Abstract
In this paper, we obtain estimates for solutions for a class of fractional order elliptic equations in different domains and boundary conditions, and prove some regularity results. Then, we study the qualitative properties of solutions with prescribed Q-curvature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Brezis, F. Merle, “Uniform estimates and blow-up behavior for solutions of −Δu = V(x)eu in two dimensions”, Comm. P. D.E., 16, 1223–1253, 1991.
W. X. Chen, C. M. Li, “Classification of solutions some nonlinear elliptic equation”, Duke Math. J., 63, 615–622, 1991.
W. X. Chen, C. M. Li, “Qualitative properties of solutions to some nonlinear elliptic equations in R2”, Duke Math. J., 71, 427–439, 1993.
X. Cabré, J. Tan, “Positive solutions of nonlinear problems involving the square root of the Laplacian”, Adv. Math., 224, 2052–2093, 2010.
A. Hyder, “Structre of conformal metrics on Rn with constant Q−curvature”, arXiv:1504.07095, 2015.
T. Jin, A. Maalaoui, L. Martinazzi, J. Xiong, “Existence and asymptotics for solutions of nonlocal Q-curvature equation in dimension three”, Calc.Var. Partial Differential Equations, 52 (3-4), 469–488, 2015.
C. S. Lin, “A classification of solutions of a conformally invariant fourth order equation in Rn”, Comment. Math. Helv., 73, 206–231, 1998.
A. Adimurthi, F. Robert, M. Struwe, “Concentration phenomena for Liouville’s equation in dimension four”, Journal EMS, 8, 171–180, 2006.
F. Robert, J. C. Wei, “Asymptotic behavior of a fourth order mean field equation with Dirichlet boundary condition”, Indiana Univ.Math. J., 57, 2039–2060, 2008.
L. Martinazzi, “Classification of solutions to the higher order Liouville’s equation on R2m,” Math. Z., 263, 307–329, 2009.
L. Martinazzi, “Concentration-compactness phenomena in higher order Liouville’s equation”, J. Funct. Anal., 256, 3743–3771, 2009.
L. Martinazzi, M. Petrache, “Asymptotics and quantization for a mean field equation of higher order”, Comm. Partial Differential Equations, 35, 1–22, 2010.
L. Silvestre, “Regularity of the obstacle problem for fractional power of the Laplace operator”, Comm. Pure Appl.Math., 60, 67–112, 2007.
J. C. Wei, X. W. Xu, “Classification of solutions of higher order conformally invariant equations”, Math. Ann., 313, 207–228. 1999.
J. C. Wei, X. W. Xu, “Prescribing Q-curvature problem on Sn”, J. Funct. Anal., 257, 1995–2023, 2009.
J. C. Wei, “Asymptotic behavior of nonlinear 4 order eigenvalue problem”, Comm. PDE 21, 1451–1467. 1996.
X. W. Xu, “Classification of solutions of certain fourth-order nonlinear elliptic equation in R4”, Pacific J. Math., 225, 361–378, 2006.
X. W. Xu, “Uniqueness and non-existence theorems for conformally invariant equations”, J. Funct. Anal., 222, 1–28, 2005.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © R. Pei, C. Ma, 2017, published in Izvestiya Natsional’noi Akademii Nauk Armenii, Matematika, 2017, No. 5, pp. 67–75.
This study was supported by the the National NSF (Grant No. 11661070) and the National NSF (Grant Nos. 11361054, 11561059) of China, Natural Science Foundation of Gansu Province China (Grant No. 1506RJZE114) and Planned Projects for Postdoctoral Research Funds of Jiangsu Province (Grant No.1301038C).
About this article
Cite this article
Pei, R., Ma, C. Priori estimates and asymptotic properties of solutions for some fractional order elliptic equations. J. Contemp. Mathemat. Anal. 52, 221–226 (2017). https://doi.org/10.3103/S1068362317050028
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1068362317050028