Characterizing the formation of singularities in a superlinear indefinite problem related to the mean curvature operator

The aim of this paper is characterizing the development of singularities by the positive solutions of the quasilinear indefinite Neumann problem egin{equation*} label{P} -(u'/sqrt{1+(u')^2})' = lambda a(x) f(u) ; ; ext{in } (0,1), quad u'(0)=0,;u'(1)=0, end{equation*} where $lambdain R$ is a parameter, $ain L^infty(0,1)$ changes sign once in $(0,1)$ at the point $zin(0,1)$, and $f in mc{C}(R)cap mc{C}^1[0, +infty)$ is positive and increasing in $(0,+infty)$ with a potential, $ int_0^{s}f(t),dt$, superlinear at $+infty$. oindent In this paper, by providing a precise description of the asymptotic profile of the derivatives of the solutions of the problem as $l o 0^+$, we can characterize the existence of singular bounded variation solutions solutions of the problem in terms of the integrability of this limiting profile, which is in turn equivalent to the condition medskip egin{center} $left( int_x^z a(t),dt ight)^{- rac{1}{2}}in L^1(0,z) $quad and quad $ left( int_x^z a(t),dt ight)^{- rac{1}{2}}in L^1(z,1).$ end{center} medskip No previous result of this nature is known in the context of the theory of superlinear indefinite problems.