One-dimensional elliptic equation with concave and convex nonlinearities. (English) Zbl 0955.34013

Sánchez, Justino; Ubilla, Pedro

Electron. J. Differ. Equ. 2000, Paper No. 50, 9 p. (2000).

The authors study the Dirichlet boundary value problem \[ -(|u'|^{m- 2}u')'= \lambda u^q+ u^p,\quad u(0)= u(1),\tag{1} \] with \(0\leq q< m-1< p\), \(\lambda> 0\), and using shooting methods, they obtain the exact number of solutions to (1) as well as their asymptotic behavior for small parameter \(\lambda\). For a particular case of (1) such results were reached by I. Addou and A. Benmezaï [Electron. J. Differ. Equ. 1999, Paper No. 9, 29 p. (1999; Zbl 0923.34024)]. The techniques developed there are used in the paper under review, as well.

Reviewer: I.Rachůnková (Olomouc)

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations

Keywords:

\(m\)-Laplacian; exact number of positive solutions; time maps; Dirichlet problem

Citations:

Zbl 0923.34024