Existence of solutions to boundary-value problems governed by general non-autonomous nonlinear differential operators

This article concerns the existence and non-existence of solutions to the strongly nonlinear non-autonomous boundary-value problem $$displaylines{ (a(t,x(t))Phi(x'(t)))' = f(t,x(t),x'(t)) quad hbox{a.e. } tin mathbb{R} \ x(-infty)=u^- ,quad x(+infty)= u^+ }$$ with $u^-<u^+$, where $Phi:mathbb{R} o mathbb{R}$ is a general increasing homeomorphism, with $Phi(0)=0$, a is a positive, continuous function and f is a Caratheodory nonlinear function. We provide sufficient conditions for the solvability which result to be optimal for a wide class of problems. In particular, we focus on the role played by the behaviors of $f(t,x,cdot)$ and $Phi(cdot)$ as $yo 0$ related to that of $f(cdot,x,y)$ and $a(cdot,x) $ as $|t|o +infty$