Abstract
In this paper, we establish uniform a priori estimates for positive solutions to higher critical order fractional Lame-Emden equations for all large exponents by adopting a new idea of using the combination of Green’s representations with suitable cut-off functions. We obtain two key ingredients needed in the estimates. One is the monotonicity of solutions in the inward normal direction near the boundary by using the method of moving planes in a local way, and the other is the integration by parts formula for higher order fractional operators. Both of them will become useful tools in the analysis of higher order fractional equations. It is well-known that, with such a priori estimate, one will be able to obtain the existence of solutions via topological degree or continuation arguments.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barlow, M., Bass, R., Chen, Z., Kassmann, M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Amer. Math. Soc. 361, 1963–1999 (2009)
Bertoin, J.: Lévy Processes, Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996)
Bogdan, K., Grzywny, T., Ryznar, M.: Heat kernel estimates for the fractional Laplacian with Dirichlet conditions. Ann. of Prob. 38, 1901–1923 (2010)
Bouchard, J., Georges, A.: Anomalous diffusion in disordered media: Statistical mechanics, models and physical applications. Phys. Rep. 195, 127–293 (1990)
Chen, Z., Kim, P., Kumagai, T.: Global heat kernel estimates for symmetric jump processes. Trans. Amer. Math. Soc. 363, 5021–5055 (2011)
Chen, W., Li, C.: A priori estimate for the Nirenberg problem. Discrete Contin. Dyn. Syst. Ser. S 1, 225–233 (2008)
Chen, W., Li, C., Li, Y.: A direct blowing-up and rescaling argument on nonlocal elliptic equations, Internat. J. Math. 27, 1650064, (2016) 20 pp
Chen, W., Li, C., Li, Y.: A direct method of moving planes for the fractional Laplacian. Adv. Math. 308, 404–437 (2017)
Chen, W., Li, Y., Ma, P.: The Fractional Laplacian. World Scientific Publishing Co, Singapore (2019)
Chen, W., Qi, S.: Direct methods on fractional equations. Disc. Cont. Dyna. Sys. 39, 1269–1310 (2019)
Chen, W., Dai, W., Qin, G.: Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy-Hénon equations in \({\mathbb{R}}^{n}\), (2018) arXiv 1808.06609v2
David, A.: Lévy Processes-From Probability to Finance and Quantum Groups, Notices of the American Mathematical Society (American Mathematical Society, 2014), pp. 1336-1347
Chen, W., Zhu, J.: Indefinite fractional elliptic problem and Liouville theorems. J. Differential Equations 260, 4758–4785 (2016)
Dai, W., Duyckaerts, T.: Uniform a priori estimates for positive solutions of higher order Lane-Emden equations in \({\mathbb{R}}^{n},\) arXiv 1905.10462v1 (2019)
Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order. Springer-Verlag, Berlin (1977)
Han, Z-C.: Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Annales de l’Institut Henri Poincare (C) Non Linear Analysis. Vol. 8. No. 2. (1991)
Jarohs, S., Weth, Tobias: Symmetry via antisymmetric maximum principles in nonlocal problems of variable order. Annali di Mat. Pura Appl. 195, 273–291 (2016)
Kamburov, N., Sirakov, B.: Uniform a priori estimates for positive solutions of the Lane-Emden equation in the plane, Calc. Var. Partial Differential Equations 57, Art. 164, 8 pp (2018)
Ken-Iti, S.: Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2011)
Lei, Y., Lv, Z.: Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality, Disc. Cont. Dyn, Sys. 33, 1987–2005 (2013)
Lu, G., Zhu, J.: The axial symmetry and regularity of solutions to an integral equation in a half space. Pacific J. Math. 253, 455–473 (2011)
Lu, G., Zhu, J.: Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality. Cal. Var. & PDEs 42, 563–577 (2011)
Lu, G., Zhu, J.: An overdetermined problem in Riesz-potential and fractional Laplacian. Nonlinear Analysis 75, 3036–3048 (2012)
Zaslavsky, G.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, England (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. H. Lin.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Leyun Wu is partially supported by NSFC-12031012 and Hong Kong Scholars Program-XJ2020035
Wenxiong Chen: partially supported by NSFC 12071229.
Rights and permissions
About this article
Cite this article
Chen, W., Wu, L. Uniform a priori estimates for solutions of higher critical order fractional equations. Calc. Var. 60, 102 (2021). https://doi.org/10.1007/s00526-021-01968-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-021-01968-w