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

Abstract

In this paper we prove mesh independent a priori $L^{\infty }$-bounds for positive solutions of the finite difference boundary value problem $$ -Δ_h u = f(x,u) \mbox{ in } Ω_h, \quad u=0 \mbox{ on } \partialΩ_h, $$ 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 $n\ge 2$. 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.