I have a question that is related to finding some boundary layer estimates. I am sure there is a general method for this but I don't know it.
I will pose the problem in two dimensions (but here there is really no problem since one doesn't need this approach). Let's assume we have a bunch of smooth solutions of $$-\Delta u^m(x,y) + u^m = (u^m)^p $$ in $B_1^+$ (the upper half ball) and we suppose that $u^m_y(x,y) \le 0$ in the domain or at least near the bottom boundary. Lets assume that $ t_m = u^m(0,0) = \max u^m \rightarrow \infty$ (so here since the dimension is $N=2$ we can directly rule this out… but let's assume we can't… or pose this same problem in higher dimensions). I want to rescale the problem but I only want to rescale in $y$ (if I rescale both I get a Laplacian and then i am capable of doing stuff). So consider $$v^m(x,y)= \frac{u^m(x, \epsilon_m y)}{t_m}.$$ Then $v^m$ solves $$-v^m_{yy} - \epsilon_m^2 v^m_{xx} + \epsilon_m^2 v^m = (\epsilon_m^2 t_m^{p-1}) (v^m)^p$$ for some domain.
So we take $\epsilon_m$ such that the term in the brackets on the right is one. Now if we could show that $ \epsilon_m^2 v^m_{xx} \rightarrow 0$ then we should have our limiting equation just depending on $y$ (or at least its an ode in y with a parameter $x$). So we'd roughly get $ -v_{yy}(y)= v^p$ for $ y>0$ with $ v(0)=1$ and $ v \le 1$ and then we can proceed to get a contradiction.
So my question is: how does one normally get enough estimates on this $u_{xx}^m $ term to show we can take it going to zero?
In some sense this convergence could be quite weak. For instance let's assume that we knew $ v^m \rightarrow v$ uniformly and hence this is sufficient that we have $ v(0,0)=1$ then we can test the pde on some test functions and put all the derivatives of $v^m_{xx}$ on the test function and that would allow me to do what I want.
I do see I can get $H^1$ estimates on $v^m$ with epsilon terms; but I am not sure this will be sufficient to get the uniform convergence I want.
