Existence of solutions for $p$-Laplacian discrete equations

This work is devoted to the study of the existence of at least one (non-zero) solution to a problem involving the discrete $p$-Laplacian. As a special case, we derive an existence theorem for a second-order discrete problem, depending on a positive real parameter $α$, whose prototype is given by $\left\{ \begin{array}{l} -Δ^2u({k-1})=αf(k,u(k)),\quad \forall\; k \in {\mathbb{Z}}[1,T] \\ {u(0)=u({T+1})=0.}\\ \end{array} \right.$ Our approach is based on variational methods in finite-dimensional setting.

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