Critical and noncritical regions on the critical hyperbola

Abstract

We consider the Hamiltonian elliptic system $$\displaystyle{\left \{\begin{array}{@{}l@{\quad }l@{}} \begin{array}{l} -\varDelta u = \vert v\vert ^{p-1}v \mbox{ in} -\varDelta v =\mu \vert u\vert ^{s-1}u + \vert u\vert ^{q-1}u \mbox{ in} u,v > \mbox{ in} \varOmega, u,v = \mbox{ on} \partial \varOmega,\end{array} \quad \end{array} \right.}$$ where \(\varOmega \subset \mathbb{R}^{N}\) is a bounded smooth domain, N ≥ 3 and μ > 0. We assume that the point (p, q) lies on the critical hyperbola $$\displaystyle{ \frac{1} {p + 1} + \frac{1} {q + 1} = \frac{N - 2} {N} \mbox{ and that $s$ satisfies} \frac{p + 1} {p} \leq s + 1 < q + 1.}$$ 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 cat Ω (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).