Entropy and renormalized solutions for some nonlinear anisotropic elliptic equations with variable exponents and L1-data

Mostafa El Moumni; Deval Sidi Mohamed

doi:10.2478/mjpaa-2021-0018

.blurhash-client-img { display: none !important; }

Entropy and renormalized solutions for some nonlinear anisotropic elliptic equations with variable exponents and L1-data

Moroccan Journal of Pure and Applied Analysis

Volume 7 (2021): Issue 2 (May 2021)

By: Mostafa El Moumni and Deval Sidi Mohamed

Open Access

|Jan 2021

Abstract

We prove in this paper some existence and unicity results of entropy and renormalized solutions for some nonlinear elliptic equations with general anisotropic diffusivities and variable exponents. The data are assumed to be merely integrable.

References

[1] A. Aissaoui Fqayeh, A. Benkirane and M. El Moumni. Entropy solutions for strongly nonlinear unilateral parabolic inequalities in Orlicz-Sobolev spaces. Applicationes Mathematicae, 41,2–3, (2014)pp: 185–193.10.4064/am41-2-6
Search in Google Scholar Back to article
[2] A. Aissaoui Fqayeh, A. Benkirane, M. El Moumni and A. Youssfi. Existence of renormalized solutions for some strongly nonlinear elliptic equations in Orlicz spaces. Georgian Math. J. Volume 22, Number 3,(2015), pp:305–321.10.1515/gmj-2015-0038
Search in Google Scholar Back to article
[3] Y. Akdim, M. Belayachi and M. El Moumni. L^∞-bounds of solutions for strongly nonlinear elliptic problems with two lower order terms. Anal. Theory Appl., Vol. 33, No. 1, (2017) pp. 46–58.
Search in Google Scholar Back to article
[4] Y. Akdim, A. Benkirane, S. M. Douiri and M. El Moumni. On a quasilinear degenerated elliptic unilateral problems with L¹ data. Rend. Circ. Mat. Palermo, II. Ser (2018)67:43–57.10.1007/s12215-016-0291-5
Search in Google Scholar Back to article
[5] Y. Akdim, A. Benkirane and M. El Moumni. Solutions of nonlinear elliptic problems with lower order terms. Annals of Functional Analysis (AFA), Volume 6, Number 1, (2015), pp: 34–53.10.15352/afa/06-1-4
Search in Google Scholar Back to article
[6] Y. Akdim, A. Benkirane and M. El Moumni. Existence results for nonlinear elliptic problems with lower order terms. International Journal of Evolution Equations (IJEE), Volume 8, Number 4, (2014) pp: 1–20.
Search in Google Scholar Back to article
[7] Y. Akdim, A. Benkirane, M. El Moumni and A. Fri. Strongly nonlinear variational parabolic initial-boundary value problems. Annals of the University of Craiova - Mathematics and Computer Science Series. Volume 41, Number 2(2014), pp: 1–13.
Search in Google Scholar Back to article
[8] Y. Akdim, A. Benkirane, M. El Moumni and H. Redwane. Existence of renormalized solutions for nonlinear parabolic equations. Journal of Partial Differential Equations (JPDE), Volume 27, Number 1, (2014)pp: 28–49.10.4208/jpde.v27.n1.2
Search in Google Scholar Back to article
[9] Y. Akdim, A. Benkirane, M. El Moumni and H. Redwane. Existence of renormalized solutions for strongly nonlinear parabolic problems with measure data. Georgian Math. J. Volume 23, Issue 3, (2016)pp: 303321.10.1515/gmj-2016-0011
Search in Google Scholar Back to article
[10] Y. Akdim, M. El Moumni and A. Salmani. Existence Results for Nonlinear Anisotropic Elliptic Equation. Adv. Sci. Technol. Eng. Syst. J. 2(5), (2017)160-166.10.25046/aj020523
Search in Google Scholar Back to article
[11] F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo. Existence and uniqueness for a degenerate parabolic equation with L¹-data. Trans. Amer. Math. Soc. 351 (1999) no.1, 285–306.
Search in Google Scholar Back to article
[12] M. Bendahmane, K.H. Karlsen and M. Saad. Nonlinear anisotropic elliptic and parabolic equations with variable exponents and L¹-data. Comm. Pure and Appl. Anal. 12 (2013) no.3, 1201–1220.
Search in Google Scholar Back to article
[13] M. Bendahmane, M. Langlais and M. Saad. On some anisotropic reaction-diffusion systems with L¹-data modeling the propagation of an epidemic disease. Nonlinear Anal., 54(4) (2003), 617–636.10.1016/S0362-546X(03)00090-7
Search in Google Scholar Back to article
[14] M. Bendahmane and P. Wittbold. Renormalized solutions for nonlinear elliptic equations with variable exponents and L¹-data. Nonlinear Analysis TMA 70 (2009) no.2, 567–583.
Search in Google Scholar Back to article
[15] Ph. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, and J. L. Vázquez. An L¹-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 22(2) (1995), 241–273.
Search in Google Scholar Back to article
[16] M. B. Benboubker, H. Chrayteh, M. El Moumni and H. Hjiaj. Entropy and renormalized solutions for nonlinear elliptic problem involving variable exponent and measure data. Acta Mathematica Sinica, English Series. Volume 31, Number 1,(2015), pp: 151–169.10.1007/s10114-015-3555-7
Search in Google Scholar Back to article
[17] A. Benkirane, Y. El Hadfi and M. El Moumni. Renormalized solutions for nonlinear parabolic problems with L¹- data in orlicz-sobolev spaces. Bulletin of Parana’s Mathematical Society (3s.) Volume 35, Number 1, (2017), pp: 57–84.10.5269/bspm.v35i1.27425
Search in Google Scholar Back to article
[18] A. Benkirane, B. El Haji and M. El Moumni. On the existence of solutions for degenerate parabolic equations with singular terms. Pure and Applied Mathematics Quaterly. Volume 14, Number 3–4, (2018), 591–606.10.4310/PAMQ.2018.v14.n3.a8
Search in Google Scholar Back to article
[19] A. Benkirane, M. El Moumni and A. Fri. An approximation of Hedberg’s type in Sobolev spaces with variable exponent and application. Chinese Journal of Mathematics, Volume 2014, Article ID 549051, (2014)7 pages.10.1155/2014/549051
Search in Google Scholar Back to article
[20] A. Benkirane, M. El Moumni and A. Fri. Renormalized solution for strongly nonlinear elliptic problems with lower order terms and L¹-data. Izvestiya RAN: Ser. Mat. (2017)81:3 3–20.10.4213/im8292
Search in Google Scholar Back to article
[21] A. Benkirane, M. El Moumni and K. Kouhaila. Solvability of strongly nonlinear elliptic variational problems in weighted Orlicz-Sobolev spaces. SeMA Journal (2020) 77:119–142.10.1007/s40324-019-00205-x
Search in Google Scholar Back to article
[22] D. Blanchard and F. Murat. Renormalised solutions of nonlinear parabolic problems with L¹ data: existence and uniqueness. Proc. Roy. Soc. Edinburgh Sect. A, 127(6) (1997), 1137–1152.10.1017/S0308210500026986
Search in Google Scholar Back to article
[23] L. Boccardo, J. I. Diaz, D. Giachetti, and F. Murat. Existence of a solution for a weaker form of a nonlinear elliptic equation. In Recent advances in nonlinear elliptic and parabolic problems (Nancy, 1988), volume 208 of Pitman Res. Notes Math. Ser., pages 229-246. Longman Sci. Tech., Harlow, 1989.
Search in Google Scholar Back to article
[24] L. Boccardo, T. Gallouët, and P. Marcellini. Anisotropic equations in L¹. Differential Integral Equations, 9(1) (1996), 209–212.
Search in Google Scholar Back to article
[25] M. Bourahma and J. Bennouna and M. El Moumni. L^∞-Bounds of solutions for a nonlinear degenerate elliptic equations in Musielak spaces. Moroccan J. of Pure and Appl. Anal. (MJPAA) Volume 6(1), 2020, Pages 16–33.10.2478/mjpaa-2020-0002
Search in Google Scholar Back to article
[26] Y. Chen, S. Levine and M. Rao. Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66 (2006), 1383–1406.10.1137/050624522
Search in Google Scholar Back to article
[27] G. Dal Maso, F. Murat, L. Orsina and A. Prignet. Renormalized solutions of elliptic equations with general measure data. Ann. Sc. Norm. Super. Pisa Cl. Sci.(4) 28 (4) (1999), 741-808.
Search in Google Scholar Back to article
[28] R. J. DiPerna and P.-L. Lions. On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. of Math. (2), 130(2) (1989), 321–366.10.2307/1971423
Search in Google Scholar Back to article
[29] B. El Haji, M. El Moumni and K. Kouhaila. On a nonlinear elliptic problems having large monotonocity with L¹-data in weighted Orlicz-Sobolev spaces. Moroccan J. of Pure and Appl. Anal. (MJPAA) Volume 5(1), (2019), Pages 104-116.10.2478/mjpaa-2019-0008
Search in Google Scholar Back to article
[30] M. El Moumni. Nonlinear elliptic equations without sign condition and L¹-data in Musielak-Orlicz-Sobolev spaces. Acta. Appl. Math. 159:95–117(2019).10.1007/s10440-018-0186-x
Search in Google Scholar Back to article
[31] M. El Moumni. Renormalized solutions for strongly nonlinear elliptic problems with lower order terms and measure data in Orlicz-Sobolev spaces. Iran. J. Math. Sci. Inform. Vol 14, No 1, (2019)pp 95–119.
Search in Google Scholar Back to article
[32] M. El Moumni. Entropy solution for strongly nonlinear elliptic problems with lower order terms and L¹-data. Annals of the University of Craiova - Mathematics and Computer Science Series. Volume 40, Number 2 (2013), pp: 211–225.
Search in Google Scholar Back to article
[33] X.L. Fan, J. Shen and D. Zhao. Sobolev embedding theorems for spaces W^k,p⁽^x⁾(Ω). J. Math. Anal. Appl. 262 (2001), 749–760.10.1006/jmaa.2001.7618
Search in Google Scholar Back to article
[34] X.L. Fan and D. Zhao. On the spaces L^p⁽^x⁾(𝒰) and W^m,p⁽^x⁾(𝒰). J. Math. Anal. Appl. 263 (2001), 424–446.10.1006/jmaa.2000.7617
Search in Google Scholar Back to article
[35] T. Harjulehto, P. Hästö, M. Koskenoja and S. Varonen. The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values. Potential Anal. 25 (2006) no.3, 205–222.
Search in Google Scholar Back to article
[36] F. Li and H. Zhao. Anisotropic parabolic equations with measure data. J. Partial Differential Equations, 14(1) (2001), 21–30.
Search in Google Scholar Back to article
[37] P.-L. Lions. Mathematical topics in fluid mechanics. Vol. 1, volume 3 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, New York, 1996.
Search in Google Scholar Back to article
[38] J.-M. Rakotoson. Generalized solutions in a new type of sets for problems with measures as data. Differential Integral Equations, 6(1) (1993), 27–36.
Search in Google Scholar Back to article
[39] J.-M. Rakotoson. Uniqueness of renormalized solutions in a T-set for the L¹-data problem and the link between various formulations. Indiana Univ. Math. J., 43(2) (1994), 685–702.10.1512/iumj.1994.43.43029
Search in Google Scholar Back to article
[40] M. Sanchon and M. Urbano. Entropy solutions for the p(x)-Laplace equation. To appear in Trans. American Math. Soc.
Search in Google Scholar Back to article
[41] M. Troisi. Teoremi di inclusione per spazi di sobolev non isotropi. Ricerche. Mat., 18 (1969), 3–24.
Search in Google Scholar Back to article
[42] A. Youssfi, A. Benkirane and M. El Moumni. Bounded solutions of unilateral problems for strongly nonlinear equations in Orlicz spaces. Electronic Journal of Qualitative Theory of Differential Equations (EJQTDE), Number 21, (2013)pp: 1–25.10.14232/ejqtde.2013.1.21
Search in Google Scholar Back to article
[43] A. Youssfi, A. Benkirane and M. El Moumni. Existence result for strongly nonlinear elliptic unilateral problems with L¹-data. Complex Variables and Elliptic Equations, Volume 59, Issue 4, (2014) pp: 447–461.10.1080/17476933.2012.725166
Search in Google Scholar Back to article