Solutions of systems of elliptic differential equations on circular domains (Q703431)

It is known the P. Rabinowitz global bifurcation result for an equation with compact operators: If the characteristic value \(\lambda_0\) has an odd algebraic multiplicity, then there is a maximal subcontinuum \(\mathcal{E}_{\lambda_0}\) of the solution set \({\mathcal N}\) such that \((0,\lambda_0)\in \mathcal{E}_{\lambda_0}\), which is either unbounded or meets some other characteristic values \((0,\lambda_i), i=\overline{1,k}\) with \(\sum_{i=0}^k\mathcal{B}\mathcal{I}\mathcal{F}_{LS}(\lambda_i)=0\), where \(\mathcal{B}\mathcal{I}\mathcal{F}_{LS}(\lambda_i)\in \mathbb{Z}\) is the bifurcation index at \((0,\lambda_i)\). Many authors have tried to exclude the second possibility for the case of bifurcation problems for nonlinear equations. The authors study global bifurcation of solutions of a system of elliptic differential equations \(-\Delta u=\nabla _u F(u,\lambda)\) in \(\Omega,\; u|_{\partial \Omega} \)=0, which possesses variational structure. Therefore on an \(SO(2)\)-symmetric domain \(\Omega,\) they use the degree theory for \(SO(2)\)-equivariant gradient maps, developed by the second author. In generalization of the results [\textit{S. Rybicki}, J. Math. Anal. Appl. 217, No. 1, 115--128 (1998; Zbl 0908.35015)] the sufficient conditions are proved for the existence of unbounded continua of nontrivial solutions emanating from trivial ones and the necessary conditions for the existence of bounded continua.