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On a planar non-autonomous Schrödinger–Poisson system involving exponential critical growth

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Abstract

In this paper, we investigate the existence of solutions to the planar non-autonomous Schrödinger–Poisson system

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta u+V(|x|)u+\gamma \phi K(|x|)u = \lambda Q(|x|)f(u),\ \&x\in {\mathbb {R}}^{2}, \\&\Delta \phi =K(|x|) u^{2}, \ \&x\in {\mathbb {R}}^{2}, \end{aligned} \right. \end{aligned}$$

where \(\gamma ,\lambda \) are positive parameters, V, K, Q are continuous potentials, which can be unbounded or vanishing at infinity. By assuming that the nonlinearity f(s) has exponential critical growth, we derive the existence of a ground state solution to the system. A key feature of our approach is a new weighted Trudinger–Moser type inequality proved here.

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References

  1. Adimurthi, Sandeep, K.: A singular Moser–Trudinger embedding and its applications. Nonlinear Differ. Equ. Appl. 13, 585–603 (2007)

  2. Albuquerque, F.S.B., Ferreira, M.C., Severo, U.B.: Ground state solutions for a nonlocal equation in \({\mathbb{R} }^2\) involving vanishing potentials and exponential critical growth, preprint, (2019). arXiv:org/abs/1911.05707

  3. Albuquerque, F.S.B., Alves, C.O., Medeiros, E.S.: Nonlinear Schrödinger equation with unbounded or decaying radial potentials involving exponential critical growth in \({\mathbb{R}}^2\). J. Math. Anal. Appl. 409, 1021–1031 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Alves, C.O., Figueiredo, G.M.: Existence of positive solution for a planar Schrödinger-Poisson system with exponential growth. J. Math. Phys. 60, 011503 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ambrosetti, A., Malchiodi, A., Ruiz, D.: Radial solutions concentrating on spheres of NLS with with vanishing potentials. Proc. R. Soc. Edinb. Sect. A 136, 889–907 (2006)

    Article  MATH  Google Scholar 

  6. Azzollini, A., Pomponio, A.: Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J. Math. Anal. Appl. 345, 90–108 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Badiale, M., Zaccagni, F.: Radial nonlinear elliptic problems with singular or vanishing potentials. Adv. Nonlinear Stud. 18, 409–428 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Badiale, M., Pisani, L., Rolando, S.: Sum of weighted Lebesgue spaces and nonlinear elliptic equations. Nonlinear Differ. Equ. Appl. 18, 369–405 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Badiale, M., Guida, M., Rolando, S.: Compactness and existence results in weighted Sobolev spaces of radial functions. Part I: compactness. Calc. Var. Partial Differ. Equ. 54, 1061–1090 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bartsch, T., Weth, T., Willem, M.: Partial symmetry of least energy nodal solutions to some variational problems. J. Anal. Math. 96, 1–18 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Benci, V., Fortunato, D.: An eigenvalue problem for the Schrödinger–Maxwell equations. Topol. Methods Nonlinear Anal. 11, 283–293 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Benci, V., Fortunato, D.: Solitary waves of the nonlinear Klein–Gordon equation coupled with Maxwell equations. Rev. Math. Phys. 14, 409–420 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Benguria, R., Brezis, H., Lieb, E.H.: The Thomas–Fermi-von Weizsäcker theory of atoms and molecules. Commun. Math. Phys. 79, 167–180 (1981)

    Article  MATH  Google Scholar 

  14. Bonheure, D., Mercuri, C.: Embedding theorems and existence results for nonlinear Schrödinger–Poisson systems with unbounded and vanishing potentials. J. Differ. Equ. 251, 1056–1085 (2011)

    Article  MATH  Google Scholar 

  15. Calanchi, M., Terraneo, E.: Non-radial maximizers for functionals with exponential non-linearity in \({\mathbb{R}}^2\). Adv. Nonlinear Stud. 5, 337–350 (2005)

    MathSciNet  MATH  Google Scholar 

  16. Cao, D.M.: Nontrivial solution of semilinear elliptic equation with critical exponent in \({\mathbb{R}}^2\). Commun. Partial Differ. Equ. 17, 407–435 (1992)

    MATH  Google Scholar 

  17. Catto, I., Lions, P.: Binding of atoms and stability of molecules in Hartree and Thomas–Fermi type theories. Commun. Partial Differ. Equ. 18, 1149–1159 (1993)

    Article  MATH  Google Scholar 

  18. Cerami, G., Molle, R.: Positive bound state solutions for some Schrödinger–Poisson systems. Nonlinearity 29(10), 3103–3119 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cerami, G., Vaira, G.: Positive solutions for some non-autonomous Schrödinger–Poisson systems. J. Differ. Equ. 248, 521–543 (2010)

    Article  MATH  Google Scholar 

  20. Cingolani, S., Weth, T.: On the planar Schrödinger–Poisson system. Ann. Inst. H. Poincaré Anal. Non Linéaire 33, 169–197 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. D’Aprile, T., Mugnai, D.: Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations. Proc. R. Soc. Edinb. Sect. A 134, 893–906 (2004)

    Article  MATH  Google Scholar 

  22. D’Aprile, T., Mugnai, D.: Non-existence results for the coupled Klein–Gordon–Maxwell equations. Adv. Nonlinear Stud. 4, 307–322 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. d’Avenia, P.: Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations. Adv. Nonlinear Stud. 2, 177–192 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. doÓ, J.M., Sani, F., Zhang, J.: Stationary nonlinear Schrödinger equations in \({\mathbb{R}}^2\) with potentials vanishing at infinity. Ann. Mat. Pura Appl. 196, 363–393 (2017)

    MathSciNet  Google Scholar 

  25. Figueiredo, G.M., Nunes, F.B.M.: Existence of positive solutions for a class of quasilinear elliptic problems with exponential growth via the Nehari manifold method. Rev. Math. Complut. 32, 1–18 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition

  27. Ianni, I., Vaira, G.: On concentration of positive bound states for the Schrödinger–Poisson problem with potentials. Adv. Nonlinear Stud. 8, 573–595 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1976/1977)

  29. Lieb, E.H.: Thomas–Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53, 263–301 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lions, P.L.: Solutions of Hartree–Fock equations for Coulomb systems. Commun. Math. Phys. 109, 33–97 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195, 455–467 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Markowich, P., Ringhofer, C., Schmeiser, C.: Semiconductor Equations. Springer, New York (1990)

    Book  MATH  Google Scholar 

  33. Masaki, S.: Energy solution to a Schrödinger–Poisson system in the two-dimensional whole space. SIAM J. Math. Anal. 43, 2719–2731 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Moroz, V., Schaftingen, J.V.: Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265, 153–184 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  36. Palais, R.S.: The principle of symmetric criticality. Commun. Math. Phys. 69, 19–30 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ruiz, D.: The Schrödinger–Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237, 655–674 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149–162 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  39. Stubbe, J.: Bound States of Two-Dimensional Schrödinger-Newton Equations. preprint, (2008). arxiv.org/abs/0807.4059

  40. Su, J., Wang, Z.Q., Willem, M.: Nonlinear Schrödinger equations with unbounded and decaying radial potentials. Commun. Contemp. Math. 9, 571–583 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. Su, J., Wang, Z.Q., Willem, M.: Weighted Sobolev embedding with unbounded and decaying radial potentials. J. Differ. Equ. 238, 201–219 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Trudinger, N.S.: On the embedding into Orlicz spaces and some applications. J. Math. Mech. 17, 473–484 (1967)

    MathSciNet  MATH  Google Scholar 

  43. Yunyan, Y.: Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space. J. Funct. Anal. 262, 1679–1704 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to express their sincere gratitude to the anonymous referees for their valuable suggestions and comments which provided insights that helped to improve the paper.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Estadual da Paraíba, Campina Grande, PB, 58700-070, Brazil

    F. S. Albuquerque

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

    J. L. Carvalho & E. Medeiros

  3. Departamento de Matemática, Universidade de Brasília, Brasília, DF, 70910-900, Brazil

    G. M. Figueiredo

Authors
  1. F. S. Albuquerque
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  2. J. L. Carvalho
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  3. G. M. Figueiredo
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  4. E. Medeiros
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Corresponding author

Correspondence to G. M. Figueiredo.

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Communicated by P. Rabinowitz.

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J. L. Carvalho was supported by CAPES/BRAZIL.

G. M. Figueiredo was supported by CNPQ, FAPDF, CAPES/BRAZIL.

E. Medeiros was partially suppoted by Grant 2019/2014 Paraíba State Research Foundation (FAPESQ) and Grant 308900/2019-7 CNPq.

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Albuquerque, F.S., Carvalho, J.L., Figueiredo, G.M. et al. On a planar non-autonomous Schrödinger–Poisson system involving exponential critical growth. Calc. Var. 60, 40 (2021). https://doi.org/10.1007/s00526-020-01902-6

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  • DOI: https://doi.org/10.1007/s00526-020-01902-6

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