  In this paper, we first answer Chen-Zhang's problem on $p$-Bergman metric
proposed in \cite{CZ22}. Second, we prove the off-diagonal p-Bergman kernel
function $K_p(z,w)$ is H\"older continuous of order (1-$\varepsilon$) about the
second component when $p>1$ for any $\varepsilon>0$, which improves the
corresponding result of Chen-Zhang. Moreover, we prove the asymptotic behavior
of the maximizer of $p$-Bergman kernel as $p\rightarrow 1^-$. Finally, we give
a characterization of a class of holomorphic functions on $\mathbb{B}^1$ to be
$L^p$-integrable.
