Three positive solutions to an indefinite Neumann problem: a shooting method

We deal with the Neumann boundary value problem u'' + ( λa^+(t) - μa^-(t) ) g(u) = 0, 0 < u(t) < 1, ∀ t∈[0,T], u'(0) = u'(T) = 0, where the weight term has two positive humps separated by a negative one and g: [0,1]→ℝ is a continuous function such that g(0)=g(1)=0, g(s) > 0 for 0<s<1 and lim_{s→0^+} g(s)/s = 0. We prove the existence of three solutions when λ and μ are positive and sufficiently large.