Consider the half space $\Omega=\{x=(x_1,\ldots,x_N)\in\mathbb{R}^N:x_N>0\}$. Let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ a positive bounded solution of $$ \begin{eqnarray*} \Delta u+f(x_N,u)=0, & \mbox{in} & \Omega\\ u=0, & \mbox{on} & \partial\Omega \end{eqnarray*} $$ Here $f:[0,\infty]\times[0,M]\rightarrow\mathbb{R}$ is continuous, for any t-finite interval $f$ is Lipschitz in $u$ on $[0,M]$, $M=\sup_{\Omega}u$, and $(y^{j})_{j}\subset\Omega$ such that $u(y^{j})\rightarrow M$. In $|x|<\delta$, define $u^{j}(x)=u(y^{j}+x)$.
How can I prove by elliptic estimates up to the boundary that there exists a subsequence of the $u^{j}$ such that the $u^{j}$ converge uniformly in $|x|<\delta/2$ to a function $u_{0}$?