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Liouville-type theorems for fully nonlinear elliptic equations and systems in half spaces. (English) Zbl 1298.35021
Summary: The main purpose of this paper is to establish Liouville-type theorems and decay estimates for viscosity solutions to a class of fully nonlinear elliptic equations or systems in half spaces without the boundedness assumptions on the solutions. Using the blow-up method and doubling lemma of [P. Poláčik et al., Duke Math. J. 139, No. 3, 555–579 (2007; Zbl 1146.35038)], we remove the boundedness assumption on solutions which was often required in the proof of Liouville-type theorems in the literature.
We also prove the Liouville-type theorems for supersolutions of a system of fully nonlinear equations with Pucci extremal operators in half spaces. Liouville theorems and decay estimates for high order elliptic equations and systems have also been established by the authors and P. Wang in an earlier work [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 29, No. 5, 653–665 (2012; Zbl 1255.35064)] when no boundedness assumption was given on the solutions.

35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35J60 Nonlinear elliptic equations
35B44 Blow-up in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35D40 Viscosity solutions to PDEs
Full Text: arXiv