  We study the electro-diffusion properties of a domain containing a
cusp-shaped structure in three dimensions when one ionic specie is dominant.
The mathematical problem consists in solving the steady-state
Poisson-Nernst-Planck (PNP) equation with an integral constraint for the number
of charges. A non-homogeneous Neumann boundary condition is imposed on the
boundary. We construct an asymptotic approximation for certain singular limits
that agree with numerical simulations. Finally, we analyse the consequences of
non-homogeneous surface charge density. We conclude that the geometry of
cusp-shaped domains influences the voltage profile, specifically inside the
cusp structure. The main results are summarized in the form of new
three-dimensional electrostatic laws for non-electroneutral electrolytes. We
discuss applications to dendritic spines in neuroscience.
