EXISTENCE FOR 2-POINT BOUNDARY-VALUE-PROBLEMS IN 2 ION ELECTRODIFFUSION

In this paper we consider the two-point boundary value problem y'' = y{lambda - (y(0)2 - y2)/2 + [llambda + (y(0)2 - y(1)2)/2]x} -[llambda + (y(0)2 - y(1)2)/2]D, x is-an-element-of [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 less-than-or-equal-to x less-than-or-equal-to 1. The parameters l, lambda, and D are functions of the physical parameters of the problem and the range of physical interest is l, lambda > 0, -1 < D < 1. We consider the case l, lambda, D > 0. Using the maximum principle we show that positive solutions are strictly decreasing and satisfy y(0) less-than-or-equal-to 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 lambda greater-than-or-equal-to 2l(1-1/(1+l)2)D2. 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. (C) 1994 Academic Press, Inc.