Yan, Dongliang Three positive solutions of fourth-order problems with clamped beam boundary conditions. (English) Zbl 1476.34078 Rocky Mt. J. Math. 50, No. 6, 2235-2244 (2020). In this paper, the author investigates the existence of three positive solutions of fourth order problems \[ u^{\prime \prime \prime \prime }\left( x\right) =\lambda f\left( u\left( x\right) \right) ,x\in \left( -1,1\right) \] with \[ u\left( 1\right) =u\left( -1\right) =u^{\prime }\left( 1\right) =u^{\prime }\left( -1\right) =0, \] where \(\lambda >0\) is a parameter, \(f\in C \left[ 0,\infty \right) \) is nondecreasing, \(f\left( 0\right) =0\) and \( f\left( s\right) >0\) for all \(s>0.\)The proof is based on the directions of a bifurcation as the author studied the existence and multiplicity of positive solutions and the shape of unbounded connected components in the positive solutions set of the considered problem.Moreover, the author made the following assumptions:(H1) There exist \(\alpha >0,\) \(f_{0}>0\) and \(f_{1}>0\) such that \( \lim\limits_{s\rightarrow 0^{+}}\frac{f\left( s\right) -f_{0}s}{s^{1+\alpha } }=-f_{1}.\)(H2) There exists \(f_{\infty }:=\lim\limits_{s\rightarrow \infty }\frac{ f\left( s\right) }{s}=0.\)(H3) There exists \(s_{0}>0\) such that \(\min\limits_{s\in \left[ s_{0},\frac{K }{m}s_{0}\right] }\frac{f\left( s\right) }{s}\geq \frac{f_{0}}{\lambda _{1}} \eta _{1},\)where \(K\) and \(m\) are suitable chosen bounds of the related Green’s function and \(\eta _{1}\) is the first positive eigenvalue of the following linear problem:\(\Psi ^{\prime \prime \prime \prime }\left( x\right) =\eta \Psi \left( x\right) ,x\in \left( -\frac{1}{2},\frac{1}{2}\right) \) with \(\Psi \left( \frac{1}{2}\right) =\Psi \left( -\frac{1}{2}\right) =\Psi ^{\prime }\left( \frac{1}{2}\right) =\Psi ^{\prime }\left( -\frac{1}{2}\right) =0.\)The main result of the paper is as follows. Assume that the above conditions (H1)-(H3) hold. Then there exist \(\lambda _{\ast }\in \left( 0,\frac{\lambda _{1}}{f_{0}}\right) \) and \(\lambda ^{\ast }>\frac{\lambda _{1}}{f_{0}}\) such that(i) the considered problem has at least one positive solution if \(\lambda =\lambda _{\ast }\);(ii) the considered problem has at least two positive solutions if \(\lambda _{\ast }<\lambda \leq \frac{\lambda _{1}}{f_{0}}\);(iii) the considered problem has at least three positive solutions if \(\frac{ \lambda _{1}}{f_{0}}<\lambda <\lambda ^{\ast }\);(iv) the considered problem has at least two positive solutions if \(\lambda =\lambda ^{\ast }\);(v) the considered problem has at least one positive solution if \(\lambda >\lambda ^{\ast }\). Reviewer: Nikolay Dimitrov (Ruse) Cited in 2 Documents MSC: 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.) Keywords:shape of connected component; symmetric positive solutions; bifurcation; principal eigenvalue PDFBibTeX XMLCite \textit{D. Yan}, Rocky Mt. J. Math. 50, No. 6, 2235--2244 (2020; Zbl 1476.34078) Full Text: DOI Euclid References: [1] D. Arcoya, J. I. Diaz, and L. Tello, “S-shaped bifurcation branch in a quasilinear multivalued model arising in climatology”, J. Differential Equations 150:1 (1998), 215-225. · Zbl 0921.35198 · doi:10.1006/jdeq.1998.3502 [2] K. J. Brown, M. M. A. Ibrahim, and R. Shivaji, “\(S\)-shaped bifurcation curves”, Nonlinear Anal. 5:5 (1981), 475-486. · Zbl 0458.35036 · doi:10.1016/0362-546X(81)90096-1 [3] A. Cabada and L. Saavedra, “Existence of solutions for \(n\) th-order nonlinear differential boundary value problems by means of fixed point theorems”, Nonlinear Anal. Real World Appl. 42 (2018), 180-206. · Zbl 1414.34018 · doi:10.1016/j.nonrwa.2017.12.008 [4] A. Castro and R. Shivaji, “Uniqueness of positive solutions for a class of elliptic boundary value problems”, Proc. Roy. Soc. Edinburgh Sect. A 98:3-4 (1984), 267-269. · Zbl 0558.35025 · doi:10.1017/S0308210500013445 [5] R. Dalmasso, “Symmetry properties of solutions of some fourth order ordinary differential equations”, Bull. Sci. Math. 117:4 (1993), 441-462. · Zbl 0798.34022 [6] R. Dalmasso, “Uniqueness of positive solutions for some nonlinear fourth-order equations”, J. Math. Anal. Appl. 201:1 (1996), 152-168. · Zbl 0856.34024 · doi:10.1006/jmaa.1996.0247 [7] E. N. Dancer, “Global solution branches for positive mappings”, Arch. Rational Mech. Anal. 52 (1973), 181-192. · Zbl 0275.47043 · doi:10.1007/BF00282326 [8] U. Elias, “Eigenvalue problems for the equations \(Ly+\lambda p(x)y=0\)”, J. Differential Equations 29:1 (1978), 28-57. · Zbl 0369.34008 · doi:10.1016/0022-0396(78)90039-6 [9] P. Korman and Y. Li, “On the exactness of an S-shaped bifurcation curve”, Proc. Amer. Math. Soc. 127:4 (1999), 1011-1020. · Zbl 0917.34013 · doi:10.1090/S0002-9939-99-04928-X [10] B. P. Rynne, “Infinitely many solutions of superlinear fourth order boundary value problems”, Topol. Methods Nonlinear Anal. 19:2 (2002), 303-312. · Zbl 1017.34015 · doi:10.12775/TMNA.2002.016 [11] T. Shibata, “S-shaped bifurcation curves for nonlinear two-parameter problems”, Nonlinear Anal. 95 (2014), 796-808. · Zbl 1296.34100 · doi:10.1016/j.na.2013.10.015 [12] R. Shivaji, “Remarks on an S-shaped bifurcation curve”, J. Math. Anal. Appl. 111:2 (1985), 374-387. · Zbl 0583.35008 · doi:10.1016/0022-247X(85)90223-9 [13] I. Sim and S. Tanaka, “Three positive solutions for one-dimensional \(p\)-Laplacian problem with sign-changing weight”, Appl. Math. Lett. 49 (2015), 42-50. · Zbl 1342.35122 · doi:10.1016/j.aml.2015.04.007 [14] X. · Zbl 1361.34018 · doi:10.1016/j.jmaa.2016.12.073 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.