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)Φ(x′(t)))′=f(t,x(t),x′(t)),a.e.t∈R governed by a nonlinear differential operator Φ extending the classical p-Laplacian, with right-hand side f having the critical rate of decay -1 as |t| → +∞, that is f(t,⋅,⋅)≈1/t. 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').