  We show scattering versus blow-up dichotomy below the ground state energy for
the focusing nonlinear Klein-Gordon equation, in the spirit of Kenig-Merle for
the $H^1$ critical wave and Schr\"odinger equations. Our result includes the
$H^1$ critical case, where the threshold is given by the ground state for the
massless equation, and the 2D square-exponential case, where the mass for the
ground state may be modified, depending on the constant in the sharp
Trudinger-Moser inequality. The main difficulty is lack of scaling invariance
both in the linear and nonlinear terms.

Analysis of PDEs