Threshold solutions in the case of mass-shift for the critical   Kline-Gordon equation
  We study global dynamics for the focusing nonlinear Klein-Gordon equation with the energy-critical nonlinearity in two or higher dimensions when the energy equals the threshold given by the ground state of a mass-shifted equation, and prove that the solutions are divided into scattering and blowup. In short, the Kenig-Merle scattering/blowup dichotomy extends to the threshold energy in the case of mass-shift for the critical nonlinear Klein-Gordon equation. 
Slim Ibrahim, Nader Masmoudi, Kenji Nakanishi
math.AP