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In this paper we prove mesh independent a priori $L^\infty$-bounds forpositive solutions of the finite difference boundary value problem $$ -\Delta_hu = f(x,u) \mbox{ in } \Omega_h, \quad u=0 \mbox{ on } \partial\Omega_h, $$where $\Delta_h$ is the finite difference Laplacian and $\Omega_h$ is adiscretized $n$-dimensional box. On one hand this completes a result of [10] onthe asympotic symmetry of solutions of finite difference boundary valueproblems. On the other hand it is a finite difference version of a criticalexponent 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 proofdiffer substantially in these two cases. In the 1-dimensional case our methodresembles ode-techniques. In the higher dimensional case the growth rate of thenonlinearity has to be bounded by an exponent $p<\frac{n}{n-1}$ where webelieve that $\frac{n}{n-1}$ plays the role of a critical exponent. Our methodin 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.

Mesh-independent a priori bounds for nonlinear elliptic finite difference boundary value problems