Dark matter is one of the most fascinating open problems in modern astrophysics. Since it cannot be directly observed, modeling it requires a balanced mix of physical intuition, mathematical deduction, and comparison with indirect experimental data.In this talk, I will briefly introduce the physical context motivating my research, specifically the problem of dark matter distributions around galaxies. Starting from the Schrödinger-Poisson system, the most commonly used model for dark matter dynamics, I will outline the main directions my work has taken.
I will focus on two key aspects. First, I will discuss the issue of stationary states, whose numerical study paves the way for comparison with experimental data. Then, I will propose a relativistic generalization of the model, the Klein-Gordon-Wave system. Its treatment by Hamiltonian perturbative techniques shows the potential of mathematical physics tools in building a comprehensive and reliable model.