Ma, Ruyun Nodal solutions for a fourth-order two-point boundary value problem. (English) Zbl 1085.34015 J. Math. Anal. Appl. 314, No. 1, 254-265 (2006). Summary: We consider boundary value problems of fourth-order differential equations of the form \[ u''''+\beta u''-\alpha u=\mu h(x) f(u),\qquad 0< x< r, \]\[ u(0)= u(r)= u''(0)= u''(r)= 0, \] where \(\mu\) is a parameter, \(\beta\in(-\infty, \infty)\), \(\alpha\in [0,\infty)\) are constants with \[ {r^2\beta\over\pi^2}+ {r^4\alpha\over\pi^4}< 1, \] \(h\in C(0, r], [0,\infty))\) with \(h\not\equiv 0\) on any subinterval of \([0, r]\), \(f\in C(\mathbb{R}, \mathbb{R})\) satisfies \(f(u)u> 0\) for all \(u\neq 0\), and \[ \lim_{u\to-\infty} {f(u)\over u}= 0,\quad \lim_{u\to+\infty} {f(u)\over u}= f_{+\infty},\quad \lim_{u\to 0} {f(u)\over u}= f_0, \] for some \(f_{+\infty}\), \(f_0\in (0,\infty)\). We use bifurcation techniques to establish existence and multiplicity results on nodal solutions to the problem. Cited in 1 ReviewCited in 26 Documents MSC: 34B15 Nonlinear boundary value problems for ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations Keywords:multiplicity results; eigenvalues; bifurcation methods; nodal zeros PDFBibTeX XMLCite \textit{R. Ma}, J. Math. Anal. Appl. 314, No. 1, 254--265 (2006; Zbl 1085.34015) Full Text: DOI References: [1] Bai, Z.; Wang, H., On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270, 357-368 (2002) · Zbl 1006.34023 [2] Coppel, W. A., Disconjugacy, Lecture Notes in Math., vol. 220 (1971), Springer: Springer Berlin · Zbl 0224.34003 [3] Del Pino, M. A.; Manasevich, R. F., Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. Amer. Math. Soc., 112, 81-86 (1991) · Zbl 0725.34020 [4] Elias, U., Eigenvalue problems for the equation \(L y + \lambda p(x) y = 0\), J. Differential Equations, 29, 28-57 (1978) · Zbl 0351.34014 [5] Eloe, P. W.; Henderson, J., Singular boundary value problems for quasi-differential equations, Internat. J. Math. Math. Sci., 18, 571-578 (1995) · Zbl 0886.34016 [6] Lazer, A. C.; McKenna, P. J., Global bifurcation and a theorem of Tarantello, J. Math. Anal. Appl., 181, 648-655 (1994) · Zbl 0797.34021 [7] Ma, R.; Wang, H., On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal., 59, 225-231 (1995) · Zbl 0841.34019 [8] Ma, R.; Thompson, B., Nodal solutions for nonlinear eigenvalue problems, Nonlinear Anal., 59, 707-718 (2004) · Zbl 1059.34013 [9] Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7, 487-513 (1971) · Zbl 0212.16504 [10] Rynne, B. P., Infinitely many solutions of superlinear fourth-order boundary value problems, Topol. Methods Nonlinear Anal., 19, 303-312 (2002) · Zbl 1017.34015 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.