Electron. J. Diff. Eqns., Vol. 2000(2000), No. 66, pp. 1-34.

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

I. Addou, A. Benmezai, S. M. Bouguima , & M. Derhab

We consider the boundary-value problem
$ -(\varphi _{p}(u'))' =\varphi _{\alpha }(u)
 +\lambda \varphi _{\beta }(u)$ in (0, 1)
$u(0) =u(1)=0$,
where $\varphi _{p}(x)=\left| x\right| ^{p-2}x$, and $\lambda \in \mathbb{R}^*$. We give the exact number of solutions for all $\lambda$ and most values of . In the particular case where , 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.

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Idris Addou
119-8230, Rue Sherbrooke Est,
Montreal, Quebec, H1L-1A9, Canada
email: idrisaddou@yahoo.com
Abdelhamid Benmezai
U.S.T.H.B.-Institut de Mathematiques
El-Alia, BP 32, Bab-Ezzouar
16111, Alger, Algeria
email: abenmzai@hotmail.com
Sidi Mohammed Bouguima
Department of Mathematics, Faculty of sciences
University of Tlemcen
B.P.119, Tlemcen 13000, Algeria
email: bouguima@yahoo.fr
Mohammed Derhab
Institut des Sciences Exactes, Universite Djilali Liabes
BP. 89, Sidi Bel Abbes
22000, Algeria
email: derhab@yahoo.fr

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