Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지

EXISTENCE AND MULTIPLICITY RESULTS FOR SOME FOURTH ORDER SEMILINEAR ELLIPTIC PROBLEMS

  • Received : 2009.10.13
  • Published : 2009.12.30
Download PDF
⟨ Previous Next ⟩

Abstract

We prove the existence and multiplicity of nontrivial solutions for a fourth order problem ${\Delta}^2u+c{\Delta}u={\alpha}u-{\beta}(u+1)^-$ in ${\Omega}$, ${\Delta}u=0$ and $u=0$ on ${\partial}{\Omega}$, where ${\lambda}_1{\leq}c{\leq}{\lambda}_2$ (where $({\lambda}_i)_{i{\geq}1}$ is the sequence of the eigenvalues of $-{\Delta}$ in$H_0^1({\Omega})$) and ${\Omega}$ is a bounded open set in $R^N$ with smooth boundary ${\partial}{\Omega}$. The results are proved by applying minimax arguments and linking theory.

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. 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
  3. L. Humphreys, Numerical and theoretical results on large amplitude periodic solutions of a suspension bridge equation. ph.D. thesis, University of Connecticut (1994).
  4. 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
  5. 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
  6. S. Li, A. Squlkin, Periodic solutions of an asymptotically linear wave equation. Nonlinear Analysis, 1 (1993), 211-230.
  7. 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
  8. Tarantello G, A note on a semilinear elliptic problem. Differential and Integral Equations. 5 (1992), 561-566.
  9. Yinghua Jin, Q-Heung Choi and Tacksun Jung, Existence of three solutions for some elliptic equations with jumping nonlinearities. Nonlinear Analysis, 65 (2005), e1935-e1941.