Existence and nonexistence results for a fourth-order discrete Neumann boundary value problem (Q2875470)

The authors consider the discrete boundary value problem (BVP) NEWLINENEWLINE\[NEWLINE\begin{aligned} &\Delta^2(p_{n-1}\Delta^2 u_{n-2})-\Delta(q_n\Delta u_{n-1})=f(n,u_{n+1},u_n,u_{n+1}),\quad n\in\{1,2,\dots,k\},\\ NEWLINE&\Delta u_{-1}=\Delta u_0=0,\quad \Delta u_k=\Delta u_{k+1}=0,\end{aligned}NEWLINE\]NEWLINE NEWLINEwhere \(p_n\neq 0\), \(q_n\) are real and \(f\in C(\mathbb R^4,\mathbb R)\). The critical point theory is used to establish sufficient conditions for the existence and nonexistence of solutions to the above BVP. The superlinear case is studied. Two examples are given.