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

2012

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 …