Mixed eigenvalues of p-Laplacian on trees

We consider the principal eigenvalue of discrete p-Laplacian ($p\geqslant2$) on the set of trees with unique root $o$. Alternatively, it is study the optimal constant of a class of weighted Hardy inequality. The main goal is estimating the eigenvalue (i.e., the optimal constant of the Hardy inequality) with Dirichlet boundary at unique root $o$. Three kinds of variational formulas for the eigenvalue are presented. As applications of these formulas, we obtain a criterion for positivity of the eigenvalue on a tree with finite vertexes.