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The local Gan-Gross-Prasad conjecture of unitary groups, which is now settledby 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 intheir associated Vogan $L$-packets which produces the Bessel model. In thispaper, 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 varingtempered $L$-parameters of $U(n)$. We observed that there still exist aGan-Gross-Prasad type formulae depending on the choice of $L$-parameter of$U(n)$.

The local Gan-Gross-Prasad conjecture for $U(n+1) \times U(n)$ : a non-generic case