Trapping Spin-$0$ particles on $p-$balls in $(D,1)$ dimensions

p-Balls are topological defects in $(D,1)$ dimensions constructed with $\mathcal{M}\ge 1$ scalar fields which depend radially on only $2 \le p\le D-2$ spatial dimensions. Such defects are characterized by an action that breaks translational invariance and are inspired on the physics of a brane with $D-p$ extra dimensions. Here we consider the issue of localization of bosonic states described by a scalar field $\Phi$ sufficiently weak to not disturb sensibly the defect configuration. After describing the general formalism, we consider some specify examples with $\mathcal{M}=1,2$ and $3$, looking for some region of parameters where bound and resonant bosonic states can be found. We investigate the way the influence of the defect structure, number of radial dimensions and coupling between the fields are related to the occurrence of bound and resonant states.