Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advanced Nonlinear Studies

Editor-in-Chief: Ahmad, Shair


IMPACT FACTOR 2017: 1.029
5-year IMPACT FACTOR: 1.147

CiteScore 2017: 1.29

SCImago Journal Rank (SJR) 2017: 1.588
Source Normalized Impact per Paper (SNIP) 2017: 0.971

Mathematical Citation Quotient (MCQ) 2017: 1.03

Online
ISSN
2169-0375
See all formats and pricing
More options …
Volume 16, Issue 3

Issues

Weighted Fractional Sobolev Inequality in ℝN

Xiaoli Chen / Jianfu Yang
  • Corresponding author
  • Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, P. R. China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-03-12 | DOI: https://doi.org/10.1515/ans-2015-5002

Abstract

In this paper, we show that the minimizing problem

Λs,N,k,α=infuH˙s(N),u0N|(-Δ)s2u(x)|2𝑑x(N|u(x)|2s,α*|y|α𝑑x)22s,α*

is achieved by a positive, cylindrically symmetric and strictly decreasing function u(x) provided 0<s<N2, 0<α<2s, where x=(y,z)k×N-k and 2s,α*=2(N-α)N-2s. Decaying laws for the minimizer u are also established.

Keywords: Sobolev–Hardy Inequality; Minimizer; Cylindrical Symmetry; Decaying Law

MSC 2010: 35A15; 35J20; 35J61; 36J75

References

  • [1]

    Abramowitz M. and Stegun I., Handbook of Mathematical Function with Formula, Graphs, and Mathematical Tables, Dover, New York, 1992. Google Scholar

  • [2]

    Aubin T., Problèmes isopérimétrique et espaces de Sobolev, J. Differential Geom. 11 (1976), 573–598. Google Scholar

  • [3]

    Badiale M. and Tarantello G., A Sobolev–Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal. 163 (2002), 259–293. Google Scholar

  • [4]

    Bertin G., Dynamics of Galaxies, Cambridge University Press, Cambridge, 2000. Google Scholar

  • [5]

    Brézis H. and Lieb E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490. Google Scholar

  • [6]

    Brézis H. and Nirenberg L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477. Google Scholar

  • [7]

    Caffarelli L. and Silvestre L., An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260. Google Scholar

  • [8]

    Chen W., Li C. and Ou B., Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), 330–343. Google Scholar

  • [9]

    Ciotti L., Dynamical Models in Astrophysics, Scuola Normale Superiore, Pisa, 2001. Google Scholar

  • [10]

    Cotsiolis A. and Travoularis N. K., Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl. 295 (2004), 225–236. Google Scholar

  • [11]

    Di Nezza E., Palatucci G. and Valdinoci E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 229 (2012), 521–573. Google Scholar

  • [12]

    Fabes E., Kenig C. and Serapioni R., The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), 77–116. Google Scholar

  • [13]

    Frank R. L. and Seiringer R., Nonlinear ground state representations and sharp Hardy inequalities, J. Funct. Anal. 255 (2008), 3407–3430. Google Scholar

  • [14]

    Han Q. and Lin F., Elliptic Partial Differential Equations, American Mathematical Society, Providence, 2000. Google Scholar

  • [15]

    Lieb E. H. and Loss M., Analysis, Grad. Stud. Math. 14, American Mathematical Society, Providence, 2001. Google Scholar

  • [16]

    Lions P. L., The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoam. 1 (1985), no. 1, 145–201. Google Scholar

  • [17]

    Lions P. L., The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoam. 1 (1985), no. 2, 45–121. Google Scholar

  • [18]

    Mancini G., Fabbri I. and Sandeep K., Classification of solutions of a critical Hardy–Sobolev operator, J. Differential Equations, 224 (2006), 258–276. Google Scholar

  • [19]

    Mancini G. and Sandeep K., Cylindrical symmetry of extremals of a Hardy–Sobolev inequality, Ann. Mat. Pura Appl. (4) 183 (2004), 165–172. Google Scholar

  • [20]

    Palatucci G. and Pisante A., Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations 50 (2014), 799–829. Google Scholar

  • [21]

    Swayer E. and Wheeden R. L., Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813–874. Google Scholar

  • [22]

    Talenti G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. Google Scholar

  • [23]

    Tan J. and Xiong J., A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst. 31 (2011), 975–983. Google Scholar

  • [24]

    Yafaev D., Sharp constants in the Hardy–Rellich inequalities, J. Funct. Anal. 168 (1999), 121–144. Google Scholar

  • [25]

    Yang J., Fractional Sobolev–Hardy inequality in N, Nonlinear Anal. 119 (2015), 179–185. Google Scholar

About the article

Received: 2015-05-27

Revised: 2015-09-20

Accepted: 2015-09-27

Published Online: 2016-03-12

Published in Print: 2016-08-01


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11461033

Award identifier / Grant number: 11401269

Award identifier / Grant number: 11271170

Award identifier / Grant number: 11361029

X. Chen is supported by the NNSF of China (no. 11461033 and 11401269) and the NNSF of Jangxi Province (no. 20142BAB201003). J. Yang is supported by the NNSF of China (no. 11271170 and 11361029) and the GAN PO 555 program of Jiangxi.


Citation Information: Advanced Nonlinear Studies, Volume 16, Issue 3, Pages 623–641, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2015-5002.

Export Citation

© 2016 by De Gruyter.Get Permission

Comments (0)

Please log in or register to comment.
Log in