Abstract and Applied Analysis

Multiple Results to Some Biharmonic Problems

Abstract

We study a nonlinear elliptic problem defined in a bounded domain involving biharmonic operator together with an asymptotically linear term. We establish at least three nontrivial solutions using the topological degree theory and the critical groups.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 267052, 6 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412279735

Digital Object Identifier
doi:10.1155/2014/267052

Mathematical Reviews number (MathSciNet)
MR3198170

Zentralblatt MATH identifier
07022051

Citation

Tang, Xingdong; Zhang, Jihui. Multiple Results to Some Biharmonic Problems. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 267052, 6 pages. doi:10.1155/2014/267052. https://projecteuclid.org/euclid.aaa/1412279735

References

• P. J. McKenna and W. Walter, “Nonlinear oscillations in a suspension bridge,” Archive for Rational Mechanics and Analysis, vol. 98, no. 2, pp. 167–177, 1987.
• A. M. Micheletti and A. Pistoia, “Nontrivial solutions for some fourth order semilinear elliptic problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 4, pp. 509–523, 1998.
• J. Zhang, “Existence results for some fourth-order nonlinear elliptic problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 45, no. 1, pp. 29–36, 2001.
• S. T. Kyritsi and N. S. Papageorgiou, “On superquadratic periodic systems with indefinite linear part,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 946–954, 2010.
• A. X. Qian and S. J. Li, “Multiple solutions for a fourth-order asymptotically linear elliptic problem,” Acta Mathematica Sinica, vol. 22, no. 4, pp. 1121–1126, 2006.
• R. Ma and H. Wang, “On the existence of positive solutions of fourth-order ordinary differential equations,” Applicable Analysis, vol. 59, no. 1–4, pp. 225–231, 1995.
• Z. Wei and C. Pang, “Positive solutions and multiplicity of fourth-order $m$-point boundary value problems with two parameters,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 5, pp. 1586–1598, 2007.
• J. Xu and Z. Yang, “Positive solutions for a fourth order $p$-Laplacian boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 7, pp. 2612–2623, 2011.
• M. Zhang and Z. Wei, “Existence of positive solutions for fourth-order $m$-point boundary value problem with variable parameters,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1417–1431, 2007.
• H. Hofer, “Variational and topological methods in partially ordered Hilbert spaces,” Mathematische Annalen, vol. 261, no. 4, pp. 493–514, 1982.
• G. Q. Zhang, “Variational methods and sub- and supersolutions,” Scientia Sinica A: Mathematical, Physical, Astronomical & Technical Sciences, vol. 26, no. 12, pp. 1256–1265, 1983.
• T. Bartsch, K.-C. Chang, and Z.-Q. Wang, “On the Morse indices of sign changing solutions of nonlinear elliptic problems,” Mathematische Zeitschrift, vol. 233, no. 4, pp. 655–677, 2000.
• K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, vol. 6 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Boston, Mass, USA, 1993.
• Z. Liu and J. Sun, “An elliptic problem with jumping nonlinearities,” Nonlinear Analysis; Theory, Methods & Applications, vol. 63, no. 8, pp. 1070–1082, 2005.
• H. Amann, “Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces,” SIAM Review, vol. 18, no. 4, pp. 620–709, 1976.
• J. Chabrowski and J. Marcos do Ó, “On some fourth-order semilinear elliptic problems in ${\mathbb{R}}^{n}$,” Nonlinear Analysis: Theory, Methods & Applications, vol. 49, no. 6, pp. 861–884, 2002.
• X. Xu, “Multiple sign-changing solutions for some m-point boundary-value problems,” Electronic Journal of Differential Equations, vol. 89, pp. 1–14, 2004.
• F. Li and Y. Li, “Multiple sign-changing solutions to semilinear elliptic resonant problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 9-10, pp. 3820–3827, 2010.
• Z. Liu, “Positive solutions of superlinear elliptic equations,” Journal of Functional Analysis, vol. 167, no. 2, pp. 370–398, 1999.
• G. Q. Zhang, “A variant mountain pass lemma,” Scientia Sinica A: Mathematical, Physical, Astronomical & Technical Sciences, vol. 26, no. 12, pp. 1241–1255, 1983.
• J. Su, “Multiplicity results for asymptotically linear elliptic problems at resonance,” Journal of Mathematical Analysis and Applications, vol. 278, no. 2, pp. 397–408, 2003. \endinput