- Open Access
The role of boundary data on the solvability of some equations involving non-autonomous nonlinear differential operators
© Marcelli; licensee Springer. 2013
- Received: 13 May 2013
- Accepted: 24 September 2013
- Published: 22 November 2013
The paper deals with the existence and non-existence of solutions of the following strongly nonlinear non-autonomous boundary value problem:
with , where is a general increasing homeomorphism, with , a is a positive, continuous function and f is a Caratheódory nonlinear function.
The same problem was already studied in the case when as in the recent paper (Marcelli in Electron. J. Differ. Equ. 2012: 171, 2012), where sharp sufficient conditions for the existence or non-existence of solutions were established. In particular, it was proved that neither the behavior of the functions and nor the boundary data , influence the solvability of problem (P).
We herein study the critical case when as , focusing on the role played by the dependence on x of the functions a and f and by the boundary data , by means of an explicit link between them and the other parameters of the differential equation.
MSC:34B40, 34C37, 34B15, 34L30.
- boundary value problems
- unbounded domains
- heteroclinic solutions
- nonlinear differential operators
- p-Laplacian operator
- Φ-Laplacian operator
governed by nonlinear differential operators such as the classical p-Laplacian or its generalizations. Various types of differential operators, even singular or non-surjective, have been considered due to many applications in different fields. We now quote just some of the papers devoted to this study, such as for the scalar case Bereanu and Mawhin [1, 2], Cabada and Pouso [3, 4], Cabada and Cid , Cid and Torres , Calamai , Garcia-Huidobro et al. , Dang and Oppenheimer , Ferracuti and Papalini , O’Regan , Papageorgiou and Papalini . In  Manásevich and Mawhin treated systems of equations with periodic boundary conditions. Finally, in the framework of differential inclusions, we quote  and the papers by Kyritsi, Matzakos and Papageorgiou [15, 16] for systems of differential inclusions involving maximal monotone operators and with various boundary conditions.
has assumed a certain interest.
In  a periodic problem on a compact interval for a vectorial inclusion with a differential operator of the type is studied, where is a positive, continuous function. Moreover, in  a Dirichlet problem driven by a more general differential operator, having the structure , is investigated.
have been studied in , where existence and non-existence of solutions was put in relation to the behavior of Φ and at 0 and at infinity, while the presence of the function a does not influence the existence of solutions. Subsequently, in  a critical case was considered in which also the dependence on the state variable x of the functions a and f and the value of the boundary data are relevant for the solvability of the boundary value problem.
with given constants, where is a general increasing homeomorphism, with , and a is a positive, continuous function, but with possibly null infimum. It was shown that also the dependence on t of the function a plays a central role for the existence and non-existence of solutions and some sufficient criteria for the existence and non-existence of solutions were established. However, in  the case when as was considered, and in this setting neither the behavior with respect to x, nor the boundary data influence the existence or non-existence of solutions.
The aim of this paper is to complete this study, investigating the critical case as for problem (1.1) governed by non-autonomous differential operators.
We provide sharp sufficient conditions guaranteeing the solvability of problem (1.1) together with conditions implying the non-existence of solutions, closely related to the former ones, involving the asymptotic behaviors of and as , the asymptotic behaviors of Φ and as , and the maxima/minima of the functions , in the interval defined by the boundary data.
We present general existence and non-existence results (see Theorems 2.1, 2.2 and 2.3) together with operative criteria (see Propositions 3.1-3.6) useful when the functions a and f appearing in the differential equation have a product structure. Some examples of application complete the paper.
Throughout the paper, Φ is a general increasing homeomorphism on ℝ such that , is a positive continuous function and is a Carathéodory function.
Of course, for every , with possibly null.
As we have mentioned in Introduction, in the present paper, we treat problems for which, roughly speaking, as . But also the rate of growth of Φ at ∞ has a great relevance, and we separately consider the case of superlinear growth from that of linear or sublinear growth.
We first state an existence result for differential operators growing at most linearly at infinity.
(with if ).
Proof The scheme of the proof is the same as in [, Theorem 3.1]. We sketch now the main points and prove in detail the parts which differ from that proof. Notice that there are only two differences between the present statement and that of [, Theorem 3.1]; that is, we here take and modify the definition of the auxiliary function .
Finally, for every , put .
Following the same argument in the proof of [, Theorem 3.1], it is possible to prove that problem (2.11) admits a solution for every , such that for all . Moreover, is increasing in and in and if for some , then whenever (see Steps 1-2 in the proof of [, Theorem 3.1]). Finally, as in Step 3 of the same proof, one can show that there exists a suitable constant C such that for every . Notice that till this point in the proof of [, Theorem 3.1], the definition of , or the fact that , were not used.
Now our goal is to show that also for every .
implying that for every (see (2.2), (2.3) and (2.8)), a contradiction when . So, and the claim is proved. The same argument works in the interval too.
Now, following the same argument as in [, Theorem 3.1], it is possible to prove that the sequence of the functions , continued in a constant way in the whole ℝ, converges to a solution x of problem (1.1), satisfying all the properties stated in the assertion. □
Similarly to what was done in , one can prove a result for differential operators having superlinear growth at infinity, provided that condition (2.7) is strengthened requiring that the Nagumo function has sublinear growth at infinity, as the following result states, whose proof is just the same as that of [, Theorem 3.2], taking account of the modifications due to the different auxiliary function , we showed in the proof of Theorem 2.1.
Then the assertion of Theorem 2.1 follows.
Of course, the operators here considered as having superlinear growth are quite general and extend the classical p-Laplacian. Nevertheless, when dealing just with the p-Laplacian, the results can be slightly improved by using the positive homogeneity of the operator, as we will show in a forthcoming paper.
The key tools in the previous existence theorems is the summability of function (condition (2.8)) joined with assumption (2.9). Such conditions are not improvable in the sense that if (2.9) is satisfied with the reversed inequality and is not summable, then problem (1.1) does not admit solutions, as the following result states.
does not belong to .
Then problem (1.1) does not admit solutions such that , that is, no function , with almost everywhere differentiable, exists satisfying the conditions of problem (1.1).
Proof Also this proof follows the scheme of that of [, Theorem 3.3]. More in detail, it is possible to show that if with and almost everywhere differentiable (not necessarily belonging to ) is a solution of problem (1.1), then the function x is monotone increasing in and in with .
Then if , necessarily we have in contradiction with the above inequality. Therefore, and again, by the above inequality, we deduce since by (2.15) the function on the right-hand side in not summable by assumption. Therefore, , implying that , for every and, consequently, .
Similarly, using (2.14) one can show that , a contradiction. □
We will highlight how the local behaviors of at and of , at infinity, related to the maximum and the minimum of the functions β, g in the interval , play a relevant role for the existence or non-existence of solutions.
where recall that and .
3.1 Case of Φ growing at most linearly
The first two existence theorems are an application of Theorem 2.1.
Then problem (1.1) admits solutions.
Proof Put for . From (3.3) and (3.8), it is immediate to verify the validity of conditions (2.6) and (2.7).
for a.e. , every and every . Then condition (2.9) of Theorem 2.1 holds.
implying that by (3.12). Then (2.8) holds too.
that is, condition (2.10). It remains to prove that .
implying that by assumption (3.9).
Therefore, Theorem 2.1 applies and guarantees the assertion of the present result. □
Remark 3.2 The introduction of the constants σ and δ serves to state the result in the most general form, but often they can be taken both equal to 1, in such a way that assumption (3.9) is trivially verified.
If (see (3.1)), condition (3.6) can be weakened, requiring that it holds only for small enough, as the following result states.
Moreover, assume that . Then problem (1.1) admits solutions.
for and for .
As it is immediate to verify, whenever and . So, (2.9) holds since for every .
From now on, the proof proceeds as that of Proposition 3.1. □
We state now two non-existence results, obtained applying Theorem 2.3.
where recall that for .
Then problem (1.1) does not admit solutions.
Proof First of all, notice that assumption (3.15) implies condition (2.16) and assumptions (3.22) and (3.23) respectively imply conditions (2.17), (2.18).
Finally, assumption (3.21) implies that is not summable in ℝ and the assertion follows as an application of Theorem 2.3. □
Remark 3.5 As for the validity of conditions (3.22), (3.23) in the previous non-existence theorem, notice that when dealing with autonomous operators, that is, for , they are trivially satisfied. However, also in the non-autonomous case, they hold in many relevant situations. For instance, they are satisfied if one the following conditions is satisfied:
is decreasing in and increasing in ;
α is uniformly continuous in ℝ and ;
as for some .
When condition (3.19) does not hold, we can use the following non-existence result.
then problem (1.1) does not admit solutions.
Proof With the same notations of the proof of Proposition 3.4, notice that under condition (3.25), by (3.24), we have for large enough, implying that is not summable and the assertion follows from Theorem 2.3. □
Let us now provide some examples of applications of the previous results.
with β, g positive continuous functions.
It is easy to show that all the assumptions of Proposition 3.1 are satisfied with , , , , , , , and L large enough (depending on ).
By applying Propositions 3.1 and 3.4, we deduce that if , then problem (1.1) admits solutions, whereas if , then (1.1) does not admit solutions. Recalling that () for , the existence or non-existence of solutions depends on the boundary data , . For instance, if and and the boundary data are symmetric, that is, , then , and . So, if , problem (1.1) admits solutions, whereas if , it does not admit solutions. Notice that for every , problem (1.1) is solvable for ν small enough.
with β, g positive continuous functions.
As one can immediately verify, assumptions (3.2)-(3.9) and (3.11) of Proposition 3.1 hold with , , , , , , , , , and L, H large enough. Therefore, if , both conditions (3.10) and (3.12) are satisfied and problem (1.1) admits solutions. Instead, if , then (1.1) has no solutions as a consequence of Proposition 3.4.
So, as in the previous example, the above conditions for the existence and non-existence of solutions become conditions on the boundary data , .
3.2 Case of Φ having superlinear growth
We handle now operators Φ having possibly superlinear growth at infinity, that is, we now remove condition (2.4). The non-existence Propositions 3.4 and 3.6 hold also in this case, since they do not require condition (2.4). As for the existence results, we now use Theorem 2.2 instead of Theorem 2.1 by assuming (2.12). As it will be clear after the proof of the next result, condition (2.12) is not satisfied when , so from now on we assume .
Then problem (1.1) admits solutions.
Hence, condition (2.12) holds and the proof proceeds as that of Proposition 3.3, applying Theorem 2.2 instead of Theorem 2.1. □
Note that condition (3.26) is not compatible with (3.6). For this reason, in the case of superlinear growth, we only treat the case .
with β, g positive continuous functions.
In this case, we can apply Proposition 3.9 since and condition (3.26) is trivially satisfied. Moreover, all the other assumptions of Proposition 3.9 hold with , , , , , , , . Hence, since (3.12) is satisfied whatever may be, problem (1.1) admits solutions for every boundary data , .
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