On the Multiplicity of Solutions for the Discrete Boundary Problem Involving the Singular ϕ-Laplacian

Abstract

In this paper, we consider the multiplicity of solutions for a discrete boundary value problem involving the singular ϕ-Laplacian. In order to apply the critical point theory, we extend the domain of the singular operator to the whole real numbers. Instead, we consider an auxiliary problem associated with the original one. We show that, if the nonlinear term oscillates suitably at the origin, there exists a sequence of pairwise distinct nontrivial solutions with the norms tend to zero. By our strong maximum principle, we show that all these solutions are positive under some assumptions. Moreover, the solutions of the auxiliary problem are solutions of the original one if the solutions are appropriately small. Lastly, we give an example to illustrate our main results.