We consider a spatially nonautonomous discrete sine‐Gordon equation with constant forcing and its continuum limit(s) to model a 0‐π Josephson junction with an applied bias current. The continuum limits correspond to the strong coupling limit of the discrete system. The nonautonomous character is due to the presence of a discontinuity point, namely, a jump of π in the sine‐ Gordon phase. The continuum model admits static solitary waves which are called π‐kinks and are attached to the discontinuity point. For small forcing, there are three types of π‐kinks. We show that one of the kinks is stable and the others are unstable. There is a critical value of the forcing beyond which all static π‐kinks fail to exist. Up to this value, the (in)stability of the π‐kinks can be established analytically in the strong coupling limits. Applying a forcing above the critical value causes the nucleation of 2π‐kinks and ‐antikinks. Besides a π‐kink, the unforced system also admits a static 3π‐kink. This state is unstable in the continuum models. By combining analytical and numerical methods in the discrete model, it is shown that the stable π‐kink remains stable and that the unstable π‐kinks cannot be stabilized by decreasing the coupling. The 3π‐kink does become stable in the discrete model when the coupling is sufficiently weak.