In recent years, the $R$-Process Alliance (RPA) has conducted a successful search for stars that are enhanced in elements produced by the rapid neutron-capture ($r$-)process. In particular, the RPA has uncovered a number of stars that are strongly enriched in light $r$-process elements, such as Sr, Y, and Zr. These so-called limited-$r$ stars were investigated to explore the astrophysical production site(s) of these elements. We investigate the possible formation sites for light neutron-capture elements by deriving detailed abundances for neutron-capture elements from high-resolution spectra with a high signal-to-noise ratio of three limited-$r$ stars. We conducted a kinematic analysis and a 1D local thermodynamic equilibrium spectroscopic abundance analysis of three stars. Furthermore, we calculated the lanthanide mass fraction ($ X_ La $) of our stars and of limited-$r$ stars from the literature. We found that the abundance pattern of neutron-capture elements of limited-$r$ stars behaves differently depending on their $ Ba/Eu $ ratios, and we suggest that this should be taken into account in future investigations of their abundances. Furthermore, we found that the $ X_ La $ of limited-$r$ stars is lower than that of the kilonova AT2017gfo. The latter seems to be in the transition zone between limited-$r$ $ X_ La $ and that of $r$-I and $r$-II stars. Finally, we found that unlike $r$-I and $r$-II stars, the current sample of limited-$r$ stars is largely born in the Galaxy and is not accreted.
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