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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
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