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The main result of the paper establishes the existence of a bounded weak solution for a nonlinear Dirichlet problem exhibiting full dependence on the solution u and its gradient in the reaction term, which is driven by a p-Laplacian-type operator with a coefficient that can be unbounded. Through a special Moser iteration procedure, it is shown that the solution set is uniformly bounded. A truncated problem is formulated that drops that be unbounded. The existence of a bounded weak solution to the truncated problem is proven via the theory of pseudomonotone operators. It is noted that the bound of the solution for the truncated problem coincides with the uniform bound of the original problem. This estimate allows us to deduce that for an appropriate choice of truncation, one actually resolves the original problem.
In this paper, we study the following Dirichlet problem:
on a bounded domain in with a Lipschitz boundary . In (1) we have a continuous function , with , a number with , and a Carathéodory function (i.e., is measurable on for each and is continuous on for almost all ). The notation stands for the gradient of u in the distributional sense. It is seen that the driving operator in Equation (1) is the p-Laplacian with a coefficient depending on the solution u. The notation in Equation (1) means the composition of the functions and , that is, for . The main point is that can be unbounded from above, which does not permit to apply any standard method. It is also worth mentioning that problem (1) is not in variational form.
The space underlying the Dirichlet problem (1) is the Banach space endowed with the norm
The dual space of is denoted . Since it was supposed that , the critical Sobolev exponent is . Refer to  for the background related to the space .
The (negative) p-Laplacian is the nonlinear operator (linear for ) defined by
Due to the unbounded function , one cannot build a definition as in (2) corresponding to the term in (1). A major tool in our arguments is the first eigenvalue of , which is positive and isolated in the spectrum of , and is given by
For the the rest of the paper, in order to simplify the notation we make the notational convention that for any real number we denote (the Hölder conjugate of r).
The Carathéodory function determining the reaction term is subject to the following hypotheses.
Hypothesis 1 (H1).
There exist constants , , , and such that
Hypothesis 2 (H2).
There exist constants and with , and a function such that
where denotes the first eigenvalue of .
The main result of this paper is stated as follows.
Assume that , with , is a continuous function and is a Carathéodory function satisfying the conditions (H1) and (H2). Then problem (1) has at least a bounded weak solution in the following sense:
Under hypothesis (H1), the integrals in (4) exist. The proof of Theorem 1 is presented in Section 3. In order to see the effective applicability of Theorem 1, we provide an example.
On a bounded domain Ω in with a Lipschitz boundary , we state the Dirichlet problem
with constants , , , , provided that and , where is given by (3). We readily check that (5) fits into the framework of problem (1) taking for all and
Indeed, one has for all ,
Assumption (H1) is verified with , , while assumption (H2) holds with , , . Theorem 1 applies because .
The inspiration for the present work comes from the recent paper  that deals with the Dirichlet problem
for a positive , a continuous function , with , and a Carathéodory function . The standing point in that work was to use the theory of weighted Sobolev spaces in  (see also ) with the weight requiring the condition
If we consider our problem (1) as a particular case of (6) taking on and apply the result in , the issue is that one obtains a solution of (1) belonging to the space with
and not to the space as it would be natural according to the statement of (1). In this respect, by (7) we note that , so is strictly contained in . Moreover, the assumptions admitted therein for the reaction in (6) are more restrictive than here because they are formulated in terms of corresponding to some s and not with p as in conditions (H1)–(H2) for . All of this shows that the treatment in  does not provide the right approach to obtain Theorem 1. For this reason, we develop a direct study for problem (1) relying just on the classical Sobolev space . The present paper is the first work studying problem (1) with unbounded coefficient in the Soboleev space . Certainly, we use some previous ideas but with substantial modifications and in a different functional setting. The technique relies on truncation, which is needed because the coefficient in the principal part of Equation (1) is unbounded. Other important tools in our study are a special version of Moser iteration and the surjectivity theorem for pseudomonotone operators.
We mention a few relevant works in the area of our paper. A large amount of results in the field is based on variational smooth or nonsmooth methods for which we refer to the recent publications [5,6,7]. They cannot be applied to problem (1) taking into account the lack of variational structure. Nonvariational problems with convection terms have been investigated in recent years through theoretic operator techniques, sub-supersolution and approximation (see, e.g., [8,9,10,11,12]). The main point in these works lies in the dependence of the reaction term with respect to the gradient of the solution without weakening the ellipticity condition of the driving operator. In this connection, we also cite papers dealing with the equations and inclusions driven by the -Laplacian operators, such as, for instance [13,14]. As an extension of this setting, the paper  deals with degenerate -Laplacian problems, but without dependence on the solution u in the principal part of the equation. An advance in this direction is ref. , where there is dependence on solution u in the principal part of the equation of type (6) subject to a weight . Here, we drop the dependence on weight and allow to have a unbounded coefficient in problem (1).
Regarding the rest of the paper, Section 2 focuses on the bounded solutions to problem (1), and Section 3 contains the proof of Theorem 1.
2. Bounded Solutions to Problem (1)
Our first goal is to estimate the solutions in .
Assume that condition (H2) holds. Then the set of solutions to problem (1) is bounded in with a bound that depends on function G only through the lower bound of G.
Since by hypothesis , the stated result is true. □
We are now able to find a uniform bound for the solutions of (1).
Assume that conditions (H1) and (H2) are satisfied. Then the solution set of problem (1) is uniformly bounded, that is, there exists a constant such that for every weak solution to problem (1). The dependence of the uniform bound C on the data in problem (1) and hypotheses (H1) and (H2) is indicated as . In particular, the uniform bound C depends on G only through its lower bound .
Given a weak solution to problem (1), we have the representation with (the positive part of u) and (the negative part of u). We prove the uniform boundedness separately for and . We only give the proof for , noting that we can argue similarly in the case of .
We proceed by using in (4) the test function , where with arbitrary constants and . The fact that follows from and is bounded, while the distributional partial derivatives
belong to because and is bounded. This gives
The left-hand side of (8) can be estimated as follows:
For the right-hand side of (8), by hypothesis (H1), we obtain
By Young’s inequality, for each there is a constant such that
It is clear that
and, since and ,
where denotes the Lebesgue measure of .
If is sufficiently small, we deduce from (8), in conjunction with (9), (10), (11), (12), and (13) that
with a constant . The last integral exists because .
On the other hand, by Bernoulli’s inequality and since on , we derive
with a constant . Through (20), we are able to carry on a Moser iteration, setting inductively with the initial step . Applying repeatedly (20), it turns out that
The series converges because and as . Consequently, we can obtain (19) letting in (21).
It remains to handle the case when the number is such that and for all . In this case, the Moser iteration reads as with the initial step if and if . In any case, we are led to (21) from which (19) can be established as before.
We can pass to the limit as in (19) obtaining for each weak solution to problem (1). Analogously, we can prove that for all weak solutions to problem (1). Altogether, we have the uniform bound for the solution set of problem (1).
A careful reading of the above proof reveals the dependence of the uniform bound C on the data in problem (1) and on the coefficients, entering assumptions (H1) and (H2). Precisely, we have to check how the constants b, q, K, , , and arising in the proof depend on the data given in (1), (H1), and (H2). Collecting all these renders the dependence . This completes the proof. □
3. Truncation Problem and Proof of Theorem 1
The method of proof relies on the truncation of the coefficient of the p-Laplacian in problem (1) to drop its unboundedness. This idea was used in  in the context of the degenerate p-Laplacian. Specifically, for any number , we introduce the truncation
By (22), we obtain a continuous function . We also consider the associated operator given by
The notation in Equation (23) means the composition of the functions and , that is for . The next proposition discusses the properties of .
The nonlinear operator in (23) is well defined, bounded (i.e., it maps bounded sets into bounded sets), continuous, and satisfies the property, that is, any sequence with in and
fulfills in .
The continuity of the function G combined with (22), (23), and Hölder’s inequality ensures
for all . It follows that the operator is well-defined and bounded.
In order to show the continuity of let in . By the continuity of G, (22), (23), Hölder’s inequality, and (2), we find
for all . We infer that
The continuity of the p-Laplacian implies that in . By Lebesgue’s dominated convergence theorem, we derive
whence in , so the continuity of is proven.
Now we show the property for the operator . Let a sequence satisfy in and (24). It is seen that
Taking into account (23) and the monotonicity of , we have
We claim that
To this end, by Hölder’s inequality and since the sequence is bounded in , we find a constant such that
By Lebesgue’s dominated convergence theorem, it holds
This is true because is continuous, in and there is the domination
Equation (30) results in . Recalling that space is uniformly convex, we conclude that in , which proves the property of the operator . The proof is thus complete. □
For any and the truncation in (22), let us consider the auxiliary problem
The solvability and a priori estimates for problem (31) are now studied.
Assume that is a continuous function with , and that is a Carathéodory function satisfying the conditions (H1) and (H2). Then, for every , the auxiliary problem (31) has a weak solution in the sense that
Moreover, the solution is uniformly bounded and fulfills the a priori estimate with the constant provided by Theorem 2.
Through hypothesis (H1) and Hólder’s inequality, we find
for all and . We deduce that the mapping
is well-defined and bounded. Furthermore, by Krasnoselskii’s theorem for Nemytskii operators, the mapping in (34) is continuous from to , so continuous from to due to the continuous embedding .
Let us define the mapping by
On account of Proposition 1 and on what was said regarding the mapping in (34), we are entitled to assert that introduced in (35) is well-defined, bounded and continuous.
The next step in the proof is to show that the mapping is a pseodomonotone operator, which means that if in and
To this end, let be a sequence as above. By the Rellich–Kondrachov theorem, we derive from in that in . As noted before, the sequence is bounded in . Therefore, we have
Then (36) entails that (24) holds true. As Proposition 1 guarantees that has the property, we can conclude that in . From here, it can be readily shown (37) thanks to the continuity and boundedness properties stated in Proposition 1 and those related to (34). This amounts to saying that is a pseudomonotone operator.
In the following, we prove that the operator is coercive, that is
Toward this we infer from (35), (33), (22), (3), Hölder’s inequality and hypothesis (H2) that
for all . Since and as known from hypothesis (H2), we confirm the validity of (38).
We showed on the reflexive Banach space that the operator defined in (35) is bounded, pseudomonotone and coercive. According to the main theorem for pseudomonotone operators (see, for example, , Th. 2.99), we can conclude that the mapping is surjective. So, in particular, there exists such that , which is exactly (32). Therefore is a weak solution of auxiliary problem (31).
Let us point out that the function G and its truncation take values in the same set , and function F is the same in both problems (1) and the (31). Consequently, Theorem 2 can be applied to the auxiliary problem (31) and provides the same uniform bound of the solution set as for the original problem (1). This ensures that , which completes the proof. □
Relying on Theorem 3, we are now able to prove Theorem 1.
Proof of Theorem 1.
It was established in Theorem 2 that the solution set of problem (1) is uniformly bounded by a constant , where is a lower bound of the function G. Since the truncated function has the lower bound too for all (see (22)) and the reaction term is unchanged in problems (1) and (31) and is subject to the same hypotheses (H1)-(H2), Theorem 2 applies to the truncated problem (31) and provides the same bound C for its solution set whenever . In particular, the solution of problem (31) provided by Theorem 3 satisfies the estimate .
Owing to the crucial information that C is independent of , we can choose . Hence, the estimate and (22) render that the functions and G coincide along the values for all . According to Theorem 3, solves problem (31), and thus it becomes a bounded weak solution of the original problem (1). The conclusion of Theorem 1 is achieved. □
This research received no external funding.
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Conflicts of Interest
The author declares no conflict of interest.
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