Existence results for singular nonlinear BVPs in the critical regime

Abstract

We study the existence of solutions for a class of boundary value problems on the half line, associated to a third order ordinary differential equation of the type ( Φ ( k ( t , u ′ ( t ) ) u ″ ( t ) ) ) ′ ( t ) = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) , a.a.  t ∈ R 0 + . The prototype for the operator Φ is the Φ -Laplacian; the function k is assumed to be continuous and it may vanish in a subset of zero Lebesgue measure, so that the problem can be singular; finally, f is a Carathéodory function satisfying a weak growth condition of Winter–Nagumo type. The approach we follow is based on fixed point techniques combined with the upper and lower solutions method.