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 − Δ h u = f ( x , u ) in Ω h , u = 0 on ∂ Ω h , where Δ h is the finite difference Laplacian and Ω h 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 < n n − 1 where we believe that 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 Moserʹs iteration method. We point out that our a priori bounds are (in principal) explicit.