Abstract
We investigate the existence of heteroclinic solutions to a class of nonlinear differential equations
governed by a nonlinear differential operator Φ extending the classical pLaplacian, with righthand side f having the critical rate of decay 1 as t → +∞, that is
Mathematical subject classification: Primary: 34B40; 34C37; Secondary: 34B15; 34L30.
Keywords:
boundary value problems; unbounded domains; heteroclinic solutions; nonlinear differential operators; pLaplacian operator; ΦLaplacian operator1. Introduction
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 wellknown pLaplacian and its generalization to the generic ΦLaplacian 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 ΦLaplacian operators, and recently also the study of singular or nonsurjective differential operators has become object of an increasing interest (see, i.e., [110]).
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 [nonnegative] (as in the diffusion [degenerate] processes), or a changing sign function, as in the diffusionaggregation models (see [7], [1113]).
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 nonexistence of heteroclinic solutions. Such criteria are based on the comparison between the behavior of the righthand 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 righthand side and the differential operator does not influence in any way the existence or the nonexistence 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 ν_{1}, ν_{2 }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 righthand 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 righthand 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 [ν_{1}, ν_{2}] determined by the prescribed boundary values.
In Section 2 we study the existence/nonexistence of solutions for general righthand sides f(t, x(t), x'(t)) (see Theorems 2.32.5); more operative criteria are stated in the subsequent section for f of product type.
We conclude the article with some examples (see Examples 3.83.10), 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.
2. Existence and nonexistence theorems
Let us consider the equation
where a : ℝ → ℝ is a positive continuous function, and f : ℝ^{3 }→ 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 ∈ C^{1}(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 welldefined, in fact
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 }∈ C^{1}(I_{n}) be such that:
Assume that there exist two functions H, γ ∈ L^{1}(ℝ) such that
Then, the sequence (x_{n})_{n }⊂ C^{1}(ℝ) defined by
admits a subsequence uniformly convergent in ℝ to a function x ∈ C^{1}(ℝ), with (a ○ x) (Φ ○ x') ∈ W^{1,1}(ℝ), solution to equation (2.1).
Moreover, if
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 ∈ C^{1}(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
Finally, suppose that for every C > 0 there exist a function η_{C }∈ L^{1}(ℝ) and a function
such that:
and put
we have
Then, there exists a function x ∈ C^{1}(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
for some constant K > 0.
Moreover, by (2.5), there exists a constant
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 : ℝ^{2 }→ 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
hence
(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
Step 1. We have
Indeed, since u_{n }∈ C^{1}(I_{n}) and
Assume, by contradiction, the existence of an interval (τ_{1}, τ_{2}) ⊂ (L, L) such that
Since
Therefore, using a change of variable and the Hölder inequality, we get
Moreover, since
Therefore, by (2.10), from the previous chain of inequalities we deduce
in contradiction with (2.11). Thus, we get
Step 2. We have
Define
so, by (2.8) we have
Then, recalling that K_{C }(L) = 0 and
implying
since
Summarizing, since
Observe now that condition (2.3) implies that
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
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 pLaplacian 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^{p2}y, and assume that there exists a pair of lower and upper solutions α, β ∈ C^{1}(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 }∈ L^{1}(ℝ) and a function
and put
we have
Then, there exists a function x ∈ C^{1}(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^{p2}y, 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
The proof of Step 1 begins as previously, determining an interval J = (τ_{1}, τ_{2}) ⊂ (L, L) such that
Therefore, put
we get
in contradiction with (2.24). Thus, we get
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 pLaplacian] 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
where
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 ∈ C^{1}(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 ∈ C^{1}(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
Indeed, since x(+∞) = ν^{+ }∈ ℝ, we have
Taking into account that Φ is an increasing homeomorphism with Φ(0) = 0, if
we deduce that a(x(t))Φ(x'(t)) is decreasing in [t_{1}, t_{2}] and then
a contradiction. Hence, necessarily
We claim that x'(t) ≥ 0 for every t ≥ t*. Indeed, if
Since a is positive, then Φ(x'(t)) < 0 for every
in contradiction with the boundedness of x. Thus, the claim is proved.
Let us define
Let us assume by contradiction that
So, assuming without loss of generality ρ ≤ 1, we get
where
so by the Gronwall's inequality we deduce a(x(t))Φ(x'(t)) ≤ 0, i.e. x'(t) ≤ 0 in the same interval, in contradiction with the definition of T. Hence T = +∞.
Therefore, since 0 < x'(t) < ρ and ν ^{ }≤ x(t) ≤ ν^{+ }in
for a.e.
and then
where
Therefore, x'(t) ≡ 0 in
Remark 2.7. 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
then it is possible to prove that
3. Criteria for righthand side of the type f(t, x, y) = b(t, x)c(x, y)
In this section we present some operative criteria useful when the righthand side has the following product structure
As we will show, there is a strict link between the local behaviors of c(x, ·) at y = 0 and of b(·, x) at infinity which plays a key role for the existence or nonexistence of solutions.
In what follows we assume that b is a Carathéodory function and c is a continuous function satisfying
Notice that in this framework, the constant functions α(t) :≡ ν^{ }and β(t) :≡ ν^{+ }are a pair of wellordered, monotone, lower and upper solutions. Consequently, according to the notations given after Definition 2.1, in this case we have
and again
According to the results of the previous section, the first three results provide sufficient conditions for the existence of solutions for our special f split in the product of b and c. Then we will deal with sufficient conditions for the nonexistence of solutions.
Theorem 3.1. Let there exists a function
Suppose that there exist positive constants h_{1}, h_{2}, k_{1}, k_{2}, ρ, H, L, ε, with ε ≤ 1, and a constant
Finally, let conditions (2.2) and (2.3) hold with
Then, problem (P) admits solutions.
Proof. Put
Since c(x, y) > 0 for y ≠ 0, denoted by
Consider the following functions:
Observe that by assumption (3.3) we have γ(t) > 0 for a.e. t ≥ L and
and let N_{C }be the function defined in (2.7).
By the positivity of the function γ, K_{C }is strictly increasing for t ≥ L. Moreover, by condition (3.1), we have
Observe that by (3.2) and the definition of ψ_{C}, we obtain
and
for a.e.
and
for a.e.
Now, from (3.3) it follows that
Then, by the upper bound on the exponent μ we get
and condition (2.6) follows.
Finally, let us define
By (3.3) and (3.4), for every y ∈ ℝ such that y ≤ N_{C}(t) for a.e. t ∈ ℝ and every x ∈ [ν^{},ν^{+}], it results
that is condition (2.9), so it remains to prove that η_{C }∈ L^{1}(ℝ). To this purpose, notice that by (3.1) and the continuity of c we have
Since
For differential operators having superlinear growth at infinity, the following result can be applied, whose proof is a consequence of Theorem 2.4.
Theorem 3.2. Let all the assumptions of Theorem 3.1 be satisfied with the exception of (2.2) and with (3.5) replaced by
Then, if (2.3) holds true with a positive
Proof. Set
Observe that θ is a continuous function on [0, +∞), such that
hence (2.4) holds. Moreover, by (3.7), for every ε > 0 there exists a real c_{ε }such that
Hence, for every s ≥ M max{Φ(c_{ε}), Φ(c_{ε})} we have
Hence, the proof proceeds as that of Theorem 3.1, applying Theorem 2.4 instead of Theorem 2.3. □
Finally, in the case of pLaplacian operators, the following result holds, as a consequence of Theorem 2.5, by the same proof of Theorem 3.1.
Theorem 3.3. Consider Φ(y) = y^{p2}y, p > 1, and let all the assumptions of Theorem 3.1 be satisfied with the exception of (2.2) and condition (3.5) replaced by
Then, if
Proof. Define
In the previous results the requirement
Theorem 3.4. Suppose that (3.2) holds for a.e. t ∈ ℝ and let there exist a real constant Λ > 0 and a positive function λ ∈ L^{1}(0, Λ) such that
Moreover, assume that there exist positive constants h, k, ρ such that
If (2.25) holds with a positive
Proof. Put
by the lower bound on the exponent μ. So, the assertion follows from Theorem 2.6. □
The following results are immediate consequences of Theorems 3.1, 3.2, and 3.4.
Corollary 3.5. Let f(t, x, y) = h(t)g(x)c(y), with
Assume that t · h(t) ≤ 0 for every t ∈ ℝ and c(y) > 0 for every y ≠ 0. Moreover, suppose that
Let (2.2) holds and
Then, if (2.3) holds with an exponent μ such that
Corollary 3.6. Let all the assumptions of Corollary 3.5 be satisfied, apart (2.2) and with (3.12) replaced by the following condition
Then, the same conclusions of Corollary 3.5 hold.
Finally, for the pLaplacian operator we can state the following criterium, consequence of Theorems 3.3 and 3.4.
Corollary 3.7. Let f(t, x, y) = h(t)g(x)c(y), with
Assume that t · h(t) ≤ 0 for every t ∈ ℝ and c(y) > 0 for every y ≠ 0. Moreover, suppose that there exist
for a positive constant h_{1}.
Then, if
We conclude with some examples in which the previous corollaries apply.
Example 3.8. Let
If
Example 3.9. Let
Example 3.10. Let
Remark 3.11. Note that in [11] the existence of heteroclinic solutions was proved when in assumption (2.8) one has Φ(y)^{γ}, instead of Φ(y), for some γ > 1. Of course, for small y we have Φ(y)^{γ }< Φ(y) for each γ > 1, hence the present condition (2.8) implies the validity of the analogous condition with γ > 1, assumed in [11] (see condition (8)). But, on the other hand, taking γ > 1 one can lose the summability of the function K_{C }required in assumption (7) of [11]. In fact, in the following example the present Theorems 2.3 and 2.4 are applicable, whereas the results established in [11] do not work.
Consider the problem, already discussed in Example 4 [7]:
where a(x) ≡ 1 and m : ℝ → ℝ is the function defined by
for some α > 0. As it easy to check, the best function K_{C }satisfying condition (8) in [11] is K_{C}(t) := [αlog t]_{+}, but condition (7) of [11] does not hold, whatever γ > 1 may be. Hence, the existence results proved in [11] are not applicable. Instead, notice that condition (2.6) herein considered holds whenever α > μ (see (2.3)) and Theorem 2.3 (or Theorem 2.4) applies, provided that the operator Φ also satisfies the other required assumptions. Similar considerations can be done for the pLaplacian operator too, using Theorem 2.5.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors wrote this article in collaboration and with same responsibility. All authors read and approved the final manuscript.
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