  We study the existence of radially symmetric solutions of the following
nonlinear scalar field equations in ${\mathbb R}^N$ ($N\geq 2$):
  $$ (*)_m \left\{
  \eqalign{
  -&\Delta u = g(u) -\mu u \quad \hbox{in}\ {\mathbb R}^N, \cr
  &\| u\|_{L^2({\mathbb R}^N)} = m, \cr
  &u \in H^1({\mathbb R}^N), \cr} \right.
  $$ where $g(\xi)\in C({\mathbb R},{\mathbb R})$, $m>0$ is a given constant
and $\mu\in {\mathbb R}$ is a Lagrange multiplier.
  We introduce a new approach using a Lagrange formulation of the problem
$(*)_m$. We develop a new deformation argument under a new version of the
Palais-Smale condition. For a general class of nonlinearities related to [BL1,
BL2, HIT], it enables us to apply minimax argument for $L^2$ constraint
problems and we show the existence of infinitely many solutions as well as
mountain pass characterization of a minimizing solution of the problem:
  $$ \inf\left\{ \int_{{\mathbb R}^N} {1\over 2}|\nabla u|^2 - G(u)\, dx;\, \|
u\|_{L^2({\mathbb R}^N)}^2 = m \right\},
  \quad G(\xi)=\int_0^\xi g(\tau)\, d\tau.
  $$
