Existence results for singular nonlinear BVPs in the critical regime - Francesca Anceschi; Giuseppina Autuori; Francesca Papalini (Electronic Journal of Qualitative Theory of Differential Equations)
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 $$\left(Φ(k(t, u'(t))u''(t))\right)'(t)=f \left(t, u(t), u'(t), u''(t) \right), \quada.a. t\in\mathbb{R}^+_0.$$ 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.