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Existence of non-trivial solutions to a class of fractional p-Laplacian equations of Schrödinger-type with a combined nonlinearity

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Abstract

In this paper, we consider the following nonlinear fractional p-Laplacian equation of Schrödinger-type

$$\begin{aligned} (-\Delta )_p^s u + V(x)|u|^{p-2}u =k(x){|u|}^ {\eta -2}u+ f(x,u), \ \ \qquad \ x \in \mathbb {R}^N \end{aligned}$$

where \(0<s<1<\eta <p\), \(N\ge 2\), V(x) is a real continuous function on \(\mathbb {R}^N\). Based on some assumptions on k, V and f we obtain the existence of non-trivial solutions using of the variational methods.

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The datasets generated and/or analysed during the current study is available from the correspond author on reasonable request.

References

  1. Autuori, G., and P. Pucci. 2013. Elliptic problems involving the fractional Laplacian in \(\mathbb{R^N} \). J. Differ. Equ. 255: 2340–2362.

    Article  Google Scholar 

  2. Khoutir, S., and H. Chen. 2016. Existence of infinitely many high energy solutions for a fractional Schrödinger equation in \(\mathbb{R^N} \). Appl. Math. Lett. 61: 156–162.

    Article  MathSciNet  Google Scholar 

  3. Teng, K.M. 2015. Multiple solutions for a class of fractional Schrödinger equations in \(\mathbb{R^N} \). Nonlinear Anal. Real World Appl. 21: 76–86.

    Article  MathSciNet  Google Scholar 

  4. Ge, B. 2016. Multiple solutions of nonlinear Schrödinger equation with the fractional Laplacian. Nonlinear Anal. Real World Appl. 30: 236–247.

    Article  MathSciNet  Google Scholar 

  5. Gou, T., and H. Sun. 2015. Solutions of nonlinear Schrödinger equation with fractional Laplacian without the Ambrosetti-Rabinowitz condition. Appl. Math. Comput. 257: 409–416.

    MathSciNet  Google Scholar 

  6. Perera, K., M. Squassina, and Y. Yang. 2015. Critical fractional \(p\)-Laplacian problems with possibly vanishing potentials. J. Math. Anal. Appl. 433 (2): 818–831.

    Article  MathSciNet  Google Scholar 

  7. Secchi, S. 2013. Ground state solutions for nonlinear fractional Schrödinger equations in \(\mathbb{R^N} \). J. Math. Phys. 54: 031501.

    Article  MathSciNet  Google Scholar 

  8. Secchi, S. 2013. Perturbation results for some nonlinear equations involving fractional operators. Differ. Equ. Appl. 5: 221–236.

    MathSciNet  Google Scholar 

  9. Secchi, S., and M. Squassina. 2014. Soliton dynamics for fractional Schrödinger equations. Appl. Anal. 93: 1702–1729.

    Article  MathSciNet  Google Scholar 

  10. Soluki, M., G.A. Afrouzi, and S.H. Rasouli. 2024. Existence and multiplicity of non-trivial solutions for fractional Schrödinger-Poisson systems with a combined nonlinearity. J. Ellip. Parab. Equ.https://doi.org/10.1007/s41808-023-00258-0.

    Article  Google Scholar 

  11. Soluki, M., S.H. Rasouli, and G.A. Afrouzi. 2019. On a class of nonlinear fractional Schrödinger-Poisson systems. Int. J. Nonlinear Anal. Appl. 10: 123–132.

    Google Scholar 

  12. Soluki, M., S.H. Rasouli, and G.A. Afrouzi. 2023. Solutions of a Schrödinger-Kirchhoff-Poisson system with concave-convex nonlinearities. J. Ellip. Parab. Equ. 9 (2): 1233–1244.

    Article  Google Scholar 

  13. Chang, X.J., and Z.Q. Wang. 2013. Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity. 26: 479–494.

    Article  MathSciNet  Google Scholar 

  14. Felmer, P., A. Quaas, and J. Tan. 2012. Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A. 142: 1237–1262.

    Article  Google Scholar 

  15. Xiang, M., V.D. Rădulescu, and B. Zhang. 2019. Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity. Calc. Var. Partial Differ. Equ. 58 (2): 57.

    Article  MathSciNet  Google Scholar 

  16. Sun, M., J. Su, and L. Zhao. 2016. Solutions of a Schrödinger-Poisson system with combined nonlinearities. J. Math. Anal. Appl. 442: 385–403.

    Article  MathSciNet  Google Scholar 

  17. Li, K. 2017. Existence of non-trivial solutions for nonlinear fractional Schrödinger-Poisson equations. Appl. Math. Lett. 72: 1–9.

    Article  MathSciNet  Google Scholar 

  18. Goyal, S., and K. Sreenadh. 2015. Nehari manifold for non-local elliptic operator with concave-convex nonlinearities and sign-changing weight functions. Proc. Indian Acad. Sci. Math. Sci. 125 (4): 545–558.

    Article  MathSciNet  Google Scholar 

  19. Chen, Q., C. Chen, and Y. Shi. 2020. Multiple solutions for fractional \(p\)-Laplace equation with concave-convex nonlinearities. Bound. Value Probl. 2020 (1): 1–3.

    Article  MathSciNet  Google Scholar 

  20. Di Nezza, E., G. Palatucci, and E. Valdinoci. 2012. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (5): 521–573.

    Article  MathSciNet  Google Scholar 

  21. Franzina, G., and G. Palatucci. 2014. Fractional \(p\)-eigenvalues. Riv. Mat. Univ. Parma. 5: 315–328.

    MathSciNet  Google Scholar 

  22. Xiang, M., B. Zhang, and V.D. Rădulescu. 2016. Existence of solutions for perturbed fractional \(p\)-Laplacian equations. J. Differ. Equ. 260: 1392–1413.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

    M. Soluki & G. A. Afrouzi

  2. Department of Mathematics, Faculty of Basic Science, Babol Noshirvani University of Technology, Babol, Iran

    S. H. Rasouli

Authors
  1. M. Soluki
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  2. G. A. Afrouzi
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  3. S. H. Rasouli
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All authors contributed to analysis, drafting, revising and proofs of the study. All authors read and approved the final manuscript.

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Correspondence to M. Soluki.

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Communicated by S. Ponnusamy.

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Soluki, M., Afrouzi, G.A. & Rasouli, S.H. Existence of non-trivial solutions to a class of fractional p-Laplacian equations of Schrödinger-type with a combined nonlinearity. J Anal (2024). https://doi.org/10.1007/s41478-024-00775-8

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  • DOI: https://doi.org/10.1007/s41478-024-00775-8

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