Skip to main content
SpringerLink
Log in

Infinitely many solutions for a nonlinear Schrödinger equation with general nonlinearity

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

Abstract

We prove the existence of infinitely many solutions for

$$\begin{aligned} - \Delta u + V(x) u = f(u) \quad \text { in } \mathbb {R}^N, \quad u \in H^1(\mathbb {R}^N), \end{aligned}$$

where V(x) satisfies \(\lim _{|x| \rightarrow \infty } V(x) = V_\infty >0\) and some conditions. We require conditions on f(u) only around 0 and at \(\infty \).

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. Azzollini, A., Pomponio, A.: On the Schrödinger equation in \({\mathbb{R}}^N\) under the effect of a general nonlinear term. Indiana Univ. Math. J. 58(3), 1361–1378 (2009)

    Article  MathSciNet  Google Scholar 

  2. Bahri, A., Lions, P.-L.: On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 14(3), 365–413 (1997). (English, with English and French summaries)

    Article  MathSciNet  Google Scholar 

  3. Cerami, G., Devillanova, G., Solimini, S.: Infinitely many bound states for some nonlinear scalar field equations. Calc. Var. Partial Differ. Equ. 23(2), 139–168 (2005)

    Article  MathSciNet  Google Scholar 

  4. Cerami, G., Molle, R., Passaseo, D.: Multiplicity of positive and nodal solutions for scalar field equations. J. Differ. Equ. 257(10), 3554–3606 (2014)

    Article  MathSciNet  Google Scholar 

  5. Cerami, G., Passaseo, D., Solimini, S.: Nonlinear scalar field equations: existence of a positive solution with infinitely many bumps. Ann. Inst. H. Poincaré Anal. Non Linéaire 32(1), 23–40 (2015)

    Article  MathSciNet  Google Scholar 

  6. Cerami, G., Passaseo, D., Solimini, S.: Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients. Commun. Pure Appl. Math. 66(3), 372–413 (2013)

    Article  MathSciNet  Google Scholar 

  7. Esteban, M.J., Lions, P.-L.: Existence and nonexistence results for semilinear elliptic problems in unbounded domains. Proc. R. Soc. Edinb. Sect. A 93(1–2), 1–14 (1982/83)

    Article  MathSciNet  Google Scholar 

  8. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften, vol. 224. Springer, Berlin (1977)

    MATH  Google Scholar 

  9. Hirata, J.: A positive solution of a nonlinear Schrödinger equation with \(G\)-symmetry. Nonlinear Anal. 69(9), 3174–3189 (2008)

    Article  MathSciNet  Google Scholar 

  10. Jeanjean, L., Tanaka, K.: Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc. Var. Partial Differ. Equ. 21(3), 287–318 (2004)

    Article  MathSciNet  Google Scholar 

  11. Jeanjean, L., Tanaka, K.: A positive solution for a nonlinear Schrödinger equation on \({\mathbb{R}}^N\). Indiana Univ. Math. J. 54(2), 443–464 (2005)

    Article  MathSciNet  Google Scholar 

  12. Li, Y., Wang, Z.-Q., Zeng, J.: Ground states of nonlinear Schrödinger equations with potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire 23(6), 829–837 (2006)

    Article  MathSciNet  Google Scholar 

  13. Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(4), 223–283 (1984)

    Article  MathSciNet  Google Scholar 

  14. Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43(2), 270–291 (1992)

    Article  MathSciNet  Google Scholar 

  15. Sato, Y.: Multi-peak solutions for nonlinear Schrödinger equations. In: Thesis, Waseda University 14, 1–113 (2007)

  16. Sato, Y., Shibata, M.: Existence of a positive solution for nonlinear Schrödinger equations with general nonlinearity. Adv. Nonlinear Anal. 3(suppl. 1), s55–s67 (2014)

    MATH  Google Scholar 

  17. Struwe, M.: Variational Methods. Springer, Berlin (1990)

    Book  Google Scholar 

  18. Tanaka, K.: Morse indices at critical points related to the symmetric mountain pass theorem and applications. Commun. Partial Differ. Equ. 14(1), 99–128 (1989)

    Article  MathSciNet  Google Scholar 

  19. Tang, X.H.: Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity. J. Math. Anal. Appl. 401(1), 407–415 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the unknown referees for their valuable comments which improved this paper. This work was supported by JSPS KAKENHI Grant Number JP15K17567.

Author information

Authors and Affiliations

  1. Department of Mathematics, Saitama University, Shimo-Okubo 255, Sakura-ku, Saitama-shi, 338-8570, Japan

    Yohei Sato

  2. Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo, 152-8551, Japan

    Masataka Shibata

Authors
  1. Yohei Sato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masataka Shibata
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yohei Sato.

Additional information

Communicated by A. Malchiodi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sato, Y., Shibata, M. Infinitely many solutions for a nonlinear Schrödinger equation with general nonlinearity. Calc. Var. 57, 137 (2018). https://doi.org/10.1007/s00526-018-1413-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-018-1413-z

Mathematics Subject Classification

Search

Navigation