Let us consider the equation
(2.1)
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') ∈ W1,1(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 , since (a ○ α)(Φ ○ α'), (a ○ β)(Φ ○ β') belong to W1,1(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: , the sequences (u
n
(0))
n
and 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') ∈ W1,1(ℝ), solution to equation (2.1).
Moreover, if and , 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
(2.2)
and
(2.3)
for some positive constant μ.
Assume that there exist a constant H > 0, a continuous function θ : ℝ+ → ℝ+ and a function λ ∈ Lq([-L, L]), with 1 ≤ q ≤ ∞, such that
(2.4)
(2.5)
(with if q = +∞).
Finally, suppose that for every C > 0 there exist a function η
C
∈ L1(ℝ) and a function , null in [0, L] and strictly increasing in [L, +∞),
such that:
(2.6)
and put
(2.7)
we have
(2.8)
(2.9)
Then, there exists a function x ∈ C1(ℝ), with (a ○ x)(Φ ○ x') ∈ W1,1(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 and
(2.10)
for some constant K > 0.
Moreover, by (2.5), there exists a constant such that
(2.11)
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 : W1,1(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 admits a solution u
n
such that
(2.12)
hence and w(t, u
n
(t)) ≡ 0. Moreover, it is possible to prove that
(2.13)
(2.14)
(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 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 , 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 , or viceversa.
Since for every t ∈ (τ1, τ2), we have for every t ∈ (τ1, τ2). Then, by the definition of and assumption (2.4), for a.e. t ∈ (τ1, τ2) we have
Therefore, using a change of variable and the Hölder inequality, we get
(2.15)
Moreover, since has constant sign in (τ1, τ2), using (2.12) we have
Therefore, by (2.10), from the previous chain of inequalities we deduce
(2.16)
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, and by (2.13), (2.14) we deduce that in . Moreover, by (2.12) and the definition of we get
so, by (2.8) we have
Then, recalling that K
C
(L) = 0 and for every , we infer
implying
since . Therefore, for every , in contradiction with the definition of . The same argument works in the interval [-n, -L] and the claim is proved.
Summarizing, since for every t ∈ I
n
, by the definition of 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
(2.17)
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 such that
(2.18)
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 ∈ Lq([-L, L]), with 1 ≤ q ≤ +∞, such that
(2.19)
(2.20)
Finally, suppose that for every C > 0 there exist a function η
C
∈ L1(ℝ) and a function , null in [0, L] and strictly increasing in [L, +∞), such that:
(2.21)
and put
we have
(2.22)
(2.23)
Then, there exists a function x ∈ C1(ℝ), with (a ○ x)(Φ ○ x') ∈ W1,1(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 and we choose , in such a way that
(2.24)
The proof of Step 1 begins as previously, determining an interval J = (τ1, τ2) ⊂ (-L, L) such that in J, and , 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
(2.25)
for some positive constant μ. Moreover, assume that there exist two constants L ≥ 0, ρ > 0 and a positive strictly increasing function satisfying
(2.26)
where , such that one of the following pair of conditions holds:
(2.27)
or
(2.28)
Moreover, assume that
(2.29)
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 W1,1(ℝ)), be a solution of problem (P). First of all, let us prove that .
Indeed, since x(+∞) = ν+ ∈ ℝ, we have and .
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], . 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 . We can prove in a similar way that . So, 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 , since a(x(t))Φ(x'(t)) is decreasing in [t*, +∞) by (2.29), we get
(2.30)
Since a is positive, then Φ(x'(t)) < 0 for every . Hence, from (2.30) we get , and so
in contradiction with the boundedness of x. Thus, the claim is proved.
Let us define . We now prove that x'(t) = 0 for every .
Let us assume by contradiction that for some . Put ; we claim that T = +∞. Indeed, if T < +∞, since 0 < x'(t) < ρ in , by (2.27) we have
(2.31)
So, assuming without loss of generality ρ ≤ 1, we get
where . Then, integrating in [t, T] with t < T we obtain (taking into account that x'(T) = 0)
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 , we get
for a.e., where . The above inequalities imply that for a.e.
and then
where . By virtue of (2.25) and (2.26), since , we get , in contradiction with the boundedness of x.
Therefore, x'(t) ≡ 0 in and by the definition of this implies . So, x'(t) ≡ 0 in [t*, +∞) and by the definition of t* this implies t* = L. □
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
(2.32)
then it is possible to prove that and x'(t) ≤ 0 for |t| ≥ L. So, since ν - < ν+, when L = 0 problem (P) does not admit solutions.