Normalized Solutions for Schrödinger Equations with Local Superlinear Nonlinearities

In this paper, we consider the following Schrödinger equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\sigma f(u) +\lambda u, &{}\text {in}\quad \mathbb {R}^{N},\\ \int _{\mathbb {R}^{N}}|u|^{2}~\textrm{d}x =a, &{} u\in H^1(\mathbb {R}^{N}), \end{array}\right. } \end{aligned}$$

where \( N \ge 3 \), \( a>0 \), \(\sigma >0\), and \( \lambda \in \mathbb {R}\) appears as a Lagrange multiplier. Assume that the nonlinear term f satisfies conditions only in a neighborhood of zero. For f has a subcritical growth, we prove the existence of the positive normalized solution for the equation with sufficiently small \(\sigma >0\). For f has a supercritical growth, we derive the existence of the positive normalized solution for the equation with \(\sigma >0\) large enough. In addition, we also obtain infinitely many normalized solutions with sufficiently small \(\sigma >0\) for the subcritical case.