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: (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 Carathe ́dory nonlinear function. We provide some sufficient conditions for the solvability, 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.