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Fractional Schrödinger equations involving potential vanishing at infinity and supercritical exponents

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Abstract

This work studies the existence of positive solutions for fractional Schrödinger equations of the form

$$\begin{aligned} (-\Delta )^s u + V(x) u = g(u)+\lambda |u|^{q-2}u\quad \text{ in }\quad {\mathbb {R}}^N, \end{aligned}$$

where \(s\in (0,1)\), \(N>2s\), \(V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) is a potential function which can vanish at infinity, \(g:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is superlinear and has subcritical growth, the exponent \(q\ge 2^*_s:=2N/(N-2s)\) and \(\lambda \) is a nonnegative parameter. Our approach is based on a truncation argument in combination with variational techniques and the Moser iteration method.

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References

  1. Alves, C.O., Souto, M.A.S.: Existence of solutions for a class of elliptic equations in \({\mathbb{R}}^N\) with vanishing potentials. J. Differ. Equ. 252, 5555–5568 (2012)

    Article  Google Scholar 

  2. Alves, C.O., Miyagaki, O.H.: Existence and concentration of solution for a class of fractional elliptic equation in \({\mathbb{R}}^N\) via penalization method. Calc. Var. Partial Differ. Equ. 55, Art. 47, 19 pp (2016)

  3. Ambrosetti, A., Felli, V., Malchiodi, A.: Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Eur. Math. Soc. (JEMS) 7, 117–144 (2005)

    Article  MathSciNet  Google Scholar 

  4. Ambrosetti, A., Malchiodi, A., Ruiz, D.: Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Anal. Math. 98, 317–348 (2006)

    Article  MathSciNet  Google Scholar 

  5. Ambrosio, V.: Multiplicity and concentration results for a fractional choquard equation via penalization method. Potential Anal. 50, 55–82 (2019)

    Article  MathSciNet  Google Scholar 

  6. Ambrosio, V.: Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method. Ann. Mat. Pura Appl. 196, 2043–2062 (2017)

    Article  MathSciNet  Google Scholar 

  7. Bisci, G.M., Radulescu, V.D., Servadei, R.: Variational Methods for Nonlocal Fractional Problems. Encyclopedia of Mathematics and Its Applications, vol. 162. Cambridge University Press, Cambridge (2016)

    Book  Google Scholar 

  8. Bucur, C., Valdinoci, E.: Nonlocal Difusion and Applications. Lecture Notes of the Unione Matematica Italiana. Springer, Unione Matematica Italiana, Bologna (2016)

    Book  Google Scholar 

  9. Bonheure, D., Van Schaftingen, J.: Groundstates for the nonlinear Schrödinger equation with potential vanishing at infinity. Ann. Mat. Pura Appl. 189, 273–301 (2010)

    Article  MathSciNet  Google Scholar 

  10. Brasco, L., Lindgren, E., Parini, E.: The fractional Cheeger problem. Interfaces Free Bound. 16, 419–458 (2014)

    Article  MathSciNet  Google Scholar 

  11. Chang, X., Wang, Z.-Q.: Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity 26, 479–494 (2013)

    Article  MathSciNet  Google Scholar 

  12. Cheng M.: Bound state for the fractional Schrödinger equation with unbounded potential. J. Math. Phys. 53, 043507, 7pp (2012)

  13. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman and Hall/CRC Financial Mathematics Series. Chapman Hall/CRC, Boca Raton (2004)

    MATH  Google Scholar 

  14. del Pino, M., Felmer, P.: Local mountain pass for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4, 121–137 (1996)

    Article  Google Scholar 

  15. 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 

  16. do Ó, J.M., Miyagaki, O.H., Squassina, M.: Critical and subcritical fractional problems with vanishing potentials. Commun. Contemp. Math. 18, 1550063, 20pp (2016)

  17. do Ó, J.M., Sani, F., Zhang, J.: Stationary nonlinear Schrödinger equations in R2 with potentials vanishing at infinity. Ann. Mat. Pura Appl. 196, 363–393 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  19. Fife, P.C.: An Integrodifferential Analog of Semilinear Parabolic PDEs. Partial Differential Equations and Applications. Lecture Notes in Pure and Applied Mathematics, vol. 177, pp. 137–145. Dekker, New York (1996)

    MATH  Google Scholar 

  20. Figueiredo, G.M., Pimenta, M.T.: Existence of ground state solutions to Dirac equations with vanishing potentials at infinity. J. Differ. Equ. 262, 486–505 (2017)

    Article  MathSciNet  Google Scholar 

  21. Hutson, V., Martinez, S., Mischaikow, K., Vickers, G.T.: The evolution of dispersal. J. Math. Biol. 47, 483–517 (2003)

    Article  MathSciNet  Google Scholar 

  22. Jarohs, S., Weth, T.: On the strong maximum principle for nonlocal operators. Math. Z. (2018). https://doi.org/10.1007/s00209-018-2193-z

    Article  MATH  Google Scholar 

  23. Laskin, N.: Fractional quantum mechanics. Phys. Rev. E 62, 3135–3145 (2000)

    Article  Google Scholar 

  24. Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E (3) 66, 056108 (2002)

    Article  MathSciNet  Google Scholar 

  25. Li, Q., Teng, K., Wu, X.: Existence of positive solutions for a class of critical fractional Schrödinger equations with potential vanishing at infinity. Mediterr. J. Math. 14, Art. 80, 14 pp (2017)

  26. Li, Q., Teng, K., Wu, X., Wang, W.: Existence of nontrivial solutions for fractional Schrödinger equations with critical or supercritical growth. Math. Methods Appl. Sci. 42, 1480–1487 (2019)

    Article  MathSciNet  Google Scholar 

  27. Perera, K., Squassina, M., Yang, Y.: Critical fractional p-Laplacian problems with possibly vanishing potentials. J. Math. Anal. Appl. 433, 818–831 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Shang, X., Zhang, J.: Ground states for fractional Schrödinger equations with critical growth. Nonlinearity 27, 187–207 (2014)

    Article  MathSciNet  Google Scholar 

  30. Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, 67–112 (2007)

    Article  MathSciNet  Google Scholar 

  31. Shang, X., Zhang, J., Yang, Y.: On fractional Schrödinger equation in \({\mathbb{R}}^N\) with critical growth. J. Math. Phys. 54, 121502 (2013)

    Article  MathSciNet  Google Scholar 

  32. Stein, E.M., Shakarchi, R.: Real Analysis. Measure Theory, Integration, and Hilbert Spaces. Princeton Lectures in Analysis, vol. 3. Princeton University Press, Princeton (2005)

    MATH  Google Scholar 

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Acknowledgements

We would like to thank the referees for careful reading of the paper with many useful comments and important suggestions, which substantially helped improving the quality of the paper.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão, SE , 49100-000, Brazil

    J. A. Cardoso & D. S. dos Prazeres

  2. Departamento de Matemática, Universidade Federal da Paraíba, João Pessoa, PB 58051-900, Brazil

    U. B. Severo

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  1. J. A. Cardoso
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  2. D. S. dos Prazeres
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  3. U. B. Severo
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Correspondence to U. B. Severo.

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The first and second authors were partially supported by FAPITEC/CAPES and CNPq. The third author was partially supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001 and CNPq grant 310747/2019-8.

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Cardoso, J.A., Prazeres, D.S.d. & Severo, U.B. Fractional Schrödinger equations involving potential vanishing at infinity and supercritical exponents. Z. Angew. Math. Phys. 71, 129 (2020). https://doi.org/10.1007/s00033-020-01354-0

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  • DOI: https://doi.org/10.1007/s00033-020-01354-0

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