A continuation and existence result for a boundary value problem on an unbounded domain arising for the electrical potential in a cylindrical double layer (Q884360)

The authors investigate a nonlinear boundary value problem for the classical Poisson-Boltzmann equation of the type \(\Delta_r u(r) = f(u(r)), r \in [R,\infty), \;u(R) = u_0, \;\lim_{r \rightarrow \infty} u(r) = 0, \) where \(\Delta_r u(r) = \frac{1}{r} \frac{d}{dr} \left ( r \frac{du}{dr} (r) \right ).\) These problems arise in the study of the electric potential distribution for the case of an infinitely long cylindrical surface. The authors state and prove a general continuation result for contractions and formulate the special case of the Leray-Schauder alternative. Then, this is used to provide the existence, uniqueness and approximation method for the previous problem, with an special emphasis on the case \(f(u) = \alpha \sinh(au).\)