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.

Keywords: 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

@article{AIHPC_2012__29_5_653_0, author = {Lu, Guozhen and Wang, Peiyong and Zhu, Jiuyi}, title = {Liouville-type theorems and decay estimates for solutions to higher order elliptic equations}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {653--665}, publisher = {Elsevier}, volume = {29}, number = {5}, year = {2012}, doi = {10.1016/j.anihpc.2012.02.004}, zbl = {1255.35064}, mrnumber = {2971025}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.02.004/} }

TY - JOUR AU - Lu, Guozhen AU - Wang, Peiyong AU - Zhu, Jiuyi TI - Liouville-type theorems and decay estimates for solutions to higher order elliptic equations JO - Annales de l'I.H.P. Analyse non linéaire PY - 2012 DA - 2012/// SP - 653 EP - 665 VL - 29 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2012.02.004/ UR - https://zbmath.org/?q=an%3A1255.35064 UR - https://www.ams.org/mathscinet-getitem?mr=2971025 UR - https://doi.org/10.1016/j.anihpc.2012.02.004 DO - 10.1016/j.anihpc.2012.02.004 LA - en ID - AIHPC_2012__29_5_653_0 ER -

%0 Journal Article %A Lu, Guozhen %A Wang, Peiyong %A Zhu, Jiuyi %T Liouville-type theorems and decay estimates for solutions to higher order elliptic equations %J Annales de l'I.H.P. Analyse non linéaire %D 2012 %P 653-665 %V 29 %N 5 %I Elsevier %U https://doi.org/10.1016/j.anihpc.2012.02.004 %R 10.1016/j.anihpc.2012.02.004 %G en %F AIHPC_2012__29_5_653_0

Lu, Guozhen; Wang, Peiyong; Zhu, Jiuyi. Liouville-type theorems and decay estimates for solutions to higher order elliptic equations. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 5, pp. 653-665. doi : 10.1016/j.anihpc.2012.02.004. http://www.numdam.org/articles/10.1016/j.anihpc.2012.02.004/

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