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 $$\left(\Phi(k(t, u'(t))u''(t))\right)'(t)=f \left(t, u(t), u'(t), u''(t) \right), \quad\mbox {a.a. } t\in\mathbb{R}^+_0.$$ The prototype for the operator $\Phi$ is the $\Phi$-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.

Keywords

• boundary value problems on unbounded domains
• heteroclinic solutions
• $\phi$-laplacian operator
• singular equations
• wintner–nagumo condition
• third order odes