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

Abstract

We investigate the solvability of the following strongly non-linear non-autonomous boundary value problemis a generic continuous positive function and f is a Carathéodory non-linear 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.

On the solvability of a boundary value problem on the real line

“…In view of what observed in Remark 6 [13], if the sign condition in (2.29) is satisfied with the reverse inequality, i.e., if tf (t, x, y) ≥ 0 for a.e. |t| ≥ L, every x ∈ R, |y| < ρ, (2:32) then it is possible to prove that lim x→±∞ x (t) = 0 and x'(t) ≤ 0 for |t| ≥ L. So, since ν -< ν + , when L = 0 problem (P) does not admit solutions.…”

“…On the other hand, in many applications the dynamic is described by a differential operator also depending on the state variable, like (a(x)x')' for some sufficiently regular function a(x), which can be everywhere positive [non-negative] (as in the diffusion [degenerate] processes), or a changing sign function, as in the diffusion-aggregation models (see [7], [11][12][13]). …”

Heteroclinic solutions of boundary value problems on the real line involving singular Φ-Laplacian operators

“…The same problem was addressed recently by Cupini, Marcelli and Papalini in [7], in the case in which Φ : R → R is a generic increasing homeomorphism, and by Marcelli and Papalini in [9] when Φ(y) ≡ y.…”

“…In the papers [6,7,9] the authors linked the solvability of (P) to the relative behaviors of f (t, x, ·) and Φ(·) as y → 0, and of f (·, x, y) as |t| → +∞. Moreover, it was shown that (apart the special case when f (t, x, y) ∼ 1 t as |t| → +∞) the presence of the function a in the differential operator and the dependence on x in the right-hand side do not play any role for the solvability of (P).…”

Existence of bounded solutions of integral boundary value problems for singular differential equations on whole lines

“…Φ is singular. Marcelli and Papalini [20], Liu [17] and Cupini, Marcelli and Papalini [8,9] discussed the solvability of the following strongly nonlinear BVP:…”