Abstract :
In this paper we consider the two-point boundary value problemy″=y{λ−(y(0)2−y2)/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, −10. 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.
