Multiplicity of Normalized Solutions for Schrödinger Equation with Mixed Nonlinearity

Abstract

In this paper, we explore the multiplicity of normalized solutions for Schrödinger equation with mixed nonlinearities \[ \begin{cases} -\Delta u + V(\epsilon x)u = \lambda u + \mu |u|^{q-2} u + |u|^{p-2} u &\textrm{in $\mathbb{R}^{N}$}, \\ \int_{\mathbb{R}^{N}} |u|^{2} \, dx = c, \end{cases} \] where $\mu \gt 0$, $c \gt 0$, $2 \lt q \lt 2+4/N \lt p \lt 2N/(N-2)$, $N \geq 3$, $\epsilon \gt 0$ is a parameter and $\lambda \in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier. The potential $V$ is a bounded and continuous nonnegative function, satisfying some suitable global conditions. By employing the minimization techniques and the truncated argument, we obtain that the number of normalized solutions is not less than the number of global minimum points of $V$ when the parameter $\epsilon$ is sufficiently small.