KK-mode contribution to the crossover scale for the brane-induced force

We discuss contributions of the KK modes to the crossover length scale $r_{\rm c}$ for the brane-induced force when the brane is given by a solitonic background field. We work in a 5D scalar model with a domain-wall background that mimics the DGP model. In spite of the infinite number of the KK modes, the crossover scale remains finite due to the warping effect on the ambient space of the domain wall. The inclusion of the KK modes relaxes the hierarchy among the model parameters that is required to realize a phenomenologically viable size of $r_{\rm c}$. We also discuss whether a nontrivial dilaton background enlarges $r_{\rm c}$ or not.