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Existence to Fractional Critical Equation with Hardy-Littlewood-Sobolev Nonlinearities


In this paper, we consider the following new Kirchhoff-type equations involving the fractional p-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity:

$$\begin{array}{*{20}{c}} {{{\left( {a + b\iint_{{\mathbb{R}^N}} {\frac{{{{\left| {u(x) - u(y)} \right|}^p}}}{{{{\left| {x - y} \right|}^{N + ps}}}}\text{d}x\text{d}y}} \right)}^{p - 1}}}&{( - \Delta )_p^su + \lambda V(x){{\left| u \right|}^{p - 2}}u} \\ { = \left( {\int\limits_{{\mathbb{R}^N}} {\frac{{{{\left| u \right|}^{p_{\mu ,s}^*}}}}{{{{\left| {x - y} \right|}^\mu }}}\text{d}y} } \right){{\left| u \right|}^{p_{\mu ,{s^{ - 2}}}^*}}u,\;x \in {\mathbb{R}^N}}&\; \end{array}\;$$

where (−Δ) ps is the fractional p-Laplacian with 0 < s < 1 < p, 0 < μ < N, N > ps, a, b > 0, λ > 0 is a parameter, V: ℝN → ℝ+ is a potential function, θ ∈ [1, 2 *μ,s ) and \(^{p_{\mu,s}^ * = {{pN - p{\mu \over 2}} \over {N - ps}}}\) is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We get the existence of infinitely many solutions for the above problem by using the concentration compactness principle and Krasnoselskii’s genus theory. To the best of our knowledge, our result is new even in Choquard-Kirchhoff-type equations involving the p-Laplacian case.

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  1. [1]

    Ambrosetti A, Rabinowitz P H. Dual variational methods in critical point theory and applications. J Functional Analysis, 1973, 14: 349–381

    MathSciNet  Article  Google Scholar 

  2. [2]

    Brézis H. Functional Analysis, Sobolev Spaces and Partial Differential Equations//Universitext. New York: Springer, 2011

    MATH  Google Scholar 

  3. [3]

    d’Avenia P, Siciliano G, Squassina M. On fractional Choquard equations. Math Models Methods Appl Sci, 2014, 25(8): 1447–1476

    MathSciNet  Article  Google Scholar 

  4. [4]

    Devillanova G, Carlo Marano G. A free fractional viscous oscillator as a forced standard damped vibration. Fractional Calculus and Applied Analysis, 2016, 19(2): 319–356

    MathSciNet  Article  Google Scholar 

  5. [5]

    Fiscella A, Pucci P. p-Fractional Kirchhoff equations involving critical nonlinearities. Nonlinear Anal RWA, 2017, 35: 350–378

    MathSciNet  Article  Google Scholar 

  6. [6]

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

    MathSciNet  Article  Google Scholar 

  7. [7]

    Guo L, Hu T. Existence and asymptotic behavior of the least energy solutions for fractional Choquard equations with potential well. arXiv preprint. 2017, arXiv:1703.08028

  8. [8]

    Gao F, Shen Z, Yang M. On the critical Choquard equation with potential well. arXiv preprint, 2017, arXiv:1703.01737

  9. [9]

    Gao F, Yang M. On the Brézis-Nirenberg type critical problem for nonlinear Choquard equation. arXiv:1604.00826v4

  10. [10]

    Gao F, Yang M. On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents. J Math Anal Appl, 2017, 448(2): 1006–1041

    MathSciNet  Article  Google Scholar 

  11. [11]

    Gao F, Yang M. On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation. Sci China Math, 2018, 61: 1219–1242

    MathSciNet  Article  Google Scholar 

  12. [12]

    Goel D, Sreenadh K. Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity. 2019, arXiv:1901.11310v1

  13. [13]

    Lan F, He X. The Nehari manifold for a fractional critical Choquard equation involving sign-changing weight functions. Nonlinear Anal, 2019, 180: 236–263

    MathSciNet  Article  Google Scholar 

  14. [14]

    Li A, Wang P, Wei C. Multiplicity of solutions for a class of Kirchhoff type equations with Hardy-Littlewood-Sobolev critical nonlinearity. Appl Math Lett, 2020, 102: 106105. DOI:

    MathSciNet  Article  Google Scholar 

  15. [15]

    Lieb E, Loss M. Analysis. 2nd Ed//Grad Stud Math 14. Providence: American Mathematical Society, 2001

    MATH  Google Scholar 

  16. [16]

    Lions P L. The concentration-compactness principle in the calculus of variations. the limit case, part 1. Rev Mat Iberoam, 1985, 1: 145–201

    Article  Google Scholar 

  17. [17]

    Lü D. A note on Kirchhoff-type equations with Hartree-type nonlinearities. Nonlinear Anal, 2014, 99: 35–48

    MathSciNet  Article  Google Scholar 

  18. [18]

    Ma P, Zhang J. Existence and multiplicity of solutions for fractional Choquard equations. Nonlinear Anal, 2017, 164: 100–117

    MathSciNet  Article  Google Scholar 

  19. [19]

    Molica Bisci G, Radulescu V, Servadei R. Variational methods for nonlocal fractional problems. Encyclopedia of Mathematics and its Applications, 162, 2016. Cambridge University Press, ISBN 9781107111943

  20. [20]

    Moroz V, Van Schaftingen J. Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics. J Functional Anal, 2013, 265(2): 153–184

    MathSciNet  Article  Google Scholar 

  21. [21]

    Mukherjee T, Sreenadh K. Fractional Choquard equation with critical nonlinearities. Nonlinear Differ Equat Appl, 2017, 24: 63

    MathSciNet  Article  Google Scholar 

  22. [22]

    Mukherjee T, Sreenadh K. On Dirichlet problem for fractional p-Laplacian with singular nonlinearity. Adv Nonlinear Anal, 2016.

  23. [23]

    Mukherjee T, Sreenadh K. Fractional choquard equation with critical nonlinearities. Nonlinear Differential Equations Appl, 2017, 24(6): 63, 34 pp

    MathSciNet  Article  Google Scholar 

  24. [24]

    Mukherjee T, Sreenadh K. Positive solutions for nonlinear Choquard equation with singular nonlinearity. Compl Var Ellip Equat, 2017, 62(8): 1044–1071

    MathSciNet  Article  Google Scholar 

  25. [25]

    Nyamoradi N, Zaidan L I. Existence and multiplicity of solutions for fractional p-Laplacian Schrödinger-Kirchhoff type equations. Complex Variables and Elliptic Equations, 2017, 63(2): 1–14

    MATH  Google Scholar 

  26. [26]

    Pekar S. Untersuchung über die Elektronentheorie der Kristalle. Berlin: Akademie Verlag, 1954

    MATH  Google Scholar 

  27. [27]

    Pucci P, Xiang M, Zhang B. Existence results for Schrödinger-Choquard-Kirchhoff equations involving the fractional p-Laplacian. Advances in Calculus of Variations, 2017. DOI:

  28. [28]

    Rabinowitz P H. Minimax methods in critical point theory with applications to differential equations//CBMS Reg Conf Series in Math Vol 65. Amer Math Soc Providence, RI, 1986

  29. [29]

    Servadei R, Valdinoci E. The Brezis-Nirenberg result for the fractional Laplacian. Trans Amer Math Soc, 2015, 367: 67–102

    MathSciNet  Article  Google Scholar 

  30. [30]

    Servadei R, Valdinoci E. A Brezis-Nirenberg result for nonlocal critical equations in low dimension. Commun Pure Appl Anal, 2013, 12: 2445–464

    MathSciNet  Article  Google Scholar 

  31. [31]

    Song Y, Shi S. Existence and multiplicity of solutions for Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity. Appl Math Lett, 2019, 92: 170–175

    MathSciNet  Article  Google Scholar 

  32. [32]

    Song Y, Shi S. Multiplicity results for Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity. J Dynamical and Control Systems, 2020, 26: 469–480

    MathSciNet  Article  Google Scholar 

  33. [33]

    Tan J. The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc Var Partial Differential Equations, 2011, 36: 21–41

    MathSciNet  Article  Google Scholar 

  34. [34]

    Wang F, Xiang M. Multiplicity of solutions to a nonlocal Choquard equation involving fractional magnetic operators and critical exponent. Elec J Differ Equat, 2016, 306: 1–11

    MathSciNet  MATH  Google Scholar 

  35. [35]

    Wang F, Xiang M. Multiplicity of solutions for a class of fractional Choquard-Kirchhoff equations involving critical nonlinearity. Anal Math Phys, 2017.

  36. [36]

    Wang Y, Yang Y. Bifurcation results for the critical Choquard problem involving fractional p-Laplacian operator. Boundary Value Problems, 2018, 132. DOI:

  37. [37]

    Wang J, Zhang J, Cui Y. Multiple solutions to the Kirchhoff fractional equation involving Hardy-Littlewood-Sobolev critical exponent. Boundary Value Problems. 2019, 124. doi:

  38. [38]

    Willem M. Minimax theorems. Boston: Birkhäuser, 1996

    Book  Google Scholar 

  39. [39]

    Xiang M Q, Zhang B L, Zhang X. A nonhomogeneous fractional p-Kirchhoff type problem involving critical exponent in ℝN. Adv Nonlinear Stud, 2017, 17(3): 611–640

    MathSciNet  Article  Google Scholar 

  40. [40]

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

    MathSciNet  Article  Google Scholar 

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Correspondence to Nemat Nyamoradi.

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Nyamoradi, N., Razani, A. Existence to Fractional Critical Equation with Hardy-Littlewood-Sobolev Nonlinearities. Acta Math Sci 41, 1321–1332 (2021).

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Key words

  • Hardy-Littlewood-Sobolev inequality
  • concentration-compactness principle
  • variational method
  • Fractional p-Laplacian operators
  • multiple solutions

2010 MR Subject Classification

  • 35B33
  • 35A15
  • 35R11