Blow-up criterion of attractive Bose-Einstein condensates passing obstacles

In this paper, we consider a constraint minimization problem describing the trapped attractive Bose-Einstein condensates passing obstacles. The domain we consider here is an exterior domain $ \Omega = \mathbb{R}^2 \setminus \omega $, where $ \omega $ is a bounded domain and denotes the region of obstacles. In [3], Deng, Guo, and Xu studied the problem passing an obstacle and obtained the blow-up behavior of the minimizer as the interaction strength $ a $ approaches a threshold value $ a^* $ in which $ \omega $ is a bounded convex domain, and the trapping potential $ V(x) $ attains its global minima only along the whole boundary $ \partial\Omega $. Here, we extend their result and consider that the trapping potential $ V(x) $ attains its global minima not only along the whole boundary $ \partial\Omega $ but also at an interior point of $ \Omega $. We provide a blow-up criterion regarding the blow-up behavior of minimizer that would occur near the boundary $ \partial \Omega $ or the interior point. Furthermore, we obtain a blow-up result for the problem passing two obstacles.