The singular perturbation analysis for one-dimensional poisson-nernst-planck equation

We investigate the qualitative property for one-dimensional steady-state Poisson-Nernst-Equation by matched asymptotic expansions, and find that the inner solution is the Weierstrass's elliptic function, so the inner and outer solutions are not matched at the boundary layer. Therefore, perhaps the long channel limit is not suitable for actual problem, otherwise this prompts us to try the other asymptotic expansions method.