Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Lu, Guozhen; Wang, Peiyong; Zhu, Jiuyi

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 5, pp. 653-665.

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma developed recently in Poláčik et al. (2007) [20], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations and elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouville-type theorems in the recent literature. Moreover, we also investigate the singularity and decay estimates of higher order elliptic equations.

Classification : 35B53, 35J40, 35J47, 35B45
Mots clés : Higher order elliptic equations, Polyharmonic operators on half spaces, Dirichlet problem, Navier boundary condition, Liouville-type theorem, Decay estimates for solutions, Doubling property, Without boundedness assumptions

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