  We discuss the solution of the Neumann problem associated with the CR Yamabe
operator on a subset $\Omega$ of the CR manifold $\mathbb{S}^3$ bounded by the
Clifford torus $\Sigma$. We also discuss the Yamabe-type problem of finding a
contact form on $\Omega$ which has zero Tanaka--Webster scalar curvature and
for which $\Sigma$ has constant $p$-mean curvature.
