The higher order quasilinear elliptic equation $-\Delta(\Delta_{p}(\Delta u))= f(x,u)$ subject to Dirichlet boundary
conditions may have unique and regular positive solution. If the
domain is a ball, we obtain a priori estimate to the radial
solutions via blowup. Extensions to systems and general domains
are also presented. The basic ingredients are the maximum
principle, Moser iterative scheme, an eigenvalue problem, a
priori estimates by rescalings, sub/supersolutions, and
Krasnosel′skiĭ fixed point theorem.