Differential Evolution (DE) is a simple yet powerful evolutionary algorithm, whose performance highly depends on the setting of some parameters. In this paper, we propose an adaptive DE algorithm for multi-objective optimization problems. Firstly, a novel tree neighborhood density estimator is proposed to enforce a higher spread between the non-dominated solutions, while the Pareto dominance strength is used to promote a higher convergence to the Pareto front. These two metrics are then used by an original replacement mechanism based on a three-step comparison procedure; and also to port two existing adaptive mechanisms to the multi-objective domain, one being used for the autonomous selection of the operators, and the other for the adaptive control of DE parameters CR and F. Experimental results confirm the superior performance of the proposed algorithm, referred to as Adap-MODE, when compared to two state-of-the-art baseline approaches, and to its static and partially-adaptive variants.