  We study the Wu metric for the non-convex domains of the form \[ E_{2m} =
\big\{ z \in \mathbb{C}^n : \vert z_1 \vert^{2m} + \vert z_2 \vert^2 + \ldots +
\vert z_{n-1} \vert^2 + \vert z_n \vert^{2} <1 \big \}, \] where $ 0 < m <
1/2$. Explicit expressions for the Kobayashi metric and the Wu metric on such
pseudo-eggs $E_{2m}$ are obtained. The Wu metric is then verified to be a
continuous Hermitian metric on $ E_{2m} $ which is real analytic everywhere
except along the complex hypersurface $ Z = \{ (0, z_2, \ldots, z_n ) \in
E_{2m} \} $. We also show that the holomorphic sectional curvature of the Wu
metric for this non-compact family of pseudoconvex domains is bounded above in
the sense of currents by a negative constant independent of $m$. This verifies
a conjecture of S. Kobayashi and H. Wu for such $E_{2m}$.
