Qualitative and spectral analysis of a partial discrete Dirichlet problem

Abstract

In this article, we study a partial discrete Dirichlet boundary value problem together with its associated eigenvalue problem. More precisely, in the first part, we identify two eigenvalues \(\lambda _{1}(m)\) and \(\lambda ^{*}(m)\) of the associated eigenvalue problem and show that there is no eigenvalue between them. In the second part, we determine intervals of the parameter \(\lambda \) ensuring the existence of solutions, and we investigate the sign of these solutions in relation to \(\lambda _{1}(m)\) and \(\lambda ^{*}(m)\).