A seven-positive-solutions theorem for a superlinear problem

We consider the superlinear boundary value problem u'' +a_μ (t)u^(g+1)u=0, u(0) = 0, u(1) = 0, where g> 0 and a_μ(t) is a sign indefinite weight of the form a+(t)−μa−(t). We prove, for μ positive and large, the existence of 2k − 1 positive solutions where k is the number of positive humps of aμ(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.