Pergamon h’on,;near Analysrs, Theory, Merhods & Applmlrons, Vol. 21. No. 6, PP. 679-697, 1996 Copyright ii, 1996 Elsevier Science Ltd Printed I” Great Britain. All rights reserved 0362-546X/96 $15.00+0.00 0362-546X(95)00071-2 FUcfK SPECTRUM OF A SINGULAR STURM-LIOUVILLE PROBLEM? M. ARIAS and J. CAMPOS Departamento de Matemtitica Aplicada, Universidad de Granada, 18071 Granada, Spain (Received 20 October 1994; received in revised form 3 March 1995: received for publication 21 April 1995) Key words and phrases: FuEik spectrum, singular Sturm-Liouville problem, Sturm comparison theory, zeros of a solution. 1. INTRODUCTION Let L: D(L) C H + H be a linear operator defined on a Banach space H. Since FuEik began to study the set A0 = {(a, b) E IR*: Lu + au+ - bu- = 0 has a nontrivial solution], where u+ = max(u, 0), U- = max(-u, 0), when L is the operator associated to the Dirichlet problem for -u” (see [l]), many authors deal with this topic (see [2-51). The study of the set A,, usually known as the FuEik spectrum of the operator L, is essential to understand the behaviour of some nonlinear problems associated to L. In [4], Dancer obtains some properties of A, when L is the operator associated to the Dirichlet problem for the Laplace operator, -A, on a domain D c IRN and H = L’(sZ), but one does not have a complete description of this set even in this apparently easy case. Sometimes, when we deal with operators associated to second-order elliptic equations, we can ask for special solutions (see examples in Section 4) of the equation Lu + auf - bu- = 0, obtaining a possibly singular Sturm-Liouville problem for an ordinary differential equation as (p(t)u’)’ + q(t)(auf - lx-1 = 0, where p and q are analytic and positive functions on a interval (T, , r,) c IR. Together with the previous equation appear boundary conditions. Normally these conditions are either a Dirichlet condition, u(T) = 0, or a bounded condition, lim supI+ T,lu(t)l < foe. We only impose hypotheses of geometrical nature about the solutions of the associated linear equation (La) (p(t)u’)’ + aq(t)u = 0, which satisfies the boundary condition at one of the end points T, i = 1 or i = 2. (See (Hi) and (H,) in the next Section.) These hypotheses depend on the behaviour of the functions p and q in a neighbour of the end points. I-Supported by DGCYT PB92-0953. M.E.C., Spain. 679