Liu, Zhaoli Exact number of solutions of a class of two-point boundary value problems involving concave and convex nonlinearities. (English) Zbl 1003.34020 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 46, No. 2, 181-197 (2001). The author studies Dirichlet boundary value problems for the equations \[ -u''= \lambda u^q+ u^p\quad\text{and}\quad -u''= \lambda|u|^{q-1}u+|u|^{p-1} u. \] For the first equation, nonnegative solutions are considered. It is assumed that \(0< q< 1< p\), \(\lambda> 0\). Properties and the exact number of solutions are studied. The research is motivated by results of A. Ambrosetti, H. Brézis and G. Cerami [J. Funct. Anal. 122, No. 2, 519-543 (1994; Zbl 0805.35028)]. Reviewer: Sergei A.Brykalov (Ekaterinburg) Cited in 1 ReviewCited in 19 Documents MSC: 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:concave and convex nonlinearities; nonlinear two-point boundary value problems; number of solutions Citations:Zbl 0805.35028 PDFBibTeX XMLCite \textit{Z. Liu}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 46, No. 2, 181--197 (2001; Zbl 1003.34020) Full Text: DOI References: [1] Ambrosetti, A.; Brezis, H.; Cerami, G., Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122, 519-543 (1994) · Zbl 0805.35028 [2] Ambrosetti, A.; Prodi, G., On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Math., 93, 231-246 (1972) · Zbl 0288.35020 [3] Berger, M. S.; Podolak, E., On the solution of a nonlinear Dirichlet problem, Indiana Univ. Math. J., 24, 837-846 (1975) · Zbl 0329.35026 [4] Gaete, S.; Manasevich, R. F., Existence of a pair of periodic solutions of an O.D.E. generalizing a problem in nonlinear elasticity, via variational methods, J. Math. Anal. Appl., 134, 257-271 (1988) · Zbl 0672.34030 [5] Z. Liu, Positive solutions of superlinear elliptic equations, J. Funct. Anal., to appear.; Z. Liu, Positive solutions of superlinear elliptic equations, J. Funct. Anal., to appear. · Zbl 0951.35051 [6] Korman, P.; Ouyang, T., Exact multiplicity results for two classes of boundary value problems, Differential Integral Equations, 6, 1507-1517 (1993) · Zbl 0780.34013 [7] Korman, P.; Ouyang, T., Multiplicity results for two classes of boundary value problems, SIAM J. Math. Anal., 26, 180-189 (1995) · Zbl 0824.34028 [8] Tineo, A., Existence of two periodic solutions for the periodic equation \(x^″=g(t,x)\), J. Math. Anal. Appl., 156, 588-596 (1991) · Zbl 0734.34034 [9] Smoller, J.; Wasserman, A., Global bifurcation of steady-state solutions, J. Differential Equations, 39, 269-290 (1981) · Zbl 0425.34028 [10] Zhang, L., Uniqueness of positive solutions of Δ \(u+u+u^p=0\) in a ball, Comm. Partial Differential Equations, 17, 1141-1164 (1992) · Zbl 0782.35025 [11] Zhang, L., Uniqueness of positive solutions of semilinear elliptic equations, J. Differential Equations, 115, 1-23 (1995) · Zbl 0812.35049 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.