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The role of boundary data on the solvability of some equations involving non-autonomous nonlinear differential operators
Boundary Value Problemsvolume 2013, Article number: 252 (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.
A wide literature has been devoted to the study of boundary value problems for differential equations involving various types of nonlinear differential operators. The most known are differential equations of the type
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.
Besides differential operators acting on the derivative , in many models such as reaction-diffusion equations (see, e.g., ) or porous media equations, differential operators involving the state variable x also occur, and the study of mixed differential equations like
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.
More recently, boundary value problems on the whole real line of the type
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.
Finally, in the recent paper , non-autonomous differential operators were also considered, introducing the dependence on t to the function a, that is, the following problem was proposed:
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.
All the present results extend the ones contained in  to the case of non-autonomous differential operators. However, according to our knowledge, the results here presented are new even for , that is, for differential equations of the type (see )
2 Existence and non-existence theorems
Throughout the paper, Φ is a general increasing homeomorphism on ℝ such that , is a positive continuous function and is a Carathéodory function.
Dealing with the nonlinear differential equation
we will adopt the following notations:
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.
Theorem 2.1 Let Φ be such that
and suppose that there exist constants , a continuous function and a function , with , such that
(with if ).
Finally, suppose that for every , there exist a function and a function , null in and positive in , such that
Then there exists a function such that belongs to and
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 .
Fix , , and put . Consider the truncation operator defined by
and for every , put
Finally, for every , put .
Let us consider the following auxiliary boundary value problem on the compact interval :
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 .
Since for every , to prove the claim it remains to show that for every . To this aim, let and assume, by contradiction, . By the definition of T and , we have
Since in , by (2.9) we have
for a.e. . Then, integrating in for , we obtain
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.
Therefore, we have for every implying that
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.
Theorem 2.2 Suppose that all the assumptions of Theorem 2.1 are satisfied, with the exception of (2.4), and with (2.7) replaced by
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.
Theorem 2.3 Suppose that there exist a constant and a positive function such that the following pair of conditions holds:
and for every constant C, the function
does not belong to .
Moreover, assume that
and there exist two constants such that
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 .
Let us now define and assume, by contradiction, that . Put , so that in . By (2.13), for every , we get
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. □
3 Some asymptotic criteria
We devote this section to state some operative criteria which can usefully applied to operators and right-hand sides having the product structure
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.
In what follows, we assume that α, β are continuous positive functions, b is a Carathéodory function and c is a continuous function such that
In this framework, putting and , we have
where recall that and .
Finally, we put
3.1 Case of Φ growing at most linearly
In this subsection, we deal with differential operators Φ satisfying condition (2.4) that is such that whenever every for some . With this class of operators, we cover differential equations of the type
The first two existence theorems are an application of Theorem 2.1.
Proposition 3.1 Suppose that
for some , and there exists a function , , such that
Moreover, assume that there exists a real constant p (not necessarily positive) such that for every , we have
for certain positive constants , , , , , , σ, δ, ρ, H such that ,
where recall that for . Finally, let condition (2.4) be satisfied and assume that
for some positive constant μ satisfying
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).
and for . By condition (3.3), we have and by (3.2) we have for . Observe that by (3.6) it follows that
for a.e. , every and every . Then condition (2.9) of Theorem 2.1 holds.
Now, from (3.5) it follows that for a.e. and by (3.4), recalling that , we deduce that
Hence, for every fixed , the function defined in (2.8) satisfies
By (3.10) we get as , and therefore by (3.11) we deduce
implying that by (3.12). Then (2.8) holds too.
Since , a constant exists such that for every . Let us define and
By (3.5) and (3.7), for a.e. , for every and every such that , we have
that is, condition (2.10). It remains to prove that .
By (3.3) and the continuity of the function c, we have . Moreover, when , by (3.13), we have
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.
Proposition 3.3 Let all the assumptions of Proposition 3.1 be satisfied, with the exception of (3.6) replaced by
Moreover, assume that . Then problem (1.1) admits solutions.
Proof For every , put
and finally put
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.
Proposition 3.4 Suppose that
and let there exist a constant , a constant and a positive function such that
for some positive constants , , ρ. Moreover, assume that (3.4) holds for some constants , , p such that
where recall that for .
Furthermore, suppose that
for some positive constant μ satisfying
Finally, suppose that there exist two constants such that
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).
we have that Λ is a positive function belonging to and one can easily verify that conditions (3.16), (3.17) and (3.18) guarantee the validity of (2.13) and (2.14). Moreover, by (3.4) we get
Hence, if is the function defined in (2.15), we have
By assumption (3.19), we get as and by virtue of (3.20), we obtain
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.
Proposition 3.6 Let all the assumptions of Proposition 3.4 be satisfied with the exception of (3.20), (3.21) and with assumption (3.19) replaced by the opposite one,
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.
Example 3.7 Let us consider the differential equation
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.
Example 3.8 Let us consider the differential equation
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 .
Proposition 3.9 Let all the assumptions of Proposition 3.3 hold true, with the exception of (3.8) replaced by
Then problem (1.1) admits solutions.
it is immediate to check that θ is a continuous function on such that
hence (2.6) holds. Moreover, by (3.26), for every , there exists a real such that
Hence, for every , we have , that is,
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 .
Example 3.10 Let us consider the following differential equation:
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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The author declares that she has no competing interests.
About this article
- boundary value problems
- unbounded domains
- heteroclinic solutions
- nonlinear differential operators
- p-Laplacian operator
- Φ-Laplacian operator