Multiplicity of Normalized Solutions for Schrödinger Equations

In this paper, we consider the following nonlinear Schrödinger equation with an \(L^2\) $L^2$ -constraint: \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\lambda u+\mu |u|^{q-2}u+|u|^{p-2}u \ \ \ \textrm{in}~\mathbb {R}^{N}, \\ \int _{\mathbb {R}^{N}}|u|^{2}dx=a^2, \ \ u\in H^1(\mathbb {R}^{N}), \end{array}\right. }\end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\mathrm{\Delta }u=\lambda u+{\mu |u|}^{q-2}u+{|u|}^{p-2}u\text{in}{\mathbb{R}}^N,\hfill \\ {\int }_{{\mathbb{R}}^N}{|u|}^{2}dx=a^2,u\in H^1({\mathbb{R}}^N),\hfill \end{array}\right)\end{array}$$ where \(N\ge 3\) $N\ge 3$ , \(a,\mu >0\) $a,\mu >0$ , \(2<q<2+\frac{4}{N}<p<2^*\) $2<q<2+\frac{4}{N}<p<2^{\ast }$ , \(2q+2N-pN<0\) $2q+2N-pN<0$ and \(\lambda \in \mathbb {R}\) $\lambda \in \mathbb{R}$ arises as a Lagrange multiplier. We deal with the concave and convex cases of energy functional constraints on the \(L^2\) $L^2$ sphere, and prove the existence of infinitely solutions with positive energy levels.