Skip to main content
SpringerLink
Log in

A Uniqueness Result for Strong Singular Kirchhoff-Type Fractional Laplacian Problems

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

Abstract

In this paper, we study the following Kirchhoff-type fractional Laplacian problem with strong singularity:

$$\begin{aligned} \left\{ \begin{array}{ll} (a+b\Vert u\Vert ^2) (-\Delta )^{s} u =f(x)u^{-\gamma }-k(x)u^q &{}\quad \text {in } \Omega , \\ u>0 &{}\quad \text {in } \Omega ,\\ u =0&{}\quad \text {in }\mathbb {R}^3\backslash \Omega , \end{array}\right. \end{aligned}$$

where \((-\Delta )^{s}\) is the fractional Laplace operator, \(a, b \ge 0, a+b>0\), \(\Omega \) is a bounded smooth domain of \(\mathbb {R}^3\), \(k \in L^{\infty }(\Omega )\) is a non-negative function, \(q \in (0,1), \gamma > 1\) and \(f \in L^1 (\Omega )\) is positive almost everywhere in \(\Omega \). Using variational method and Nehari method, we obtain a uniqueness result. A novelty is that the Kirchhoff coefficient may vanish at zero.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abdellaoui, B., Medina, M., Peral, I., Primo, A.: The effect of the Hardy potential in some Calderón–Zygmund properties for the fractional Laplacian. J. Differ. Equ. 260, 8160–8206 (2016)

    Article  Google Scholar 

  2. Anello, G.: A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problem. J. Math. Anal. Appl. 373, 248–251 (2011)

    Article  MathSciNet  Google Scholar 

  3. Autuori, G., Fiscella, A., Pucci, P.: Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal. 125, 699–714 (2015)

    Article  MathSciNet  Google Scholar 

  4. Banks, H.T.: Modeling and Control in the Biomedical Sciences. Springer, Berlin (1975)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Binlin, Z., Fiscella, A., Liang, S.: Infinitely many solutions for critical degenerate Kirchhoff type equations involving the fractional \(p\)-Laplacian. Appl. Math. Optim. 80, 63–80 (2019)

    Article  MathSciNet  Google Scholar 

  7. Callegari, A., Nachman, A.: Some singular nonlinear differential equations arising in boundary layer theory. J. Math. Anal. Appl. 64, 96–105 (1978)

    Article  MathSciNet  Google Scholar 

  8. Canino, A., Montoro, L., Sciunzi, B., Squassina, M.: Nonlocal problems with singular nonlinearity. Bull. Sci. Math. 141, 223–250 (2017)

    Article  MathSciNet  Google Scholar 

  9. Chen, S., Zhang, B., Tang, X.: Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity. Adv. Nonlinear Anal. 9, 148–167 (2020)

    Article  MathSciNet  Google Scholar 

  10. Crandall, M.G., Rabinowitz, P.H., Tartar, L.: On a Dirichlet problem with a singular nonlinearity. Commun. Part. Differ. Equ. 2, 193–222 (1977)

    Article  MathSciNet  Google Scholar 

  11. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    Article  MathSciNet  Google Scholar 

  12. Figueiredo, G., Molica Bisci, G., Servadei, R.: On a fractional Kirchhoff-type equation via Krasnoselskii’s genus. Asymptot. Anal. 94, 347–361 (2015)

    Article  MathSciNet  Google Scholar 

  13. Fiscella, A.: A fractional Kirchhoff problem involving a singular term and a critical nonlinearity. Adv. Nonlinear Anal. 8, 645–660 (2019)

    Article  MathSciNet  Google Scholar 

  14. Fiscella, A., Mishra, P.K.: The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms. Nonlinear Anal. 186, 6–32 (2019)

    Article  MathSciNet  Google Scholar 

  15. Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014)

    Article  MathSciNet  Google Scholar 

  16. Ghergu, M., Rădulescu, V.: Multi-parameter bifurcation and asymptotics for the singular Lane–Emden–Fowler equation with a convection term. Proc. R. Soc. Edinb. Sect. A 135, 61–83 (2005)

    Article  MathSciNet  Google Scholar 

  17. Giacomoni, J., Saoudi, K.: Multiplicity of positive solutions for a singular and critical problem. Nonlinear Anal. 71, 4060–4077 (2009)

    Article  MathSciNet  Google Scholar 

  18. Keller, H.B., Cohen, D.S.: Some positone problems suggested by nonlinear heat generation. J. Math. Mech. 16, 1361–1376 (1967)

    MathSciNet  MATH  Google Scholar 

  19. Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)

    MATH  Google Scholar 

  20. Lazer, A.C., Mckenna, P.J.: On a singular nonlinear elliptic boundary value problem. Proc. Am. Math. Soc. 111, 721–730 (1991)

    Article  MathSciNet  Google Scholar 

  21. Lei, C., Liao, J., Tang, C.: Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents. J. Math. Anal. Appl. 421, 521–538 (2015)

    Article  MathSciNet  Google Scholar 

  22. Liao, J., Zhang, P., Liu, J., Tang, C.: Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity. J. Math. Anal. Appl. 430, 1124–1148 (2015)

    Article  MathSciNet  Google Scholar 

  23. Liao, J., Ke, X., Lei, C., Tang, C.: A uniqueness result for Kirchhff type problems with singularity. Appl. Math. Lett. 59, 24–30 (2016)

    Article  MathSciNet  Google Scholar 

  24. Lions, J.L.: On some equations in boundary value problems of mathematical physics. In: Proceedings of the Internat Symposium Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Inst. Mat. Univ. Fed. Rio de Janeiro, Rio de Janeiro, (1997), vol. 30, pp. 284–346, North-Holland Mathematics Studies, Amsterdam (1978)

  25. Liu, X., Sun, Y.: Multiple positive solutions for Kirchhoff type problems with singularity. Commun. Pure Appl. Anal. 12, 721–733 (2013)

    MathSciNet  MATH  Google Scholar 

  26. Liu, R., Tang, C., Liao, J., Wu, X.: Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. Commun. Pure Appl. Anal. 15, 1841–1856 (2016)

    Article  MathSciNet  Google Scholar 

  27. Mingqi, X., Molica Bisci, G., Tian, G., Zhang, B.: Infinitely many solutions for the stationary Kirchhoff problems involving the fractional \(p\)-Laplacian. Nonlinearity 29, 357–374 (2016)

    Article  MathSciNet  Google Scholar 

  28. Molica Bisci, G., Rădulescu, V., Servadei, R.: Variational Methods for Nonlocal Fractional Problems. Cambridge University Press, Cambridge (2016)

    Book  Google Scholar 

  29. Nachman, A., Callegari, A.: A nonlinear singular boundary value problem in the theory of pseudoplastic fluids. SIAM J. Appl. Math. 38, 275–281 (1980)

    Article  MathSciNet  Google Scholar 

  30. Perry, W.L.: A monotone iterative technique for solution of pth order (\(p<0\)) reactiondiffusion problems in permeable catalysis. J. Comput. Chem. 5, 353–357 (1984)

    Article  MathSciNet  Google Scholar 

  31. Pucci, P., Xiang, M., Zhang, B.: Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional \(p\)-Laplacian in \(\mathbb{R}^N\). Calc. Var. Partial Differ. Equ. 54, 2785–2806 (2015)

    Article  Google Scholar 

  32. Pucci, P., Xiang, M., Zhang, B.: Existence and multiplicity of entire solutions for fractional \(p\)-Kirchhoff equations. Adv. Nonlinear Anal. 5, 27–55 (2016)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  34. Sun, Y.: Compatibility phenomena in singular problems. Proc. R. Soc. Edinb. Sect. A 143, 1321–1330 (2013)

    Article  MathSciNet  Google Scholar 

  35. Sun, Y., Wu, S., Long, Y.: Combined effects of singular and superlinear nonlinearities in some singular boundary value problems. J. Differ. Equ. 176, 511–531 (2001)

    Article  MathSciNet  Google Scholar 

  36. Wang, D., Yan, B.: A uniqueness result for some Kirchhoff-type equations with negative exponents. Appl. Math. Lett. 92, 93–98 (2019)

    Article  MathSciNet  Google Scholar 

  37. Xiang, M., Zhang, B., Rădulescu, V.: Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional \(p\)-Laplacian. Nonlinearity 29, 3186–3205 (2016)

    Article  MathSciNet  Google Scholar 

  38. Xiang, M., Zhang, B., Rădulescu, V.: Superlinear Schrödinger–Kirchhoff type problems involving the fractional \(p\)-Laplacian and critical exponent. Adv. Nonlinear Anal. 9, 690–709 (2020)

    Article  MathSciNet  Google Scholar 

  39. Zhang, Q.: Existence of positive solution to Kirchhoff–Schrödinger–Poisson system with strong singular term. J. Math. Phys. 60, 041504 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to express their gratitude to the anonymous referees for valuable comments and suggestions.

Author information

Authors and Affiliations

  1. College of Science, East China Jiaotong University, Nanchang, 330013, People’s Republic of China

    Li Wang

  2. Department of Information Engineering, Jingdezhen Ceramic Institute, Jingdezhen, 333001, People’s Republic of China

    Kun Cheng

  3. College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Binlin Zhang

Authors
  1. Li Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kun Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Binlin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Binlin Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

L. Wang was supported by National Natural Science Foundation of China (No. 11701178, 11561024). K. Chen was supported by Natural Science Foundation of Jiangxi Educational Committee (GJJ180737). B. Zhang was supported by the National Natural Science Foundation of China (No. 11871199).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, L., Cheng, K. & Zhang, B. A Uniqueness Result for Strong Singular Kirchhoff-Type Fractional Laplacian Problems. Appl Math Optim 83, 1859–1875 (2021). https://doi.org/10.1007/s00245-019-09612-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-019-09612-y

Keywords

Mathematics Subject Classification

Search

Navigation