Electron. J. Diff. Eqns., Vol. 2000(2000), No. 52, pp. 1-42.

Multiplicity results for classes of one-dimensional p-Laplacian boundary-value problems with cubic-like nonlinearities

Idris Addou

Abstract:
We study boundary-value problems of the type
$$\displaylines{ -(\varphi_{p}( u') ) ' =\lambda f( u) ,\hbox{ in }(0,1) \cr u( 0) =u( 1) =0, }$$
where p>1, $\varphi_{p}( x) =\left| x\right| ^{p-2}x$, and $\lambda$ is positive. We provide multiplicity results when f behaves like a cubic with three distinct roots, at which it satisfies Lipschitz-type conditions involving a parameter q>1. We shall show how changes in the position of q with respect to p lead to different behavior of the solution set. When dealing with sign-changing solutions, we assume that f is half-odd; a condition generalizing the usual oddness. We use a quadrature method.

Submitted April 16, 1999. Revised May 1, 2000. Published July 3, 2000.
Math Subject Classifications: 34B15.
Key Words: p-Laplacian, time-maps, multiplicity results, cubic-like nonlinearities.

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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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