Breaking symmetry in focusing nonlinear Klein-Gordon equations with potential

We study the dynamics for the focusing nonlinear Klein-Gordon equation, utt - Δu + m2u = V (x)|u|p-1u, with positive radial potential V and initial data in the energy space. Under suitable assumption on the potential, we establish the existence and uniqueness of the ground state solution. This enables us to define a threshold size for the initial data that separates global existence and blow-up. An appropriate Gagliardo-Nirenberg inequality gives a critical exponent depending on V. For subcritical exponent and subcritical energy global existence vs blow-up conditions are determined by a comparison between the nonlinear term of the energy solution and the nonlinear term of the ground state energy. For subcritical exponents and critical energy some solutions blow-up, other solutions exist for all time due to the decomposition of the energy space of the initial data into two complementary domains.