Normalized solutions to Schrödinger equations in the strongly sublinear regime

We look for solutions to the Schrödinger equation $$-\mathrm{\Delta }u+\lambda u=g(u)\text{in }{\mathbb{R}}^N$$ coupled with the mass constraint ${\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx={\rho }^2$, with $N\ge 2$. The behaviour of $g$ at the origin is allowed to be strongly sublinear, i.e., ${\lim }_{s\to 0}g(s)\mathrm{/}s=-\mathrm{\infty }$, which includes the case $$g(s)=\alpha s\ln s^2+\mu \mathrm{\mid }s{\mathrm{\mid }}^{p-2}s$$ with $\alpha >0$ and $\mu \in \mathbb{R}$, $2<p\le 2^{\ast }$ 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.