A weak solution for a class of quasilinear elliptic Systems with nonstandard growth In musielak-Orlicz space.
Résumé
Here we study existence of a weak solutions for some nonlinear elliptic systems like
$$
\left\{\begin{aligned}
-\operatorname{div} \sigma(x, u, D u)= & f(x, u, D u)
\quad \text { in } \Omega \\
u(x)= & 0 \quad \text { on } \partial \Omega,
\end{aligned}\right.
$$
where $\Omega \subset \mathbb{R}^{n}$ is a bounded open domain. The Term $f$ satisfy the growth and sign condition. We establish the existence solution by using the idea of Young measure.
Téléchargements
Références
A. Benkirane, B. El Haji and M. El Moumni. On the existence of solution for degenerate parabolic equations with singular terms. Pure and Applied Mathematics Quarterly Volume 14, Number 3-4, (2018), 591–606
A. Benkirane, B. El Haji and M. El Moumni; Strongly nonlinear elliptic problem with measure data in Musielak-Orlicz spaces. Complex Variables and Elliptic Equations, 1-23, https://doi.org/10.1080/17476933.2021.1882434 .
A. Benkirane, B. El Haji and M. El Moumni; On the existence solutions for some Nonlinear elliptic problem, Boletim da Sociedade Paranaense de Matem´atica,(3s.) v. 2022 (40) : 1?8, DOI: https://doi.org/10.5269/bspm.53111.
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, Journal of Nonlinear Functional Analysis, Vol. 2021 (2021), Article ID 8, pp. 1-17.
B. El Haji and M. El Moumni and K. Kouhaila; Existence of entropy solutions for nonlinear elliptic problem having large monotonicity in weighted Orlicz-Sobolev spaces , LE MATEMATICHE Vol. LXXVI (2021), Issue I, pp. 37?61, https://doi.org/10.4418/2021.76.1.3.
B. El Haji and 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. Vol 5(1), (2019), 104-116.
J. Musielak. Modular spaces and Orlicz spaces, Lecture Notes in Math. (1983), 10–34
N. El Amarti, B. El Haji and M. El Moumni. Entropy solutions for unilateral parabolic problems with L1-data in Musielak-Orlicz-Sobolev spaces Palestine Journal of Mathematics, Vol. 11(1)(2022) , 504?523.
N. El Amarti, B. El Haji and M. El Moumni. Existence of renomalized solution for nonlinear Elliptic boundary value problem without Δ2 -condition SeMA Journal, SeMA Journal 77 (4), 389-414.
O. Azraibi, B.EL haji, M. Mekkour; Nonlinear parabolic problem with lower order terms in Musielak-Sobolev spaces without sign condition and with Measure data, Palestine Journal of Mathematics,Vol. 11(3)(2022) , 474-503.
Dong, G.: An existence theorem for weak solutions for a class of elliptic partial differential systems in general Orlicz? Sobolev spaces. Nonlinear Anal. 69, 2049?2057 (2008)
Gossez JP. Orlicz-Sobolev spaces and nonlinear elliptic boundary value problems. Nonlinear Analysis, function spaces and applications. Proc. Spring School, Horni Bradlo, 1978, Leipzig: Teubner; 1979, 59-94.
Mustonen V, Tienar M. On monotone-like mappings in Orlicz-Sobolev spaces. Math Bohem. 1999;124(2-3): 255-271.
Faria, L.F.O., Miyagaki, O. H., Motreanu, D., Tanaka, M.: Existence results for nonlinear elliptic equations with Leray?Lions operator and dependence on the gradient. Nonlinear Anal. 96, 154?166 (2014)
Fuchs, M.: Regularity theorems for nonlinear systems of partial differential equations under natural ellipticity conditions. Analysis 7, 83?93 (1987)
Gossez JP. Boundary value problems for quasilinear elliptic equations with rapidly increasing coefficients. American Mathematical Society. 1972 Sept; 78(5) : 753-758.
Hungerbuhler N. A refinement of Ball’s theorem on Young measures. NY J Math. 1997;3:48-53.
Hungerbuhler N. Young measures and nonlinear PDEs Habilitationsschrift ETH Zurich. 2001.
M´alek J, Necas J, Rokyta M, Ru?Icka M. Weak and measure-valued solutions to evolutionary PDEs. London: Chapman & Hall; 1996.
Minty G. Monotone (nonlinear) operators in Hilbert space. Duke Math J. 1962;29:341-346.
Pucci, P., Servadei, R.: Regularity of weak solutions of homogeneous or inhomogeneous quasilinear elliptic equations. Indiana Univ. Math. J. 57(7), 3329?3363 (2008)
Young, L. C.: Generalized curves and the existence of an attained absolute minimum in the calculus of variations. Comptes Rendus de la Societe des Sciences et des Lettres de Varsovie 30, 212-234 (1937)
Yosida, K.: Functional Analysis. Springer, Berlin (1980)
Yongqiang, F., Zengfu, D., Yan, Y.: On the existence of weak solutions for a class of elliptic partial differential systems. Nonlinear Anal. 48, 961?977 (2002)
Zhang, K. W.: On the Dirichlet problem for a class of quasilinear elliptic systems of partial differential equations in divergence form. Partial Differ. Equ. 1306, 262?277 (1988)
Valadier M. A course on Young measures. Workshop on Measure Theory and Real Analysis (Grado, 1993). Rend Istit Mat Univ Trieste. 1994;26(suppl):349-394.
Copyright (c) 2025 Boletim da Sociedade Paranaense de Matemática
Ce travail est disponible sous la licence Creative Commons Attribution 4.0 International .
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
