  We look for solutions to the Schr\"odinger equation \[ -\Delta u + \lambda u
= g(u) \quad \text{in } \mathbb{R}^N \] coupled with the mass constraint
$\int_{\mathbb{R}^N}|u|^2\,dx = \rho^2$, with $N\ge2$. The behaviour of $g$ at
the origin is allowed to be strongly sublinear, i.e., $\lim_{s\to0}g(s)/s =
-\infty$, which includes the case \[ g(s) = \alpha s \ln s^2 + \mu |s|^{p-2} s
\] with $\alpha > 0$ and $\mu \in \mathbb{R}$, $2 < p \le 2^*$ properly chosen.
We consider a family of approximating problems that can be set in
$H^1(\mathbb{R}^N)$ and the corresponding least-energy solutions, then we prove
that such a family of solutions converges to a least-energy one to the original
problem. Additionally, under certain assumptions about $g$ that allow us to
work in a suitable subspace of $H^1(\mathbb{R}^N)$, we prove the existence of
infinitely many solutions.
