DYNAMICS OF THRESHOLD SOLUTIONS FOR ENERGY CRITICAL NLS WITH INVERSE SQUARE POTENTIAL

We consider the focusing energy critical nonlinear Schrodinger equation (NLS) with inverse square potential in dimension d = 3, 4, 5 with the details given in d = 3 and remarks on results in other dimensions. Solutions on an energy surface of the ground state are characterized. We prove that solutions with kinetic energy less than that of the ground state must scatter to zero or belong to the stable/unstable manifolds of the ground state. In the latter case they converge to the ground state exponentially in the energy space as t -> infinity or t -> -infinity. (In three-dimensions without radial assumption, this holds under the compactness assumption of nonscattering solutions on the energy surface.) When the kinetic energy is greater than that of the ground state, we show that all radial H-1 solutions blow up in finite time, with the only two exceptions being in the case of five-dimensions which belong to the stable/unstable manifold of the ground state. The proof relies on detailed spectral analysis, local invariant manifold theory, and a global Virial analysis.

Automated summary

This article examines the focusing energy critical nonlinear Schrodinger equation with inverse square potential in dimensions 3, 4, and 5. The characteristics of solutions on the energy surface of the ground state are described and proved. It is shown that solutions with kinetic energy less than that of the ground state must converge to the ground state, while solutions with greater kinetic energy will blow up in finite time.