Debye Layer in Poisson–Boltzmann Model with Isolated Singularities

We show the existence of solutions to the charge conserved Poisson–Boltzmann equation with a Dirichlet boundary condition on ∂ Ω . Here Ω is a smooth simply connected bounded domain in R n with n ⩾ 2 . When n = 2 , the solutions can have isolated singularities at prescribed points in Ω , in which case they are essentially weak solutions of the charge conserved Poisson–Boltzmann equations with Dirac measures as source terms. By contrast, for higher dimensional cases n ⩾ 3 , all the isolated singularities are removable. As a small parameter ϵ tends to zero, and the solutions develop a Debye boundary layer near the boundary ∂ Ω . In the interior of Ω , the solutions converge to a unique constant. The limiting constant is explicitly calculated in terms of a novel formula which depends only on the supplied Dirichlet data on ∂ Ω . In addition we also give a quantitative description on the asymptotic behaviour of the solutions as ϵ → 0 .