\}\). Then the degree \(\text{DEG}(\text{grad}f,D_\alpha(V))\) is defined as an element in \(\mathbb{Z}\oplus\bigoplus_{n=1}^\infty\mathbb{Z}\) . On the one hand, this degree is a powerful tool in an equivariant situation; on the other hand, the definition is fairly involved and computing the degree is extremely complicated. The author's objective in this article is twofold: First, he computes the equivariant degree for \(S^1\)-equivariant gradient maps, in particular product maps, in a variety of special situations. He then applies these results to obtain sufficient conditions for the existence of bifurcation points of \(S^1\)-equivariant variational bifurcation problems. In a similar way, the author deduces sufficient criteria implying the existence of nontrivial orbits of zeros of \(S^1\)-equivariant gradient maps.
scientific article; zbMATH DE number 1130754