Electronic Journal of Differential Equations,
Vol. 2000(2000), No. 52, pp. 1-42.
Title: Multiplicity results for classes of one-dimensional p-Laplacian
boundary-value problems with cubic-like nonlinearities
Author: Idris Addou (USTHB Institut de Mathematiques, Alger, Algerie)
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
$\lambda >0$. 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 {\it 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.