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

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.