Existence of Solutions for Unbounded Elliptic Equations with Critical Natural Growth
Existence of Solutions for Unbounded Elliptic Equations with Critical Natural Growth
Aziz Bouhlal,1 Abderrahmane El Hachimi,2 Jaouad Igbida,3 El Mostafa Sadek,4 and Hamad Talibi Alaoui1
1Labo Math Appli, Faculty of Sciences, B. P. 20, El Jadida, Morocco
2Laboratory of Mathematical Analysis and Applications, Mohammed V University, Faculty of Sciences, Rabat, Morocco
3Labo DGTIC, department of Mathematics, CRMEF El Jadida, Morocco
4Laboratory LabSIPE, ENSA, d'EL Jadida, University Chouaib Doukkali, Morocco
Correspondence should be addressed to El Mostafa Sadek; firstname.lastname@example.org
Received 29 June 2018; Accepted 26 October 2018; Published 26 November 2018
Academic Editor: Peiguang Wang
Copyright © 2018 Aziz Bouhlal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We investigate existence and regularity of solutions to unbounded elliptic problem whose simplest model is , where , and belongs to some appropriate Lebesgue space. We give assumptions on with respect to and to show the existence and regularity results for the solutions of previous equation.
In this paper, we consider the Dirichlet problem for some nonlinear elliptic problems such asunder the following assumptions: is a bounded open subset of , where , , and with and is a measurable function satisfying the following conditions:for almost every , where and are positive real constants. is a Carathéodory-type function satisfying to:for some .
In , Arcoya, Boccardo, and Leonor obtained the existence and regularity results for the following elliptic problem with degenerate coercivity:where , with , and is a bounded subset of , .
The purpose of the present paper is to study the same kind of lower order terms as in problem (4) in the case of an elliptic operator with unbounded coefficients such as (1).
There are several papers concerned with existence and regularity of the solution for the following problem:We refer the intersting articles: Boccardo, Murat and Puel , Bensoussan, Boccardo and Murat , and Boccardo, Gallout . In all these works is a nonlinear lower term having natural growth with respect to , data in suitable Lebesgue spaces, and is a Carathéodory-type bounded function subject to certain structural inequalities.
Another motivation for studying these problem arises from the calculus of variations in the case where with andwhere , which is considered by Puel in .
We point out that in  the authors considered as a bounded function andwhere . The function is symmetric, measurable with respect to and continuous with respect to with the following uniform ellipticity condition: for , and ,
We shall prove the following main results on existence and regularity of solutions for problem (1).
Theorem 1. Let Assuming that the functions and satisfy (2) and (3) then, if belong to , withthere exists a distributional solution of problem (1) such thatFurthermore, any solution of the problem (1) belongs to .
In the next result, we consider the case where has a high summability.
Theorem 2. Let , and assume that (2) and (3) hold true. If the solution given by Theorem 1 and belongs to , withthen belongs to .
The rest of the paper is organized as follows: Section 2 is devoted to give some a priori estimates for the approximated problem associated with problem (1); while in Section 3, we give the detailed proofs of Theorems 1 and 2.
2. The Approximated Problem
In this section, we use the hypotheses (2) and (3) and we suppose thatwhere holds true. To prove Theorem 1 and Theorem 2, we will use the following approximating problems associated with problem (1):whereandBy the results of [2, 4] there exists a weak solution in of problem (13) in the sense thatfor every .
The following lemma will be very useful, as it gives us an a priori estimate on the summability of the solutions to problems (13).
Lemma 3. If is a solution to problem (13), then for every ,Moreover, there exist depending on , , , and , such that
Remark 4. (i) Let be a sequence of solutions of (13). As a consequence of Lemma 3, there exists such that, up to a subsequence, converges weakly to in and a.e. in .
(ii) By the previous lemma we deduce from (3) that
Proof of Lemma 3. In order to prove (17), we claim that by assumption (2) and , there exist positive constant such thatChoosing in (16) and using (20), we obtainThus, joining the terms involving the gradient, we getUsing (12) we deduce thatand the Hölder inequality on the right hand side yieldswhich implies (17).
Let us choose now as a test function in (16), and we obtainSince , the previous calculations implyUsing (17) with , (18) follows.
Lemma 5. Let be the sequence of solutions to problems (13) and let the function given by Remark 4. Then strongly converges to in . Moreover strongly converges to in .
Remark 6. Note that (25) implies that there exists independent of such thatBy using the previous lemma, we deduce that
Proof of Lemma 5. We use (17) written for :Since almost everywhere converges to , we have from Fatou’s lemma thatHence belongs to . Using assumption (17), for any we haveAs before, we first choose such that the second integral is small, uniformly with respect to , and then the measure of small enough such that the first term is small. The almost everywhere convergence of to and Vitali’s theorem imply that strongly converges to in .
For the second convergence, we will follow the same technique as in  (see also ). Let . In the sequel will denote a constant independent of . Let us consider as a test function in problems (16). Then,Moreover, thanks to the convergence of , the second integral in (32) converges (as n diverges) to a positive number. Thus, it yields toLet , observing that if , thenSince converges to weakly in and strongly converges to in , we havethus, yieldingwhere denote any quantity that vanishes as diverges. Hence, by Hölder’s inequality, we deduce thatFix, now, there exist such that, for , we haveThanks to the weak convergence of in and the absolute continuity of the integral, there exists independent from such that, for , we haveIn addition, by Dunford Pettis Theorem, we deduce that there exists such that, for , we haveWe can writeUsing (37), (39), and (40), for and , we haveThis proves the strong convergence of to in .
The following lemma yields some a priori estimate on .
Lemma 7. Let be the function given by Remark 4. Then belongs to , for every
Proof. For every , we take as a test function in (16). Droping positive terms yields Hence, using , it follows thatOn the other hand, for every ; we haveThen, we obtainLet us choose such that , that isSince , we then have an estimate on in , for every
The next result will be used in the proof of Theorem 2.
Lemma 8. Suppose that (2), (3), and (11) hold true. Let and be a solution of (13) with for every . Then the norms of in and in are bounded by a constant which depends on and on the norm of in .
Proof. Since , we have . Let us choose such thatThe use ofas test function in (16), (3), and (20), implies thatwhereBy Young and Hölder’s inequalities, we findThen, using Sobolev’s inequality gives where denotes the best constant in Sobolev inequality. Now, we setand and the fact that ., the last inequality givesNote that implies that . Then Stampacchia’s technique implies the following relation for some positive constant ,that is, is bounded.
3. Proof of the Main Results
We are now ready to prove the main result of this paper. We first observe that condition (9) implies (12). Hence the results of the previous section hold true. In order to prove the result, we have to pass to the limit in (16). To this aim, let be a function in such thatwhereObserve that, by (9), is positive, increasing, and it verifiesWe will use, for and ,to define a test function. Remark that , andFirst of all, note that the a.e. convergence of (see Lemma 5), Remark 6, and (20) imply both thatandwhere is defined in (59).
The proof of the result will be achieved in two steps.
Step 1 (The first inequality). We fix , with , and takeAs test function in (16), we have thatRemark now that, by the assumptions on , , relation (60) and the fact that , then we haveTherefore, using the almost everywhere convergence of both and , and applying Fatou’s lemma, we get Furthermore, by using Lebesgue’s theorem and (63), we obtainandSimilarly, using the convergence in , we haveNow, from (62), we getPassing to the limit in (66) when tends to infinity and gathering together (68)-(72), weobtain Choosing in (16), we getBy Fatou’s lemma, we haveIn order to pass to the limit as tends to infinity in the inequality (73), we recall that and ,. We obtainfor every , with ; that is, is a subsolution of problem (1).
Step 2 (The second inequality). Let be in , with , and be given by (58), and chooseasa test function in (16). We obtainWe observe that, by (60) and the fact that , we haveApplying the same argument of Step 1 and using (64) instead of (63), we deduce thatfor every , with .
Consequently, summarizing Steps 1 and 2, we havefor every .
Finally, interchanging and we conclude thatfor every .
The authors do not have data available.
This paper has been presented at the 5th International Congress of the Moroccan Society of Applied Mathematics (SM2A-2017) and at the second edition of the “Journe doctorale de l’ENSAJ’’ in the “Ecole Nationale des Sciences Appliques d’El Jadida’’- Morocco. We are gratefull to all people who helped in any way in this work.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
The authors are grateful to all people who helped in this work.
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