Positive solutions of higher order quasilinear elliptic equations. (English) Zbl 1044.35021

Summary: 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 blow-up. 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.


35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J60 Nonlinear elliptic equations
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