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 $$ \left\{ \begin{array}{l} -\Delta^2u({k-1})=\alpha f(k,u(k)),\quad \forall\; k \in {\mathbb{Z}}[1,T] \\ {u(0)=u({T+1})=0.}\\ \end{array} \right. $$ Our approach is based on variational methods in finite-dimensional setting. 
Giovanni Molica Bisci and Du\v{s}an Repov\v{s}
math.AP