Scattering threshold for the focusing nonlinear Klein-Gordon equation

We show scattering versus blow-up dichotomy below the ground state energy for the focusing nonlinear Klein-Gordon equation, in the spirit of Kenig and Merle for the H¹ critical wave and Schrödinger equations. Our result includes the H¹ 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 the lack of scaling invariance in both the linear and the nonlinear terms.