Taiwanese Journal of Mathematics

Nonlocal Elliptic Systems Involving Critical Sobolev-Hardy Exponents and Concave-convex Nonlinearities

Jinguo Zhang and Tsing-San Hsu

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Open access

Abstract

In this paper, a system of fractional elliptic equation is investigated, which involving fractional critical Sobolev-Hardy exponent and concave-convex terms. By means of variational methods and analytic techniques, the existence and multiplicity of positive solutions to the system is established.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 32 pages.

Dates
First available in Project Euclid: 26 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1551150033

Digital Object Identifier
doi:10.11650/tjm/190109

Subjects
Primary: 35J50: Variational methods for elliptic systems 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx] 35B65: Smoothness and regularity of solutions

Keywords
fractional Laplacian Hardy potential multiple positive solutions fractional critical Sobolev-Hardy exponent concave-convex nonlinearities

Citation

Zhang, Jinguo; Hsu, Tsing-San. Nonlocal Elliptic Systems Involving Critical Sobolev-Hardy Exponents and Concave-convex Nonlinearities. Taiwanese J. Math., advance publication, 26 February 2019. doi:10.11650/tjm/190109. https://projecteuclid.org/euclid.twjm/1551150033


Export citation

References

  • C. O. Alves, D. C. de Morais Filho and M. A. S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal. 42 (2000), no. 5, 771–787.
  • B. Barrios, M. Medina and I. Peral, Some remarks on the solvability of non-local elliptic problems with the Hardy potential, Commun. Contemp. Math. 16 (2014), no. 4, 1350046, 29 pp.
  • G. M. Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications 162, Cambridge University Press, Cambridge, 2016.
  • D. Cao and P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations 205 (2004), no. 2, 521–537.
  • Y. Cao and D. Kang, Solutions of a quasilinear elliptic problem involving a critical Sobolev exponent and multiple Hardy-type terms, J. Math. Anal. Appl. 333 (2007), no. 2, 889–903.
  • A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl. 295 (2004), no. 1, 225–236.
  • E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573.
  • M. M. Fall, Semilinear elliptic equations for the fractional Laplacian with Hardy potential, Nonlinear Analysis, (In Press), https://doi.org/10.1016/j.na.2018.07.008.
  • M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations 39 (2014), no. 2, 354–397.
  • V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations 31 (2006), no. 1-3, 469–495.
  • R. Filippucci, P. Pucci and F. Robert, On a $p$-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl. (9) 91 (2009), no. 2, 156–177.
  • R. L. Frank, E. H. Lieb and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc. 21 (2008), no. 4, 925–950.
  • N. Ghoussoub, F. Robert, S. Shakerian and M. Zhao, Mass and asymptotics associated to fractional Hardy-Schrödinger operators in critical regimes, Comm. Partial Differential Equations 43 (2018), no. 6, 859–892.
  • N. Ghoussoub and S. Shakerian, Borderline variational problems involving fractional Laplacians and critical singularities, Adv. Nonlinear Stud. 15 (2015), no. 3, 527–555.
  • I. W. Herbst, Spectral theory of the operator $(p^2+m^2)^{1/2} - Ze^2/r$, Comm. Math. Phys. 53 (1977), no. 3, 285–294.
  • T.-S. Hsu, Multiple positive solutions for semilinear elliptic equations involving Hardy terms and critical Sobolev-Hardy exponents, J. Math. Sci. Adv. Appl. 3 (2009), no. 2, 243–266.
  • ––––, Multiplicity results for $p$-Laplacian with critical nonlinearity of concave-convex type and sign-changing weight functions, Abstr. Appl. Anal. 2009 (2009), Art. ID 652109, 24 pp.
  • ––––, Multiple positive solutions for a quasilinear elliptic problem involving critical Sobolev-Hardy exponents and concave-convex nonlinearities, Nonlinear Anal. 74 (2011), no. 12, 3934–3944.
  • T.-S. Hsu and H.-L. Lin, Multiple positive solutions for singular elliptic equations with weighted Hardy terms and critical Sobolev-Hardy exponents, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), no. 3, 617–633.
  • T.-S. Hsu, H.-L. Lin and C.-C. Hu, Multiple positive solutions of quasilinear elliptic equations in $\mathbb{R}^{N}$, J. Math. Anal. Appl. 388 (2012), no. 1, 500–512.
  • D. Kang, On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms, Nonlinear Anal. 68 (2008), no. 7, 1973–1985.
  • R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), no. 2, 887–898.
  • ––––, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015), no. 1, 67–102.
  • S. Shakerian, Multiple positive solutions for Nonlocal elliptic problems involving the Hardy potential and concave-convex nonlinearities, arXiv:1708.01369v1.
  • L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), no. 1, 67–112.
  • Y. Song and S. Shi, On a degenerate $p$-fractional Kirchhoff equations involving critical Sobolev-Hardy nonlinearities, Mediterr. J. Math. 15 (2018), no. 1, Art. 17, 18 pp.
  • S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations 1 (1996), no. 2, 241–264.
  • C. Wang, J. Yang and J. Zhou, Solutions for a nonlocal elliptic equation involving critical growth and Hardy potential, arXiv:1509.07322v1.
  • J. Yang, Fractional Sobolev-Hardy inequality in $\mathbb{R}^{N}$, Nonlinear Anal. 119 (2015), 179–185.
  • J. Yang and X. Yu, Fractional Hardy-Sobolev elliptic problems, arXiv:1503.00216.
  • J. Zhang, X. Liu and H. Jiao, Multiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity, Topological Methods in Nonlinear Analysis, (In Press).