Multiplicity results for discrete anisotropic equations

Abstract

In this article we continue the study of discrete anisotropic equations and we will provide a new multiplicity results of the solutions for a discrete anisotropic equation. The procedure viewed here is according to variational methods and critical point theory. In fact, using a consequence of the local minimum theorem due Bonanno and mountain pass theorem we look into the existence results for our problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by mingling two algebraic conditions on the nonlinear term employing two consequences of the local minimum theorem due Bonanno we guarantee the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for our problem.

An Existence Result for Discrete Anisotropic Equations

“…Stegliński in [25], obtained the existence of infinitely many solutions for a parametric version the problem (1.1). In [8], a parametric version of the problem (1.1) was studied and a new multiplicity results have been established combining, first, the Bonanno local minimum theorem with the mountain pass theorem, next, a two local minimum theorem of Bonanno again with the well known three critical point theorem of Pucci and Serrin, see also [4], for a complete overview on these topics. In [13] the existence of at least three distinct solutions for a perturbed anisotropic discrete Dirichlet problem was studied.…”