Standing waves for nonlinear Klein-Gordon equations with nonnegative potentials

This paper is concerned with the standing waves for nonlinear Klein-Gordon equations with nonnegative potentials. First, the existence of standing waves associated with the ground states is obtained by using variational calculus as well as a compactness lemma. Next, a series of sharp conditions for global existence of nonlinear Klein-Gordon equations with nonnegative potentials are established in terms of the characteristics of the ground state and the local theory. Then, that how small the initial data are, the global solutions exist is given. Finally, the instability of standing wave is shown by combining those results.