The geometry of finite difference discretizations of semilinear elliptic operators (Q2882661)

This paper is concerned with the study of various discretizations by finite differences of some semilinear elliptic equations described by nonlinear convex diagonal perturbations of symmetric matrices. The main result is described in the following. Let \(A\) be a symmetric matrix and let \(a\) and \(b\) satisfy one of the following assumptions:NEWLINENEWLINE(i) \( a < 0,\;b > 0\), where \(|a|\) and \(b\) are sufficiently large;NEWLINENEWLINE(ii) \(A\) is an irreducible Stieltjes matrix, \(a < \lambda_1\) and \(b > 0\) is sufficiently large.NEWLINENEWLINEThen any \(a\ell\)-admissible map \(F(u) = Au-f(u)\) with asymptotic parameters \(a\) and \(b\) satisfies the following property: for any \(y, p \in {\mathbb R}^n\), there exists \(t_0\) so that, for \(t > t_0\), \(F(u) =y- tp\) has exactly one solution in each orthant.