Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지

EXISTENCE OF NONTRIVIAL SOLUTIONS OF A NONLINEAR BIHARMONIC EQUATION

  • Received : 2009.10.11
  • Published : 2009.12.30
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Abstract

We consider the existence of solutions of a nonlinear biharmonic equation with Dirichlet boundary condition, ${\Delta}^2u+c{\Delta}u=f(x, u)$ in ${\Omega}$, where ${\Omega}$ is a bounded open set in $R^N$ with smooth boundary ${\partial}{\Omega}$. We obtain two new results by linking theorem.

Keywords

Acknowledgement

Supported by : Jiangnan University

References

  1. Q. H. Choi and Yinghua Jin, Nonlinearity and nontrivial solutions of fourth order semilinear elliptic equations. Journ. Math. Anal. Appl. 29 (2004),224-234.
  2. Q. H. Choi, T. Jung and P. J. McKenna, The study of a nonlinear suspension bridge equation by a variational reduction method. Appl. Anal. 50 (1993),71-90.
  3. H. Hofer, On strongly indefinite functionals with applications. Trans. Amer. Math. Soc. 275 (1983), 185-214. https://doi.org/10.1090/S0002-9947-1983-0678344-2
  4. L. Humphreys, Numerical and theoretical results on large amplitude periodic solutions of a suspension bridge equation. ph. D. thesis, University of Connecticut (1994).
  5. P. J. McKenna and W. Walter, Nonlinear Oscillations in a Suspension Bridge. Arch. Rational Mech. Anal. 98 (1987), 167-177.
  6. P. J. McKenna and W. Walter, Global bifurcation and a Theorem of Tarantello. Journ. Math. Anal. Appl. 181 (1994), 648-655. https://doi.org/10.1006/jmaa.1994.1049
  7. A. M. Micheletti and C. Saccon, Multiple nontrivial solutions for a floating beam via critical point theory. J. Differential Equations, 170 (2001), 157-179. https://doi.org/10.1006/jdeq.2000.3820
  8. S. Li and A. Squlkin, Periodic solutions of an asymptotically linear wave equation. Nonlinear Analysis, 1 (1993), 211-230.
  9. J. Q. Liu, Free vibrations for an asymmetric beam equation. Nonlinear Analysis, 51 (2002), 487-497. https://doi.org/10.1016/S0362-546X(01)00841-0
  10. Tarantello G, A note on a semilinear elliptic problem. Differential and Integral Equations. 5 (1992), 561-566.