Skip to main content
SpringerLink
Log in

Existence of three nontrivial solutions for semilinear elliptic equations on ℝN

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

We establish the existence theorem of three nontrivial solutions for a class of semilinear elliptic equation on ℝN by using variational theorems of mixed type due to Marino and Saccon and linking theorem.

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. Ambrosetti A, Rabinowitz P H. Dual variational methods in critical point theory and applications. J Funct Anal, 1973, 14: 349–381

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Capozzi A, Lupo D, Solimini S. On the existence of a nontrivial solution to nonlinear problems at resonance. Nonlinear Anal, 1989, 13: 151–163

    Article  MathSciNet  MATH  Google Scholar 

  4. Costa D G, Silva E A. On a class of resonant problems at higher eigenvalues. Differential Integral Equations, 1995, 8: 663–671

    MathSciNet  MATH  Google Scholar 

  5. Ding Y H, Li S J. Some existence results of solutions for the semilinear elliptic equations on RN. J Differential Equations, 1995, 119: 401–425

    Article  MathSciNet  MATH  Google Scholar 

  6. Jeanjean L, Tanaka K. A positive solution for a nonlinear schrödinger equation on RN. Indiana Univ Math J, 2005, 54: 443–464

    Article  MathSciNet  MATH  Google Scholar 

  7. Li S J, Willem M. Applications of local linking to critical point theory. J Math Anal Appl, 1995, 189: 6–32

    Article  MathSciNet  MATH  Google Scholar 

  8. Magrone P, Mugnai D, Servadei R. Multiplicity of solutions for semilinear variational inequalities via linking and theorems. J Differential Equations, 2006, 228: 191–225

    Article  MathSciNet  MATH  Google Scholar 

  9. Marino A, Mugnai D. Asymptotical multiplicity and some reversed variational inequalities. Topol Methods Nonlinear Anal, 2002, 20: 43–62

    MathSciNet  MATH  Google Scholar 

  10. Marino A, Saccon C. Some variational theorems of mixed type and elliptic problems with jumping nonlinearities. Ann Sc Norm Super Pisa Cl Sci, 1997, 4: 631–665

    MathSciNet  MATH  Google Scholar 

  11. Marino A, Saccon C. Asymptotically critical points and multiple elastic bounce trajectories. Topol Methods Nonlinear Anal, 2007, 30: 351–395

    MathSciNet  MATH  Google Scholar 

  12. Mugnai D. On a reversed variational inequality. Topol Methods Nonlinear Anal, 2001, 17: 321–358

    MathSciNet  MATH  Google Scholar 

  13. Mugnai D. Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem. NoDEA Nonlinear Differential Equations Appl, 2004, 11: 379–391

    Article  MathSciNet  MATH  Google Scholar 

  14. Mugnai D. Four nontrivial solutions for subcritical exponential equations, Calc Var Partial Differential Equations, 2008, 32: 481–497

    Article  MathSciNet  MATH  Google Scholar 

  15. Mugnai D. Existence and multiplicity results for the fractional Laplacian in bounded domains. Adv Calc Var, DOI: 10.1515/acv-2015-0032

  16. Ou Z Q, Li C. Existence of three nontrivial solutions for a class of superlinear elliptic equations. J Math Anal Appl, 2012, 390: 418–426

    Article  MathSciNet  MATH  Google Scholar 

  17. Rabinowitz P H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg Conf Ser Math, Vol 65. Providence: Amer Math Soc, 1986

    Book  MATH  Google Scholar 

  18. Stuart C A, Zhou H S. Applying the mountain pass theorem to an asymptotically linear elliptic equation on RN. Comm Partial Differential Equations, 1999, 24: 1731–1758

    Article  MathSciNet  MATH  Google Scholar 

  19. Tehrani H T. A note on asymptotically linear elliptic problems in RN. J Math Anal Appl, 2002, 271: 546–554

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang F. Multiple solutions for some nonlinear Schrödinger equations with indefinite linear part. J Math Anal Appl, 2007, 331: 1001–1022

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang F. Multiple solutions for some Schrodinger equations with convex and critical nonlinearities in RN. J Math Anal Appl, 2008, 342: 255–276

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang W, Zang A, Zhao P. Multiplicity of solutions for a class of fourth elliptic equations. Nonlinear Anal, 2009, 70: 4377–4385

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Tianshui Normal University, Tianshui, 741001, China

    Ruichang Pei

  2. Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University, Nanjing, 210097, China

    Jihui Zhang

Authors
  1. Ruichang Pei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jihui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ruichang Pei.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pei, R., Zhang, J. Existence of three nontrivial solutions for semilinear elliptic equations on ℝN . Front. Math. China 11, 723–735 (2016). https://doi.org/10.1007/s11464-016-0538-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-016-0538-7

Keywords

MSC

Search

Navigation