Critical point theory for indefinite functionals with symmetries
Let \(X\) be a Hilbert space and \(G\) a compact Lie group acting orthogonally on \(X\). Let \(\phi \in C^1(X,\mathbb{R})\) be a strongly indefinite functional, invariant with respect to the action of \(G\). In order to find critical points of \(\phi\), the authors introduce an equivariant version of the limit relative category of \textit{G. Fournier, D. Lupo, M. Ramos}, and \textit{M. Willem} [Limit relative category and critical point theory, Dynamics Rep. 3, 1-24 (1994; Zbl 0794.58007)]. This category is employed in order to obtain generalizations of the symmetric mountain pass theorem and the symmetric linking theorem. In the mountain pass case it is shown that \(\phi\) has an unbounded sequence of critical values. The hypotheses made here are weaker than the usual ones, even in the case of \(G = Z_2\). In the linking theorem a lower bound for the number of critical orbits is given in terms of the asymptotic behaviour of \(\phi\) at 0 and \(\infty\). The proofs are obtained by reduction to equivariant (relative) Borsuk-Ulam theorems. An application of the mountain pass theorem is given to the elliptic system \(\Delta v = F_v(x,v,w)\), \(-\Delta w = F_w(x,v,w)\) in \(\Omega\), \(v = w = 0\) on \(\partial \Omega\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) and \(F\) is superquadratic (in a weaker sense than the usual). It is shown that if \(F\) is even in \(w\) or in \((v,w)\), then this system has infinitely many solutions. An application of the linking theorem is also sketched. The problem considered is that of existence of periodic solutions to an asymptotically linear Hamiltonian system with symmetric Hamiltonian.
scientific article; zbMATH DE number 889770