Conditions necessary for a boundary layer to exist

Question

Determine values of $a$ for which the problem:

$\epsilon y^{''} + y^{'}+ae^y=0,$ $ y(0)=y(1)=0$

has a solution with a boundary layer structure.

I am familiar with the procedure for tackling this problem having already known there is a boundary layer at either of the endpoints. My thought process is that I can assume a boundary layer at some point within $(0,1)$ and then apply the matching conditions to the inner and outer solutions. Then hopefully that would yield some constraints for the paramater $a$.

Is this the right approach?

Answer 1

$e^y$ never goes close enough to infinity to influence the balancing, so you always get $e^{-(x-x_0)/ϵ}$ as inner solution. It is only finite to the right, $x>x_0$, which pushes the boundary layer to the left boundary of the interval.

So you only have to care that the outer solution $y=-\ln(1-a+ax)$ exists on the whole interval and does not have the value $y(0)=0$.