Normalized solutions to Schrödinger equations in the strongly sublinear regime

Abstract

We look for solutions to the Schrödinger equation

$$\begin{aligned} -\Delta u + \lambda u = g(u) \quad \text {in } \mathbb {R}^N \end{aligned}$$

coupled with the mass constraint \(\int _{\mathbb {R}^N}|u|^2\,dx = \rho ^2\), with \(N\ge 2\). The behaviour of g at the origin is allowed to be strongly sublinear, i.e., \(\lim _{s\rightarrow 0}g(s)/s = -\infty \), which includes the case

$$\begin{aligned} g(s) = \alpha s \ln s^2 + \mu |s|^{p-2} s \end{aligned}$$

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.