Entire solutions to a quasilinear purely critical competitive system

We establish the existence of a fully nontrivial solution with nonnegative components for a weakly coupled competitive system for the $p$-Laplacian in ${\mathbb{R}}^N$ whose nonlinear terms are purely critical. We also show that the purely critical equation for the $p$-Laplacian in ${\mathbb{R}}^N$ has infinitely many nodal solutions.