  In any dimension $N\geq1$ and for given mass $m>0$, we revisit the nonlinear
scalar field equation with an $L^2$ constraint:
  $$ -\Delta u=f(u)-\mu u, \quad u \in H^1(\mathbb{R}^N) \quad \text{with}
\quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ where $\mu\in\mathbb{R}$ will arise as
a Lagrange multiplier. Assuming only that the nonlinearity $f$ is continuous
and satisfies weak mass supercritical conditions, we show the existence of
ground states and reveal the basic behavior of the ground state energy $E_m$ as
$m>0$ varies. In particular, to overcome the compactness issue when looking for
ground states, we develop robust arguments which we believe will allow treating
other $L^2$ constrained problems in general mass supercritical settings. Under
the same assumptions, we also obtain infinitely many radial solutions for any
$N\geq2$ and establish the existence and multiplicity of nonradial
sign-changing solutions when $N\geq4$. Finally we propose two open problems.
