Conditions necessary for a boundary layer to exist

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?

$$e^y$$ 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 $$y(0)=0$$.
• It can, for $a=0$ with $y=0$. But then you do not have a boundary layer, as there is no jump from the outer solution to the boundary condition. The more serious case is "no outer solution at all" for $a>1$. – LutzL Oct 24 '18 at 15:10
• does $x_0$ here represent the arbitrary location of the boundary layer? – John Simpleton Oct 24 '18 at 15:25
• Yes, it is the point that is tested for the viability of a boundary layer. Set $x=x_0+ϵX$ into the equation, $Y(X)=y(x)$, to get $ϵY''+ϵY'+ϵ^2e^Y=0$ and solve to get $Y=Ce^{-X}+D+O(ϵ)$. – LutzL Oct 24 '18 at 15:30