Existence and multiplicity of semiclassical states for Gross–Pitaevskii equation in dipolar quantum gases

Abstract

In this paper, we study the singularly perturbed Gross–Pitaevskii equation

$$\begin{aligned} -\epsilon ^2\Delta u+V(x)u+\lambda _1|u|^2u+\lambda _2(K*|u|^2)u=0,\ u\in H^1({\mathbb {R}}^3), \end{aligned}$$

where $\epsilon >0$ is a parameter, the potential V is a positive function which possesses global minimum points, $\lambda _1,\lambda _2\in {\mathbb {R}}$, $*$ denotes the convolution, $K(x)=\frac{1-3\cos ^2{\theta }}{|x|^3}$ and $\theta =\theta (x)$ is the angle between the dipole axis determined by (0, 0, 1) and the vector x. Using variational methods, we show the existence of ground states for $\epsilon$ small, and describe the concentration phenomena of ground states as $\epsilon \rightarrow 0$. We also investigate the relationship between the number of positive solutions and the profile of the potential V.