On the fractional p-Laplacian equations with weight and general datum
-
,
Abstract
The aim of this paper is to study the following problem:
where Ω is a smooth bounded domain of
with
1 Introduction and motivations
We consider the following problem:
where
Ω is a smooth bounded domain containing the origin and f belongs to a suitable Lebesgue space.
This class of operators appear in a natural way when dealing with the improved Hardy inequality, namely, for all
where
In the same way, we can consider
then it satisfies a weak Harnack inequality.
More precisely, there exists a positive constant
where
We refer to [16, 19] and the references therein for a complete discussion and the proof of the Harnack inequality and its generalization of admissible weights.
Our objective in this work is to analyze the properties of the operator
The case of p-Laplacian equations is well known in the literature; we refer for example to [8, 7] where the authors proved the existence and the uniqueness of an entropy solution for
The local case with weight was considered in [3]. The authors proved the existence and the uniqueness of an entropy solution for datum in
For the operator
The case
For general datum, based on some generalization of the Wolff potential theory, Kuusi, Mingione and Sire succeeded in [21] in obtaining the existence of a weak solution belonging to a suitable fractional Sobolev space.
In this paper, we will treat the case
Our approach is more simple and it is based on a suitable choice of a test function’s family and on some algebraic inequalities.
In the first part of the present paper, we will consider the case
Theorem 1.1.
Assume that
and
If
It is clear that for
Next, assuming that
Theorem 1.2.
Assume that
with
In the second part of the paper, we consider the case
The paper is organized as follows. In Section 2, we introduce some useful tools and preliminaries that we will use throughout the paper, like the weighted fractional Sobolev spaces and some related inequalities, a weak comparison principle and some algebraic inequalities. We also specify the sense in which the solutions to problem (1.1) are defined.
In Section 3, we begin by proving Theorem 1.1, namely, the case where
In Section 4, we study the case where
2 Functional setting and main tools
In this section, we give some functional settings that will be used below. We refer to [15, 23] for more details.
Let
Let
Furthermore,
In the same way, we define the space
As in [5] (see also [15]) we can prove the following extension result.
Lemma 2.1.
Assume that
where
The following weighted Sobolev inequality is obtained in [1] and will be used systematically in this paper.
Theorem 2.2 (Weighted fractional Sobolev inequality).
Assume that
where
Moreover, if
for all
Remark 2.3.
As in the case
Now, for
It is clear that for all
In the case where
The following comparison principle extends the classical one obtained by Brezis–Kamin in [10]. See [22, 1] for the proof.
Lemma 2.4.
Let Ω be a bounded domain and let h be a non-negative
continuous function such that
Then
The following algebraic inequalities can be proved using a suitable rescaling argument.
Lemma 2.5.
Assume that
and
In the case where
Since we are considering a solution with datum in
Define
and
where
In the same way, we will use the classical weighted Marcinkiewicz spaces.
Definition 2.6.
For a measurable function u we set
where
We say that u is in the Marcinkiewicz space
Since Ω is a bounded domain,
for all
Since we are considering problems with general datum, we need to specify the concept of solution. We begin by the following definitions.
Definition 2.7.
Let u be a measurable function. We say that
Now, we are able to state what we mean by a solution to problem (1.1).
Definition 2.8.
Assume that
Following [6], we define the notion of entropy solution as follows.
Definition 2.9.
Consider
where
and for all
Remark 2.10.
Notice that for
Since
in
and
we reach that
and
It is clear that
and
3 Existence results: Proofs of Theorems 1.1 and 1.2
In this section, we consider the problem
where
The main goal of this section is to show that problem (3.1) has a weak solution u in the sense of Definition 2.8. As in the local case, the main idea is to proceed by approximation and then pass to the limit by using suitable a priori estimates.
Before proving the main existence results, we need several lemmas.
Let
Notice that the existence and the uniqueness of
The first a priori estimate is given by the following Lemma.
Lemma 3.1.
Let
Proof.
Using
Thus,
Recall that
Now, using the weighted Sobolev inequality in Theorem 2.2, we get
Since
Hence,
Setting
As a consequence we easily get that the sequence
As in the local case, we prove now that the sequence
Lemma 3.2.
Assume that
Proof.
Let
Define
In what follows, we denote by
Using
Hence,
Let
by (3.5), it follows that
Now, using the fact that
Defining
Now, using the algebraic inequality (2.3), we have
Hence, taking into consideration that
So, going back to (3.7), we reach that
By inequality (2.1), we have
Therefore,
It is clear that
hence we just have to estimate the first term. We have
Since Ω is a bounded domain,
where
Let
Then
We set
Defining
as in [17], we find
Notice that
Therefore, there holds
Recall that
Now, if
where
Since
using the fact that
Now, going back to (3.10), there holds
Since
Hence by (3.8) and (3.11), we conclude that
In the same way, by using
Combining the above estimates, we reach that
and the result follows. ∎
Remark 3.3.
As a consequence we get the existence of a measurable function u
such that
It is clear that
Notice that by Lemma 3.1 we conclude that
Let
Since Ω is a bounded domain, by the result of Lemma 3.2 and using Vitali’s lemma, we obtain that
We are now able to prove the first existence result.
Proof of Theorem 1.1.
It is clear that estimate (1.2) follows by using Lemma 3.2 and Fatou’s lemma.
Let
We set
It is clear that
We claim that
Since
Using the fact that
Thus,
Using Lemma 3.2 and Remark 3.3, we easily find that
We now deal with
Since
we have
Using Lemma 3.1 and Remark 3.3, we get
Thus by the dominated convergence theorem we arrive to
Therefore, passing to the limit in (3.13), we arrive at
Remark 3.4.
(i) It is clear that the same existence result holds if we replace f by a bounded Radon measure ν.
(ii) In the case where
3.1 The case of positive datum: Existence and uniqueness of the positive entropy solution
If
Lemma 3.5.
Let
The proof of Lemma 3.5 will be a consequence of the following more general compactness result.
Lemma 3.6.
Let
Proof.
Since
Thus,
Define
and
For simplicity of notation, we set
We have
In the same way, using Young inequality, we obtain that
Combining the above estimates and going back to (3.15), we find
Define
We claim that
Then
In
Thus,
Assume that
Since
by (3.17), it follows that
Going back to (3.16), there results that
Since
Remark 3.7.
(i) As a consequence of the previous strong convergence we reach that
(ii) Letting
For
It is clear that for
Since
(iii) From (3.18) we conclude that
Hence by Fatou’s lemma, we reach that
Now, we are in the position to prove the existence and the uniqueness of the entropy solution.
Proof of Theorem 1.2: Existence part.
It is clear that the existence of u follows by using Theorem 1.1, however the strong convergence of
Let us begin by proving that (2.5) holds.
Since
Using
It is not difficult to show that, for
Thus, using Fatou’s lemma and by (3.19), we conclude that
It is clear that for all
Therefore, using the fact that
and by (3.20), we conclude that
Hence (2.5) holds.
Recall that
Let
One immediately sees that
We now deal with the first term. We have
where
and
It is clear that
as
We now deal with
We set
Then
We claim that
We divide the proof of the claim into two cases according to the value of p.
The singular case:
In this case, we have
Thus,
Since
as
as
The degenerate case:
Thus,
The term
We define
where
strongly in
Now, consider the set
where
Hence we just have to deal with sets of the form
or
We will use Remark 3.7 and a duality argument.
It is clear that for
From Remark 3.7, we know that
Notice that
It is clear that
Thus by (3.25), (3.26) and using a duality argument, we find that
strongly in
We can treat the set
Therefore, combining the above estimates and using the dominated convergence theorem, we conclude that
and the claim follows.
Hence by (3.21)–(3.23), we conclude that
and the result follows at once. ∎
It is clear that if u is an entropy solution of (3.1), then for all
where
Moreover, we can prove that (3.28) holds for all
Lemma 3.8.
Assume that u is an entropy solution to (3.1) with
Proof.
Let
Since u is an entropy solution to (3.1), by (3.27), for k fixed such that
It is clear that
Notice that for
Define
Then
It is clear that
Since
Using the properties of w, we have
As above, since
For
Using the fact that
by the dominated convergence theorem, we conclude that
Hence by (3.31) and (3.32), we obtain that
as
We can treat
We now deal with
Now, for
It is clear that for
Thus,
Therefore, we conclude that
Let
Thus,
In the same way, we can prove that
As a conclusion, and going back to (3.30), we have proved that
Substituting w by
which is the desired result. ∎
Now we are in a position to prove the uniqueness result in Theorem 1.2.
Proof of Theorem 1.2: Uniqueness part.
Let u be the entropy positive solution defined in Theorem 1.1. Recall that
Assume that v is another entropy positive solution to problem (3.1). We claim that
where
Thus,
Using the fact that
it follows that
Let us now prove that
Since u and v are entropy solutions to (3.1), for
and
It is clear that
Thus from (3.33) and (3.34), we have
Let
and
Then
and
We claim that
Let us begin by proving that
If
On the other hand, by (2.5), we find
Hence,
Notice that for
Then by (3.36) there results that
In the same way, we can prove that
Thus combining (3.37) and (3.38), we arrive at
We now deal with
As above, we have
where
and
Let
If
Hence we can assume that
Therefore, we conclude that
Let us begin by proving that
Then for
If
Now taking into consideration that
Therefore, we conclude that
If
It then remains to consider the case
If
If
Now, by (2.7), we get
We now deal with the set
Let us assume that
So for
Now, by using again (2.7), it follows that
Therefore, combining the above estimates, we obtain
For
Thus,
Hence,
We now deal with
If
Now, assume
Hence following the same discussion as in case (iii) in the analysis of
Notice that in a symmetric way we can prove that
Going back to (3.39), it holds that
Therefore, going back to the definition of
It is clear that
We claim that
Notice that
where
Since
we conclude that
For
(I) If
(II) If
(III) Now consider the case where
If
Thus, we deal with the set
It is clear that if
Assume that
Therefore, using (2.7), we obtain
We now consider the set
Thus using again (2.7), we obtain
Therefore, we conclude that
Hence
In conclusion, we have proved that
Let
Thus
4 Problem with reaction term and general datum
In this section, we consider the problem
where
Let us begin with the case
Theorem 4.1.
Assume that
Proof.
Without loss of generality we can assume that
Notice that the existence of
However, the uniqueness follows by using the comparison result in Lemma 2.4. It is clear that by the same comparison principle we obtain that
We claim that
Then
We set
Then
Since
Now taking
as
Therefore,
Since
Since
We now prove that u is the minimal solution of (4.1).
Let
It is clear that the sequence
In the case where
As in the case
Now, we can formulate our existence result.
Theorem 4.2.
Assume that
To prove Theorem 4.2, we need the following classical regularity result.
Lemma 4.3.
Let u be the unique solution to the problem
where
Proof.
We follow closely the Stampacchia argument given in [24]. Using
where
By the Weighted Fractional Sobolev Inequality in Theorem 2.2, it follows that
where
Thus,
Let
Hence,
We set
Since
By the classical result of Stampacchia (see [24]) we get the existence of
Proof of Theorem 4.2.
We follow closely the argument used in the proof of Theorem 4.1.
Define
Notice that, since
However, the uniqueness follows by using the comparison result in Lemma 2.4. It is clear that by using the same comparison principle we obtain that
We claim that
as
Then
We set
Since
We claim that
Hence by using inequality (2.2), it follows that
Using the weighted Sobolev inequality in Theorem 2.2, we reach that
Now we set
Now consider the sequence
Let us now consider the case
More precisely, if
We are able to prove the following result.
Theorem 4.4.
Assume that
Proof.
Recall that by the results of Lemmas 3.1 and 3.2 we know that
Let v be the minimal solution to the problem
It is not difficult to show that
Then v is a
supersolution to (4.1) for
By an induction argument, we can prove that
Funding source: Ministerio de Economía y Competitividad
Award Identifier / Grant number: MTM2013–40846-P
Funding statement: Work partially supported by Project MTM2013–40846-P, MINECO, Spain.
Acknowledgements
The authors would like to express their gratitude to the anonymous referee for their comments and suggestions that improved the last version of the manuscript.
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