1 Introduction and statement of the main results
Let () be a bounded domain with smooth boundary . We are interested in the following quasilinear system:
which exhibits a singularity at zero through logarithm function. The variable exponents , are positive, the constants γ and θ are both greater than 0, and (resp. ) stands for the -Laplacian (resp. -Laplacian) differential operator on (resp. ) with ,
where and . In the sequel, we set
Throughout this paper, we denote by the pair of functions such that there is a constant , which depends on u and v, verifying
A weak solution of (1.1) is a pair with u, v being positive a.e. in Ω and satisfying
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 -growth condition, we refer to [1, 2, 4, 6, 7, 14, 19, 21, 20, 23, 22].
Elliptic problems involving the logarithmic nonlinearity appear in some physical models like in the dynamic of thin films of viscous fluids; see for instance . An interesting point regarding these problems comes from the fact that is sign changing and behaving at the origin like the power function for with a slow growth. In addition, the logarithmic function is not invariant by scaling, which does not hold for the power function. These facts motivated the recent studies in [16, 8, 12], where de Queiroz et al. considered the scalar semilinear case of (1.1) (that is, ) with constant exponents and by essentially using the linearity of the principal part. We also mention , focusing on problems with constant exponents involving nonlinear operators.
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.
Assume (1.2) holds.
Assume (1.2) and that
holds for all , where e denotes the Euler number. Then problem (1.1) has a positive solution
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 . This gives rise to a regularized system for (1.1) depending on , whose study is relevant for our initial problem. We construct a sub-supersolution pair for the regularized system independent on ε, and we show the existence of a positive family of solutions , for certain , through a new result regarding sub-supersolutions for quasilinear competitive (noncooperative) systems involving variable exponents growth conditions (see Section 3). Then a (positive) solution of (1.1) is obtained by passing to the limit as essentially relying on the independence on ε of the upper and lower bounds of the approximate solutions and on Arzelà–Ascoli’s Theorem. An important part of our result lies in obtaining the sub and supersolution, which cannot be constructed easily. Precisely, this is due to the fact that the -Laplacian operator is inhomogeneous and, in general, it has no first eigenvalue, that is, the infimum of the eigenvalues of the -Laplacian equals 0 (see ). At this point, the choice of suitable functions with an adjustment of adequate constants is crucial.
The proof of Theorem 1.2 is done in Section 5. It is chiefly based on a theorem by Rabinowitz (see ) which establishes, for each , the existence of positive solutions for the regularized problem of (1.1) in . The solution of (1.1) under assumption (1.7) is obtained by passing to the limit as . This is based on a priori estimates, the Hardy–Sobolev inequality and Lebesgue’s dominated convergence theorem.
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 .
Let with in Ω. Consider the Lebesgue space
which is a Banach space with the Luxemburg norm
The Banach space is defined as
equipped with the norm
The space is defined as the closure of in with respect to the norm. The space is a separable and reflexive Banach space when . For a later use, we recall that the embedding
is compact with .
The following result gives important properties related to the logarithmic nonlinearity.
For each , there is a constant C that depends only on α and θ such that
for all .
For each , there is a constant C that depends only on ε and θ such that
for all .
Let γ, θ and δ be real numbers. If and , then the function , , attains a positive global minimum.
With respect to the inequalities, we only prove (i) because (ii) can be justified similarly. A simple computation provides
Thus, there is a small such that
On the other hand, the limit
implies that there is such that
Since the function , , is continuous for all , there is a constant which depends on α and θ such that in . Therefore, for all , where the constant C depends only on α and θ.
In order to show (iii), observe that . Then f has a unique critical point at . Thus, by solving the inequalities and for , it follows that f is increasing on the interval and decreasing on . By noticing that
the condition implies that , which proves the result. ∎
3 Sub-supersolution theorem
Let us introduce the quasilinear system
where are Carathéodory functions satisfying the following assumption:
Given , there is a constant such that
The following result is a key point in the proof of Theorem 1.1.
Assume that H and G satisfy (I), and let and , with in Ω and such that
for all nonnegative functions . Then problem (3.1) has a (positive) solution
The proof is chiefly based on pseudomonotone operator theory. Define the functions
In what follows, we fix with and set
Using the above functions, we introduce the auxiliary problem
where E is the Banach space endowed with the norm
Let us show that the function B satisfies the hypotheses of the Minty–Browder theorem.
(i) B is continuous.
Let be a sequence that converges to in E. We need to prove that . To this end, let with . By the Hölder inequality, one has
Up to a subsequence, we can assume that a.e in Ω and that there exists a function such that a.e in Ω. Therefore, Lebesgue’s dominated convergence theorem yields
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 a.e in Ω and that there exists such that a.e in Ω. Arguing as before, we get
Hence, the previous reasoning provides
which justify the continuity of B.
(ii) B is bounded.
Let us show that if is a bounded set, then is bounded. To this end, consider a bounded set U and such that . Then for the Hölder inequality gives
Since is bounded, we derive that
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
where C is a positive constant. The triangular inequality and the fact that for nonnegative numbers a and b with give
Gathering the last inequality with the embeddings
where C is a positive constant. In the same manner, we can see that
If and , then
If and , then
Consider in E a sequence such that . Thus,
Suppose that the first possibility happens and that for all . Then we consider two cases:
and for . In this case, we have
and for . In this second case, we have
Consequently, in both cases studied above, one has
The other situations regarding and can be handled in much the same way.
(iv) B is pseudomonotone.
We recall that B is a pseudomonotone operator if in E and
for all .
If , then and in and , respectively. Since and are bounded, we must have
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
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 such that
and, in particular, is a solution of (3.2).
It remains to prove that
We only prove the first inequalities in (3.10) because the second ones can be justified similarly. Set . From the definition of we obtain
wich implies that in Ω. Using a quite similar argument for , we get in Ω. This completes the proof. ∎
4 Proof of Theorem 1.1
For every , let us introduce the auxiliary problem
Let be a bounded domain in with smooth boundary such that , and set . In [24, Lemma 3.1], Yin and Zang have proved that, for small enough and for constants , the function
is a subsolution of the problem
where is a number that does not depend on η, and
with a fixed number and is a number depending only on δ, τ, and p. Note that
Given , let and in be the unique solutions of the problems
where σ is a real constant.
If , considering the corresponding function w for and applying the weak maximum principle, we get
where and are constants that do not depend on λ. If from [11, Lemma 2.1] and for λ large one has
where , , and are positive constants independent of λ. Moreover, by the strong maximum principle there is a constant (that can depend on λ) such that
Now, let and in be the unique solutions of the homogeneous Dirichlet problems
By the weak maximum principle, we have and in for sufficiently large.
We state the following existence result for the regularized problem (4.1).
for all . Moreover,
for all , provided that is sufficiently large.
for all , provided that λ is large enough.
for small enough, for all and all , provided that λ is sufficiently large.
for all small and all , provided that is large enough.
for all with . This shows that is a supersolution for (4.1) for all .
for some positive constants and . Then, thanks to [3, Lemma 2], we deduce that
for certain . ∎
Proof of Theorem 1.1.
Set for . By Theorem 4.1, we know that there exists a positive solution bounded in , for certain , for the problem
5 Proof of Theorem 1.2
This section is devoted to the proof of Theorem 1.2. For , let us consider the regularized problem
Our demonstration strategy will be to show, by applying the well-known result due to Rabinowitz , that for each system (5.1) possesses a positive solution in , and then derive a solution of (1.1) by taking the limit .
5.1 Existence result for the regularized system
Fix , and for each pair let us consider the auxiliary problem
Observe the following facts:
: Indeed, consider such that for all . By Lemma 2.1, one has
From (1.2) the claim follows.
: By (1.2), notice that
Since , the claim is proved.
In the same manner, we have and for all . Then, on account of the above remarks, the unique solvability of in (5.2) is readily derived from the Minty–Browder theorem. Therefore, the solution operator
is well defined.
The operator is continuous and compact.
Consider a sequence in , and . By using as a test function, one gets
Since is bounded in , by Lemma 2.1, is bounded in . Let . Using as a test function, we have
where the constant C does not depend on .
In the sequel, up to a subsequence, we can assume that a.e in Ω and a.e in Ω for some . Then, by Lemma 2.1 and the Lebesgue theorem, we have
A similar reasoning leads to
Since is bounded in , from (5.4) we deduce that in . This proves that is continuous.
In order to show that is compact, it suffices to prove that is compact for all bounded. At this point, a quite similar argument as above produces the desired conclusion. This completes the proof. ∎
On the other hand, by Lemma 2.1, the function , for , attains a strictly positive minimum if . Since , we obtain the following assertions:
If , then
If , then
Therefore, , where , and, with a quite similar reasoning, we get for some . Thus, by the maximum principle, must be constituted by strictly positive functions.
Next, we show that the component is unbounded with respect to . By contradiction, suppose that there is such that implies that . Fix . Using u as a test function, we get
where C depends on , ε, γ and . Note that
where C depends on θ and ε. Now we will estimate the integral . We have . In order to prove this, note that
The last function belongs to because . Thus, by the Hölder inequality we obtain
By the Hölder inequality and considering all the possibilities for the norms
A similar reasoning leads to
Since and , it follows that the component is bounded, which is absurd. Consequently, crosses the set , and this implies that there is a solution of (5.1). The proof is completed. ∎
5.2 Passage to the limit
The sequences and are bounded in and , respectively, and the weak limits (that exist up to a subsequence) are strictly positive in Ω.
First of all, we know that , where . If denotes the unique positive solution of
the maximum principle gives
By the strong maximum principle (see [11, Theorem 1.2]), we have , where η is the inward normal vector of . Let be an eigenfunction associated to the first eigenvalue of the operator . Note that
where is a positive constant that does not depend on .
Denote by an eigenfunction associated to the first eigenvalue of the operator . Reasoning as above, we also have and
where is a positive constant that does not depend on , with being the unique positive solution of
where the constant C does not depend on .
a.e in Ω,
a.e in Ω
for some pair . From the previous pointwise convergence and the relations between , and , , we conclude that and , which proves the claim.
From the previous strong convergence of and , combined with Lebesgue’s dominated convergence theorem, we obtain
for all , and the existence of a solution is proved.
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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About the article
Published Online: 2017-11-27
Funding Source: Conselho Nacional de Desenvolvimento Científico e Tecnológico
Award identifier / Grant number: 304036/2013-7
Award identifier / Grant number: 402792/2015-7
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).
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 928–945, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0200.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0