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

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