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

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.

### MSC:

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

Zbl 0923.34024
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