Abstract: In this paper, we concern with the existence of constrained minimizers for the variational problem (1.2). We give a classification of the exponent p determining the existence and nonexistence of minimizers. For any fixed a > 0, (1.2) admits minimizers if 0 < p < (4/N) and there is no minimizer of (1.2) if p > (4/N). Specially, if p=(4/N), the existence of minimizers is then proved if and only if a satisfies 0 < a ≤ a^*:=||φ||₂^(4/N), where φ(x) is the unique (up to translations) positive radial solution of -Δ u(x)+u(x)=u^1+(4/N)(x) in R^N. Moreover, there is no minimizer of (1.2) if a > a^*.

Key words: Schrödinger-Poisson Equations, Constrained minimizers, Existence