Ground state and non-ground state solutions of some strongly coupled elliptic systems

Abstract

We study an elliptic system of the form L u = | v | p − 1 v Lu = \left | v\right |^{p-1} v and L v = | u | q − 1 u Lv=\left | u\right |^{q-1} u in Ω \Omega with homogeneous Dirichlet boundary condition, where L u := − Δ u Lu:=-\Delta u in the case of a bounded domain and L u := − Δ u + u Lu:=-\Delta u + u in the cases of an exterior domain or the whole space R N \mathbb {R}^N . We analyze the existence, uniqueness, sign and radial symmetry of ground state solutions and also look for sign changing solutions of the system. More general non-linearities are also considered.