Boundary value problems for third-order differential equations involving singular Phi-Laplacian operators

Abstract

We study strongly nonlinear, third-order differential equations of type (Φ(k(t)u''(t)))' = f(t, u(t), u'(t), u''(t)), a.e. t ∈ J, where Φ is the singular Φ-Laplacian operator. That is, Φ : (–r, r) -> R, r > 0, is a generic strictly increasing homeomorphism with bounded domain, which generalizes the relativistic operator Φ(u) := u (r² – u²)^–1/2. Moreover, k is a nonnegative continuous function, which can vanish on a set of zero measure, so such equations can be singular, and f is a general Carathédory function. For these equations, we investigate boundary value problems both in compact intervals (when J = [a; b]) and in a half-line (with J = [a;+∞)), and we prove existence results under mild assumptions. Our approach is based on fixed point techniques.