Nonexistence and existence results for a 2\textit{n}th-order discrete Dirichlet boundary value problem (Q740771)

The authors consider a second-order nonlinear difference equation of the form \[ \Delta^n( \gamma_{i-n+1}\Delta^n u_{i-n} ) = (-1)^n f(i,u_{i+1},u_i,u_{i-1}), \quad n\in \mathbb{Z}(1),\, i\in \mathbb{Z}(1,k), \] with boundary conditions \[ u_{1-n}=u_{2-n}= \dots= u_0 =0, \quad u_{k+1}=u_{k+2}=\dots=u_{k+n}=0, \] where \( \Delta \) is the forward difference operator, \( \gamma_i \) is nonzero and real-valued for each \( i\in \mathbb{Z}(2-n,k+1)\), \(f\in C(\mathbb{R}^4, \mathbb{R} ) \). Using the critical point theory they give some sufficient conditions for nonexistence and existence of the solutions of the boundary value problem (BVP), by using the transfer of existence of the solutions of the BVP to the existence of the critical points of some functional. The proof of the results is based on the mountain pass lemma in combination with a variational technique. The authors provide a large number of references related with the subject and interesting numerical examples which are illustrating the support of the introduced conditions.