Discrete Elliptic Dirichlet Problems and Nonlinear Algebraic Systems

Abstract

In this paper we are interested in the existence of infinitely many solutions for a partial discrete Dirichlet problem depending on a real parameter. More precisely, we determine unbounded intervals of parameters such that the treated problems admit either an unbounded sequence of solutions, provided that the nonlinearity has a suitable behaviour at infinity, or a pairwise distinct sequence of solutions that strongly converges to zero if a similar behaviour occurs at zero. Finally, the attained solutions are positive when the nonlinearity is supposed to be nonnegative thanks to a discrete maximum principle.