Normalized Solutions for Schrödinger Equations with Local Superlinear Nonlinearities

In this paper, we consider the following Schrödinger equation: \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\sigma f(u) +\lambda u, &{}\text {in}\quad \mathbb {R}^{N},\\ \int _{\mathbb {R}^{N}}|u|^{2}~\textrm{d}x =a, &{} u\in H^1(\mathbb {R}^{N}), \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u=\sigma f(u)+\lambda u,\hfill & \text{in}{\mathbb{R}}^N,\hfill \\ {\int }_{{\mathbb{R}}^N}{|u|}^2\text{d}x=a,\hfill & u\in H^1({\mathbb{R}}^N),\hfill \end{array}\right)\end{array}$$ where \( N \ge 3 \) $N\ge 3$ , \( a>0 \) $a>0$ , \(\sigma >0\) $\sigma >0$ , and \( \lambda \in \mathbb {R}\) $\lambda \in \mathbb{R}$ 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 \(\sigma >0\) $\sigma >0$ . For f has a supercritical growth, we derive the existence of the positive normalized solution for the equation with \(\sigma >0\) $\sigma >0$ large enough. In addition, we also obtain infinitely many normalized solutions with sufficiently small \(\sigma >0\) $\sigma >0$ for the subcritical case.