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

for Φ-Laplacian operators, and recently also the study of singular or non-surjective differential operators has become object of an increasing interest (see, i.e., [1–10]).

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], [11–13]).

obtaining results on both existence and non-existence of heteroclinic solutions. Such criteria are based on the comparison between the behavior of the right-hand side f(t, x, x') as |t| → +∞ and x' → 0, combined to the infinitesimal order of the differential operator Φ(x') as x' → 0. Rather surprisingly, the presence of the state variable x inside the right-hand side and the differential operator does not influence in any way the existence or the non-existence of solutions, but it only entails a more technical proof and a sligthly stronger set of assumptions on the operator Φ. Roughly speaking, if a(x) is positive and f(t, x, x') = g(t, x')h(x) for some positive continuous function h, then the solvability of the boundary value problem depends neither on a, nor on h. Moreover, even the prescribed boundary values ν₁, ν₂ are not involved on the existence of solutions.

A crucial assumption in [11] is a limitation on the rate of the possible decay of f(·, x, x') as |t| → +∞; precisely, we assumed that f(t, x, x') ≈ |t| ^δ for some δ > -1 (possibly positive).

In the present article we focus our attention on right-hand sides having the critical rate of decay δ = -1 and show that, contrary to the situation studied in [11], now the solvability of the boundary value problem is influenced by the behavior of the right-hand side and of the differential operator with respect to the state variable x. For instance, when f(t, x, x') = g(x)h(t, x') the existence of solutions depends on the amplitude of the range of the values assumed by the functions a and g in the interval [ν₁, ν₂] determined by the prescribed boundary values.

In Section 2 we study the existence/non-existence of solutions for general right-hand sides f(t, x(t), x'(t)) (see Theorems 2.3-2.5); more operative criteria are stated in the subsequent section for f of product type.

We conclude the article with some examples (see Examples 3.8-3.10), useful to have a quick glance on the role played by the behavior with respect to x.

The study of the solvability of the boundary value problem for rates of decay δ < - 1 is still open.

where a : ℝ → ℝ is a positive continuous function, and f : ℝ³ → ℝ is a given Carathéodory function. From now on we will take into consideration increasing homeomorphisms Φ : ℝ → ℝ, with Φ(0) = 0.

Our approach is based on fixed point techniques suitably combined to the method of upper and lower solutions, according to the following definition.

Note that the value d is well-defined, in fact $\underset{\mid t\mid \to +\infty }{lim}{\alpha }^{\prime }\left(t\right)=\underset{\mid t\mid \to +\infty }{lim}{\beta }^{\prime }\left(t\right)=0$, since (a ○ α)(Φ ○ α'), (a ○ β)(Φ ○ β') belong to W^1,1(ℝ) and m > 0.

Moreover, in what follows [x]⁺ and [x]^- will respectively denote the positive and negative part of the real number x, and we set x ∧ y := min{x, y}, x ∨ y := max{x, y}.

The next result proved in [11] concerns the convergence of sequences of functions correlated to solutions of the previous equation.

admits a subsequence uniformly convergent in ℝ to a function x ∈ C¹(ℝ), with (a ○ x) (Φ ○ x') ∈ W^1,1(ℝ), solution to equation (2.1).

The first existence result concerns differential operators growing at most linearly at infinity.

Theorem 2.3. Assume that there exists a pair of lower and upper solutions α, β ∈ C¹(ℝ) of the equation (2.1), satisfying α(t) ≤ β(t), for every t ∈ ℝ, with α increasing in (-∞, -L), β increasing in (L, +∞), for some L > 0.

for some positive constant μ.

(with $\frac{1}{q}=0$ if q = +∞).

Finally, suppose that for every C > 0 there exist a function η _C ∈ L¹(ℝ) and a function $K_C\in W_{loc}^{1,1}\left(\left[0,+\infty \right)\right)$, null in [0, L] and strictly increasing in [L, +∞),

Proof. In some parts the proof is similar to that of Theorem 3.2 [11]. So, we provide here only the arguments which differ from those used in that proof.

for some constant K > 0.

Fix n ∈ ℕ, n > L, and put I _n := [-n, n].

and finally w : ℝ² → ℝ is the penalty function defined by w(t, x) := [x - β(t)]⁺ - [x -α(t)]^-.

(see Steps 3 and 4 in the proof of Theorem 3.2 [11]).

Now our goal is to prove an a priori bound for the derivatives, that is $\mid u_n^{\prime }\left(t\right)\mid \le N_C\left(t\right)$ for a.e. t ∈ I _n . We split this part into two steps.

Step 1. We have $\mid u_n^{\prime }\left(t\right)\mid <C\le N_C\left(t\right)$ for every t ∈ [-L, L].

Assume, by contradiction, the existence of an interval (τ₁, τ₂) ⊂ (-L, L) such that $H<\mid u_n^{\prime }\left(t\right)\mid <C$ in (τ₁, τ₂) and $\mid u_n^{\prime }\left({\tau }_1\right)\mid =H$, $\mid u_n^{\prime }\left({\tau }_2\right)\mid =C$ or viceversa.

in contradiction with (2.11). Thus, we get $\mid u_n^{\prime }\left(t\right)\mid <C$ for every t ∈ [-L, L] and the claim is proved.

Step 2. We have $u_n^{\prime }\left(t\right)<N_C$(t) for every t ∈ I _n \ [-L, L].

since $u_n^{\prime }\left(L\right)<C$. Therefore, $u_n^{\prime }\left(t\right)\le N_C\left(t\right)$ for every $t\in \left[L,\widehat{t}\right]$, in contradiction with the definition of $\widehat{t}$. The same argument works in the interval [-n, -L] and the claim is proved.

Observe now that condition (2.3) implies that $\underset{\xi \to 0^{+}}{limsup}\frac{{\Phi }^{-1}\left(\xi \right)}{{\xi }^{1∕\mu }}<+\infty$. Hence, by assumption (2.6) we get N _C ∈ L¹(ℝ) and applying Lemma 2.2 with H(t) = N _C (t) and γ(t) = η _C (t) we deduce the existence of a solution x to problem (P). □

In order to deal with differential operators having superlinear growth at infinity, we need to strengthen condition (2.5), taking a Nagumo function with sublinear growth at infinity, as in the statement of the following result.

Then, the assertion of Theorem 2.3 follows.

Proof. The proof is quite similar to that of the previous Theorem. Indeed, notice that assumptions (2.2) and (2.5) of Theorem 2.3 have been used only in the choice of the constant C (see (2.11)) and in the proof of Step 1. Hence, we now present only the proof of this part, the rest being the same.

in contradiction with (2.18). From here on, the proof proceeds in the same way. □

In the particular case of p-Laplacian operators, one can use the positive homogeneity for weakening assumption (2.17) of Theorem 2.4 and widening the class of the admissible Nagumo functions, as we show in the following result.

Theorem 2.5. Let Φ : ℝ → ℝ, Φ(y) = |y|^p-2y, and assume that there exists a pair of lower and upper solutions α, β ∈ C¹(ℝ) to equation (2.1), satisfying α(t) ≤ β(t), for every t ∈ ℝ, with α increasing in (-∞, -L), β increasing in (L, +∞), for some constant L > 0.

Moreover, assume that there exist a positive constant H, a continuous function

Proof. The proof is quite similar to that of Theorem 2.3. Indeed, notice that the present statement has the same assumptions of Theorem 2.3, written for Φ(y) = |y|^p-2y, with the exception of conditions (2.2) and (2.5), which were used only in the proof of Step 1. Hence, as in the proof of the previous Theorem 2.4, we now provide only the proof of Step 1, the rest being the same.

in contradiction with (2.24). Thus, we get $\mid u_n^{\prime }\left(t\right)\mid <C$ for every t ∈ [-L, L] and Step 1 is proved. □

As we mentioned in Section 1, the assumptions of the previous existence Theorems are not improvable in the sense that if conditions (2.3) and (2.8) are satisfied with the reversed inequalities and the summability condition (2.6) [respectively (2.21) for the case of p-Laplacian] does not hold, then problem (P) does not admit solutions, as the following results state.

Then, problem (P) can only admit solutions which are constant in [L, +∞) (when (2.27) holds) or constant in (-∞, -L] (when (2.28) holds). Therefore, if both (2.27) and (2.28) hold and L = 0, then problem (P) does not admit solutions. More precisely, no function x ∈ C¹(ℝ), with (a○x)(Φ○x') almost everywhere differentiable, exists satisfying the boundary conditions and the differential equation in (P).

Proof. Suppose that (2.27) holds (the proof is the same if (2.28) holds).

Let x ∈ C¹(ℝ), with (a ○ x)(Φ○x') almost everywhere differentiable (not necessarily belonging to W^1,1(ℝ)), be a solution of problem (P). First of all, let us prove that $\underset{t\to +\infty }{lim}\Phi \left(x^{\prime }\left(t\right)\right)=0$.

Indeed, since x(+∞) = ν⁺ ∈ ℝ, we have $\underset{t\to +\infty }{\text{lim sup}}{x}^{\prime }\left(t\right)\ge 0$ and $\underset{t\to +\infty }{\text{lim inf}}{x}^{\prime }\left(t\right)\le 0$.

Taking into account that Φ is an increasing homeomorphism with Φ(0) = 0, if $\underset{t\to +\infty }{\text{lim inf}}{x}^{\prime }\left(t\right)<0$, then there exists an interval [t₁, t₂] ⊂ [L, +∞) such that -ρ < Φ (x'(t)) < 0 in [t₁, t₂], $\Phi \left(x^{\prime }\left(t_2\right)\right)>\frac{m}{M}\Phi \left(x^{\prime }\left(t_1\right)\right)$. But by virtue of assumption (2.29)