Heteroclinic solutions of boundary-value problems on the real line involving general nonlinear differential operators

We discuss the solvability of the following strongly nonlinear non-autonomous boundary-value problem: (P) (a(x(t))Φ(x′(t)))′ = f(t,x(t),x′(t)) a.e. t ∈ R, x(−∞) = ν−, x(+∞) = ν+ with ν− < ν+, where Φ : R → R is a general increasing homeomorphism, with Φ(0) = 0, a is a positive, continuous function and f is a Caratheodory nonlinear function. We provide some sufficient conditions for the solvability of (P) which turn out to be optimal for a large class of problems. In particular, we highlight the role played by the behavior of f(t,x,·) and Φ(·) as y → 0 related to that of f(·,x,y) as |t| → +∞. We also show that the dependence on x, both of the differential operator and of the right-hand side, does not influence in any way the existence or non-existence of solutions.