  In this paper we prove mesh independent a priori $L^\infty$-bounds for
positive solutions of the finite difference boundary value problem $$ -\Delta_h
u = f(x,u) \mbox{ in } \Omega_h, \quad u=0 \mbox{ on } \partial\Omega_h, $$
where $\Delta_h$ is the finite difference Laplacian and $\Omega_h$ is a
discretized $n$-dimensional box. On one hand this completes a result of [10] on
the asympotic symmetry of solutions of finite difference boundary value
problems. On the other hand it is a finite difference version of a critical
exponent problem studied in [11]. Two main results are given: one for dimension
$n=1$ and one for the higher dimensional case $n\geq 2$. The methods of proof
differ substantially in these two cases. In the 1-dimensional case our method
resembles ode-techniques. In the higher dimensional case the growth rate of the
nonlinearity has to be bounded by an exponent $p<\frac{n}{n-1}$ where we
believe that $\frac{n}{n-1}$ plays the role of a critical exponent. Our method
in this case is based on the use of the discrete Hardy-Sobolev inequality as in
[3] and on Moser's iteration method. We point out that our a priori bounds are
(in principal) explicit.
