On the solvability of a boundary value problem on the real line

We investigate the existence of heteroclinic solutions to a class of nonlinear differential equations (a(x)F (x'(t)))' = f (t, x(t), x (t)), a.e. t ∈ R governed by a nonlinear differential operator F extending the classical p-Laplacian, with right-hand side f having the critical rate of decay -1 as |t| goes to +∞, that is f (t, ·, ·) ≈ 1. We prove general existence and non-existence results, as well as some simple criteria useful for right-hand side having the product structure f(t, x, x’) = b(t,x)c(x, x’).