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 $\alpha$, whose prototype is given by $$\{\begin{array}{l}-{\mathrm{\Delta }}^{2}u(k-1)=\alpha f(k,u(k)),\mathrm{\forall }k\in \mathbb{Z}[1,T]\\ u(0)=u(T+1)=0.\end{array}$$ Our approach is based on variational methods in finite-dimensional setting.