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}Δu+μu+V(\varepsilon x)u + λ_1 |u|^{2}u + λ_2(K\ast|u|^{2})u + λ_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 $a,\epsilon >0$, $2<p<\frac{10}{3}$, $\hat{I}¼\in \mathbb{R}$ denotes the Lagrange multiplier, $\hat{I}{»}_3<0$, $(\hat{I}{»}_1,\hat{I}{»}_2)\in \left\{(\hat{I}{»}_1,\hat{I}{»}_2)\in {\mathbb{R}}^2:\hat{I}{»}_1<\frac{4\ddot{I}€}{3}\hat{I}{»}_2\le 0\text{or}\hat{I}{»}_1<-\frac{8\ddot{I}€}{3}\hat{I}{»}_2\le 0\right\}$, $V(x)$ is an external potential, $\ast$ stands for the convolution, $K(x)=\frac{1-3co{s}^2\hat{I}¸(x)}{|x{|}^3}$ and $\hat{I}¸(x)$ 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 $\epsilon >0$ is sufficiently small.