  In this paper, we prove the existence of normalized solutions for the
following Schr\"odinger equation \begin{equation*}
  \left\{
  \begin{array}{ll}
  -\Delta u-\lambda u=f(u), & x\in \R^N,
  \int_{\R^N}u^2\mathrm{d}x=c
  \end{array}
  \right.
  \end{equation*} with $N\ge3$, $c>0$, $\lambda\in \R$ and $f\in
\mathcal{C}(\R,\R)$ in the Sobolev subcritical case with weaker
$L^2$-supercritical conditions and in the Sobolev critical case when $f(u)=\mu
|u|^{q-2}u+|u|^{2^*-2}u$ with $\mu>0$ and $2<q<2^*=\f{2N}{N-2}$ allowing to be
$L^2$-subcritical, critical or supercritical. Our approach is based on several
new critical point theorems on a manifold, which not only help to weaken the
previous $L^2$-supercritical conditions in the Sobolev subcritical case, but
present an alternative scheme to construct bounded (PS) sequences on a manifold
when $f(u)=\mu |u|^{q-2}u+|u|^{2^*-2}u$ technically simpler than the Ghoussoub
minimax principle involving topological arguments, as well as working for all
$2<q<2^*$. In particular, we propose new strategies to control the energy level
in the Sobolev critical case which allow to treat, in a unified way, the
dimensions $N=3$ and $N\ge 4$, and fulfill what were expected by Soave and by
Jeanjean-Le . We believe that our approaches and strategies may be adapted and
modified to attack more variational problems in the constraint contexts.
