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 ν− ≤x(t)≤ν+, x(−∞)=ν−, x(+∞)=ν+ with ν− < ν+ given constants, where a(x) is a generic continuous positive function and f is a Caratheodory nonlinear function. We show that the solvability of (P ) is strictly connected to a sharp relation between the behaviors of f(t, x, ·) as |x′| → 0 and f(·, x, x′) as |t| → +∞. Such a relation is optimal for a wide class of problems, for which we prove that (P ) is not solvable when it does not hold.