  The local Gan-Gross-Prasad conjecture of unitary groups, which is now settled
by the works of Plessis, Gan and Ichino, says that for a pair of generic
$L$-parameters of $(U(n+1),U(n))$, there is a unique pair of representations in
their associated Vogan $L$-packets which produces the Bessel model. In this
paper, we examined the conjecture for a pair of $L$-parameters of $\big(U(n+1),
U(n)\big)$ as fixing a special non-generic parameter of $U(n+1)$ and varing
tempered $L$-parameters of $U(n)$. We observed that there still exist a
Gan-Gross-Prasad type formulae depending on the choice of $L$-parameter of
$U(n)$.
