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Fractional p-Laplacian Equations with Subcritical and Critical Exponential Growth Without the Ambrosetti–Rabinowitz Condition

  • Ruichang Pei
Article
  • 123 Downloads

Abstract

The main purpose of this paper is to investigate the existence of nontrivial solutions to a class of quasilinear non-local problems which do not satisfy the Ambrosetti–Rabinowitz (AR) condition where the nonlinear terms are superlinear at 0 and of subcritical or critical exponential growth (subcritical polynomial growth) at \(\infty \). Some existence results for nontrivial solution are obtained using mountain pass theorem combined with the fractional Moser–Trudinger inequality.

Keywords

Fractional p-Laplacian problems subcritical or critical exponential growth mountain pass theorem without the (AR)condition 

Mathematics Subject Classification

35A15 35J92 46E30 

Notes

Acknowledgements

This research is supported by the NSFC (Nos. 11661070, 11764035 and 11571176).

References

  1. 1.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brasco, L., Parini, E.: The second eigenvalue of the fractional \(p\)-Laplacian (preprint) Google Scholar
  3. 3.
    Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana Series, vol. 20. Springer (2016)Google Scholar
  4. 4.
    Costa, D.G., Magalhães, C.A.: Variational elliptic problems which are nonquadratic at infinity. Nonlinear Anal. 23, 1401–1412 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Costa, D.G., Miyagaki, O.H.: Nontrivial solutions for perturbations of the \(p\)-Laplacian on unbounded domains. J. Math. Anal. Appl. 193, 737–755 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    de Figueiredo, D.G., doÓ, J.M., Ruf, B.: Elliptic equations in \({\mathbb{R}}^2\) with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equ. 3, 139–153 (1995)CrossRefGoogle Scholar
  8. 8.
    doÓ, J.M.: Semilinear Dirichlet problems for the \(N\)-Laplacian in \({\mathbb{R}}^{N}\) with nonlinearities in the critical growth range. Differ. Integral. Equ. 9, 967–979 (1996)Google Scholar
  9. 9.
    Franzina, G., Palatucci, G.: Fractional \(p\)-eigenvalues. Riv. Mat. Univ. Parma 5, 373–386 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fiscella, A., Molica Bisci, G., Servadei, R.: Bifurcation and multiplicity results for critical nonlocal fractional problems. Bull. Sci. Math 140, 14–35 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Brezis, H., Nirenberg, L.: positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Iannizzotto, A., Liu, S.B., Perera, K., Squassina, M.: Existence results for fractional \(p\)-Laplacian problems via Morse theory. Adv. Calc. Var. (2014).  https://doi.org/10.1515/acv-2014-0024
  13. 13.
    Iannizzotto, A., Mosconi, S., Squassina, M.: Global Hölder regularity for the fractional \(p\)-Laplacian. arXiv:1411.2956 (preprint)
  14. 14.
    Iannizzotto, A., Squassina, M.: Weyl-type laws for fractional \(p\)-eigenvalue problems. Asymptot. Anal. 88, 233–245 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Iannizzotto, A., Squassina, M.: \(1/2\)-Laplacian problems with exponential nonlinearity. J. Math. Anal. Appl. 414, 372–385 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jeanjean, L.: On the existence of bounded Palais–Smale sequences and applications to a Landesman–Lazer-type problem set on \({\mathbb{R}}^N\). Proc. R. Soc. Edinb. 129, 787–809 (1999)CrossRefzbMATHGoogle Scholar
  17. 17.
    Lam, N., Lu, Guozhen: N-Laplacian equations in \({\mathbb{R}}^N\) with subcritical and critical growth without the Ambrosetti–Rabinowitz condition. Adv. Nonlinear Stud. 13, 289–308 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Liu, Z.L., Wang, Z.Q.: On the Ambrosetti–Rabinowitz superlinear condition. Adv. Nonlinear Stud. 4, 563–574 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Li, G.B., Zhou, H.S.: Asymptotically linear Dirichlet problem for the \(p\)-Laplacian. Nonlinear Anal. 43, 1043–1055 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Liu, S.B., Li, S.J.: Infinitely many solutions for a superlinear elliptic equation. Acta. Math. Sin. 46, 625–630 (2003)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Martinazzi, L.: Fractional Adams–Moser–Trudinger type inequalities. arXiv:1506.00489 (preprint)
  22. 22.
    Molica Bisci, G.: Sequence of weak solutions for fractional equations. Math. Res. Lett. 21, 241–253 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Molica Bisci, G., Rǎdulescu, D.V.: Ground state solutions of scalar field fractional Schrödinger equations. Calc. Var. Partial Differ. Equ. 54, 2985–3008 (2015)CrossRefzbMATHGoogle Scholar
  24. 24.
    Molica Bisci, G., Rǎdulescu, D.V., Servadei, R.: Variational methods for nonlocal fractional problems, Encyclopedia of Mathematics and its Applications, vol. 162. Cambridge University Press, Cambridge (2016)Google Scholar
  25. 25.
    Parini, E., Ruf, B.: On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces (preprint) Google Scholar
  26. 26.
    Perera, K., Agarwal, R.P., O’Regan, D.: Morse theoretic aspects of \(p\)-Laplacian operators. Mathematical Surveys and Monographs, vol. 161. American Mathematical Society, Providence (2010)Google Scholar
  27. 27.
    Perera, K., Squassina, M., Yang, Y.: Bifurcation and multiplicity results for critical fractional \(p\)-Laplacian problems. arXiv:1407.8061 (preprint)
  28. 28.
    Schechter, M., Zou, W.M.: Superlinear problems. Pac. J. Math. 214, 145–160 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Servadei, R., Valdinoci, E.: Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33, 2105–2137 (2013)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Xiang, M.Q., Molica Bisci, G., Tian, G.H., Zhang, B.L.: Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian. Nonlinearity 29, 357–374 (2015)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Zhang, B.L., Ferrara, M.: Multiplicity of soutions for a class of superlinear non-local fractional equations. Complex Var. Elliptic Equ. 60, 583–595 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zhang, B.L., Molica Bisci, G., Servadei, R.: Superlinear nonlocal fractional problems with infinitely many solutions. Nonlinearity 28, 2247–2264 (2015)Google Scholar
  34. 34.
    Zhang, Y.M., Shen, Y.T.: Existence of solutions for elliptic equations without superquadraticity condition. Front. Math. China 7, 587–595 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTianshui Normal UniversityTianshuiPeople’s Republic of China

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