论文标题
非线性schrödinger方程的最小质量爆炸溶液具有反向电位
Minimal mass blow-up solutions for nonlinear Schrödinger equations with an inverse potential
论文作者
论文摘要
我们考虑以下具有反向电位的非线性schrödinger方程：\ [i \ frac {\ partial u} {\ partial t}+ΔU+| u | u |^{\ frac {\ frac {4} {n} {n}} {n}} {n}} u \ pm pm \ pm \ pm \ frac {1} {1} {1} {| x | x | x |^| x |^| x |^{2σ} $ \ mathbb {r}^n $。从经典参数中，具有亚临界质量的解决方案（$ \ | U \ | _2 <\ | q \ | _2 $）是全局的，并以$ h^1（\ Mathbb {r}^n）$限制。在这里，$ Q $是质量关键问题的基础状态。因此，我们对阈值的爆破解决方案的存在和行为感兴趣（$ \ weft \ | u_0 \ right \ | _2 = \ left \ | q \ | q \ right \ | _2 $）。先前的研究调查了当电势平滑或无界但代数易于处理时，关键质量爆炸解决方案的存在和行为。当无法使用经典方法时，没有结果，例如逆功率类型电位。但是，我们构建了一个关键的质量初始值，相应的解决方案在有限的时间内对其进行了爆炸。此外，我们表明，相应的爆破解决方案会收敛到病毒空间中的某些爆破轮廓。
We consider the following nonlinear Schrödinger equation with an inverse potential: \[ i\frac{\partial u}{\partial t}+Δu+|u|^{\frac{4}{N}}u\pm\frac{1}{|x|^{2σ}}u=0 \] in $\mathbb{R}^N$. From the classical argument, the solution with subcritical mass ($\|u\|_2<\|Q\|_2$) is global and bounded in $H^1(\mathbb{R}^N)$. 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 ($\left\|u_0\right\|_2=\left\|Q\right\|_2$). 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 can not be used, such as the inverse power type potential. However, we construct a critical-mass initial value for which the corresponding solution blows up in finite time. Moreover, we show that the corresponding blow-up solution converges to a certain blow-up profile in virial space.
