论文标题
在具有局部和非本地非线性的质量 - 临界NLS上
On mass - critical NLS with local and non-local nonlinearities
论文作者
论文摘要
我们考虑使用以下非线性schrödinger方程，并具有双$ l^2 $ - 临界非线性\ begin {align*} iu_t+Δu+| u | u |^\ |^\ frac {4} {3} {3} u+μ+μ\ lest（| $ \ mathbb {r}^3 $，} \ end {align*}其中$μ> 0 $足够小。我们的第一个目标是证明基态$q_μ$的存在和非分类。特别是，我们开发了一种适当的扰动方法来证明径向非脱位属性，然后获得基态$q_μ$的一般非分类。然后，我们以最小的质量$ \ | | | _ {l^2} = \ |q_μ\ | _ {l^2} $显示有限的时间爆炸解决方案的存在。更确切地说，我们构建了通过能量$e_μ（u_0）> 0 $和动量$p_μ（u_0）$参数化的最小质量爆炸解决方案。此外，非分类财产在该建筑中起着至关重要的作用。
We consider the following nonlinear Schrödinger equation with the double $L^2$-critical nonlinearities \begin{align*} iu_t+Δu+|u|^\frac{4}{3}u+μ\left(|x|^{-2}*|u|^2\right)u=0\ \ \ \text{in $\mathbb{R}^3$,} \end{align*} where $μ>0$ is small enough. Our first goal is to prove the existence and the non-degeneracy of the ground state $Q_μ$. In particular, we develop an appropriate perturbation approach to prove the radial non-degeneracy property and then obtain the general non-degeneracy of the ground state $Q_μ$. We then show the existence of finite time blowup solution with minimal mass $\|u_0\|_{L^2}=\|Q_μ\|_{L^2}$. More precisely, we construct the minimal mass blowup solutions that are parametrized by the energy $E_μ(u_0)>0$ and the momentum $P_μ(u_0)$. In addition, the non-degeneracy property plays crucial role in this construction.
