Abstract

The aim of this paper is characterizing the development of singularities by the positive solutions of the quasilinear indefinite Neumann problem−(u′/1+(u′)2)′=λa(x)f(u)in (0,1),u′(0)=0,u′(1)=0, where λ∈R is a parameter, a∈L∞(0,1) changes sign once in (0,1) at the point z∈(0,1), and f∈C(R)∩C1[0,+∞) is positive and increasing in (0,+∞) with a potential, ∫0sf(t)dt, superlinear at +∞. In this paper, by providing a precise description of the asymptotic profile of the derivatives of the solutions of the problem as λ→0+, we can characterize the existence of singular bounded variation solutions of the problem in terms of the integrability of this limiting profile, which is in turn equivalent to the condition(∫xza(t)dt)−12∈L1(0,z)and(∫xza(t)dt)−12∈L1(z,1). No previous result of this nature is known in the context of the theory of superlinear indefinite problems.

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