# How to argue this “standard elliptic estimates” for a convergence?

I am estudying the paper: INEQUALITIES FOR SECOND-ORDER ELLIPTIC EQUATIONS WITH APPLICATIONS TO UNBOUNDED DOMAINS I - H. BERESTYCKI, L. A. CAFFARELLI, AND L. NIRENBERG

At the page $482$, we have a function $$z^j(x,y)=\frac{1}{\alpha^j}u(x,y),$$ with $\alpha^j\Rightarrow\infty$ and $$\Delta z^j+\frac{1}{\alpha^j}g(\alpha^j z^j)+\frac{1}{\alpha^j}f(0)=0, \ \ \ \mbox{in} \ \ (-2,2)\times\overline\omega$$ The function $z^j$ satisfies $$z^j\leq a, \ \ \ \mbox{in} \ \ (-2,2)\times\overline\omega,$$ where $\omega$ is a bounded domain. Then, by "standad elliptic estimates", $z^j$ converge uniformly in $[-1,1]\times\overline\omega$ to a function $\overline z$, which belongs to $C^{2,\gamma}$ there, for some $\gamma\in(0,1)$, and $\overline z$ satisfies: $$\begin{array}{rl} \Delta\overline z+\overline g(\overline z)=0, & (-1,1)\times\omega,\\ \overline z=0, & (-1,1)\times\partial\omega. \end{array}$$ This page can be viewed in: http://books.google.com.br/books?id=iCH9EGj3YkEC&pg=PA468&lpg=PA467&ots=rRKiPZbFhr&dq=INEQUALITIES+FOR+SECOND-ORDER+ELLIPTIC+EQUATIONS+WITH+APPLICATIONS+TO+UNBOUNDED+DOMAINS+I&lr=&hl=pt-PT

I don't know to explain this convergence. Someone can help me? Thank You!

-
You should specifiy what all the terms are, which ones are functions, which are variables, etc. –  Ray Yang Feb 11 '13 at 13:36