Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-23T07:25:24.077Z Has data issue: false hasContentIssue false
  • English
  • Français

Infinitely Many Solutions for Schrödinger–Kirchhoff-Type Fourth-Order Elliptic Equations

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  30 January 2017

Hongxue Song  and
Caisheng Chen
Hongxue Song*
Affiliation:
College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, People's Republic of China (songhx@njupt.edu.cn) College of Science, Hohai University, Nanjing 210098, People's Republic of China
Caisheng Chen
Affiliation:
College of Science, Hohai University, Nanjing 210098, People's Republic of China
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper deals with the class of Schrödinger–Kirchhoff-type biharmonic problems

where Δ2 denotes the biharmonic operator, and f ∈ C(ℝN × ℝ, ℝ) satisfies the Ambrosetti–Rabinowitz-type conditions. Under appropriate assumptions on V and f, the existence of infinitely many solutions is proved by using the symmetric mountain pass theorem.

Keywords

biharmonic operatorPalais–Smale sequencessymmetric mountain pass theoremKirchhoff equation

MSC classification

Secondary: 35J30: Higher-order elliptic equations 35J35: Variational methods for higher-order elliptic equations 35J62: Quasilinear elliptic equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 4 , November 2017 , pp. 1003 - 1020
Copyright
Copyright © Edinburgh Mathematical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Azzollini, A., D’Avenia, P. and Pomponio, A., Multiple critical points for a class of nonlinear functionals, Annali Mat. Pura Appl. 190(3) (2011), 507–523.CrossRefGoogle Scholar
2. Ball, J. M., Initial-boundary value problems for an extensible beam, J. Math. Analysis Applic. 42 (1973), 61–90.Google Scholar
3. Berger, H. M., A new approach to the analysis of large deflections of plates, J. Appl. Mech. 22 (1955), 465–472.Google Scholar
4. Cabada, A. and Figueiredo, G. M., A generalization of an extensible beam equation with critical growth in ℝ N , Nonlin. Analysis 20 (2014), 134–142.Google Scholar
5. Colasuonno, F. and Pucci, P., Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlin. Analysis 74(17) (2011), 5962–5974.Google Scholar
6. Corrêa, F. J. S. A. and Figueiredo, G. M., On an elliptic equation of p-Kirchhoff type via variational methods, Bull. Austral. Math. Soc. 74(2) (2006), 263–277.Google Scholar
7. Drábek, P. and Pohozaev, S. I., Positive solutions for the p-Laplacian: application of the fibering method, Proc. R. Soc. Edinb. A127(4) (1997), 703–726.Google Scholar
8. He, X. and Zou, W., Existence and concentration behavior of positive solutions for a Kirchhoff equation in ℝ3 , J. Diff. Eqns 252(2) (2012), 1813–1834.Google Scholar
9. Jeanjean, L., On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝ N , Proc. R. Soc. Edinb. A129(4) (1999), 787–809.Google Scholar
10. Kajikiya, R., A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Analysis 225(2) (2005), 352–370.Google Scholar
11. Lazer, A. C. and McKenna, P. J., Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev. 32(4) (1990), 537–578.CrossRefGoogle Scholar
12. Liu, S. B. and Li, S. J., Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica 46(4) (2003), 625–630.Google Scholar
13. Liu, W. and He, X., Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput. 39 (2012), 473–487.Google Scholar
14. Ma, T. F., Positive solutions for a nonlinear Kirchhoff type beam equation, Appl. Math. Lett. 18(4) (2005), 479–482.CrossRefGoogle Scholar
15. Ma, T. F. and A. Martinez, L. M., Positive solutions for a fourth order equation with nonlinear boundary conditions, Math. Comput. Simulat. 80(11) (2010), 2177–2184.Google Scholar
16. Rabinowitz, P. H., Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, Volume 65 (American Mathematical Society, Providence, RI, 1986).Google Scholar
17. Wang, F., Avci, M. and An, Y., Existence of solutions for fourth order elliptic equations of Kirchhoff type, J. Math. Analysis Applic. 409(1) (2014), 140–146.Google Scholar
18. Wang, W., Zang, A. and Zhao, P., Multiplicity of solutions for a class of fourth order elliptic equations, Nonlin. Analysis 70(12) (2009), 4377–4385.CrossRefGoogle Scholar
19. Yang, Y. and Zhang, J., Existence of solutions for some fourth-order nonlinear elliptic equations, J. Math. Analysis Applic. 351(1) (2009), 128–137.Google Scholar
20. Yin, Y. and Wu, X., High energy solutions and nontrivial solutions for fourth-order elliptic equations, J. Math. Analysis Applic. 375(2) (2011), 699–705.Google Scholar