|| for (+)|
or for small or radial sub-ground state energy (-)
||Energy-critical NLS on R^3, on R^4
The energy-critical NLS occurs when and . Note that the power non-linearity is smooth in dimensions (quintic NLS) and (cubic NLS).
LWP is known for all CaWe1990.
The GWP and scattering theory in the energy class is as follows.
- For small energy this is in GiVl1978, GiVl1979, Sr1981, Sr1981b.
- For radial focusing data with energy less than the ground state this is in [Kenig-Merle]
- For radial defocusing data in three and four dimensions this is in [Bourgain] (see also [Grillakis])
- For radial defocusing data in higher dimensions, see Ta2005, [Visan-Zhang]
- For defocusing data in three dimensions this is in [CKSTT]
- For four dimensions, this is in [Ryckman-Visan]
- For five and higher dimensions, this is in [Visan]