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: \[ (P) \quad \left \{ \begin{array}{ll} (a(x(t))\Phi(x'(t)))' = f(t,x(t),x'(t)) \ \ \mbox{a.e. } t\in {\mathbb R} & \\ x(-\infty)=\nu^- ,\ \ x(+\infty)= \nu^+ & \end{array} \right . \] with $\nu^-< \nu^+$, where $\Phi:{\mathbb R} \to {\mathbb R}$ is a general increasing homeomorphism, with $\Phi(0)=0$, $a$ 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 $f(t,x,\cdot)$ and $\Phi(\cdot)$ as $y\to 0$ related to that of $f(\cdot,x,y)$ as $|t|\to +\infty$. 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.