论文标题
旋转的超临界NLS的全球稳定性，爆炸和稳定性
Global well-posedness, blow-up and stability of standing waves for supercritical NLS with rotation
论文作者
论文摘要
我们考虑使用旋转\ begin {等式*}的聚焦质量超临界非线性schrödinger方程 iu_ {t} = - \ frac {1} {2}ΔU+\ \ \ \ \ frac {1} {2} v（x）u- | u- | u |^{p-1} u+l_Ωu，\ quad（x，x，x，x，x，t） $ n = 2 $或$ 3 $，$ v（x）$是各向异性的谐波潜力。这里$L_Ω$是量子机械角动量操作员。我们建立了能源空间中全球存在和爆炸的条件。此外，我们证明在某些条件下，在旋转和波频率上，立即波浪具有强烈的不稳定性。最后，我们通过考虑合适的局部最小化问题来构建轨道稳定的常规解决方案。这些结果是针对$ l^{2} $ - 超临界的非线性获得的。
We consider the focusing mass supercritical nonlinear Schrödinger equation with rotation \begin{equation*} iu_{t}=-\frac{1}{2}Δu+\frac{1}{2}V(x)u-|u|^{p-1}u+L_Ωu,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}, \end{equation*} where $N=2$ or $3$ and $V(x)$ is an anisotropic harmonic potential. Here $L_Ω$ is the quantum mechanical angular momentum operator. We establish conditions for global existence and blow-up in the energy space. Moreover, we prove strong instability of standing waves under certain conditions on the rotation and the frequency of the wave. Finally, we construct orbitally stable standing waves solutions by considering a suitable local minimization problem. Those results are obtained for nonlinearities which are $L^{2}$-supercritical.
