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.