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 $$left({u'}/{sqrt{1+(u')^2}} ight)' = lambda a(x) f(u), quadu'(0)=0,,,u'(1)=0. $$ Here $lambda in R$ is a parameter, $a in L^1(0,1) $ changes sign, and $fin mc{C}(R)$. 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 $f'(0)=+infty$, are discussed. We investigate, in some special configurations, the possible development of singularities and the corresponding appearance in this problem of bounded variation solutions.