  We investigate the best constants for the regional fractional $p$-Poincar\'e
inequality and the fractional $p$-Poincar\'e inequality in cylindrical domains.
For the special case $p=2$, the result was already known due to
Chowdhury-Csat\'{o}-Roy-Sk [Study of fractional Poincar\'{e} inequalities on
unbounded domains, Discrete Contin. Dyn. Syst., 41(6), 2021]. We addressed the
asymptotic behaviour of the first eigenvalue of the nonlocal Dirichlet
$p$-Laplacian eigenvalue problem when the domain is becoming unbounded in
several directions.
