  For the structure of the thin electrical double layer~(EDL) and the property
related to the EDL capacitance, we analyze boundary layer solutions
(corresponding to the electrostatic potential) of a non-local elliptic equation
which is a steady-state Poisson--Nernst--Planck equation with a singular
perturbation parameter related to the small Debye screening length.
Theoretically, the boundary layer solutions describe that those ions exactly
approach neutrality in the bulk, and the extra charges are accumulated near the
charged surface. Hence, the non-neutral phenomenon merely occurs near the
charged surface. To investigate such phenomena, we develop new analysis
techniques to investigate thin boundary layer structures. A series of fine
estimates combining the Poho\v{z}aev's identity, the inverse H\"{o}lder type
estimates and some technical comparison arguments are developed in arbitrary
bounded domains. Moreover, we focus on the physical domain being a ball with
the simplest geometry and gain a clear picture on the effect of the curvature
on the boundary layer solutions. In particular, for the cylindrical electrode,
our result has a same analogous measurement as the specific capacitance of the
well-known Helmholtz double layer.
