  In this paper, we study the existence of minimizers for a class of
constrained minimization problems derived from the Schr\"{o}dinger-Poisson
equations: $$-\Delta u+V(x)u+(|x|^{-1}*u^2)u-|u|^\frac{4}{3}u=\lambda
u,~~x\in\R^3$$ on the $L^2$-spheres $\widetilde{S}(c)=\{u\in
H^1(\R^3)|~\int_{\R^3}V(x)u^2dx<+\infty,~|u|_2^2=c>0\}$. If $V(x)\equiv0$, then
by a different method from Jeanjean and Luo [Z. Angrew. Math. Phys. 64 (2013),
937-954], we show that there is no minimizer for all $c>0$; If $0\leq V(x)\in
L^{\infty}_{loc}(\R^3)$ and $\lim\limits_{|x|\rightarrow+\infty}V(x)=+\infty$,
then a minimizer exists if and only if $0<c\leq c^*=|Q|_2^2$, where $Q$ is the
unique positive radial solution of $-\Delta u+u=|u|^{\frac{4}{3}}u,$
$x\in\R^3$. Our results are sharp. We also extend some results to constrained
minimization problems on $\widetilde{S}(c)$ derived from Schr\"{o}dinger
operators: $$F_\mu(u)=\frac{1}{2}\ds\int_{\R^N}|\nabla
u|^2-\frac{\mu}2\ds\int_{\R^N}V(x)u^2-\frac{N}{2N+4}|u|^\frac{2N+4}{N}$$ where
$0\leq V(x)\in L^{\infty}_{loc}(\R^N)$ and
$\lim\limits_{|x|\rightarrow+\infty}V(x)=0$. We show that if $\mu>\mu_1$ for
some $\mu_1>0$, then a minimizer exists for each $c\in(0,c^*)$.

Analysis of PDEs