In this paper, we study minimizers of the Hartree-type energy functional Ea(u)≔∫RN∇u(x)2+V(x)u(x)2dx−ap∫RNIα∗u(x)pu(x)pdx,a≥0 under the mass constraint ∫RNu2dx=1, where p=N+α+2N with α ∈ (0, N) for N ≥ 2 is the mass critical exponent. Here Iα denotes the Riesz potential and the trapping potential 0≤V(x)∈Lloc∞(RN) satisfies limx→∞V(x)=∞. We prove that minimizers exist if and only if a satisfies a<a∗=Q22(p−1), where Q is a positive radially symmetric ground state of −Δu+u=(Iα∗up)up−2u in ℝN. The uniqueness of positive minimizers holds if a > 0 is small enough. The blow-up behavior of positive minimizers as a↗a∗ is also derived under some general potentials. Especially, we prove that minimizers must blow up at the central point of the biggest inscribed sphere of the set Ω ≔ {x ∈ ℝN, V(x) = 0} if Ω>0.

In this paper, we study constraint minimizers of the following L 2 −critical minimization problem:and N denotes the mass of the particles in the Schrödinger-Poisson-Slater system. We prove that e(N) admits minimizers for N < N * ∶= ||Q|| 2 2 and, however, no minimizers for N > N * , where Q(x) is the unique positive solution of △u − u + u 7 3 = 0 in R 3 . Some results on the existence and nonexistence of minimizers for e(N * ) are also established. Further, when e(N * ) does not admit minimizers, the limit behavior of minimizers as N ↗ N * is also analyzed rigorously. KEYWORDS constraint minimizers, limit behavior, Schrödinger-Poisson-Slater system 4 3 u. From the physical point of view, we are interested in looking for solutions of Equation 1 with a prescribed L 2 -norm. Specifically, for any given constant N > 0, we look for solutions u N ∈ H 1 (R 3 ) with ||u N || 2 2 = N. Motivated by other studies, 8,14-16 taking N ∈ R in Equation 1 as a suitable Lagrange multiplier, a solution u N ∈ H 1 (R 3 ) of (1) Math Meth Appl Sci. 2017;40 7705-7721.wileyonlinelibrary.com/journal/mma