Existence for 2-Point Boundary Value Problems in Two Ion Electrodiffusion

Abstract

In this paper we consider the two-point boundary value problem y″=y{λ−(y(0) 2−y 2)/2+[lλ+(y(0) 2−y(1) 2)/2]x} −[lλ+y(0) 2−y(1) 2)/2]D, x∈[0,1], y′(0)=0=y′(1)which arises when two ions with the same valency diffuse and migrate across a liquid junction under the influence of an electric field E. Here y is proportional to the electric field E and, after scaling, the junction occupies the region 0≤x≤1. The parameters l, λ, and D are functions of the physical parameters of the problem and the range of physical interest is l, λ>0, −1< D<1. We consider the case l, λ, D>0. Using the maximum principle we show that positive solutions are strictly decreasing and satisfy y(0)≤ y(1)(1+ l), and that there are no negative solutions. Using Schauder degree theory and upper and lower solutions we show positive solutions exist if [formula]. These results extend and unify those of the author ( Acta Math. Sci. 8, 1988, 373-387). We briefly discuss the corresponding model for two ions of different valencies.