Dynamics of Threshold Solutions for Energy Critical NLS with Inverse Square Potential
Kai Yang, Chongchun Zeng, Xiaoyi Zhang
Abstract
We consider the focusing energy critical NLS with inverse square potential in\ndimension $d= 3, 4, 5$ with the details given in $d=3$ and remarks on results\nin other dimensions. Solutions on the energy surface of the ground state are\ncharacterized. We prove that solutions with kinetic energy less than that of\nthe ground state must scatter to zero or belong to the stable/unstable\nmanifolds of the ground state. In the latter case they converge to the ground\nstate exponentially in the energy space as $t\\to \\infty$ or $t\\to -\\infty$. (In\n3-dim without radial assumption, this holds under the compactness assumption of\nnon-scattering solutions on the energy surface.) When the kinetic energy is\ngreater than that of the ground state, we show that all radial $H^1$ solutions\nblow up in finite time, with the only two exceptions in the case of 5-dim which\nbelong to the stable/unstable manifold of the ground state. The proof relies on\nthe detailed spectral analysis, local invariant manifold theory, and a global\nVirial analysis.\n