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

We investigate the existence of heteroclinic solutions to a class of nonlinear differential equations

governed by a nonlinear differential operator Φ extending the classical p- Laplacian, with right-hand side f having the critical rate of decay -1 as |t| → +∞, that is f\left(t,\cdot ,\cdot \right)\approx \frac{1}{t}. We prove general existence and non-existence results, as well as some simple criteria useful for right-hand side having the product structure f(t, x, x') = b(t, x)c(x, x').

Mathematical subject classification: Primary: 34B40; 34C37; Secondary: 34B15; 34L30.

Differential equations governed by nonlinear differential operators have been extensively studied in the last decade, due to their several applications in various sciences. The most famous differential operator is the well-known p-Laplacian and its generalization to the generic Φ-Laplacian operator (an increasing homeomorphism of ℝ with Φ(0) = 0). Many articles have been devoted to the study of differential equations of the type

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]).

So, it naturally arises the interest for mixed nonlinear differential operators of the type (a(x)Φ(x'))'. In this context, in [11] we studied boundary value problems on the whole real line

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.

Let us consider the equation

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.

Definition 2.1. A lower [upper] solution to equation (2.1) is a bounded function α ∈ C¹(ℝ) such that (a ○ α)(Φ ○ α') ∈ W^1,1(ℝ) and

Throughout this section we will assume the existence of an ordered pair of lower and upper solutions α, β, i.e., satisfying α(t) ≤ β(t) for every t ∈ ℝ, and we will adopt the following notations:

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.

Lemma 2.2. For all n ∈ ℕ let I _n := [-n, n] and let u _n ∈ C¹(I _n ) be such that: \left(a\circ u_n\right)\left(\Phi \circ u_n^{\prime }\right)\in W^{1,1}\left(I_n\right), the sequences (u _n (0)) _n and {\left(u_n^{\prime }\left(0\right)\right)}_n are bounded and finally

Assume that there exist two functions H, γ ∈ L¹(ℝ) such that

Then, the sequence (x _n ) _n ⊂ C¹(ℝ) defined by

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).

Moreover, if \underset{n\to +\infty }{lim}{u}_n\left(-n\right)=u^{-} and \underset{n\to +\infty }{lim}{u}_n\left(n\right)=u^{+}, then we have that

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.

Let Φ be such that

and

for some positive constant μ.

Assume that there exist a constant H > 0, a continuous function θ : ℝ⁺ → ℝ⁺ and a function λ ∈ L^q([-L, L]), with 1 ≤ q ≤ ∞, such that

(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, +∞),

such that:

and put

we have

Then, there exists a function x ∈ C¹(ℝ), with (a ○ x)(Φ ○ x') ∈ W^1,1(ℝ), such that

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.

By (2.2), without loss of generality we assume H>\frac{\nu }{2L} and

for some constant K > 0.

Moreover, by (2.5), there exists a constant C>{\Phi }^{-1}\left(\frac{M}{m}\Phi \left(H\right)\right)\ge H such that

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

Let us consider the following auxiliary boundary value problem on the compact interval I _n :

where T : W^1,1(I _n ) → W ^1,1(I _n ) is the truncation operator defined by

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

By the same argument used in the proof of Theorem 3.2 [11], one can show, using only assumption (2.9), that for every n > L problem \left(P_n^{*}\right) admits a solution u _n such that

hence T_{u_n}\left(t\right)\equiv u_n\left(t\right) and w(t, u _n (t)) ≡ 0. Moreover, it is possible to prove that

(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 for every t ∈ [-L, L].

Indeed, since u _n ∈ C¹(I _n ) and u_n\left(\left[-L,L\right]\right)\subset \mathcal{I}, we can apply Lagrange Theorem to deduce that for some τ₀ ∈ [-L, L] we have

Assume, by contradiction, the existence of an interval (τ₁, τ₂) ⊂ (-L, L) such that 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.

Since N_C\left(t\right)={\Phi }^{-1}\left(\frac{M}{m}\Phi \left(C\right)\right)\ge C for every t ∈ (τ₁, τ₂), we have for every t ∈ (τ₁, τ₂). Then, by the definition of \left(P_n^{*}\right) and assumption (2.4), for a.e. t ∈ (τ₁, τ₂) we have

Therefore, using a change of variable and the Hölder inequality, we get

Moreover, since u_n^{\prime } has constant sign in (τ₁, τ₂), using (2.12) we have

Therefore, by (2.10), from the previous chain of inequalities we deduce

in contradiction with (2.11). Thus, we get for every t ∈ [-L, L] and the claim is proved.

Step 2. We have (t) for every t ∈ I _n \ [-L, L].

Define , and assume by contradiction that . Hence, u_n^{\prime }\left(\widehat{t}\right)=N_C\left(\widehat{t}\right)>0 and by (2.13), (2.14) we deduce that u_n^{\prime }\left(t\right)>0 in \left[L,\widehat{t}\right]. Moreover, by (2.12) and the definition of Q_{u_n} we get

so, by (2.8) we have

Then, recalling that K _C (L) = 0 and u_n^{\prime }\left(t\right)>0 for every t\in \left[L,\widehat{t}\right], we infer

implying

since . 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.

Summarizing, since \mid u_n^{\prime }\left(t\right)\mid \le N_C\left(t\right) for every t ∈ I _n , by the definition of Q_{u_n} we have

Observe now that condition (2.3) implies that . 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.

Theorem 2.4. Suppose that all the assumptions of Theorem 2.3 are satisfied, with the exception of (2.2), and with (2.5) replaced by

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.

Notice that by assumption (2.17), we have

hence, there exists a constant C>{\Phi }^{-1}\left(\frac{M}{m}\Phi \left(H\right)\right)\ge H such that

With this choice of the constant C, the proof proceeds as in Theorem 2.3. The only modification concerns formula (2.16), which becomes, taking (2.15) into account:

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

θ : ℝ⁺ → ℝ⁺ and a function λ ∈ L^q([-L, L]), with 1 ≤ q ≤ +∞, such that

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, +∞), such that:

and put

we have

Then, there exists a function x ∈ C¹(ℝ), with (a ○ x)(Φ ○ x') ∈ W^1,1(ℝ), such that

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.

At the beginning of the proof, without loss of generality we assume H>\frac{\nu }{2L} and we choose C>{\left(\frac{M}{m}\right)}^{\frac{1}{p-1}}H\ge H, in such a way that

The proof of Step 1 begins as previously, determining an interval J = (τ₁, τ₂) ⊂ (-L, L) such that in J, and \mid u_n^{\prime }\left({\tau }_1\right)\mid =H, \mid u_n^{\prime }\left({\tau }_2\right)\mid =C or vice versa. Then, as in the proof of Theorem 2.3, assumption (2.19) implies that for a.e. t ∈ J we have

Therefore, put

we get

in contradiction with (2.24). Thus, we get 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.

Theorem 2.6. Suppose that

for some positive constant μ. Moreover, assume that there exist two constants L ≥ 0, ρ > 0 and a positive strictly increasing function K\in W_{loc}^{1,1}\left(\left[L,+\infty \right)\right) satisfying

where \tilde{m}:=\underset{x\in \left[{\nu }^{-},{\nu }^{+}\right]}{min}a\left(x\right), such that one of the following pair of conditions holds:

or

Moreover, assume that

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 , 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)

we deduce that a(x(t))Φ(x'(t)) is decreasing in [t₁, t₂] and then

a contradiction. Hence, necessarily \underset{t\to +\infty }{\text{lim inf}}{x}^{\prime }\left(t\right)=0. We can prove in a similar way that \underset{t\to +\infty }{\text{lim sup}}{x}^{\prime }\left(t\right)=0. So, \underset{t\to +\infty }{lim}{x}^{\prime }\left(t\right)=0 and we can define t* := inf{t ≥ L : |x'(τ)| < ρ in [t, +∞)}.

We claim that x'(t) ≥ 0 for every t ≥ t*. Indeed, if for some \widehat{t}\ge t^{*}, since a(x(t))Φ(x'(t)) is decreasing in [t*, +∞) by (2.29), we get

Since a is positive, then Φ(x'(t)) < 0 for every t\ge \widehat{t}. Hence, from (2.30) we get M\Phi \left(x^{\prime }\left(t\right)\right)\le m\Phi \left(x^{\prime }\left(\widehat{t}\right)\right), and so