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

TL;DR

This paper proves the existence of multiple normalized solutions for a dipolar Gross-Pitaevskii equation with a mass subcritical perturbation, using variational methods under certain conditions on parameters and external potential.

Contribution

It establishes the existence of multiple solutions related to the minima of the external potential for the dipolar Gross-Pitaevskii equation with a mass subcritical perturbation.

Findings

Number of solutions ≥ number of minima of V for small ε

Existence of multiple solutions under specific parameter conditions

Solutions characterized by variational methods

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.