We consider the following nonlinear Schrödinger equation with an inverse potential: $i\frac{\partial u}{\partial t}+\Delta u+{\left|u\right|}^{\frac{4}{N}}u\pm \frac{1}{{\left|x\right|}^{2\sigma }}u=0$in ${\mathbb{R}}^N$. From the classical argument, the solution with subcritical mass (${\Vert u\Vert }_2<{\Vert Q\Vert }_2$) is global and bounded in $H^1\left({\mathbb{R}}^N\right)$. Here, $Q$ is the ground state of the mass-critical problem. Therefore, we are interested in the existence and behaviour of blow-up solutions for the threshold (${\Vert u_0\Vert }_2={\Vert Q\Vert }_2$). Previous studies investigate the existence and behaviour of the critical-mass blow-up solution when the potential is smooth or unbounded but algebraically tractable. There exist no results when classical methods cannot be used, such as the inverse power type potential. However, in this paper, we construct a critical-mass finite-time blow-up solution. Moreover, we show that the blow-up solution converges to a certain blow-up profile in the virial space.

我们考虑以下具有逆电位的非线性薛定谔方程： $\mathrm{一世}\frac{\partial 你}{\partial 吨}+\Delta 你+{\left|你\right|}^{\frac{4}{N}}你\pm \frac{1}{{\left|X\right|}^{2\sigma }}你=0$在 ${\mathbb{电阻}}^N$. 从经典论证来看，具有亚临界质量的解（${\Vert 你\Vert }_2<{\Vert 问\Vert }_2$) 是全局的并且有界 $H^1\left({\mathbb{电阻}}^N\right)$. 这里，$问$是质量临界问题的基态。因此，我们对阈值（${\Vert {你}_0\Vert }_2={\Vert 问\Vert }_2$）。以前的研究调查了当势势平滑或无界但代数易处理时临界质量爆炸解的存在和行为。当不能使用经典方法时，则没有结果，例如逆幂型势。然而，在本文中，我们构建了一个临界质量有限时间爆炸解决方案。此外，我们表明膨胀解收敛到维里空间中的某个膨胀剖面。