Normalized solutions to Schrödinger equations in the strongly sublinear regime

We look for solutions to the Schrödinger equation \(\begin{aligned} -\Delta u + \lambda u = g(u) \quad \text {in } \mathbb {R}^N \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u+\lambda u=g(u)\text{in}{\mathbb{R}}^N\end{array}$$ coupled with the mass constraint \(\int _{\mathbb {R}^N}|u|^2\,dx = \rho ^2\) ${\int }_{{\mathbb{R}}^N}{|u|}^{2}dx={\rho }^2$ , with \(N\ge 2\) $N\ge 2$ . The behaviour of g at the origin is allowed to be strongly sublinear, i.e., \(\lim _{s\rightarrow 0}g(s)/s = -\infty \) ${lim}_{s\to 0}g(s)/s=-\infty$ , which includes the case \(\begin{aligned} g(s) = \alpha s \ln s^2 + \mu |s|^{p-2} s \end{aligned}\) $$\begin{array}{c}\hfill g(s)=\alpha sln{s}^2+\mu {|s|}^{p-2}s\end{array}$$ with \(\alpha > 0\) $\alpha >0$ and \(\mu \in \mathbb {R}\) $\mu \in \mathbb{R}$ , \(2 < p \le 2^*\) $2<p\le 2^{\ast }$ properly chosen. We consider a family of approximating problems that can be set in \(H^1(\mathbb {R}^N)\) $H^1({\mathbb{R}}^N)$ 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 \(H^1(\mathbb {R}^N)\) $H^1({\mathbb{R}}^N)$ , we prove the existence of infinitely, many solutions.