Taiwanese Journal of Mathematics

MULTIPLE SOLUTIONS FOR THE NONHOMOGENEOUS FOURTH ORDER ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE

Liping Xu and Haibo Chen

Full-text: Open access

Abstract

This paper considers the following nonhomogeneous fourth order elliptic equations of Kirchhoff type:\begin{eqnarray*}\begin{cases}\displaystyle\triangle ^2u-(a+b\int_{\text R^N}|\nabla u|^2dx)\triangle u+V(x)u= f(x,u)+h(x),~\text{in}~ \text R^N,\\ \displaystyle u\in H^2(\text R^N), \end{cases}\end{eqnarray*}where constants $a\gt 0,~b\geq0$. Under certain assumptions on $V(x)$, $f(x,u)$ and $h(x)$, we show the existence and multiplicity of solutions  by the Ekeland$^{,}$s variational principle and the Mountain Pass Theorem in the critical theory.

Article information

Source
Taiwanese J. Math., Volume 19, Number 4 (2015), 1215-1226.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133697

Digital Object Identifier
doi:10.11650/tjm.19.2015.4716

Mathematical Reviews number (MathSciNet)
MR3384687

Zentralblatt MATH identifier
1357.35113

Subjects
Primary: 35J20: Variational methods for second-order elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations 35J60: Nonlinear elliptic equations

Keywords
fourth order elliptic equations of Kirchhoff type nonhomogeneous Ekeland's variational principle mountain pass theorem

Citation

Xu, Liping; Chen, Haibo. MULTIPLE SOLUTIONS FOR THE NONHOMOGENEOUS FOURTH ORDER ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE. Taiwanese J. Math. 19 (2015), no. 4, 1215--1226. doi:10.11650/tjm.19.2015.4716. https://projecteuclid.org/euclid.twjm/1499133697


Export citation

References

  • W. Zou and M. Schechter, Critical Point Theory and Its Applications, Springer, New York, 2006.
  • J. Ball, Initial-boundary value for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90.
  • H. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472.
  • F. Wang, M. Avci and Y. An, Existence of solutions for fourth order elliptic equations of Kirchhoff type, J. Math. Anal. Appl., 409 (2014), 140-146.
  • F. Wang, T. An and Y. An, Existence of solutions for fourth order elliptic equations of Kirchhoff type on $\text R^N$, Electronic Journal of Qualitative Theory of Differential Equations, 39 (2014), 1-11.
  • A. Cabada and G. M. Figueiredo, A generalization of an extensible beam equation with critical growth in $\text R^N$, Nonlinear Analysis: Real World Applications, 20 (2014), 134-142.
  • T. F. Ma, Existence results for a model of nonlinear beam on elastic bearings, Appl. Math. Lett., 13 (2000), 11-15.
  • T. F. Ma, Existence results and numerical solutions for a beam equation with nonlinear boundary conditions, Appl. Numer. Math., 47 (2003), 189-196.
  • F. Wang and Y. An, Existence and multiplicity of solutions for a fourth-order elliptic equation, Bound. Value Probl., 2012 (2012), 6.
  • Y. Yin and X. Wu, High energy solutions and nontrivial solutions for fourth-order elliptic equations, J. Math. Anal. Appl., 375 (2011), 699-705.
  • Y. Ye and C. Tang, Existence and multiplicity of solutions for fourth-order elliptic equations in $R^N$, J. Math. Anal. Appl., 406 (2013), 335-351.
  • J. Liu, S. Chen and X. Wu, Existence and multiplicity of solutions for a class of fourth order elliptic equations in $R^N$, J. Math. Anal. Appl., 395 (2012), 608-615.
  • X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $R^N$, Nonlinear Analysis:Real World Applications, 12 (2011), 1278-1287.
  • Z. Liu and S. Guo, Positive solutions for asymptotically linear Schrödinger-Kirchhoff-type equations, Math. Meth. Appl. Sci., 37 (2014), 571-580.
  • A. Ambrosetti and P. H. Rabinowitz, variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
  • I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.