Existence and concentration of solutions to the Gross–Pitaevskii equation with steep potential well

Abstract

In this paper, we consider the following Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases:

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+\lambda Q(x)u+\omega _{1}|u|^{2}u+\omega _{2}(K\star |u|^{2})u=0, &{} x\in {\mathbb {R}}^{3},\\ \displaystyle u>0,\quad u\in H^{1}({\mathbb {R}}^{3}),\\ \end{array}\right. } \end{aligned}$$

(0.1)

where $\lambda >0$, $Q\in C({\mathbb {R}}^{3},{\mathbb {R}})$ is a potential well, $\star$ 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 the vector x and the vector (0, 0, 1). When $(\omega _{1},\omega _{2})\in {\mathbb {R}}^{2}$ lies in the defined unstable regime, under some suitable conditions on Q, the existence and concentration of nontrivial solutions to problem (0.1) are proved by using variational methods. In particular, the trapping potential is allowed to be sign-changing. Moreover, when $(\omega _{1},\omega _{2})\in {\mathbb {R}}^{2}$ lies in the stable regime, we show that problem (0.1) with small bounded sign-changing potential has only trivial solutions.