Combined effects of logarithmic and superlinear nonlinearities in fractional Laplacian systems


In this paper, we consider the existence and multiplicity of solutions for the following fractional Laplacian system with logarithmic nonlinearity

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{s}u=\lambda h_1(x)u\ln |u|+\frac{p}{p+q}b(x)|v|^{q}|u|^{p-2}u\ \ &{}x\in \Omega , \\ (-\Delta )^t v=\mu h_2(x)v\ln |v|+\frac{q}{p+q}b(x)|u|^p|v|^{q-2}v\ \ &{}x\in \Omega , \\ u=v=0\ \ &{}x\in \mathbb {R}^N{\setminus }\Omega , \end{array}\right. } \end{aligned}$$

where \(s,t\in (0,1),\ N>\max \{2s,2t\}\), \(\lambda ,\mu >0\), \(2<p+q<\min \{\frac{2N}{N-2s},\frac{2N}{N-2t}\}\), \(\Omega \subset \mathbb {R}^N\) is a bounded domain with Lipschitz boundary, \(h_1,h_2,b\in C(\overline{\Omega })\) and \((-\Delta )^{s}\) is the fractional Laplacian. When \(h_1,h_2,b\) are positive functions, the existence of ground state solutions for the problem is obtained. When \(h_1,h_2\) are sign-changing functions and b is a positive function, two nontrivial and nonnegative solutions are obtained. Our results are new even in the case of a single equation.

This is a preview of subscription content, access via your institution.

Availability of data and materials

Data sharing not applicable to this paper as no datasets were generated or analyzed during the current study.


  1. 1.

    Ardila, A.H.: Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity. Nonlinear Anal. 155, 52–64 (2017)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)

    Google Scholar 

  3. 3.

    Bernini, F., Mugnai, D.: On a logarithmic Hartree equation. Adv. Nonlinear Anal. 9, 850–865 (2019)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Brown, K.J., Zhang, Y.: The Nehari manifold for a semilinear elliptic problem with a sign changing weight function. J. Differ. Equ. 193, 481–499 (2003)

    Article  Google Scholar 

  5. 5.

    Bhakta, M., Chakraborty, S., Pucci, P.: Nonhomogeneous systems involving critical or subcritical nonlinearities. Differ. Integral Equ. 33(7–8), 323–336 (2020)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Caffarelli, L.: Non-local diffusions, drifts and games, nonlinear partial differential equations. Abel Symposia 7, 37–52 (2012)

    Article  Google Scholar 

  7. 7.

    Chen, W.J., Deng, S.B.: The Nehari manifold for a fractional \(p\)-Laplacian system involving concave-convex nonlinearities. Nonlinear Anal. RWA 27, 80–92 (2016)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Chen, S., Tang, X.: Ground state sign-changing solutions for elliptic equations with logarithmic nonlinearity. Act. Math. Hungarica 157, 27–38 (2019)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Cotsiolis, A., Tavoularis, N.K.: On logarithmic Sobolev inequalities for higher order fractional derivatives. C. R. Acad. Sci. Paris Ser. I 340, 205–208 (2005)

    MathSciNet  Article  Google Scholar 

  10. 10.

    d ’Avenia, P., Squassina, M., Zenari, M.: Fractional logarithmic Schrödinger equations. Math. Methods Appl. Sci. 38, 5207–5216 (2015)

    MathSciNet  Article  Google Scholar 

  11. 11.

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

    MathSciNet  Article  Google Scholar 

  12. 12.

    Fiscella, A., Pucci, P.: (p, q) systems with critical terms in \(\mathbb{R}^N\). Nonlinear Anal. 177, 454–479 (2018)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Fiscella, A., Pucci, P., Zhang, B.: p-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities. Adv. Nonlinear Anal. 8(1), 1111–1131 (2019)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Fiscella, A., Pucci, P.: Degenerate Kirchhoff (p, q)-fractional systems with critical nonlinearities. Fract. Calc. Appl. Anal. 23(3), 723–752 (2020)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Fu, Y., Li, H., Pucci, P.: Existence of nonnegative solutions for a class of systems involving fractional (p, q)-Laplacian operators. Chin. Ann. Math. Ser. B 39(2), 357–372 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Ji, C., Fang, F., Zhang, B.: A multiplicity result for asymptotically linear Kirchhoff equations. Adv. Nonlinear Anal. 8, 267–277 (2019)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298–305 (2000)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Liu, H.L., Liu, Z.S., Xiao, Q.Z.: Ground state solution for a fourth-order nonlinear elliptic problem with logarithmic nonlinearity. Appl. Math. Lett. 79, 176–181 (2018)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Mingqi, X., Rǎdulescu, V.D., Zhang, B.L.: Nonlocal Kirchhoff problems with singular exponential nonlinearity. Appl. Math. Optim. (2020).

  20. 20.

    Mingqi, X., Rǎdulescu, V.D., Zhang, B.L.: Combined effects for fractional Schrǒdinger–Kirchhoff systems with critical nonlinearities. ESAIM COCV 24, 1249–1273 (2018)

    Article  Google Scholar 

  21. 21.

    Mingqi, X., Rǎdulescu, V.D., Zhang, B.L.: A critical fractional Choquard–Kirchhoff problem with magnetic field. Commun. Contemp. Math. 21, 1850004 (2019)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Mingqi, X., Rǎdulescu, V.D., Zhang, B.L.: Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity. Calc. Var. Partial Differ. Equ. 58, 57 (2019)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Molica Bisci, G., Rădulescu, V.: Ground state solutions of scalar field fractional for Schrödinger equations. Calc. Var. Partial Differ. Equ. 54, 2985–3008 (2015)

    Article  Google Scholar 

  24. 24.

    Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65. Amer. Math. Soc., Providence (1986)

    Google Scholar 

  25. 25.

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

    MathSciNet  Article  Google Scholar 

  26. 26.

    Tian, S.Y.: Multiple solutions for the semilinear elliptic equations with the sign-changing logarithmic nonlinearity. J. Math. Anal. Appl. 454, 816–828 (2017)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Truong, L.X.: The Nehari manifold for fractional p-Laplacian equation with logarithmic nonlinearity on whole space. Comput. Math. Appl. 78, 3931–3940 (2019)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Truong, L.X.: The Nehari manifold for The Nehari manifold for a class of Schrödinger equation involving fractional p-Laplacian and sign-changing logarithmic nonlinearity. J. Math. Phys. 60, 111505 (2019)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Vázquez, J.L.: Nonlinear diffusion with fractional Laplacian operators. In: Nonlinear Partial Differential Equations, Abel Symp., vol.7, 271–298. Springer, Heidelberg (2012)

  30. 30.

    Wu, T.F.: Multiplicity results for a semi-linear elliptic equation involving sign-changing weight function. Rocky Mt. J. Math. 39, 995–1011 (2009)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Xiang, M., Zhang, B., Ferrara, M.: Existence of solutions for Kirchhoff type problem involving the non-local fractional \(p\)-Laplacian. J. Math. Anal. Appl. 424, 1021–1041 (2015)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Xiang, M., Zhang, B., Yang, D.: Multiplicity results for variable-order fractional Laplacian equations with variable growth. Nonlinear Anal. 178, 190–204 (2019)

    MathSciNet  Article  Google Scholar 

  33. 33.

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

    MathSciNet  Article  Google Scholar 

  34. 34.

    Xiang, M., Yang, D., Zhang, B.: Degenerate Kirchhoff-type fractional diffusion problem with logarithmic nonlinearity. Asympt. Anal.

  35. 35.

    Xiang, M., Hu, D., Yang, D.: Least energy solutions for fractional Kirchhoff problems with logarithmic nonlinearity. Nonlinear Anal. (2020).

Download references


Mingqi Xiang was supported by the National Nature Science Foundation of China (No. 11601515) and the Tianjin Youth Talent Special Support Program.

Author information




All three authors contributed equally to this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Mingqi Xiang.

Ethics declarations

Conflict of interest

None of the authors has any competing interests in the manuscript.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, F., Die, H. & Xiang, M. Combined effects of logarithmic and superlinear nonlinearities in fractional Laplacian systems. Anal.Math.Phys. 11, 9 (2021).

Download citation


  • Fractional Laplacian systems
  • Nehari manifold method
  • Multiplicity of solutions
  • Ground state

Mathematics Subject Classification

  • 35K55
  • 35R11
  • 47G20