I will present some existence results for a Lane-Emden system on a bounded regular domain with Neumann boundary conditions and critical nonlinearities. We show that, under suitable conditions on the exponents in the nonlinearities, least-energy (sign-changing) solutions exist. In the proof we exploit a dual variational formulation, and we establish a compactness condition which is based on a new Cherrier type inequality.  We then prove such condition by using as test functions the solutions to the system in the whole space and performing delicate asymptotic estimates. I will also discuss convergence of solutions in dependence of the exponents of the nonlinearities, and related results. Based on joint works with A. Pistoia, A. Saldaña, and H. Tavares.