This work is concerned with the solution of some multi-objective optimal control problems for several PDEs: linear and semilinear elliptic equations and stationary Navier-Stokes systems. More precisely, we look for Pareto equilibria associated to standard cost functionals. First, we study the linear and semilinear case. We prove the existence of equilibria, we deduce appropiate optimality systems, we present some iterative algorithms and we establish convergence results. Then, in order to analyze the existence and characterization of the Pareto equilibria for the Navier-Stokes equations, we use the formalism of Dubovitskii and Milyutin. We also present a finite element approximation of the bi-objective problem and we illustrate the techniques with several numerical experiments.