Abstract. We study minimizers of the pseudo-relativistic Hartree functionalunder the mass constraint R 3 |u(x)| 2 dx = 1. Here m > 0 is the mass of particles and V ≥ 0 is an external potential. We prove that minimizers exist if and only if a satisfies 0 ≤ a < a * , and there is no minimizer if a ≥ a * , where a * is called the Chandrasekhar limit. When a approaches a * from below, the blowup behavior of minimizers is derived under some general external potentials V . Here we consider three cases of V : trapping potential, i.e. V ∈ L ∞ loc (R 3 ) satisfies lim |x|→∞ V (x) = ∞; periodic potential, i.e. V ∈ C(R 3 ) stisfies V (x+z) = V (x) for all z ∈ Z 3 ; and ring-shaped potential, e.g. V (x) = ||x|−1| p

We consider a 3D quantum system of N identical bosons in a trapping potential |x| p , with p ≥ 0, interacting via a Newton potential with an attractive interaction strength a N . For a fixed large N and the coupling constant a N smaller than a critical value a * (Chandrasekhar limit mass), in an appropriate sense, the many-body system admits a ground state. We investigate the blow-up behavior of the ground state energy as well as the ground states when a N approaches a * sufficiently slowly in the limit N → ∞. The blow-up profile is given by the Gagliardo-Nirenberg solutions.2010 Mathematics Subject Classification. 81V17, 81V70.

We study the minimizers of a magnetic 2D non-linear Schrödinger energy functional in a quadratic trapping potential, describing a rotating Bose-Einstein condensate. We derive an effective Thomas-Fermi-like model in the rapidly rotating limit where the centrifugal force compensates the confinement, and available states are restricted to the lowest Landau level. The coupling constant of the effective Thomas-Fermi functional is to linked the emergence of vortex lattices (the Abrikosov problem). We define it via a low density expansion of the energy of the corresponding homogeneous gas in the thermodynamic limit.

We consider the gravitational collapse for neutron stars in the Hartree-Fock-Bogoliubov theory. We prove that when the number particle becomes large and the gravitational constant is small such that the attractive interaction strength approaches the Chandrasekhar limit mass slowly, the minimizers develop a universal blow-up profile. It is given by the Lane-Emden solution.2010 Mathematics Subject Classification. 81V17, 35Q55, 49J40.

We study the Chandrasekhar variational model for neutron stars, with or without an external potential. We prove the existence of minimizers when the attractive interaction strength τ is strictly smaller than the Chandrasekhar limit τc and investigate the blow-up phenomenon in the limit τ ↑ τc. We show that the blow-up profile of the minimizer(s) is given by the Lane-Emden solution. 1 3 .The mass m > 0 and the spin number q ∈ N will be fixed. Moreover, V : R 3 → R stands for a general external potential; in the translation-invariant case V ≡ 0 we will 1991 Mathematics Subject Classification. 49J40.