Homoclinic solutions of singular differential equations with φ-Laplacian

摘要

A singular nonlinear initial value problem (IVP) with a φ-Laplacian of the form (p(t)φ(u 0 (t)))0 + p(t)f(φ(u(t))) = 0, u(0) = u0 ∈ [L0, 0), u 0 (0) = 0 is investigated on the half-line [0, ∞). Here, function φ is smooth and increasing on R with φ(0) = 0, function f is locally Lipschitz continuous with three zeros φ(L0) < 0 < φ(L), function p is smooth and increasing on (0, ∞), and the problem is singular in the sense that p(0) = 0 and 1/p(t) may not be integrable on [0, 1]. The main result of the paper is the existence of homoclinic solutions defined as nondecreasing solutions u of the IVP satisfying limt→∞ u(t) = L.