Electron. J. Diff. Eqns., Vol. 2000(2000), No. 01, pp. 1-26.

Exact multiplicity results for quasilinear boundary-value problems with cubic-like nonlinearities

Idris Addou Abstract:
We consider the boundary-value problem
$-(\varphi_p (u'))' =\lambda f(u)$ in (0,1)
$u(0) = u(1) =0$
where p> 1, $\lambda$ is positive and $\varphi_p (x) =| x|^{p-2}x$. The nonlinearity f is cubic-like with three distinct roots 0=a < b< c. By means of a quadrature method, we provide the exact number of solutions for all $\lambda$ is positive. This way we extend a recent result, for p=2, by Korman et al. [17] to the general case p>1. We shall prove that when &1<p\leq 2$ the structure of the solution set is exactly the same as that studied in the case p=2 by Korman et al. [17], and strictly different in the case p>2.

An addendum  was attached on May 3, 2000. There it is shown that Possibility B of Theorem 2.4 and Possibility D of Theorem 2.5 never happens. See last page of this article.

Submitted May 26, 1999. Revised October 1, 1999. Published January 1, 2000.
Math Subject Classifications: 34B15.
Key Words: One dimensional p-Laplacian, multiplicity results, time-maps.

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Idris Addou
USTHB, Institut de Mathematiques
El-Alia, B.P. no. 32 Bab-Ezzouar
16111, Alger, Algerie.
e-mail: idrisaddou@hotmail.com

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