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) , 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.
Cite this article
Guozhen Lu, Peiyong Wang, Jiuyi Zhu, Liouville-type theorems and decay estimates for solutions to higher order elliptic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 5, pp. 653–665DOI 10.1016/J.ANIHPC.2012.02.004