Skip to main content
Log in

A critical fractional elliptic equation with singular nonlinearities

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

Kingdom of Saudi Arabia

We use variational methods, in order to show the existence of multiple positive solutions to the problem (Pλ) for different values of λ.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convexe nonlinearities in some elliptic problems. J. Funct. Anal. 122 (1994), 519–543.

    Article  MathSciNet  Google Scholar 

  2. A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349–381.

    Article  MathSciNet  Google Scholar 

  3. D. Applebaum, Levy Processes and Stochastic Calculus, 2nd ed.. Ser. Cambridge Studies in Adv. Math. 116 Cambridge University Press, Cambridge (2009).

  4. D. Averna, S. Tersian, E. Tornatore, on the existence and multiplicity of solutions for Dirichlet’s problem for fractional equations. Fract. Calc. Appl. Anal. 19, No 1 (2016), 253–266; 10.1515/fca-2016-0014. https://www.degruyter.com/view7j/fca.2016.19.issue-1/issue-files/fca.2016.19.issue-1.xml

    Article  MathSciNet  Google Scholar 

  5. B. Barrios, I. De Bonis, M. Maria, I. Peral, Semilinear problems for the fractional laplacian with a singular nonlinearity. Open Math. 13 (2015), 390–407.

    Article  MathSciNet  Google Scholar 

  6. B. Barriosa, E. Coloradoc, R. Servadeid, F. Soria, A critical fractional equation with concave-convex power nonlinearities. Ann. I. H. Poincare. 32 (2015), 875–900.

    Article  MathSciNet  Google Scholar 

  7. G. Bonanno R., S. Rodrlguez-Lopez, Tersian, Existence of solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17, No 3 (2014), 717–744; 10.2478/s13540-014-0196-y. https://www.degruyter.com/view/j/fca.2014.17.issue-3/issue-files/fca.2014.17.issue-3.xml

    Article  MathSciNet  Google Scholar 

  8. H. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functinals. Proc. Amer. Math. Soc. 88, No 3 (1983), 486–490.

    Article  MathSciNet  Google Scholar 

  9. M.M. Coclite, G. Palmieri, On a singular nonlinear Dirichlet problem. Comm. Partial Differential Equations. 14, No 10 (1989), 1315–1327.

    Article  MathSciNet  Google Scholar 

  10. M.G. Crandall, P.H. Rabinowitz, L. Tartar, On a Dirichlet problem with a singular nonlinearity. Comm. Partial Differential Equations. 2, No 2 (1977), 193–222.

    Article  MathSciNet  Google Scholar 

  11. A. Cotsiolis, N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295 (2004), 225–236.

    Article  MathSciNet  Google Scholar 

  12. R. Dhanya, J. Giacomoni, S. Prashanth, K. Saoudi, Global bifurcation and local multiplicity results for elliptic equations with singular nonlinearity of super exponential growth in ℝ2. Advances in Differential Equations. 17, No 3–4 (2012), 369–400.

    MathSciNet  MATH  Google Scholar 

  13. Y. Fang, Existence, uniqueness of positive solution to a fractional laplacians with singular non linearity. arXiv Preprint. (2014) http://arxiv.org/pdf/1403.3149.pdf

    Google Scholar 

  14. P. Felmer, A. Quass, Boundary blow up solutions for fractional elliptic equations. Asymptot. Anal. 78, No 3 (2012), 123–144.

    Article  MathSciNet  Google Scholar 

  15. A. Ghanmi, K. Saoudi, The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator. Fractional Differential Calculus. 6, No 2 (2016), 201–217.

    Article  MathSciNet  Google Scholar 

  16. A. Ghanmi, K. Saoudi, A multiplicity results for a singular problem involving the fractional p-Laplacian operator. Complex Var. Elliptic Equ. 61, No 9 (2016), 1199–1216.

    Article  MathSciNet  Google Scholar 

  17. N. Ghoussoub, D. Preiss, A general mountain pass principle for locating and classifying critical points. Ann. Inst. H. Poincare Anal. Non Lineaire. 6, No 5 (1989), 321–330.

    Article  MathSciNet  Google Scholar 

  18. J. Giacomoni, K. Saoudi, Multiplicity of positive solutions for a singular and critical problem. Nonlinear Anal. 71, No 9 (2009), 4060–4077.

    Article  MathSciNet  Google Scholar 

  19. Y. Haitao, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem. J. Differential Equations. 189, No 2 (2003), 487–512.

    Article  MathSciNet  Google Scholar 

  20. N. Hirano, C. Saccon, N. Shioji, Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities. Adv. Differential Equations. 9, No 1–2 (2004), 197–220.

    MathSciNet  MATH  Google Scholar 

  21. A. Iannizzotto, S. Mosconi, M. Squassina, Hs versus C0weighted minimizers. Nonlinear Diff. Equations Appl. 22, No 3 (2015), 477–497.

    Article  Google Scholar 

  22. T. Mukherjee, K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearities. Electronic J. of Differential Equations. 2016, No 54 (2016), 1–23.

    MathSciNet  MATH  Google Scholar 

  23. R. Rodrlguez-Lopez, S. Tersian, Multiple solutions to boundary value problem for impulsive fractional equations. Fract. Calc. Appl. Anal. 17, No 4 (2014), 1016–1038; 10.2478/s13540–014-0212–2.https://www.degruyter.com/viewZj/fca.2014.17.issue-4/issue-files/fca.2014.17.issue-4.xml

    Article  MathSciNet  Google Scholar 

  24. X. Ros-Oton, J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary. J. Math. Pures Appl. 101, No 3 (2012), 275–302; 10.1016/j.matpur.2013.06.003.

    Article  MathSciNet  Google Scholar 

  25. L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, No 1 (2007), 67–112.

    Article  MathSciNet  Google Scholar 

  26. K. Saoudi, M. Kratou, Existence of multiple solutions for a singular and quasilinear equation. Complex Var. Elliptic Equ. 60, No 7 (2015), 893–925.

    Article  MathSciNet  Google Scholar 

  27. R. Servadei, E. Valdinoci, A Brezis-Nirenberg result for nonlocal critical equations in low dimension. Commun. Pure Appl. Anal. 12, No 6 (2013), 2445–2464.

    Article  MathSciNet  Google Scholar 

  28. R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33 (2013), 2105–2137.

    Article  MathSciNet  Google Scholar 

  29. R. Servadei, E. Valdinoci, Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389 (2012), 887–898.

    Article  MathSciNet  Google Scholar 

  30. P. Takáč, On the Fredholm alternative for the p-Laplacian at the first eigenvalue. Indiana Univ. Math. J. 51, No 1 (2002), 187–237.

    Article  MathSciNet  Google Scholar 

  31. S. Yijing, W. Shaoping, L. Yiming, Combined effects of singular and superlinear nonlinearities in some singular boundary value problems. J. Differential Equations. 176, No 2 (2001), 511–531.

    Article  MathSciNet  Google Scholar 

  32. Q.M. Zhou, K.Q. Wang, Existence and multiplicity of solutions for nonlinear elliptic problems with the fractional Laplacian. Fract. Calc. Appl. Anal. 18, No 1 (2015), 133–145; 10.1515/fca-2015–0009. https://www.degruyter.com/view/j/fca.2015.18.issue-1/issue-files/fca.2015.18.issue-1.xml

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kamel Saoudi.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saoudi, K. A critical fractional elliptic equation with singular nonlinearities. FCAA 20, 1507–1530 (2017). https://doi.org/10.1515/fca-2017-0079

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1515/fca-2017-0079

MSC 2010

Key Words and Phrases

Navigation