0,\\ (b) & \frac{d\eta}{dz}(0) =0 \end{array} \eqno(2.8) $$ where $\eta _H$ is the prescribed potential (or polarization). Next we scale the problem by defining dimensionless variables and parameters: Let $$ u \equiv\frac{C}{C_{0}}, \ \ x^* \equiv\frac{x}{L}, \ \ z^* \equiv\frac{z}{H} $$ and $$ v\equiv \frac{F\eta}{RT}, \ \ V\equiv \frac{F\eta_{H}}{RT}. $$ Also define the lumped parameters $$ \alpha \equiv \frac{L^{2}i_{0}s\tilde{A}}{DC_{0}nF}, \ \ \ \beta \equiv \frac{DnF^{2}C_{0}H^{2}}{RTsL^{2}\kappa}.\eqno(2.9) $$ Writing (2.1), (2.2), (2.6)-(2.8) in terms of these dimensionless variables and parameters and dropping the asterisks leads to the compact system of equations (1.1). Some typical values for the constants in (2.9) are given by Giner and Hunter. For $H=10^{-2}$ cm, $L=10^{-4}$ cm and $i_o = 10^{-8} A/$cm$^2$, those values imply that $\alpha \simeq 10^{-2}$ and $\beta \simeq 10^{-1}$. For the scaled potential, $V\simeq 10$ when $\eta _H = 300$ mV. \section{Preliminaries: the problem auxiliary $(P_1)$.} In this section we consider solutions of Problem $(P_1)$ for $p>0$ and $\lambda \geq 0$. The case $\lambda =0$ implies the trivial solution $w(x)\equiv 1$ for $0\leq x \leq 1$. We therefore concentrate on solutions for which $\lambda >0$. From the differential equation and the left boundary condition (1.3a,b), it follows that $w$ is smooth (at least $C^2$) and satisfies $$ w''>0 \mbox{ and } w'>0 \mbox{ on } \{ x\in (0,1):w(x) >0\}.\eqno(3.1) $$ There are two types of solutions of interest: positive solutions and deadcore solutions where $w=0$ in part of the domain [see for example Bandle, Sperb \& Stakgold (1984)]. In view of (3.1), the corresponding deadcore, i.e. the set where $w=0$, must be an interval of the form $[0,x_0]$ with $x_0\in [0,1)$. Assume now that $w$ is any solution. Multiplying (1.3a) by $w'$ and integrating yields $$ \frac{1}{2} (w')^2 = \frac{\lambda}{p+1} \left( w^{p+1} - w^{p+1}_0\right) \eqno(3.2) $$ with $w_0=w(0)>0$ for positive solutions and $w_0=0$ for deadcore solutions. Rearranging and integrating once more leads to the expression $$ \int^1_{w(x)} \frac{ds}{\{s^{p+1}-w^{p+1}_{0}\}^{1/2}} = \sqrt{\frac{2\lambda}{p+1}}(1-x)\ \ \ \mbox{ for } 0\leq x\leq 1.\eqno(3.3) $$ Explicitly evaluating this integral with $w_0=0$ yields the following form for the deadcore solutions provided that $0

1$ the integral converges but $w_0^{\frac{1-p}{2}} \rightarrow \infty$, and if $p=1$ the integral diverges. Hence $$ \lim\limits_{w_0\downarrow 0} F_p (w_0) = +\infty \ \ \ \forall p\geq 1.\eqno(3.7) $$ If $0

0\eqno(3.10)
$$
in (3.9), one arrives at
$$
\lim\limits_{w_{0}\uparrow 1} F'_p (w_0) = -\infty \ \ \ \forall
p>0.\eqno(3.11)
$$
Moreover for $p\ge 1$, (3.9) implies
$$
F'_p(w_0) < -\frac{1}{w_{0}\{1-w^{p+1}_{0}\}^{1/2}}<0,\ \ \ 0 (u-w_0)^{p+1}\ \ \ \mbox{ for }\ \ \ 0 0$ the function $\Phi _p:[0,\infty)\rightarrow [0,\infty)$
defined through Problem $(P_1)$ by
$$
\Phi _p(\lambda) := w'(1;\lambda ,p) ,\ \ \ \lambda \geq 0.\eqno(3.15)
$$
\paragraph{Proposition 3.2} For every $p>0$,
{
\def\labelenumi{(\roman{enumi})}
\def\theenumi{(\roman{enumi})}
\begin{enumerate}
\item $\Phi _p$ is strictly increasing on $[0,\infty)$;
\item $\Phi _p(\lambda) \leq \sqrt{\frac{2\lambda}{p+1}}$ for all $\lambda
\geq 0$ and $\lim\limits_{\lambda \rightarrow \infty}
\frac{1}{\sqrt{\lambda}} \Phi _p(\lambda )=\sqrt{\frac{2}{p+1}}$. In
particular $\Phi _p(\lambda)=\sqrt{\frac{2\lambda}{p+1}}$ for $0 0$ and
$$
\parbox{4cm}{$w_0(\cdot ;p)$ is strictly decreasing on}
\left\{
\begin{array}{lclcl}
[0,\infty) & \mbox{with} & w_0(\infty ;p)=0 & \mbox{if} & p\geq 1.\\{}
[0,\lambda (p)] & \mbox{with} & w_0(\lambda (p),p)=0 & \mbox{if} & 0 \left( \frac{1+p}{2} \right) \left( \frac{1-p}{2} \right) -
\frac{p+1}{2} w^p_0 \left\{ 1-w^{p+1}_0 \right\}^{-3/2} >0
$$
with $w_0\in (0,\delta)$ for some $\delta >0$. Thus
$$
F'_p(w_0) \mbox{ decreases monotonically as } w_0\downarrow 0.
$$
Then in view of (3.19), the only possibility is that
$$
\lim\limits_{w_{0}\downarrow 0} F'_p(w_0) =-\infty \ \ \mbox{ for } 0 w_2\}} \bigg[ \{(w_1-w_2)'\}^2 + \{g_1(w_1) -
$$
$$
g_1(w_2)\} (w_1-w_2) + \{g_1(w_2) - g_2(w_2) \} (w_1-w_2)\bigg] dz=0.
$$
Since all three terms are nonnegative on $\{w_1>w_2\}$, we must have
$$
\{w_1>w_2\} = \emptyset\ \ \ \mbox{ or }\ \ \ w_1 \leq w_2\ \ \ \mbox{ on } [0,1].
$$
\hfill$\Box$
To use this lemma, we define (see also (3.15))
$$
\Psi (v) := \beta \Phi _p (\alpha f(v)) \ \ \ (v\geq 0)\eqno(4.1)
$$
and write (1.1 $d-f$) as
$$
(P_2)\left\{
\begin{array}{lcr}
v'' = \Psi (v) & 0