  Motivated by the recent experimental realization of minimal Kitaev chains
using quantum dots, we investigate the Majorana zero modes (MZM) in $Y$-shape
Kitaev wires. We solve the associated Kitaev models analytically at the sweet
spot ($t_h=\Delta$) and derive the exact form of MZM wave-functions in this
geometry. The novelty of our result is the observation of multi-site MZMs
located near the junction center on the nearby edge sites of each leg. This
result is important for potential braiding of Majoranas and the performance of
$Y$-junctions made from arrays of quantum dots. Furthermore, we study the
stability of local (single-site) and multi-site MZMs modes in the presence of
Coulomb repulsion, using density matrix renormalization group theory. Our local
density-of-states calculation shows that these multi-site MZMs are as equally
topologically protected as the single-site MZMs when in the presence of Coulomb
repulsion or when away from the sweet-spot.
