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

Abstract

In this paper we prove mesh independent a priori L∞-bounds for positive solutions of the finite difference boundary value problem−Δhu=f(x,u)in Ωh,u=0on ∂Ωh, where Δh is the finite difference Laplacian and Ωh is a discretized n-dimensional box. On the one hand this completes a result of [9] on the asymptotic 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 [10]. Two main results are given: one for dimension n=1 and one for the higher dimensional case n≥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<nn−1 where we believe that nn−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.