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Reichel","submitted_at":"2014-04-09T07:53:05Z","abstract_excerpt":"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]. 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