Multiplicity of Normalized Solutions for Schrödinger Equations

Abstract

In this paper, we consider the following nonlinear Schrödinger equation with an $L^2$-constraint:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\lambda u+\mu |u|^{q-2}u+|u|^{p-2}u \ \ \ \textrm{in}~\mathbb {R}^{N}, \\ \int _{\mathbb {R}^{N}}|u|^{2}dx=a^2, \ \ u\in H^1(\mathbb {R}^{N}), \end{array}\right. }\end{aligned}$$

where $N\ge 3$, $a,\mu >0$, $2<q<2+\frac{4}{N}<p<2^*$, $2q+2N-pN<0$ and $\lambda \in \mathbb {R}$ arises as a Lagrange multiplier. We deal with the concave and convex cases of energy functional constraints on the $L^2$ sphere, and prove the existence of infinitely solutions with positive energy levels.