Korean J. Math. Vol. 22 No. 4 (2014) pp.633-644
DOI: https://doi.org/10.11568/kjm.2014.22.4.633

# At least two solutions for the semilinear biharmonic boundary value problem

## Abstract

We get one theorem that there exists a unique solution for the fourth order semilinear elliptic Dirichlet boundary value problem when
the number 0 and the coefficient of the semilinear part belong to the same open interval made by two successive eigenvalues of the fourth order elliptic eigenvalue problem. We prove this result by the contraction mapping principle.
We also get another theorem that there exist at least two solutions when there exist $n$ eigenvalues
of the fourth order elliptic eigenvalue problem between the coefficient of the semilinear part and the number 0.
We prove this result by the critical point theory and the variation of linking method.

## Supporting Agencies

This work was supported by Basic Science Research Program through the Na- tional Research Foundation of Korea(NRF) funded by the Ministry of Education Science and Technology (KRF-2013010343).

## References

[1] Chang, k. C., Infinite dimensional Morse theory and multiple solution problems, Birkh ̈auser (1993). Google Scholar

[2] Choi, Q. H. and Jung, T., Multiplicity of solutions and source terms in a fourth order nonlinear elliptic equation, Acta Mathematica Scientia 19 (4) (1999), 361– 374. Google Scholar

[3] Choi, Q. H. and Jung, T., Multiplicity results on nonlinear biharmonic operator, Rocky Mountain J. Math. 29 (1) (1999), 141–164. Google Scholar

[4] Choi, Q. H. and Jung, T., An application of a variational reduction method to a nonlinear wave equation, J. Differential Equations 7 (1995), 390–410. Google Scholar

[5] Jung, T. S. and Choi, Q. H., Multiplicity results on a nonlinear biharmonic equation, Nonlinear Analysis, Theory, Methods and Applications 30 (8) (1997), 5083–5092. Google Scholar

[6] Lazer, A. C. and McKenna, P. J., Multiplicity results for a class of semilinear elliptic and parabolic boundary value problems, J. Math. Anal. Appl. 107 (1985), 371–395. Google Scholar

[7] McKenna, P. J. and Walter W., On the multiplicity of the solution set of some nonlinear boundary value problems, Nonlinear Analysis, Theory, Methods and Applications, 8 (1984), 893–907. Google Scholar

[8] Micheletti, A. M. and Pistoia, A., Multiplicity results for a fourth-order semi- linear elliptic problem, Nonlinear Analysis TMA, 31 (7) (1998), 895–908. Google Scholar

[9] Rabinowitz, P. H., Minimax methods in critical point theory with applications to differential equations, CBMS. Regional conf. Ser. Math., 65, Amer. Math. Soc., Providence, Rhode Island (1986). Google Scholar

[10] Tarantello, A note on a semilinear elliptic problem, Diff. Integ. Equat. 5 (3) (1992), 561–565. Google Scholar