Positive solutions of higher order quasilinear elliptic equations

Abstract

The higher order quasilinear elliptic equation $-\Delta \left({\Delta }_p\left(\Delta u\right)\right)=f\left(x,u\right)$ 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.