Critical and noncritical regions on the critical hyperbola

Abstract

We consider the Hamiltonian elliptic system (Formula presented.) where Ω⊂ ℝN is a bounded smooth domain, N ≥ 3 and μ > 0. We assume that the point (p, q) lies on the critical hyperbola (Formula presented.) The main contributions in this paper are twofold: to indicate that the location, critical or noncritical, of the point (p, q) on the critical hyperbola can interfere on the existence of solutions of the above system; to prove that if Ω has a rich topology, described by its Lusternik-Schnirelmann category, then the system has multiple solutions, at least as many as catΩ(Ω), in case the parameter μ > 0 is sufficiently small and if s satisfies some suitable and natural conditions which depends on the critical or noncritical location of (p, q).