Debye Layer in Poisson-Boltzmann Model with Isolated Singularities

We showthe existence of solutions to the charge conserved Poisson-Boltzmann equation with a Dirichlet boundary condition on partial derivative Omega. Here Omega is a smooth simply connected bounded domain in Rn with n >= 2. When n = 2, the solutions can have isolated singularities at prescribed points in Omega, 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 epsilon tends to zero, and the solutions develop a Debye boundary layer near the boundary partial derivative Omega. In the interior of Omega, 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 partial derivative Omega. In addition we also give a quantitative description on the asymptotic behaviour of the solutions as epsilon -> 0.