Heteroclinic connections for fully non-linear non-autonomous second-order differential equations

We investigate the solvability of the following strongly non-linear non-autonomous boundary value problem(P){(a(x(t))x′(t))′=f(t,x(t),x′(t))a.e.t∈R,x(−∞)=ν−,x(+∞)=ν+ with ν−<ν+ν−<ν+ given constants, where a(x)a(x) is a generic continuous positive function and f   is a Carathéodory non-linear function. We show that the solvability of (P)(P) is strictly connected to a sharp relation between the behaviors of f(t,x,⋅)f(t,x,⋅) as |x′|→0|x′|→0 and f(⋅,x,x′)f(⋅,x,x′) as |t|→+∞|t|→+∞. Such a relation is optimal for a wide class of problems, for which we prove that (P)(P) is not solvable when it does not hold.