Regular versus singular solutions in quasilinear indefinite problems with sublinear potentials

The aim of this paper is analyzing existence, multiplicity, and regularity issues for the positive solutions of the quasilinear Neumann problem{−(u′/1+(u′)2)′=λa(x)f(u),0<x<1,u′(0)=u′(1)=0. Here, (u′/1+(u′)2)′ is the one-dimensional curvature operator, λ∈R is a parameter, which is generally taken positive, the weight a(x) changes sign, and, in most occasions, the function f(u) has a sublinear potential F(u) at ∞. Our discussion displays the manifold patterns occurring for these solutions, depending on the behavior of the potential F(u) at u=0, and, possibly, at infinity, and of the weight function a(x) at its nodal points.