×

Existence and multiplicity of positive solutions of a boundary-value problem for sixth-order ODE with three parameters. (English) Zbl 1203.34045

Summary: We study the existence and multiplicity of positive solutions of the following boundary-value problem:
\[ -u^{(6)} - \gamma u^{(4)} + \beta u'' - \alpha u = f(t,u),\quad 0<t<1, \]
\[ u(0) = u(1) = u''(0) = u''(1) = u^{(4)}(0) = u^{(4)}(1)=0, \]
where \(f:[0,1]\times {\mathbb{R}}^+\to {\mathbb{R}}^+\) is continuous, \(\alpha\), \(\beta\), and \(\gamma\to\mathbb R\) satisfy some suitable assumptions.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B08 Parameter dependent boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Gardner RA, Jones CKRT: Traveling waves of a perturbed diffusion equation arising in a phase field model.Indiana University Mathematics Journal 1990,39(4):1197-1222. 10.1512/iumj.1990.39.39054 · Zbl 0799.35106 · doi:10.1512/iumj.1990.39.39054
[2] Caginalp G, Fife PC: Higher-order phase field models and detailed anisotropy.Physical Review. B 1986,34(7):4940-4943. 10.1103/PhysRevB.34.4940 · doi:10.1103/PhysRevB.34.4940
[3] Gyulov T, Morosanu G, Tersian S: Existence for a semilinear sixth-order ODE.Journal of Mathematical Analysis and Applications 2006,321(1):86-98. 10.1016/j.jmaa.2005.08.007 · Zbl 1106.34007 · doi:10.1016/j.jmaa.2005.08.007
[4] Peletier LA, Troy WC: Spatial Patterns, Progress in Nonlinear Differential Equations and their Applications. Volume 45. Birkhäuser Boston, Boston, Mass, USA; 2001:xvi+341.
[5] Peletier LA, Rottschäfer V: Large time behaviour of solutions of the Swift-Hohenberg equation.Comptes Rendus Mathématique. Académie des Sciences. Paris 2003,336(3):225-230. · Zbl 1031.35072 · doi:10.1016/S1631-073X(03)00021-9
[6] Tersian S, Chaparova J: Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations.Journal of Mathematical Analysis and Applications 2001,260(2):490-506. 10.1006/jmaa.2001.7470 · Zbl 0984.34031 · doi:10.1006/jmaa.2001.7470
[7] Li Y: Positive solutions of fourth-order boundary value problems with two parameters.Journal of Mathematical Analysis and Applications 2003,281(2):477-484. 10.1016/S0022-247X(03)00131-8 · Zbl 1030.34016 · doi:10.1016/S0022-247X(03)00131-8
[8] Fan S: The new root formula and criterion of cubic equation.Journal of Hainan Normal University 1989, 2: 91-98.
[9] Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275. · Zbl 0661.47045
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.