Ambrosetti-Prodi type results for the discrete boundary value problems involving the singular $\phi$-Laplacian

Abstract:

      In this paper, we consider the discrete boundary value problem of the type $$\nabla(t_k^{N-1}\phi(\Delta u_k))+t_k^{N-1}f_k(t_k,u_k,\Delta u_k)=0,~(2\leq k\leq n-1),~~~\Delta u_1=0=\Delta u_{n-1},$$ where $\phi:(-a,a)\rightarrow\mathbb{R}$, $0