EXISTENCE OF THREE SOLUTIONS FOR A MIXED BOUNDARY VALUE PROBLEM WITH THE STURM-LIOUVILLE EQUATION

Abstract

Abstract. The aim of this paper is to establish the existence of threesolutions for a Sturm-Liouville mixed boundary value problem. The ap-proach is based on multiple critical points theorems. 1. IntroductionThe aim of this paper is to establish, under a suitable set of assumptions, theexistence of at least three solutions for the following Sturm-Liouville problemwith mixed boundary conditions(RS λ )ˆ−(pu ′ ) ′ +qu = λf(t,u) in I =]a,b[u(a) = u ′ (b) = 0,where λ is a positive parameter and p, q, f are regular functions. To be precise,if f : [a,b] × R→ Ris a L 2 -Carath´eodory function and p,q ∈ L ∞ ([a,b]) suchthatp 0 := essinf t∈[a,b] p(t) > 0, q 0 := essinf t∈[a,b] q(t) ≥ 0,then we prove the existence of three weak solutions for problem (RS λ ) (seeTheorems 3.1 and 3.2). Clearly, when f : [a,b] × R → R is a continuousfunction, p ∈ C 1 ([a,b]) and q ∈ C 0 ([a,b]), the solutions of (RS λ ) are actuallyclassical (see for instance Corollaries 3.1 and 3.2).The problem (RS λ ) with p = q = 1 has been studied in [5] (see also [1])but it is worth noticing that our results assure a more precise conclusion. Infact, in [5] precise values of parameters λ were not established, and in [1] anasymptotic condition at infinity was assumed (see Remark 4.2).In our main results a precise interval of real parameters λ for which theproblem (RS