Existence and blow-up behavior of constrained minimizers for Schrödinger-Poisson-Slater system

In this article, we study constrained minimizers of the following variational problem e(ρ):=inf{u∈H1(R3),‖u‖22=ρ}E(u),ρ>0,where E(u) is the Schrödinger-Poisson-Slater (SPS) energy functional E(u):=12∫R3|∇u(x)|2dx-14∫R3∫R3u2(y)u2(x)|x-y|dydx-1p∫R3|u(x)|pdxinR3,and p∈ (2, 6). We prove the existence of minimizers for the cases 2 < p < 103, ρ > 0, and p = 103, 0 < ρ < ρ *, and show that e(ρ)= − ∞ for the other cases, where ρ* = ‖ϕ‖22 and ϕ(x) is the unique (up to translations) positive radially symmetric solution of -Δu+u=u73 in R3. Moreover, when e(ρ*)=-∞, the blow-up behavior of minimizers as ρ↗ρ* is also analyzed rigorously.