Infinitely many solutions of nonlinear discrete Dirichlet boundary value problems involving ϕ -Laplacian

Abstract

In this work, we employ critical point theory to investigate the existence of multiple solutions for a class of second-order partial difference equations subject to Dirichlet boundary conditions. A main novelty of this work lies in the consideration of a general ϕ-Laplacian operator, which covers the p-Laplacian and (p,q)-Laplacian operators as special cases. Through rigorous proofs, we separately derive conditions under which the problem admits an unbounded sequence of solutions and a sequence of pairwise distinct nontrivial solutions converging to zero. We further prove that the obtained nontrivial solutions are strictly positive in the interior of the domain by establishing a new strong maximum principle adapted to the present discrete ϕ-Laplacian setting. We also provide several illustrative examples to enhance comprehension of the theoretical findings.