Existence and nonexistence of positive solutions for fourth-order elliptic problems
DOI:
https://doi.org/10.58997/ejde.2023.52Abstract
This article studies a fourth-order elliptic problem with and without an eigenvalue parameter. New criteria for the existence and nonexistence of positive solution are established under some sublinear conditions which involve the principal eigenvalues of the corresponding linear problems. The interesting point is that the nonlinear term \(f\) is involved in the
second-order derivative explicitly.
For more information see https://ejde.math.txstate.edu/Volumes/2023/52/abstr.html
References
Ravi P Agarwal, Leonid Berezansky, Elena Braverman, Alexander Domoshnitsky; Nonoscillation theory of functional differential equations with applications, Springer Science & Business Media, 2012.
D. R. Anderson, R. I. Avery; A fourth-order four-point right focal boundary value problem, The Rocky Mountain Journal of Mathematics (2006), 367-380.
N. V. Azbelev, A. Domoshnitsky; A question concerning linear differential inequalities, Differentsialanye uravnenija 27 (1991), no. 1, 257-263.
Nikolaj V. Azbelev, Viktor Petrovich Maksimov, L Rakhmatullina; Introduction to the theory of linear functional differential equations, vol. 3, World Federation Publishers, Incorporated, 1995.
N. V. Azbelev, VP Maksimov, L. F. Rakhmatullina; Introduction to the theory of functional differential equations, Nauka, Moscow, Russian, vol. 34123, 1991.
C. Bai, D. Yang, H. Zhu; Existence of solutions for fourth order differential equation with four-point boundary conditions, Applied Mathematics Letters 20 (2007), no. 11, 1131-1136.
Z. Bai; Iterative solutions for some fourth-order periodic boundary value problems, Taiwanese Journal of Mathematics 12 (2008), no. 7, 1681-1690.
Zhanbing Bai; The method of lower and upper solutions for a bending of an elastic beam equation, Journal of Mathematical Analysis and Applications 248 (2000), no. 1, 195-202.
A. Cabada, J. A. Cid, L. Sanchez; Positivity and lower and upper solutions for fourth order boundary value problems, Nonlinear Analysis: Theory, Methods & Applications 67 (2007), no. 5, 1599-1612.
A. Cabada, R. R. Enguica; Positive solutions of fourth order problems with clamped beam boundary conditions, Nonlinear Analysis: Theory, Methods & Applications 74 (2011), no. 10, 3112-3122.
S. A. Chaplygin; Foundations of new method of approximate integration of differential equations, Moscow, l919 (Collected works 1, GosTechlzdat, 1948, 348-368) (1919).
S. Chen, W. Ni, C. Wang; Positive solution of fourth order ordinary differential equation with four-point boundary conditions, Applied Mathematics Letters 19 (2006), no. 2, 161-168.
P. Drabek, G. Holubov Ìa, A. Matas, P. Necesal; Nonlinear models of suspension bridges: discussion of the results, Applications of Mathematics 48 (2003), no. 6, 497-514.
J. R. Graef, C. Qian, B. Yang; A three point boundary value problem for nonlinear fourth order differential equations, Journal of Mathematical Analysis and Applications 287 (2003), no. 1, 217-233.
C. P. Gupta; Existence and uniqueness theorems for the bending of an elastic beam equation, Applicable Analysis 26 (1988), no. 4, 289-304.
Ivan Kiguradze, Bedtrich Putza; On boundary value problems for systems of linear functional differential equations, Czechoslovak Mathematical Journal 47 (1997), no. 2, 341-373 (eng).
Ivan Kiguradze, Bedtrich Putza; Boundary value problems for systems of linear functional differential equations, Folia : math Ìematica, Masaryk University, 2003.
A. C. Lazer, P. J. McKenna; Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Review 32 (1990), no. 4, 537-578.
H. Li, Y. Liu; Multiple solutions for fourth order m-point boundary value problems with sign-changing nonlinearity, Electronic Journal of Qualitative Theory of Differential Equations 2010 (2010), no. 55, 1-10.
Y. Li, Y. Gao; The method of lower and upper solutions for the cantilever beam equations with fully nonlinear terms, Journal of Inequalities and Applications 2019 (2019), no. 1, 1-16.
Yongxiang Li; Positive solutions of fourth-order boundary value problems with two parameters, Journal of Mathematical Analysis and Applications 281 (2003), no. 2, 477-484.
Yongxiang Li; A monotone iterative technique for solving the bending elastic beam equations, Applied Mathematics and Computation 217 (2010), no. 5, 2200-2208.
Y. Lu, R. Ma; Disconjugacy conditions and spectrum structure of clamped beam equations with two parameters, Communications on Pure & Applied Analysis 19 (2020), no. 6, 3283.
Nikolai Nikolaevich Luzin; On the method of approximate integration of academician sa chaplygin, Uspekhi matematicheskikh nauk 6 (1951), no. 6, 3-27.
D. Ma, X. Z. Yang; Upper and lower solution method for fourth-order four-point boundary value problems, Journal of Computational and Applied Mathematics 223 (2009), no. 2, 543-551.
R. Ma, Z. Jihui, F. Shengmao; The method of lower and upper solutions for fourth-order two-point boundary value problems, Journal of Mathematical Analysis and Applications 215 (1997), no. 2, 415-422.
R. Ma, J. Wang, Y. Long; Lower and upper solution method for the problem of elastic beam with hinged ends, Journal of fixed point theory and applications 20 (2018), no. 1, 1-13.
R. Ma, J. Wang, D. Yan; The method of lower and upper solutions for fourth order equations with the navier condition, Boundary value problems 2017 (2017), no. 1, 1-9.
F. Minh Ìos, T. Gyulov, A. I. Santos; Existence and location result for a fourth order boundary value problem, Conference Publications, vol. 2005, American Institute of Mathematical Sciences, 2005, p. 662.
S. A. Pak; Dokl Akad. Nauk SSR 148 (1963), 265-1267.
Y. Pang, Z. Bai; Upper and lower solution method for a fourth-order four-point boundary value problem on time scales, Applied Mathematics and Computation 215 (2009), no. 6, 2243-2247.
M. Pei, S. K. Chang; Monotone iterative technique and symmetric positive solutions for a fourth-order boundary value problem, Mathematical and Computer Modelling 51 (2010), no. 9-10, 1260-1267.
M. Singh, N. Urus, A. K. Verma; A different monotone iterative technique for a class of nonlinear three-point bvps, Computational and Applied Mathematics 40 (2021), no. 8, 1-22.
Stanislaw Sedziwy; Upper and lower solutions method for even order two point boundary value problems, The Rocky Mountain journal of mathematics 31 (2001), no. 4, 1429-1434.
N. Urus, A. K. Verma; Existence of solutions for a class of nonlinear Neumann boundary value problems in the presence of upper and lower solutions, Mathematical Methods in the Applied Sciences (2022).
N. Urus, A. K. Verma, M. Singh; Some new existence results for a class of four point nonlinear boundary value problems, JNPG-The Journal of Revelations 3 (2019), 7-13.
A. K. Verma, N. Urus; Well ordered monotone iterative technique for nonlinear second order four point dirichlet bvps, Mathematical Modelling and Analysis 27 (2022), no. 1, 59-77.
A. K. Verma, N. Urus, R.P. Agarwal; Region of existence of multiple solutions for a class of robin type four-point bvps, Opuscula Mathematica 41 (2021), no. 4, 571-600.
A. K. Verma, N. Urus, M. Singh; Monotone iterative technique for a class of four point bvps with reversed ordered upper and lower solutions, International Journal of Computational Methods 17 (2020), no. 09, 1950066.
R. Vrabel; On the lower and upper solutions method for the problem of elastic beam with hinged ends, Journal of mathematical analysis and applications 421 (2015), no. 2, 1455-1468.
J. Wang, C. Gao, Y. Lu; Global structure of positive solutions for semipositone nonlinear Euler-Bernoulli beam equation with neumann boundary conditions, Quaestiones Mathematicae (2022), 1-29.
S. Weng, H. Gao, D. Jiang, X. Hou; Upper and lower solutions method for fourth-order periodic boundary value problems, Journal of Applied Analysis 14 (2008), no. 1, 53-61.
Q. Zhang, S. Chen, J. Lu; Upper and lower solution method for fourth-order four-point boundary value problems, Journal of Computational and Applied Mathematics 196 (2006), no. 2, 387-393.
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