Multiple normalized solutions for a class of dipolar Gross-Pitaveskii equation with a mass subcritical perturbation

Abstract

In this paper, we study the existence of multiple normalized solutions to the following dipolar Gross-Pitaveskii equation with a mass subcritical perturbation \begin{align*} \left\{ \begin{array}{lll} -\frac{1}{2}\Delta u+\mu u+V(\varepsilon x)u + \lambda_1 |u|^{2}u + \lambda_2(K\ast|u|^{2})u + \lambda_3|u|^{p-2}u = 0, \;&\text{in}\; \mathbb{R}^{3},\\ \int_{{\mathbb{R}}^3} |u|^{2}dx = a^{2}, \end{array}\right. \end{align*} where $a,\varepsilon>0$, $2<p<\frac{10}{3}$, $\mu \in \mathbb{R}$ denotes the Lagrange multiplier, $\lambda_3<0$, $(\lambda_1,\lambda_2) \in \left\lbrace (\lambda_1,\lambda_2) \in \mathbb{R}^{2}:\lambda_1<\frac{4\pi}{3}\lambda_2\le 0\; \text{or}\; \lambda_1<-\frac{8\pi}{3}\lambda_2\le 0 \right\rbrace$, $V(x)$ is an external potential, $\ast$ stands for the convolution, $K(x)=\frac{1-3cos^{2}\theta (x)}{|x|^{3}}$ and $\theta (x)$ is the angle between the dipole axis determined by $(0,0,1)$ and the vector $x$. Under some assumptions of $V$, we use variational methods to prove that the number of normalized solutions is not less than the number of global minimum points of $V$ if $\varepsilon> 0$ is sufficiently small.