# zbMATH — the first resource for mathematics

Exact number of solutions of a class of two-point boundary value problems involving concave and convex nonlinearities. (English) Zbl 1003.34020
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)].

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text:
##### 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. · 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+up=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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.