Home Global structure for a fourth-order boundary value problem with sign-changing weight
Article
Licensed
Unlicensed Requires Authentication

Global structure for a fourth-order boundary value problem with sign-changing weight

  • Fumei Ye
Published/Copyright: October 4, 2021
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 71 Issue 5

Abstract

We study the fourth-order boundary value problem with a sign-changing weight function:

u ⁗ = λ m ( t ) u + f 1 ( t , u , u ′ , u ″ , u ‴ , λ ) + f 2 ( t , u , u ′ , u ″ , u ‴ , λ ) , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 ,

where λ ∈ ℝ is a parameter, f1, f2 ∈ C([0, 1] × ℝ5, ℝ), f1 is not differentiable at the origin and infinity. Under some suitable conditions on nonlinear terms, we prove the existence of unbounded continua of positive and negative solutions of this problem which bifurcating from intervals of the line of trivial solutions or from infinity, respectively.

Keywords: interval bifurcation; principal eigenvalue; sign-changing weight; one-sign solutions
MSC 2010: Primary 34B15; 34F10; Secondary 47J10

This work was supported by the NSFC Grant No.11801453.

  1. (Communicated by Michal Fečkan)

References

[1] Aliyev, Z. S.—Huseynova, R. A.: Bifurcation in nonlinearizable eigenvalue problems for ordinary differential equations of fourth order with indefinite weight, Electron. J. Qual. Theory Differ. Equ. 92 (2017), pp. 12.10.14232/ejqtde.2017.1.92Search in Google Scholar

[2] Berestycki, H.: On some nonlinear Sturm-Liouville problems, J. Differential Equations 26 (1977), 375–390.10.1016/0022-0396(77)90086-9Search in Google Scholar

[3] Brown, K. J.—Lin, S. S.: On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl. 75 (1980), 112–120.10.1016/0022-247X(80)90309-1Search in Google Scholar

[4] Cabada, A.—Cid, J.Á.—Sanchez, L.: Positivity and lower and upper solutions for fourth order boundary value problems, Nonlinear Anal. 67 (2007), 1599–1612.10.1016/j.na.2006.08.002Search in Google Scholar

[5] Dai, G. Spectrum of Navier p-biharmonic problem with sign-changing weight, https://arxiv.org/abs/1207.7159v1.Search in Google Scholar

[6] Dai, G.: Global bifurcation from intervals for Sturm-Liouville problems which are not linearizable, Electron. J. Qual. Theory Differ. Equ. 65 (2013), 1–7.10.14232/ejqtde.2013.1.65Search in Google Scholar

[7] Dai, G. W.—Han, X. L.—Ma, R. Y.: Global bifurcation and nodal solutions for fourth-order problems with sign-changing weight, Appl. Math. Comput. 17 (2013), 9399–9407.10.1016/j.amc.2013.03.103Search in Google Scholar

[8] Drábek, P.—Holubová, G.: On the maximum and antimaximum principles for the beam equation, Appl. Math. Lett. 56 (2016), 29–33.10.1016/j.aml.2015.12.009Search in Google Scholar

[9] Huseynova, R. A.: Global bifurcation from principal eigenvalues for nonlinear fourth order eigenvalue problem with indefinite weight, roc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 42 (2016), 202–211.Search in Google Scholar

[10] Han, X. L.—Gao, T: A priori bounds and existence of non-real eigenvalues of fourth-order boundary value problem with indefinite weight function, Electron. J. Differential Equations 82 (2016), 1–9.Search in Google Scholar

[11] Liu, Y.—O'Regan, D.: Bifurcation techniques for fourth order m-point boundary value problems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 18 (2011), 215–234.Search in Google Scholar

[12] Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513.10.1016/0022-1236(71)90030-9Search in Google Scholar

[13] Rabinowitz, P. H.: Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161–202.10.1216/RMJ-1973-3-2-161Search in Google Scholar

[14] Rabinowitz, P. H.: On bifurcation from infinity, J. Differential Equations 14 (1973), 462–475.10.1016/0022-0396(73)90061-2Search in Google Scholar

[15] Rynne, B. P.: Bifurcation from zero or infinity in Sturm-Liouville problems which are not linearizable, J. Math. Anal. Appl. 228 (1998), 141–156.10.1006/jmaa.1998.6122Search in Google Scholar

[16] Yao, Q. L.: Existence, multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem, Nonlinear Anal. 63 (2005), 237–246.10.1016/j.na.2005.05.009Search in Google Scholar
Received: 2020-09-07
Accepted: 2020-11-09
Published Online: 2021-10-04
Published in Print: 2021-10-26

© 2021 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Regular papers
  2. Algebraic structures for pairwise comparison matrices: Consistency, social choices and Arrow’s theorem
  3. Polynomial functions on rings of dual numbers over residue class rings of the integers
  4. Sufficient conditions for p-valent functions
  5. Upper bounds for analytic summand functions and related inequalities
  6. Global structure for a fourth-order boundary value problem with sign-changing weight
  7. On the nonexistence conditions of solution of two-point in time problem for nonhomogeneous PDE
  8. Solvability of a nonlinear three-dimensional system of difference equations with constant coefficients
  9. Properties of critical and subcritical second order self-adjoint linear equations
  10. Korovkin type approximation via statistical e-convergence on two dimensional weighted spaces
  11. Approximation by a new sequence of operators involving Apostol-Genocchi polynomials
  12. Poisson like matrix operator and its application in p-summable space
  13. On the homological and algebraical properties of some Feichtinger algebras
  14. Disjoint topological transitivity for weighted translations generated by group actions
  15. On simultaneous limits for aggregation of stationary randomized INAR(1) processes with poisson innovations
  16. Marshall-Olkin Lindley-Log-logistic distribution: Model, properties and applications
  17. The shifted Gompertz-G family of distributions: Properties and applications
  18. On the testing hypothesis in uniform family of distributions with nuisance parameter
  19. Clarkson inequalities related to convex and concave functions
https://doi.org/10.1515/ms-2021-0042
Keywords for this article
interval bifurcation; principal eigenvalue; sign-changing weight; one-sign solutions

Articles in the same Issue

  1. Regular papers
  2. Algebraic structures for pairwise comparison matrices: Consistency, social choices and Arrow’s theorem
  3. Polynomial functions on rings of dual numbers over residue class rings of the integers
  4. Sufficient conditions for p-valent functions
  5. Upper bounds for analytic summand functions and related inequalities
  6. Global structure for a fourth-order boundary value problem with sign-changing weight
  7. On the nonexistence conditions of solution of two-point in time problem for nonhomogeneous PDE
  8. Solvability of a nonlinear three-dimensional system of difference equations with constant coefficients
  9. Properties of critical and subcritical second order self-adjoint linear equations
  10. Korovkin type approximation via statistical e-convergence on two dimensional weighted spaces
  11. Approximation by a new sequence of operators involving Apostol-Genocchi polynomials
  12. Poisson like matrix operator and its application in p-summable space
  13. On the homological and algebraical properties of some Feichtinger algebras
  14. Disjoint topological transitivity for weighted translations generated by group actions
  15. On simultaneous limits for aggregation of stationary randomized INAR(1) processes with poisson innovations
  16. Marshall-Olkin Lindley-Log-logistic distribution: Model, properties and applications
  17. The shifted Gompertz-G family of distributions: Properties and applications
  18. On the testing hypothesis in uniform family of distributions with nuisance parameter
  19. Clarkson inequalities related to convex and concave functions
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 26.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2021-0042/pdf
Scroll to top button