Skip to main content
SpringerLink
Log in

On the planar Schrödinger equation with indefinite linear part and critical growth nonlinearity

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

In the present paper, we develop a direct approach to find nontrivial solutions and ground state solutions for the following planar Schrödinger equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V(x)u=f(x,u), \;\;&{} x\in {{\mathbb {R}}}^{2},\\ u\in H^1({\mathbb {R}}^2), \end{array}\right. } \end{aligned}$$

where V(x) is an 1-periodic function with respect to \(x_1\) and \(x_2\), 0 lies in a gap of the spectrum of \(-\Delta +V\) , and f(x, t) behaves like \(\pm e^{\alpha t^2}\) as \(t\rightarrow \pm \infty \) uniformly on \(x\in {\mathbb {R}}^2\). Our theorems extend and improve the results of de Figueiredo-Miyagaki-Ruf (Calc Var Partial Differ Equ, 3(2):139–153, 1995), of de Figueiredo-do Ó-Ruf (Indiana Univ Math J, 53(4):1037–1054, 2004), of Alves-Souto-Montenegro (Calc Var Partial Differ Equ 43: 537–554, 2012), of Alves-Germano (J Differ Equ 265: 444–477, 2018) and of do Ó-Ruf (NoDEA 13: 167–192, 2006).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adachi, S., Tanaka, K.: Trudinger type inequalities in \( {\bf R}^N\) and their best exponents. Proc. Am. Math. Soc. 128, 2051–2057 (2000)

    Article  Google Scholar 

  2. Adimurthi, S.L., Yadava: Multiplicity results for semilinear elliptic equations in a bounded domain of \({\bf R}^2\) involving critical exponents. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17, 481–504 (1990)

    MathSciNet  MATH  Google Scholar 

  3. Alves, C.O., Germano, G.F.: Ground state solution for a class of indefinite variational problems with critical growth. J. Differ. Equ. 265, 444–477 (2018)

    Article  MathSciNet  Google Scholar 

  4. Alves, C.O., Souto, M.A.S.: Multiplicity of positive solutions for a class of problems with exponential critical growth in \({\mathbb{R}}^2\). J. Differ. Equ. 244, 1502–1520 (2008)

    Article  Google Scholar 

  5. Alves, C.O., Souto, M.A.S., Montenegro, M.: Existence of a ground state solution for a nonlinear scalar field equation with critical growth. Calc. Var. Partial Differ. Equ. 43, 537–554 (2012)

    Article  MathSciNet  Google Scholar 

  6. Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. 7, 981–1012 (1983)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  8. Cassani, D., Sani, F., Tarsi, C.: Equivalent Moser type inequalities in \({\mathbb{R}}^2\) and the zero mass case. J. Funct. Anal. 267, 4236–4263 (2014)

    Article  MathSciNet  Google Scholar 

  9. Chabrowski, J., Szulkin, A.: On a semilinear Schrödinger equation with critical Sobolev exponent. Proc. Am. Math. Soc. 130, 85–93 (2002)

    Article  Google Scholar 

  10. Chen, S.T., Tang, X.H.: Axially symmetric solutions for the planar Schrödinger-Poisson system with critical exponential growth. J. Differ. Equ. 269, 9144–9174 (2020)

    Article  Google Scholar 

  11. de Figueiredo, D.G., do Ó, J.M., Ruf, B.: On an inequality by N. Trudinger and J. Moser and related elliptic equations. Commun. Pure Appl. Math. 55, 135–152 (2002)

    Article  MathSciNet  Google Scholar 

  12. de Figueiredo, D.G., do Ó, J.M., Ruf, B.: Critical and subcritical elliptic systems in dimension two. Indiana Univ. Math. J. 53, 1037–1054 (2004)

    Article  MathSciNet  Google Scholar 

  13. de Figueiredo, D.G., Miyagaki, O.H., Ruf, B.: Elliptic equations in \({\bf R}^2\) with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equ. 3, 139–153 (1995)

    Article  Google Scholar 

  14. Ding, Y.: Variational Methods for Strongly Indefinite Problems. World Scientific Publishing Co Pvt. Ltd., Hackensack (2007)

    Book  Google Scholar 

  15. Ding, Y., Lee, C.: Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms. J. Differ. Equ. 222, 137–163 (2006)

    Article  Google Scholar 

  16. do Ó, J.M.: Semilinear Dirichlet problems for the \(N\)-Laplacian in \({\bf R}^N\) with nonlinearities in the critical growth range. Diff. Int. Equ. 9, 967–979 (1996)

    MATH  Google Scholar 

  17. do Ó, J.M., Ruf, B.: On a Schrödinger equation with periodic potential and critical growth in \({{\mathbb{R}}^2}\). NoDEA Nonlinear Differ. Equ. Appl. 13, 167–192 (2006)

    Article  Google Scholar 

  18. Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. The Clarendon Press, Oxford University Press, New York (1987)

    MATH  Google Scholar 

  19. Egorov, Y., Kondratiev, V.: On Spectral Theory of Elliptic Operators, vol. 89. Birkhäuser Verlag, Basel (1996)

    Book  Google Scholar 

  20. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)

    Book  Google Scholar 

  21. Kryszewski, W., Szulkin, A.: Generalized linking theorem with an application to a semilinear Schrödinger equation. Adv. Differ. Equ. 3, 441–472 (1998)

    MATH  Google Scholar 

  22. Lam, N., Lu, G.Z.: Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition. J. Geom. Anal. 24, 118–143 (2014)

    Article  MathSciNet  Google Scholar 

  23. Li, G., Szulkin, A.: An asymptotically periodic Schrödinger equation with indefinite linear part. Commun. Contemp. Math. 4, 763–776 (2002)

    Article  MathSciNet  Google Scholar 

  24. Schechter, M., Zou, W.: Weak linking theorems and Schrödinger equations with critical Sobolev exponent. ESAIM Control Optim. Calc. Var. 9, 601–619 (2003)

    Article  MathSciNet  Google Scholar 

  25. Szulkin, A., Weth, T.: Ground state solutions for some indefinite variational problems. J. Funct. Anal. 257, 3802–3822 (2009)

    Article  MathSciNet  Google Scholar 

  26. Szulkin, A., Weth, T.: The Method of Nehari Manifold. Handbook of Nonconvex Analysis and Applications, pp. 597–632. International Press, Somerville (2010)

    MATH  Google Scholar 

  27. Tang, X.H.: Non-Nehari manifold method for asymptotically periodic Schrödinger equations. Sci. China Math. 58, 715–728 (2015)

    Article  MathSciNet  Google Scholar 

  28. Tang, X.H., Lin, X.Y., Yu, J.S.: Existence of a bound state solution for quasilinear Schrödinger equations. J. Dyn. Differ. Equ. 31, 369–383 (2019)

    Article  Google Scholar 

  29. Tang, X.H., Chen, S.T., Lin, X., Yu, J.S.: Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions. J. Differ. Equ. 268, 4663–4690 (2020)

    Article  Google Scholar 

  30. Willem, M.: Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, 24. Birkhäuser Boston Inc., Boston (1996)

    Google Scholar 

  31. Zhang, H., Xu, J., Zhang, F.: On a class of semilinear Schrödinger equations with indefinite linear part. J. Math. Anal. Appl. 414, 710–724 (2014)

    Article  MathSciNet  Google Scholar 

  32. Zhang, H., Xu, J., Zhang, F.: Ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part. Math. Methods Appl. Sci. 38, 113–122 (2015)

    Article  MathSciNet  Google Scholar 

  33. Zhang, J., do Ó, J.M.: Standing waves for nonlinear Schrödinger equations involving critical growth of Trudinger–Moser type. Z. Angew. Math. Phys. 66, 3049–3060 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Sitong Chen & Xianhua Tang

Authors
  1. Sitong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xianhua Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xianhua Tang.

Additional information

Communicated by A. Malchiodi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is partially supported by the National Natural Science Foundation of China (No: 11971485; No: 12001542)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, S., Tang, X. On the planar Schrödinger equation with indefinite linear part and critical growth nonlinearity. Calc. Var. 60, 95 (2021). https://doi.org/10.1007/s00526-021-01963-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-021-01963-1

Keywords

Mathematics Subject Classification

Search

Navigation