Critical Point Theory for Indefinite Functionals with Symmetries

Abstract

Let X be a Hilbert space and , # C 1 (X, R) be strongly indefinite. Assume in addition that a compact Lie group G acts orthogonally on X and that , is invariant. In order to find critical points of , we develop the limit relative category of Fournier et al. in the equivariant context. We use this to prove two generalizations of the symmetric mountain pass theorem and a linking theorem. In the case of the mountain pass theorem the mountain range is allowed to lie in a subspace of infinite codimension. Also other conditions of the classical symmetric mountain pass theorem for G=ZÂ2 (due to Ambrosetti and Rabinowitz) can be weakened considerably. For example, we are able to deal with infinite-dimensional fixed point spaces. The proofs consist of a direct reduction to a relative Borsuk Ulam type theorem. This provides a new proof even for the classical mountain pass theorem. The abstract critical point theorems are applied to an elliptic system with Dirichlet boundary conditions. We only need a weak version of the usual superquadraticity condition. The linking theorem can be applied to asymptotically linear Hamiltonian systems which are symmetric with respect to a (generalized) symplectic group action.

Generalized Fountain Theorem and applications to strongly indefinite semilinear problems

“…The solutions of pP 1 q represent the steady state solutions of reaction-diffusion systems which are derived from several applications, such as mathematical biology or chemical reactions (see [19] and reference therein). Recently the existence and multiplicity of solutions for noncooperative elliptic systems of the form pP 1 q have been proved by several authors, see for instance [3,4,5,6,19] and references therein.…”

A local min-max-orthogonal method for finding multiple solutions to noncooperative elliptic systems

“…In some sense, the assumption G(x; U ) ≥ 0, ∀U ∈ R 2 in (F 3 ) is a necessity for conditions (F ± 4 ) or (F 5 )-(F 6 ). Meanwhile, other authors [3,9,15] proved that such an assumption may be weakened by, e.g., G(x; 0, v) ≥ 0, for a.e. x ∈ Ω, v ∈ R, under which the existence results can still be obtained.…”

“…In this section we consider a noncooperative system of the form (4.1) where the nonlinear term G(x; u, v) is indefinite (i.e., sign-changing). Due to this indefinite nature, none of the existence results in [3,7,8,9,15] is applicable. However, we have numerically found several solutions to such systems and discovered some interesting phenomena.…”

Infinitely Many Solutions of Nonlinear Elliptic Systems

“…Proof: We need to recall the equivariant limit category defined in [2], specialized to our situation. We set G = Z/2 which acts on E via the antipodal map.…”