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Résumé : This paper investigates a semi-relativistic model describing self-gravitating boson stars. The model is derived from the semi-relativistic Hartree equation with Newtonian gravitational potential, where the kinetic energy is represented by the operator $\sqrt{-\Delta + m^2}$. It is well known that such systems possess a critical mass, analogous to the Chandrasekhar limit for fermion stars. Our main objective is to analyze the asymptotic behavior and the mass concentration mechanism near this critical regime. These results shed light on the mathematical nature of self-gravitational collapse in bosonic systems and extend existing nonrelativistic theories to the semi-relativistic framework. In particular, we provide an alternative proof of the existence, stability, and collapse of ground states when the total mass exceeds the critical threshold, based directly on variational methods.