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Existence and asymptotic behavior of solutions for the Schrödinger–Born–Infeld system with steep potential well

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Abstract

In this paper, we study the following Schrödinger–Born–Infeld system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda V(x)u+\nu \phi u=|u|^{p-2}u, &{}\text {in}\;\mathbb {R}^3, \\ -\text {div}\left( \frac{\nabla \phi }{\sqrt{1-\left| \nabla \phi \right| ^2}}\right) =u^2, &{}\text {in}\;\mathbb {R}^3,\\ u(x)\rightarrow 0,\;\phi (x)\rightarrow 0, &{}\text {as}\;|x|\rightarrow \infty , \end{array}\right. } \end{aligned}$$

where \(\lambda >0\) and \(\nu >0\) are parameters, \(2<p<6\), and V is a radial function. Under some suitable assumptions on V, the existence and asymptotic behavior of nontrivial solutions are obtained by using variational techniques. It is worth mentioning that the potential V is allowed to be sign-changing for the case \(p\in (4,6)\).

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References

  1. Azzollini, A., Pomponio, A., Siciliano, G.: On the Schrödinger–Born–Infeld system. Bull. Braz. Math. Soc. 50, 275–289 (2019)

    Article  MathSciNet  Google Scholar 

  2. Bartsch, T., Pankov, A., Wang, Z.-Q.: Nonlinear Schrödinger equations with steep potential well. Commun. Contemp. Math. 3, 549–569 (2001)

    Article  MathSciNet  Google Scholar 

  3. Bartsch, T., Wang, Z.-Q.: Existence and multiplicity results for superlinear elliptic problems on \(\mathbb{R} ^N\). Commun. Partial Differ. Equ. 20, 1725–1741 (1995)

    Article  Google Scholar 

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

    Article  Google Scholar 

  5. Benmilh, K., Kavian, O.: Existence and asymptotic behaviour of standing waves for quasilinear Schrödinger–Poisson systems in \(\mathbb{R} ^3\). Ann. Inst. Henri Poincaré Anal. Non Linéaire 25, 449–470 (2008)

    Article  MathSciNet  Google Scholar 

  6. Bonheure, D., d’Avenia, P., Pomponio, A.: On the electrostatic Born–Infeld equation with extended charges. Commun. Math. Phys. 346, 877–906 (2016)

    Article  MathSciNet  Google Scholar 

  7. Bonheure, D., Iacopetti, A.: On the regularity of the minimizer of the electrostatic Born–Infeld energy. Arch. Ration. Mech. Anal. 232, 697–725 (2019)

    Article  MathSciNet  Google Scholar 

  8. Bonheure, D., Iacopetti, A.: A sharp gradient estimate and \(W^{2, q}\) regularity for the prescribed mean curvature equation in the Lorentz–Minkowski space. Arch. Ration. Mech. Anal. 247, 87 (2023)

    Article  Google Scholar 

  9. Born, M., Infeld, L.: Foundations of the new field theory. Nature 132, 1004 (1933)

    Article  Google Scholar 

  10. Born, M., Infeld, L.: Foundations of the new field theory. Proc. R. Soc. Lond. Ser. A 144(852), 425–451 (1934)

    Article  Google Scholar 

  11. Byeon, J., Ikoma, N., Malchiodi, A., Mari, L.: Existence and regularity for prescribed Lorentzian mean curvature hypersurfaces, and the Born–Infeld model. arXiv:2112.11283 (2022)

  12. 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  Google Scholar 

  13. d’Avenia, P., Pisani, L.: Nonlinear Klein-Gordon equations coupled with Born–Infeld type equations. Electron. J. Differ. Equ. 2002(26), 1–13 (2002)

    MathSciNet  Google Scholar 

  14. Ding, Y.H., Szulkin, A.: Bound states for semilinear Schrödinger equations with sign-changing potential. Calc. Var. 29(3), 397–419 (2007)

    Article  MathSciNet  Google Scholar 

  15. Figueiredo, G.M., Siciliano, G.: Quasi-linear Schrödinger–Poisson system under an exponential critical nonlinearity: existence and asymptotic behaviour of solutions. Arch. Math. 112, 313–327 (2019)

    Article  MathSciNet  Google Scholar 

  16. Figueiredo, G.M., Siciliano, G.: Existence and asymptotic behaviour of solutions for a quasi-linear Schrödinger–Poisson system with a critical nonlinearity. Z. Angew. Math. Phys. 71, 130 (2020)

    Article  Google Scholar 

  17. Haarala, A.: The electrostatic Born–Infeld equations with integrable charge densities. arXiv:2006.08208v2 (2021)

  18. Illner, R., Kavian, O., Lange, H.: Stationary solutions of quasi-linear Schrödinger-Poisson system. J. Differ. Equ. 145, 1–16 (1998)

    Article  Google Scholar 

  19. Jeanjean, L.: On the existence of bounded Palais–Smale sequence and application to a Landesman–Lazer type problem set on \(\mathbb{R} ^N\). Proc. R. Soc. Edinb. Sect. A 129, 787–809 (1999)

    Article  Google Scholar 

  20. Jeanjean, L., Le Coz, S.: An existence and stability result for standing waves of nonlinear Schrödinger equations. Adv. Differ. Equ. 11, 813–840 (2006)

    Google Scholar 

  21. Li, A.R., Wei, C.Q., Zhao, L.G.: Existence and asymptotic behavior of solutions for a Schrödinger–Born–Infeld system in \(\mathbb{R} ^3\) with a general nonlinearity. J. Math. Anal. Appl. 516, 126555 (2022)

    Article  Google Scholar 

  22. Li, Y.H., Li, F.Y., Shi, J.P.: Existence of a positive solution to Kirchhoff type problems without compactness conditions. J. Differ. Equ. 253, 2285–2294 (2012)

    Article  MathSciNet  Google Scholar 

  23. Liu, Z.S., Siciliano, G.: A perturbation approach for the Schrödinger-Born-Infeld system: solution in the subcritical and critical case. J. Math. Anal. Appl. 503, 125326 (2021)

    Article  Google Scholar 

  24. Mugnai, D.: Coupled Klein–Gordon and Born–Infeld type equations: looking for solitary waves. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. 460, 1519–1527 (2004)

    Article  MathSciNet  Google Scholar 

  25. Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, No. 65. AMS, Providence, RI (1986)

  26. Schechter, M.: A variation of the mountain pass lemma and applications. J. Lond. Math. Soc. 2(44), 491–502 (1991)

    Article  MathSciNet  Google Scholar 

  27. Struwe, M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60, 558–581 (1985)

    Article  MathSciNet  Google Scholar 

  28. Sun, J.-J., Tang, C.-L.: Existence and multiplicity of solutions for Kirchhoff type equations. Nonlinear Anal. 74, 1212–1222 (2011)

    Article  MathSciNet  Google Scholar 

  29. Sun, J.T., Wu, T.F.: Ground state solutions for an indefinite Kirchhoff type problem with steep potential well. J. Differ. Equ. 256, 1771–1792 (2014)

    Article  MathSciNet  Google Scholar 

  30. Sun, J.T., Wu, T.F.: On Schrödinger–Poisson systems involving concave–convex nonlinearities via a novel constraint approach. Commun. Contemp. Math. 23, 2050048 (2021)

    Article  Google Scholar 

  31. Wang, Z.P., Zhou, H.S.: Positive solutions for nonlinear Schrödinger equations with deepening potential well. J. Eur. Math. Soc. 11, 545–573 (2009)

    Article  MathSciNet  Google Scholar 

  32. Willem, M.: Analyse Harmonique Réelle. Hermann, Paris (1995)

    Google Scholar 

  33. Willem, M.: Minimax Theorems. Birkhäser, Boston (1996)

    Book  Google Scholar 

  34. Yu, Y.: Solitary waves for nonlinear Klein–Gordon equations coupled with Born–Infeld theory. Ann. Inst. Henri Poincaré Anal. Non Linéaire 27, 351–376 (2010)

    Article  MathSciNet  Google Scholar 

  35. Zhang, F.B., Du, M.: Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well. J. Differ. Equ. 269, 1–23 (2020)

    Article  MathSciNet  Google Scholar 

  36. Zhao, L.G., Liu, H.D., Zhao, F.K.: Existence and concentration of solutions for the Schrödinger–Possion equation with steep well potential. J. Differ. Equ. 255, 1–23 (2013)

    Article  Google Scholar 

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Acknowledgements

J. Sun supported by NSFC (No. 12361024) and Jiangxi Provincial Natural Science Foundation (No. 20232ACB211004), J. Chen supported by NSFC (No. 11901276) and Jiangxi Provincial Natural Science Foundation (No. 20232BAB201001). We would like to thank the referee for valuable comments and helpful suggestions which have led to an improvement of the presentation of this paper.

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  1. Department of Mathematics, Nanchang University, Nanchang, 330031, Jiangxi, China

    Fenqi Wang, Jijiang Sun & Jianhua Chen

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  1. Fenqi Wang
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Wang, F., Sun, J. & Chen, J. Existence and asymptotic behavior of solutions for the Schrödinger–Born–Infeld system with steep potential well. Z. Angew. Math. Phys. 74, 242 (2023). https://doi.org/10.1007/s00033-023-02138-y

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  • DOI: https://doi.org/10.1007/s00033-023-02138-y

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