Про деякi нелiнiйнi елiптичнi проблеми з великою монотоннiстю в просторах Мусйелака–Орлича–Соболєва

Автор(и)

  • Ouidad Azraibi Laboratory LAMA, Department of Mathematics, Faculty of Sciences Fez, Sidi Mohamed Ben Abdellah University, P.O. Box 1796 Atlas Fez, Morocco
  • Badr El Haji Laboratory LaR2A, Dpartement of Mathematics, Faculty of Sciences Tetouan, Abdel- malek Essaadi University , BP 2121, Tetouan, Morocco
  • Mounir Mekkour Laboratory LAMA, Department of Mathematics, Faculty of Sciences Fez, Sidi Mohamed Ben Abdellah University, P.O. Box 1796 Atlas Fez, Morocco

DOI:

https://doi.org/10.15407/mag18.03.332

Ключові слова:

елiптична проблема, ентропiйний розв’язок, простори Мусйелака–Орлича–Соболєва, компактне вкладення, ∆2-умова

Анотація

У цiй роботi ми вивчаємо iснування ентропiйного розв’язку деякої нелiнiйної елiптичної проблеми типу Лерея–Лiонса, пов’язану з рiвнянням −div a(x,u,∇u) = f(x)−div F(u) в Ω з умовою великої монотонностi у визначеннi просторiв Мусйелака–Орлича–Соболєва, де права частина належить L1 (Ω) i F = (F 1 ,...,F N ) задовольняє умову F ∈ (C0 (ℝ))N .

Mathematical Subject Classification 2010: 35J62, 35J25

Посилання

R. Adams, Sobolev spaces, Academic Press, New York, 1975.

Y. Akdim, E. Azroul, and M.Rhoudaf, Existence of T-solution for degenerated problem via Minty's Lemma, Acta Math. Sinica (English Ser.) 24 (2008), 431-438. https://doi.org/10.1007/s10114-007-0970-4

E. Azroul, A.Barbara, M. El Lekhlifi, and M. Rhoudaf, T-p(x)-solutions for non-linear elliptic equations with an L1-dual datum, Appl. Math. (Warsaw) 39 (2012), 339-364. https://doi.org/10.4064/am39-3-8

A.M. Ben Cheikh and O. Guibe, Nonlinear and non-coercive elliptic problems with integrable data, Adv. Math. Sci. Appl. 16 (2006), 275-297.

M.A. Khellou, A. Benkirane, and S.M. Douiri, Some properties of Musielak spaces with only the log-Hölder continuity condition and application, Ann. Funct. Anal. 11 (2020), 1062-1080. https://doi.org/10.1007/s43034-020-00069-7

P. 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 (1995), 241-273.

A. Benkirane and A. Elmahi, Almost everywhere convergence of the gradient of solutions to elliptic equations in Orlicz spaces, Nonlinear Anal. 28 (1997), 1769-1784. https://doi.org/10.1016/S0362-546X(96)00017-X

A. Benkirane and A. Elmahi, An existence theorem for a strongly nonlinear elliptic problem in Orlicz spaces, Nonlinear Anal., Ser. A: Theory Methods 36 (1999), 11-24. https://doi.org/10.1016/S0362-546X(97)00612-3

A. Benkirane and M.S. El Vally, Some approximation properties in Musielak-Orlicz-Sobolev spaces, Thai. J. Math. 10 (2012), 371-381.

A. Benkirane and M.S. El Vally, Variational inequalities in Musielak-Orlicz-Sobolev spaces, Bull. Belg. Math. Soc. Simon Stevin 21 (2014), 787-811. https://doi.org/10.36045/bbms/1420071854

A. Benkirane, B. El Haji, and M. El Moumni, On the existence of solution for degenerate parabolic equations with singular terms, Pure Appl. Math. Q. 14 (2018), 591-606. https://doi.org/10.4310/PAMQ.2018.v14.n3.a8

L. Boccardo, Some nonlinear Dirichlet problems in L1 involving lower order terms in divergence form, Progress in Elliptic and Parabolic Partial Differential Equations, Pitman Res. Notes Math. Ser., 350, Longman, Harlow, 1996, 43-57.

Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (Ed. H. Brézis,), North-Holland Mathematics Studies, 5, Elsevier, New York, 1973.

F.E. Browder, Existence theorems for nonlinear partial differential equations, Global Analysis, Proc. Sympos. Pure Math., XVI, Berkeley, Calif., 1968, Amer. Math. Soc., Providence, R.I., 1970, 1-60. https://doi.org/10.1090/pspum/016/0269962

A.Dall'Aglio, Approximated solutions of equations with L1 data. Application to the H-convergence of quasi-linear parabolic equations. Ann. Mat. Pura App1. 170 (1996), 207-240. https://doi.org/10.1007/BF01758989

J. Droniou and A.Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data. NoDEA Nonlinear Differential Equations, App1. 14 (2007), 181-205. https://doi.org/10.1007/s00030-007-5018-z

R.J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann.of Math. 130 (1989), 321-366. https://doi.org/10.2307/1971423

B. El Haji and M. El Moumni, Entropy solutions of nonlinear elliptic equations with L1-data and without strict monotonocity conditions in weighted Orlicz-Sobolev spaces, J. Nonlinear Funct. Anal. 2021 (2021), 8. https://doi.org/10.23952/jnfa.2021.8

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. 5 (2019), 104-116. https://doi.org/10.2478/mjpaa-2019-0008

N. El Amarty, B. El Haji, and M. El Moumni, Existence of renomalized solution for nonlinear Elliptic boundary value problem without ∆2-condition SeMA J. 77 (2020), 389-414. https://doi.org/10.1007/s40324-020-00224-z

R. Elarabi, M. Rhoudaf, and H. Sabiki, Entropy solution for a nonlinear elliptic problem with lower order term in Musielak-Orlicz spaces. Ric. Mat. 67 (2018), 549-579. https://doi.org/10.1007/s11587-017-0334-z

J.-P. Gossez, Some approximation properties in Orlicz-Sobolev spaces, Studia Math. 74 (1982), 17-24. https://doi.org/10.4064/sm-74-1-17-24

J.-P. Gossez and V. Mustonen, Variational inequalities in Orlicz-Sobolev spaces, Nonlinear Anal. 11 (1987), 379-392. https://doi.org/10.1016/0362-546X(87)90053-8

E.Hewitt and K.Stromberg, Real and Abstract Analysis, Springer, Berlin, 1965. https://doi.org/10.1007/978-3-642-88044-5

C. Leone and A. Porretta, Entropy solutions for nonlinear elliptic equations in L1, Nonlinear Anal. 32 (1998), 325-334. https://doi.org/10.1016/S0362-546X(96)00323-9

P.-L. Lions and F. Murat, Sur les solutions d`équations elliptiques non linéaires, C. R. Acad. Sci. Paris, to appear (French).

J.-L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaire, Dunod et Gauthier Villars, Paris, 1969 (French).

G.J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341-346. https://doi.org/10.1215/S0012-7094-62-02933-2

J. Musielak, Modular spaces and Orlicz spaces, Lecture Notes in Math. 1034, Springer-Verlag, Berlin-Heidelberg, 1983. https://doi.org/10.1007/BFb0072212

F.Petitta, A.C.Ponce, and A.Porretta, Diffuse measures and nonlinear parabolic equations. J. Evol. Equ. 11 (2011), 861-905. https://doi.org/10.1007/s00028-011-0115-1

F. Petitta, Asymptotic behavior of solutions for parabolic operators of Leray-Lions type and measure data, Adv. Differential Equations 12 (2007), 867-891. https://doi.org/10.57262/ade/1367241140

A. Prignet, Existence and uniqueness of "entropy", solutions of parabolic problems with L1 data, Nonlinear Anal. 28 (1997), 1943-1954 . https://doi.org/10.1016/S0362-546X(96)00030-2

A. Porretta and S.S. de León, Nonlinear elliptic equations having a gradient term with natural growth, J. Math. Pures Appl. 85 (2006), 465-492. https://doi.org/10.1016/j.matpur.2005.10.009

A. Porretta, Existence results for strongly nonlinear parabolic equations via strong convergence of truncations, Annali di matematica pura ed applicata (IV) CLXXVII (1999), 143-172. https://doi.org/10.1007/BF02505907

S. Polidoro and M.A. Ragusa, Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term, Rev. Mat. Iberoam. 24 (2008), 1011-1046. https://doi.org/10.4171/RMI/565

M.A. Ragusa, Elliptic boundary value problem in Vanishing Mean Oscillation hypothesis, Comment. Math. Univ. Carolin. 40 (1999), 651-663.

M.A. Ragusa, Holder regularity results for solutions of parabolic equations, Variational Analysis and Applications (Eds. F. Giannessi and A. Maugeri), Nonconvex Optimization and Its Applications, 79, 2005, 921-934. https://doi.org/10.1007/0-387-24276-7_53

B.Wang, D. Liu, and P. Zhao, Holder continuity for nonlinear elliptic problem in Musielak-Orlicz-Sobolev space, J. Differential Equations 266 (2019), 4835-4863. https://doi.org/10.1016/j.jde.2018.10.013

W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1974.

Downloads

Опубліковано

2023-01-10

Як цитувати

(1)
Azraibi, O.; Haji, B. E. .; Mekkour, M. . Про деякi нелiнiйнi елiптичнi проблеми з великою монотоннiстю в просторах Мусйелака–Орлича–Соболєва. J. Math. Phys. Anal. Geom. 2023, 18, 332-349.

Номер

Розділ

Статті

Завантаження

Дані завантаження ще не доступні.