Research Article | Open Access
Hicham Redwane, "Existence of Solution for Nonlinear Elliptic Equations with Unbounded Coefficients and Data", International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 219586, 18 pages, 2009. https://doi.org/10.1155/2009/219586
Existence of Solution for Nonlinear Elliptic Equations with Unbounded Coefficients and Data
An existence result of a renormalized solution for a class of nonlinear elliptic equations is established. The diffusion functions may not be in for a finite value of the unknown and the data belong to .
In this paper we investigate the problem of existence of a renormalized solutions for elliptic equations of the type
where is an open bounded subset of , with the data in . The operator is a Leray-Lions operator defined on the weighted sobolev spaces , but which is not restricted by any growth condition with respect to (see assumptions (2.2), (2.4), and (2.5) of Section 3). The function is controlled by a real function which blows up for a finite value (see (2.2), (2.3)).
There are mainly two types of difficulties that are studying Problem (1.1). One consists to give a sense to the flux on the set . The second one is that the data only belong to , so that proving existence of a weak solution (i.e., in the distribution meaning) seems to be an arduous task. To overcome this difficulty we use in this paper the framework of renormalized solutions. This notion was introduced by DiPerna and Lions  for the study of Boltzmann equation (see also Lions  for a few applications to fluid mechanics models). This notion was then adapted to elliptic vesion of (1.1) in Boccardo et al. , and Murat [4, 5] (see also [6, 7] for nonlinear parabolic problems). At the same time the equivalent notion of entropy solutions has been developed independently by Bénilan et al.  for the study of nonlinear elliptic problems.
In the case where is replaced by (problems with diffusion matrices that have at least one diagonal coefficient that blows up for a finite value of the unknown) and , existence and uniqueness has been established in Blanchard and Redwane [9, 10].
As far as we have the stationary and evolution equations case (1.1), the existence and a partial uniqueness of renormalized solutions have been proved in Blanchard et al.  in the case where is replaced by (where is a Carathéodory symmetric matrices, such that blows up as uniformly with respect to ). It has also been applied to the study of linear and nonlinear elliptic and parabolic equations when the diffusion coefficient has a singularity for a finite value of the unknown (see García Vázquez and Ortegón Gallego [12, 13] and Orsina ).
The paper is organized as follows. In Section 2 we will precise some basic properties of weighted Sobolev spaces. Section 3 is devoted to specify the assumptions on and needed in the present study and gives the definition of a renormalized solution of (1.1). In Section 4 (Theorem 4.1) we establish the existence of such a solution.
Throughout the paper, we assume that the following assumptions hold true. is a bounded open subset on . Let us suppose that is a real number, and is a vector of weight functions. Furthermore we suppose that every component is a measurable function which is strictly positive and satisfies
We define the weighted Lebesgue space with weight , as the space of all real-valued measurable functions for which
In order to define the weighted Sobolev space of , as the space of all real-valued functions such that the derivaties in the sense of distributions satisfy for all . This set of functions forms a Banach space under the norm To deal with the Dirichlet problem, we use the space defined as the closure of with respect to the norm . Note that, is dense in and is a reflexive Banach space. Note that the expression
is a norm defined on and is equivalent to the norm (2.3). Moreover is a reflexive Banach space, and there exist a weight function on and a parameter such that the Hardy inequality
holds for every with a constant independent of . Moreover, the imbedding is compact.
3. Assumptions on the Data and Definition of a Renormalized Solution
Throughout the paper, we assume that the following assumptions hold true. is a bounded open set on . Let , and let be a vector of weight functions.
Let now be a Leray-Lions operator defined on into and where there exists a positive function which satisfies
for almost every , for every and .
For any , for almost every , for every and , and where is a positive function in
for any , for any and for almost every
Remark 3.1. As already mentioned in the introduction, problem (1.1) does not admit a weak solution under assumptions (3.1)–(3.5) since the growth of is not controlled with respect to , the field is not, in general, defined as a distribution because the difficulty is defining the field on the subset of , (since on this set, ).
The following notations will be used throughout the paper. For any , the truncation at height is defined by , for any positive numbers and , the functions are defined by
We define for fixed and , for all .
The definition of a renormalized solution for Problem (1.1) can be stated as follows.
Definition 3.2. A measurable function defined on is a renormalized solution of Problem (1.1) if and if, for every function in such that supp is compact and satisfies
The following remarks are concerned with Definition 3.2.
Remark 3.3. Notice that, thanks to our regularity assumptions (3.8), (3.9), (3.10) and the choice of , all terms in (3.13) are well defined.
The following two identifications are made in (3.13):
(i) identifies with for almost every , where and supp. As a consequence of (3.8), (3.9), and (3.10), and of , it follows that (ii), because and .
4. Existence Result
This section is devoted to establish the following existence theorem.
Proof. The proof is divided into 7 steps. In Step 1, we introduce an approximate problem. Step 2 is devoted to establish a few a priori estimates, the limit of the approximate solutions is introduced and it is shown that satisfies (3.8) and (3.9). Step 3 is devoted to prove an energy estimate (Lemma 4.2) which is a key point for the monotonicity arguments that are developed in Step 4. Step 5 is devoted to prove that satisfies (3.11). In Step 6 we prove that satisfies (3.12). Finally, Step 7 is devoted to prove that satisfies (3.13) of Definition 3.2.
Step 2. A priori estimates and pointwise convergence of
Using as a test function in (4.4) leads to
Since satisfies (3.2), (4.2), and owing to (4.8) we have From (4.10) we deduce with a classical argument (see, e.g., ) that, for a subsequence still indexed by , as tends to 0, where is a measurable function defined on which is finite a.e. in .
Taking now as a test function in (4.4) gives
Since satisfies (3.2) and satisfies (3.1), permit to deduce from (4.13) that where is a constant independent of .
Now for a fixed , assumption (3.3) gives for ,
In view of (4.14) and (4.15), we deduce that then there exists a function such that To prove that is less or equal to is an easy task which is performed exactly as in [10, 11]. Using as a test function in (4.4) leads to which implies easily that
Then (3.2), (4.1), and (4.2) yield
With the help of Poincaré's inequality, we have where does not depend on . Then in view of (3.1), (4.11), and , we can pass to the limit in (4.21) as tends to 0, to deduce that
Let us now take as a test function in (4.4), where . We obtain
Then (3.2) yields We deduce with a classical argument that, for a subsequence still indexed by , as tends to 0, where is a measurable function defined on which is finite a.e. in .
Using the admissible test function in (4.4) leads to
As a consequence of the previous convergence results, we are in a position to pass to the limit as tends to 0 in (4.27) Using the pointwise convergence of to 0 as tends to and a.e. in independently of , since , Lebesgue's convergence theorem shows that as tends to . Passing to the limit in (4.28) we obtain
Step 3. In this step we prove the following monotonicity estimate.Lemma 4.2. The subsequence of defined in Step 1 satisfies for any Proof. Let be fixed. Equality (4.30) is split into
In the sequel we pass to the limit in (4.31) when tends to 0.Limit of
Using the admissible test function in (4.4) leads to where , pass to the limit as tends to 0 in (4.33).
Since supp and , we have for and
In view of (4.24), (4.34) we deduce that for fixed : independently of . Then there exists a function such that for fixed : Now for , we have a.e. in which implies that, through the use of (4.17), (4.25), and (4.36) and passing to the limit as tends to 0, a.e. in . As a consequence of (4.38) we have for We are now in a position to exploit (4.33), which gives together with (4.36) and (4.39) Passing to the limit as tends to in (4.40) leads to The second term of (4.33) Then (4.29) implies that In view (4.41) and (4.43), passing to the limit as tends to 0 and as tends to in (4.33) is an easy task and leads to We are now in a position to exploit (4.44).
The use of the test function in (4.4), yields
Passing to the limit as tends to 0 in (4.45), in view (4.44), we have Limit of
In view of (4.12), (4.17) and since converges to a.e. in and due to the bound a.e. in , we have Limit of
Let us remark that (3.1), (4.1), and (4.11) imply that as tends to 0, and that for a.e. in , uniformly with respect to .
It follows that when tends to 0
In view of (4.12), we conclude that As a consequence of (4.50) and (4.51) we have for all Equations (4.46), (4.46), (4.47), and (4.46) allow to pass to the limit as tends to zero in (4.31) and to obtain (4.30) of Lemma 4.2.
Step 4. In this step we identify the weak limit and we prove the weak convergence of the “truncated” energy as tends to 0.
Lemma 4.3. For fixed , one has And as tends to 0
Proof. Let be fixed. From (4.11) and (4.50) together with (4.30) of Lemma 4.2, we obtain
We remark the monotone character (with respect to ) and since converges to a.e. in and due to the bound a.e. in , we conclude that for all we have
The usual Minty's argument applies in view of (4.56). It follows that
which together with (4.20) yields (4.53) of Lemma 4.3.
In order to prove (4.54), we observe that the monotone character of (with respect to ) and (4.30) give
strongly in as tends to 0. Moreover (4.12), (4.17), (4.50), and (4.53) imply that weakly in as tends to 0 weakly in as tends to 0, and strongly in as tends to 0.
Using the above convergence results (4.59), (4.60), and (4.61) in (4.58) we obtain that for any
weakly in as tends to 0.
Step 5. In this step we prove that satisfies (3.11).
Using as a test function in (4.4) leads to
Since supp, we have In view of (4.54) of Lemma 4.3 and since converges to a.e. in and due to the bound a.e. in , we can pass to the limit as tends to 0 for fixed to obtain Taking the limit as tends to 0 and tends to in (4.63) and using the estimate (4.64) and (4.65) show that
Step 6. In this step we prove that satisfies (3.12).
Using as a test function in (4.4) leads to
where . Since supp and we have In view of (4.22), (4.36), (4.39), and (4.53), passing to the limit as tends to 0 and tends to