Existence and multiplicity of stable bound states for the nonlinear Klein-Gordon equation

We are interested in the problem of existence of soliton-like solutions for the nonlinear Klein-Gordon equation. In particular we study some necessary and sufficient conditions on the nonlinear term to obtain solitons of a given charge. We remark that the conditions we consider can be easily verified. Moreover we show that multiplicity of solitons of the same charge is guaranteed by the "shape" of the nonlinear term for equations on $\R^{N}$, hence without appealing to topological or geometrical properties of the domain.