Existence and multiplicity of positive solutions for discreteanisotropic equationsExistence and multiplicity of positive solutions for discreteanisotropic equations

Abstract

In this paper we consider the Dirichlet problem for a discrete anisotropic equation with some function α , a nonlinear term f , and a numerical parameter λ : ∆We derive the intervals of a numerical parameter λ for which the considered BVP has at least 1, exactly 1, or at least 2 positive solutions. Some useful discrete inequalities are also derived.

Multiplicity results for discrete anisotropic equations

“…If f is non-negative then the solutions ensured in Theorems 3.1, 3.2 and 3.3 is non-negative. Here we reason exactly as in [20] supposing the solution is not positive and arriving at contradiction. Now, we point out some results in which the function f has separated variables.…”

“…We easily observe from (18) that the condition ( 13) is satisfied. Moreover, by choosing δ small enough and γ =γ, one can drive the condition (10) from (17) as well as the conditions ( 12) and ( 16) from (20). Hence, the conclusion follows from Theorem 3.8.…”

“…Moreover, note that Mihailescu et al (see [33,34]) were the first authors to study anisotropic elliptic problems with variable exponent, and after that, in recent years, a great deal of work has been done in the study of the existence of solutions for discrete anisotropic boundary value problems (BVPs). For background and recent results, we refer the reader to [18,20,21,35,37,42] and the references therein.…”

On class of discrete boundary value problem via variational methods

“…Very recently, also the critical point theory has aroused the attention of many authors in the study of these problems. Far from being exhaustive, further details can be found in [1,4,6,7,[9][10][11] and the reference therein.…”

“…Now, we list some inequalities that will be are used later. For (a)-(c) see [6,9], for (d) and (e) see [6] and for (g) see [13].…”