Existence results for a model of nonlinear beam on elastic bearings. (English) Zbl 0965.74030

Summary: We study the existence of solutions of the nonlinear fourth-order equation of Kirchhoff type \(u^{(iv)}- m(\int^1_0 |u'(x) |^2 dx)u''+f (x,u)=0\) under nonlinear boundary conditions, which models the deformations of beams on elastic bearings.


74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74G25 Global existence of solutions for equilibrium problems in solid mechanics (MSC2010)
Full Text: DOI


[1] Kirchhoff, G., Vorlessungen über Mathematiche Physik: Mechanik (1876), Teubner: Teubner Leipzig
[2] Doppel, K.; Pflüger, K.; Herfort, W., A nonlinear beam equation arising in the theory of bluff bodies, Z. Anal. Anwend., 16, 945-960 (1997) · Zbl 0902.35066
[3] Medeiros, L. A., On a new class of nonlinear wave equations, J. Math. Anal. Appl., 69, 252-262 (1979) · Zbl 0407.35051
[4] Aftabizadeh, A. R., Existence and uniqueness theorems for fourth order boundary value problems, J. Math. Anal. Appl., 116, 415-426 (1986) · Zbl 0634.34009
[5] Agarwal, R. P., Boundary Value Problems for Higher Order Differential Equations (1986), World Scientific: World Scientific Singapore · Zbl 0598.65062
[6] Gupta, C. P., Existence and uniqueness theorems for a fourth order boundary value problem of Sturm-Liouville type, Diff. Int. Equations, 4, 397-410 (1991) · Zbl 0728.34019
[7] Sanchez, L., Boundary value problems for some fourth order ordinary differential equations, Applicable Analysis, 38, 161-177 (1990) · Zbl 0682.34020
[8] Feireisl, E., Nonzero time periodic solutions to an equation of Petrovsky type with nonlinear boundary conditions: Slow oscillations of beams on elastic bearings, Ann. Sc. Norm. Sup. Pisa, 20, 133-146 (1993) · Zbl 0794.73029
[9] Grossinho, M. R.; Ma, T. F., Symmetric equilibria for a beam with a nonlinear elastic foundation, Portugaliae Mathematica, 51, 375-393 (1994) · Zbl 0815.34014
[10] Grossinho, M. R.; Ma, T. F., Nontrivial solutions for a fourth order ODE with singular boundary conditions, (Bainov, D., Proceedings of the Seventh International Colloquium on Differential Equations (1997), VSP: VSP Netherlands), 123-130 · Zbl 0956.34016
[11] Mawhin, J., Critical point theory and nonlinear differential equations, (Vosmansky, J.; Zlamal, M., Differential Equations and Their Applications, Equadiff 6 Lect. Notes in Math. 1192 (1986), Springer-Verlag: Springer-Verlag New York), 49-58 · Zbl 0617.58015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.