Multiplicity Results for Classes of One-dimensional p-Laplacian Boundary-value Problems with Cubic-like Nonlinearities

Abstract

<p>We study boundary-value problems of the type</p> <pre>-(φ_p(u'))' = λƒ(u), in (0, 1)<br> u(0) = u(1) = 0,</pre> <p>where p > 1, φ_p(x) = |x|^p-2 x, and λ > 0. We provide multiplicity results when ƒ behaves like a cubic with three distinct roots, at which it satisfies Lipschitz-type conditions involving a parameter q > 1. We shall show hos 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 ƒ is <i>half-odd</i>; a condition generalizing the usual oddness. We use a quadrature method.</p>