An elliptic system with logarithmic nonlinearity
-
Claudianor Alves
Abstract
In the present paper, we study the existence of solutions for some classes of
singular systems involving the
1 Introduction and statement of the main results
Let
which exhibits a singularity at zero through logarithm function. The
variable exponents
where
Throughout this paper, we denote by
where
A weak solution of (1.1) is a pair
for all
The study of problems involving variable exponents growth conditions is
widely justified with many physical examples, and these problems arise from a variety of
nonlinear phenomena. They are used in electrorheological fluids as well as
in image restorations. For more inquiries on modeling physical phenomena
involving the
Elliptic problems involving the logarithmic nonlinearity appear in some
physical models like in the dynamic of thin films of viscous fluids; see for
instance [12]. An interesting point regarding these
problems comes from the fact that
The essential point in this work is that the singularity in system (1.1) comes through logarithmic nonlinearities involving variable exponents growth conditions. According to our knowledge, this happens for the first time when such problems are studied. Our main results provide the existence and regularity of (positive) solutions for problem (1.1). They are stated as follows.
Theorem 1.1.
Assume (1.2) holds.
If
(1.4)then problem ( 1.1 ) has a solution
If
(1.5)then problem ( 1.1 ) has a solution
If
(1.6)then problem ( 1.1 ) admits a solution
Theorem 1.2.
Assume (1.2) and that
holds for all
satisfying (1.3).
The proof of Theorem 1.1 is done in Section 4. Our approach
relies on the sub-supersolutions techniques. However, this method in its
system version (see [5, p. 269]) does not work for problem (1.1)
due to its noncooperative character, which means that the right-hand sides
of the equations in (1.1) are not necessarily increasing whenever u
(resp. v) is fixed in the first (resp. second) equation in (1.1).
Another reason this approach cannot be directly implemented is the presence
of singularities in (1.1). To overcome this difficulties, we disturb
problem (1.1) by introducing a parameter
The proof of Theorem 1.2 is done in Section 5. It is chiefly
based on a theorem by Rabinowitz (see [18]) which establishes,
for each
A significant feature of our existence results concerns the regularity part. In Theorem 1.1, the regularity of the obtained solution for problem (1.1) is derived through the weak comparison principle and the regularity result in [3].
2 Preliminaries
Let
which is a Banach space with the Luxemburg norm
The Banach space
equipped with the norm
The space
is compact with
The following result gives important properties related to the logarithmic nonlinearity.
Lemma 2.1.
For each
for all
For each
for all
Let γ, θ and δ be real numbers. If
Proof.
With respect to the inequalities, we only prove (i) because (ii) can be justified similarly. A simple computation provides
Thus, there is a small
On the other hand, the limit
implies that there is
Since the function
In order to show (iii), observe that
the condition
3 Sub-supersolution theorem
Let us introduce the quasilinear system
where
Given
The following result is a key point in the proof of Theorem 1.1.
Theorem 3.1.
Assume that H and G satisfy (I), and
let
Suppose that
and
for all nonnegative functions
satisfying
Proof.
The proof is chiefly based on pseudomonotone operator theory. Define the functions
and
In what follows, we fix
Using the above functions, we introduce the auxiliary problem
where
and
By the Minty–Browder theorem (see, e.g., [17]), problem
(3.2) has a solution
where E is the Banach space
Let us show that the function B satisfies the hypotheses of the Minty–Browder theorem.
(i) B is continuous.
Let
Up to a subsequence, we can assume that
Note that
Then the continuity and the boundedness of H, together with Lebesgue’s dominated convergence theorem and the Hölder inequality, gives
On the other hand, we can assume that
and so
Hence, the previous reasoning provides
and
which justify the continuity of B.
(ii) B is bounded.
Let us show that if
Since
On the other hand, since
the Hölder inequality ensures
From the above arguments we obtain the boundedness of B.
(iii) B is coercive.
Next, we prove that
Note that
where C is a positive constant. The triangular inequality and the fact
that
Gathering the last inequality with the embeddings
we derive
where C is a positive constant. In the same manner, we can see that
If
If
Consider in E a sequence
Suppose that the first
possibility happens and that
Consequently, in both cases studied above, one has
The other situations regarding
(iv) B is pseudomonotone.
We recall that B is a pseudomonotone operator if
Then
for all
If
and
Note that
The previous arguments can be repeated to show that
Gathering the above limits together with (3.6), we have
From the weak convergence we get
and
Therefore,
and
By using (3.8) and (3.9) in (3.7), the
for all
Finally, from properties (i)–(iv) we are in a position to apply [17, Theorem
3.3.6] which ensures that B is surjective. Thereby, there
exists
and, in particular,
It remains to prove that
We only prove the first inequalities in (3.10) because the second ones
can be justified similarly. Set
Therefore,
wich implies that
4 Proof of Theorem 1.1
For every
Our goal is to show through Theorem 3.1 that (4.1) has a positive solution
Let
defined by
is a subsolution of the problem
where
with a fixed number
Given
where σ is a real constant.
If
where
where
Now, let
By [10, Lemma 2.1] and [11], there exist positive constants
and
By the weak maximum principle, we have
We state the following existence result for the regularized problem (4.1).
Theorem 4.1.
Under the assumptions of Theorem 1.1, there
exists
for all
Proof.
First, let us show that
and
for all
Next, we will show that
and
If (1.4) holds, it follows from (4.3), (4.10), (4.11) and for
and
for all
If (1.5) is satisfied, combining Lemma 2.1 with (4.4) and (4.5), by (4.10), (4.11) and
for
and
for
Finally, if (1.6) holds, using (4.3), (4.5), (4.10) and (4.11), for
and similarly
for all
Consequently, it turns out from (4.12), (4.13), (4.14), (4.15), (4.16) and (4.17) that
and
for all
Then, owing to Theorem 3.1, we conclude that the perturbed problem (4.1) has a solution
within
and
for some positive constants
for certain
Proof of Theorem 1.1.
Set
Moreover, the property formulated in (4.9) holds true.
Employing Arzelà–Ascoli’s theorem, we may pass to the limit in
5 Proof of Theorem 1.2
This section is devoted to the proof of Theorem 1.2. For
Our demonstration strategy will be to show, by applying the well-known
result due to Rabinowitz [18], that for each
5.1 Existence result for the regularized system
Fix
Observe the following facts:
From (1.2) the claim follows.
Since
In the same manner, we have
is well defined.
Lemma 5.1.
The operator
Proof.
Consider a sequence
Since
Note that
where the constant C does not depend on
In the sequel, up to a subsequence, we can assume that
A similar reasoning leads to
Since
In order to show that
Theorem 5.2.
Under assumptions (1.2) and (1.7), problem (5.1) admits a solution
Proof.
From Lemma 5.1 and by invoking [18], there is an unbounded
continuum
On the other hand, by Lemma 2.1, the function
If
where
If
where
Therefore,
Next, we show that the component
and
where C depends on
where C depends on θ and ε. Now we will estimate the
integral
The last function belongs to
By the Hölder inequality and considering all the possibilities for the norms
we get
Using the embedding
Thus,
A similar reasoning leads to
Since
5.2 Passage to the limit
Set
Claim.
The sequences
First of all, we know that
the maximum principle gives
By the strong maximum principle (see [11, Theorem 1.2]), we have
where
Denote by
where
Let
where C is a constant that depends on γ and θ.
By the Hardy–Sobolev inequality (see [13]), together with the
embedding
where the constant C does not depend on
By (5.16) and using the reasoning that leads to (5.13) and (5.14), we obtain that
for some pair
Taking
From the previous strong convergence of
and
for all
Funding source: Conselho Nacional de Desenvolvimento Científico e Tecnológico
Award Identifier / Grant number: 304036/2013-7
Award Identifier / Grant number: 402792/2015-7
Funding statement: The first author was partially supported by Conselho Nacional de Desenvolvimento Científico (304036/2013-7) and Instituto Nacional de Ciência de Tecnologia. The second author was supported by Conselho Nacional de Desenvolvimento Científico (402792/2015-7).
Acknowledgements
The authors thank the anonymous referees and the editor, Prof. Vicentiu Rădulescu, for their valuable comments which helped to improve this work. The work was started while the second and the third authors were visiting the Federal University of Campina Grande. They thank Prof. Claudianor Alves and the other members of the department for hospitality.
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