Abstract
The paper presents existence results for nonlinear elliptic problems under a nonhomogeneous Dirichlet boundary condition. The considered elliptic equations exhibit nonlinearities containing derivatives of the solution.
MSC: 35H30, 35A16.
Keywords:
quasilinear elliptic problem; nonhomogeneous Dirichlet boundary conditions; existence result1 Introduction
The aim of the paper is twofold: first, to study nonlinear elliptic problems under nonhomogeneous Dirichlet boundary condition; second, to incorporate in the problem statement nonlinearities exhibiting derivatives of the solution. These requirements need to develop a nonstandard approach, in particular prevent the use of variational methods.
Specifically, we study two problems on a bounded domain () with Lipschitz boundary ∂Ω. We first consider the problem
where , , are Carathéodory functions (that is, they are measurable in and continuous in the other variables), , and denotes the space of sized symmetric matrices. In the following definition we make clear what we understand by solution to problem (1).
Definition 1 A (weak) solution of problem (1) is an element such that , , , , and
Next we focus on nonhomogeneous Dirichlet problems where, contrary to problem (1), the dependence with respect to the gradient ∇u of the solution u is not expressed in a divergence form, namely
Here , and g are as in problem (1), while and () are Carathéodory functions. The meaning of solution of problem (2) is as follows.
Definition 2 A (weak) solution of problem (2) is an element such that , , , for all , , and
Problems of type (1) and (2) have been investigated in settings that are different from ours (see, e.g., [16]). For instance, problem (1) is studied in [3] when and with functions and corresponding to certain physical models, as described by Reynolds equation where with , , , and . Whereas many of the previous results on problems (1) and (2) involve technical and somewhat restrictive assumptions on the data, the purpose of the present paper is to provide an elementary resolution of problems (1) and (2) in geometrically relevant situation. As an example of such a geometrically relevant situation, we mention the assumption on the term in problem (1) to vanish at two points.
Our results are stated as Theorems 1 and 2. They are existence and location theorems on problems (1) and (2), respectively, guaranteeing solutions in the sense of Definitions 1 and 2 that fulfill an estimate with given constants . This a priori estimate of the solution is derived through natural geometric hypotheses that can be directly checked. It is also worthwhile to remark that we cannot drop by translation the nonhomogeneous boundary conditions to become homogeneous because our hypotheses would be no longer verified. The arguments used in the proof are based on truncation techniques and Schauder’s fixed point theorem. We emphasize that, due to the type of assumptions we impose, it is essential in our approach to keep separate the two terms in divergence form appearing in the statement of (1) and (2). A careful inspection of our proofs shows that we rely on the linearity with respect to the gradient ∇u in the first divergence term and on the vanishing at suitable points in the second divergence term.
The rest of the paper is organized as follows. Section 2 is devoted to problem (1). Section 3 studies problem (2).
2 Result on problem (1)
Throughout the paper the notation and stands for the usual norms on (or ) and , respectively. By we denote the Euclidean norm of .
Let be the first eigenvalue of the negative Laplacian differential operator on , which is known to be positive and characterized by
We suppose the following hypotheses on the data a, b, f, and g in problem (1):
(H_{1}) There is a Carathéodory function such that
(H_{2}) There are constants with on ∂Ω such that
(H_{3}) The functions a, f are bounded on the set and
(H_{4}) There is a Carathéodory function such that
and
Remark 1 The constants and are not solutions of problem (1), unless or on ∂Ω. Thus, in general, problem (1) has no evident solution.
Remark 2 Due to their different structure and requirements, the two terms in (1) that are in divergence form cannot be combined.
Remark 3 The last part of hypothesis (H_{4}) incorporates the monotonicity condition
as well as the Lipschitz condition
and it is more general than both of them.
The result that we set forth in this section is the following theorem ensuring existence and location of solution for problem (1).
Theorem 1Assume that hypotheses (H_{1})(H_{4}) are satisfied. Then problem (1) has at least one solutionin the sense of Definition 1 satisfying
Proof Consider the set
which is a nonempty, bounded, closed, convex subset in .
Claim 1: Given , there is a unique solution of the problem
Note that Claim 1 is equivalent to solving uniquely the problem
Here and in (4) are expressed by
and
for all . Notice that the operators A and B are well defined due to our hypotheses.
With the fixed element , let us introduce the Carathéodory map by
From hypotheses (H_{1}), (H_{3}), (H_{4}), and because , it follows that satisfies the properties: there is a constant such that
and
Estimate (5) guarantees that the operator A is bounded (in the sense to be bounded on bounded sets). It is easily seen that (6) implies that A is coercive, that is,
Moreover, relations (5)(6) ensure that the operator A is maximal monotone, so pseudomonotone (see, e.g., [[7], §2.3.1]). Since A is bounded, coercive, and pseudomonotone, it is surjective (see, e.g., [[7], Theorem 2.99]), whence the existence of in Claim 1. The uniqueness of is a direct consequence of (6) (notice that ). This establishes Claim 1.
Now, taking advantage of Claim 1, we define the operator by for all , where is the unique element corresponding to as proved in Claim 1.
Claim 2: The mapping is continuous.
Let and let be a sequence such that in . Denote and . Using the definition of T and choosing as a test function in Claim 1 (written with and u), we have
Combining this formula with (H_{1}), (H_{3}), (H_{4}), (3) and the CauchySchwarz inequality, we obtain
Taking into account hypothesis (H_{4}) leads to
Set
We claim that
To this end, we show that any subsequence of possesses a subsequence converging to 0 in . Since in , we have that, along a relabeled subsequence, for a.a. . Invoking (H_{3}), we have that , with some constant . Through Lebesgue’s dominated convergence theorem, we conclude that as , so (8) holds true.
Similarly, we have
Then, in view of (7), we infer that . Since the domain Ω is bounded and , we can make use of the Poincaré inequality for , which yields , whence in . This establishes Claim 2.
With the truncation function defined by
consider the operator introduced as follows
Claim 3: The mapping has a fixed point.
Since is continuous by Claim 2 (thus a fortiori is continuous) and τ is a bounded continuous function, we infer that is continuous. We claim that is a compact operator. To this end, it suffices to check that is relatively compact in . Because of the compact embedding of in , it is sufficient to prove that is bounded in .
Let and denote . By the definition of T and inserting therein the test function , we see that
Then, as in the proof of Claim 2, from assumptions (H_{1}) and (H_{4}) we obtain that
whence, by (H_{4}),
with a constant independent of u.
Using (11), we derive
with constants independent of u. It follows that the set is bounded in , so according to what was said before, the map is compact. Consequently, Schauder’s fixed point theorem can be applied (see, e.g., [[8], p.452]), through which it follows that S admits a fixed point in C. This shows Claim 3.
Claim 4: Let be a fixed point of S. Then there holds .
The existence of a point such that is ensured by Claim 3. Fix such a point u and set . In order to deduce the desired conclusion from , it suffices to check that a.e. in Ω. We only verify the inequality a.e. in Ω because the proof of the other inequality is similar. By virtue of hypothesis (H_{2}), we have on ∂Ω (in the sense of traces), hence on ∂Ω and so the function belongs to (see, e.g., [[7], p.35]). Using as a test function in the definition of T gives
which reads as
By the assumption that and from (10) we know that for a.a. , hence a.e. in . Then hypothesis (H_{2}) implies that and a.e. in . Consequently, (12), (H_{1}), and (H_{4}) entail
whence a.e. in . On the other hand, we have in . Altogether, we obtain that in Ω. Since , we conclude that a.e. in Ω, thus a.e. in Ω. This proves Claim 4.
By Claims 3 and 4, the operator T admits a fixed point . Then the definition of T implies that , so u is a solution of problem (1). In addition, the fact that guarantees that a.e. in Ω. The proof of Theorem 1 is complete. □
3 Result on problem (2)
The hypotheses on the data a, (), f, and g in problem (2) that we suppose are as follows: (H_{1}) in Section 2,
() There exist constants such that on ∂Ω and
() There exist constants and such that
() and there exist constants , , with , such that
Remark 4 As in the case of problem (1), we note that the constant functions and are not solutions of problem (2), unless or on ∂Ω.
Now we state our result of existence and location of solutions for problem (2).
Theorem 2Assume that (H_{1}), (), (), and () are satisfied. Then problem (2) has at least one solutionin the sense of Definition 2 satisfying
Proof We follow the pattern of proof of Theorem 1. Hence, using the constants and prescribed in (), we consider
which is a nonempty, bounded, closed, convex subset of . We proceed by proving four claims regarding problem (2) that correspond to those in the proof of Theorem 1 for problem (1). We provide the proof since there are some differences with respect to the proof of Theorem 1.
Claim 1: For every , there is a unique solution of the problem
As in the proof of Theorem 1, first we note that Claim 1 is equivalent to proving that the problem
admits a unique solution, where
and
For , by the CauchySchwarz inequalities in and in , as well as () and (3), we derive the estimate
for all . Using the CauchySchwarz inequality in , the fact that , () and (14), we get
which ensures that is a continuous bilinear form. From (H_{1}), the fact that , () and (14), we have
Since (as postulated in ()), we infer that is also coercive.
On the basis of the reasoning in (14), the following estimate holds
for all . Taking into account that , (), (15) and (3), we see that
where stands for the Lebesgue measure of Ω. Therefore is linear and continuous. The properties of the mappings A and B permit to apply the LaxMilgram theorem, through which we conclude that problem (13) admits a unique solution. This establishes Claim 1.
As in the proof of Theorem 1, we introduce the operator defined by for all , with given in Claim 1.
Claim 2: The mapping is continuous.
In order to prove this assertion, we proceed as in the proof of Claim 2 in Theorem 1. Fix and consider a sequence such that in . Denoting and , we find that
A straightforward calculation entails
Combining (H_{1}), (), (), (16), (17), (3), and the CauchySchwarz inequality yields
Proceeding as in (8) we show that
Now it suffices to combine (18), (19), (20) and recall that (see ()) to conclude that . Then, because and Ω is bounded, by the Poincaré inequality, we also deduce that . This amounts to saying that in , which proves Claim 2.
Following the approach developed in the proof of Theorem 1, we introduce the operator given by (10), with the truncation function defined in (9) corresponding to the constants and in ().
Claim 3: The mapping has a fixed point.
Claim 2 readily implies that the mapping is continuous. Let us check that is a compact operator. To see this, it suffices to check that is relatively compact in . Thanks to the compactness of the embedding of into , this reduces to show that is bounded in . To this end, let and denote . We can argue as in the proof of Theorem 1 by relying now on the present hypotheses. We obtain from with the test function , in conjunction with (H_{1}), (), (), that
where is a constant independent of u. In view of hypothesis (), it follows that
with a constant independent of u. Using (21) and the definition of S, we get the estimate
with independent of u. We conclude that the set is bounded in , so relatively compact in . Therefore the map is compact. This enables us to apply Schauder’s fixed point theorem (see, e.g., [[8], p.452]), which implies that S possesses a fixed point in C. Claim 3 is thus shown.
Claim 4: If is a fixed point of S, then .
Let be a fixed point of S and set . In order to show that u is a fixed point of T, it is needed to be fulfilled a.e. in Ω. The proof is done following the pattern of the corresponding part in the proof of Theorem 1. We outline the proof of a.e. in Ω (the proof of the other inequality is similar).
Then hypothesis () implies that
Combining with (22), (H_{1}), and () entails
It turns out that a.e. in . Also, it is clear that in . Consequently, the equality in Ω is valid, which results in a.e. in Ω because . This reads as a.e. in Ω, so Claim 4 is fulfilled.
Now we can conclude the proof. Claims 3 and 4 ensure that there exists a fixed point of the operator T. This means that and u is a solution of problem (2). Moreover, since , we also have a.e. in Ω. The desired conclusion is achieved. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
DM and VVM jointly worked and obtained all the results presented in the paper and participated equally in the preparation of the paper. Both authors read and approved the final manuscript.
Acknowledgements
The second author is supported by the Marie Curie IntraEuropean Fellowship for Career Development within the European Community’s 7th Framework Program (Grant Agreement No. PIEFGA2010274519).
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