Bifurcation of positive solutions for a one-dimensional indefinite quasilinear Neumann problem

We study the structure of the set of the positive regular solutions of the one-dimensional quasilinear Neumann problem involving the curvature operator −(u′/1+(u′)2)′=λa(x)f(u),u′(0)=0,u′(1)=0. Here λ∈R is a parameter, a∈L1(0,1) changes sign, and f∈C(R). We focus on the case where the slope of f at 0, f′(0), is finite and non-zero, and the potential of f is superlinear at infinity, but also the two limiting cases where f′(0)=0, or f′(0)=+∞, are discussed. We investigate, in some special configurations, the possible development of singularities and the corresponding appearance in this problem of bounded variation solutions.