  This work investigates a nonlocal sinh-Gordon equation with a singularly
perturbed parameter in a ball. Under the Robin boundary condition, the solution
asymptotically forms a quite steep boundary layer in a thin annular region, and
rapidly becomes a flat curve outside this region. {Focusing more particularly
on the structure of the thin annular layer in this region, the pointwise
asymptotic expansion involving the domain-size is evaluated more sharply, where
the domain-size exactly appears in the second term of the asymptotic
expansion.} It should be stressed that the standard argument of matching
asymptotic expansions is limited because the model has a nonlocal coefficient
depending on the unknown~solution. A new approach relies on integrating ideas
based on a Dirichlet-to-Neumann map in an asymptotic framework. The rigorous
asymptotic expansions for the thin layer structure also matches well with the
numerical results. Furthermore, various boundary concentration phenomena of the
thin annular layer are precisely demonstrated.
