Branches of positive solutions of a superlinear indefinite problem driven by the one-dimensional curvature operator

Abstract

This paper aims at proving the existence and the localization of an unbounded connected set of positive regular solutions (λ,u) of the quasilinear Neumann problem −(u′/1+(u′)2)′=λa(x)f(u),0<x<1,u′(0)=u′(1)=0,bifurcating from u=0 as λ→+∞. Here, (u′/1+(u′)2)′ is the one-dimensional curvature operator, λ∈R is a parameter, the weight a changes sign, and the function f is superlinear at 0. A novel approach is introduced based on the explicit construction of non-ordered sub and supersolutions.