Abstract and Applied Analysis
Volume 7 (2002), Issue 8, Pages 423-452

Positive solutions of higher order quasilinear elliptic equations

Marcelo Montenegro

Universidade Estadual de Campinas, IMECC, Departamento de Matemática, Caixa Postal 6065, CEP 130813-970, Campinas, SP, Brazil

Received 25 February 2002

Copyright © 2002 Marcelo Montenegro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The higher order quasilinear elliptic equation Δ(Δp(Δu))=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.