Korean J. Math. Vol. 23 No. 3 (2015) pp.379-391
DOI: https://doi.org/10.11568/kjm.2015.23.3.379

Nonlinear biharmonic equation with polynomial growth nonlinear term

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Published: Sep 30, 2015
Keywords:
Biharmonic boundary value problem, polynomial growth, variational method, generalized mountain pass geometry, critical point theory, (P.S.) condition.
Mathematics Subject Classification:
35J35, 35J40.

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Tacksun Jung
Department of Mathematics Kunsan National University Kunsan 573-701, Korea
Q-Heung Choi
Department of Mathematics Education Inha University Incheon 402-751, Korea

Abstract

We investigate the existence of solutions of the nonlinear biharmonic equation with variable coefficient polynomial growth nonlinear term and Dirichlet boundary condition. We get a theorem which shows that there exists a bounded solution and a large norm solution depending on the variable coefficient. We obtain this result by variational method, generalized mountain pass geometry and critical point theory.


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Supporting Agencies

This work was financially supported by the Kunsan National University’s re- search program for faculty member in the year 2014.

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] Khanfir, S. and Lassoued, L., On the existence of positive solutions of a semi-linear elliptic equation with change of sign, Nonlinear Analysis, TMA, 22, No.11, 1309-1314(1994). Google Scholar

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