A seven-positive-solutions theorem for a superlinear problem

We consider the superlinear boundary value problem mu" + a(mu)(t)u(gamma+1) = 0, u(0) = 0, u(1) = 0, where gamma > 0 and a(u)(t) is a sign indefinite weight of the form a+(t) - mua-(t). We prove, for mu positive and large, the existence of 2(k) - 1 positive solutions where k is the number of positive humps of a(mu)(t) which are separated by k - 1 negative humps. For sake of simplicity, the proof is carried on for the case k = 3 yielding to 7 positive solutions. Our main argument combines a modified shooting method in the phase plane with some properties of the blow up solutions in the intervals where the weight function is negative.