Electronic Journal of Differential Equations,
Vol. 2000(2000), No. 66, pp. 1-34.

Title: Exactness results for generalized Ambrosetti-Brezis-Cerami problem and
       related one-dimensional elliptic equations

Authors: I. Addou  (Montreal, Quebec, Canada)
         A.  Benmezai (USTHB -Institut de Mathematiques, Algeria)
         S. M. Bouguima (Univ. of Tlemcen,  Algeria)
         M. Derhab (Univ. Djilali Liabes, Algeria)
         
Abstract: 
 We consider the boundary-value problem
 $$ \displaylines{
 -(\varphi _{p}(u'))' =\varphi _{\alpha }(u)
 +\lambda \varphi _{\beta }(u) \quad\hbox{in }(0, 1) \cr 
 u(0) =u(1)=0, }
 $$ 
 where $\varphi _{p}(x)=\left| x\right| ^{p-2}x$, $p,\alpha ,\beta >1$
 and $\lambda \in \mathbb{R}^*$. We give the exact number of
 solutions for all $\lambda$ and most values of 
 $\alpha ,\beta, p>1$. In the particular case where 
 $1<\beta <p=2<\alpha $,  we  resolve completely a problem suggested 
 by A. Ambrosetti, H. Brezis and G. Cerami and which was partially 
 solved by S. Villegas.

Submitted January 1, 2000. Published November 2, 2000.
Math Subject Classifications: 34B15, 34B18.
Key Words: Exactness; p-Laplacian; concave-convex nonlinearities; 
           quadrature method.