Existence of weak solutions for some quasilinear degenerated elliptic systems in weighted Sobolev spaces

• El Houcine, Rami ^[1] ; Elhoussine , Azroul ; Abdelkrim, Barbara ^[1]
  1. [1] Sidi Mohammed Ben Abdellah University.
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 41, Nº. 6, 2022, págs. 1523-1549
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-5336
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• Resumen

  • We consider, for a bounded open domain Ω in Rn; (n ≥ 1) and a function u : Ω → ℝm; (m ≥ 1) the quasilinear elliptic system:

      (0.1) Which is a Dirichlet problem. Here, v belongs to the dual space , f and g satisfy some stan- dard continuity and growth conditions. we will show the existence of a weak solution of this problem in the four following cases: σ is mono- tonic, σ is strictly monotonic, σ is quasi montone and σ derives from a convex potential.

• Referencias bibliográficas
  • J. Ball, “A version of the fundamental theorem for Young measures,” in PDEs and Continuum Models of Phase Transitions: Proceedings of an NSF-CNRS...
  • D. Kinderlehrer, P. Pedregal, “Gradient Young measures generated by sequences in Sobolev spaces”, Journal of Geometric Analysis, vol. 4, no....
  • F. E. Browder, “Existence theorems for nonlinear partial differential equations”, Global Analysis, pp. 1–60, 1970. https://doi.org/10.1090/pspum/016/0269962
  • G. Dolzmann, N. Hungerbühler and S. Muller, “Nonlinear elliptic systems with measure-valued right hand side”, Mathematische Zeitschrift, vol....
  • G. J. Minty, “Monotone (nonlinear) operators in Hilbert space”, Duke Mathematical Journal, vol. 29, pp. 341-346, 1962. https://doi.org/10.1215/s0012-7094-62-02933-2
  • H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Amsterdam: Elsevier, 1973.
  • E. Hewitt and K.R. Stromberg, Real and abstract analysis. Berlin: Springer, 1969.
  • I. Fonseca, S. Muller and P. Pedregal, “Analysis of concentration and oscillation effects generated by gradients”, SIAM Journal on Mathematical...
  • J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites non Lineaires. Paris: Dunod, Gauthier-Villars, 1969.
  • J. Kristensen, “Lower semicontinuity in space of weakly differentiable functions”, Mathematische Annalen, vol. 313, no. 4, pp. 653-710, 1999....
  • L. Boccardo and F. Murat, “Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations”, Nonlinear Analysis:...
  • M. Candela, E. Medeiros, G. Palmieri and K. Perera, “Weak solutions of quasilinear elliptic systems via the cohomological index”, Topological...
  • M. I. Visik, “Quasi-linear strongly elliptic systems of differential equations of divergence form”, Trudy Moskovskogo Matematicheskogo Obshchestva,...
  • N. Hungerbuhler, “Quasilinear elliptic systems in divergence form with weak monotonicity”, New York Journal of Mathematics , vol. 5, pp. 83-90,...
  • P. Drabek, A. Kufnen and V. Mustonen, “Pseudo-Monotinicity and degenerated, a singular operators”, Bulletin of the Australian Mathematical...
  • R. Landes and V. Mustonen, “On pseudomonotone operators and nonlinear noncoercive variational problems on unbounded domains”, Mathematische...
  • Y. Akdim and E. Azroul, “Pseudo-monotonicity and degenerated elliptic operator of second order”, Electronic Journal of Differential Equations,...
  • Y. Akdim, E.H. Azroul and A. Benkirane, “Existence of solutions for quasilinear degenerate elliptic equations”, Electronic Journal of Differential...
  • E. Zeidler, Nonlinear functional Analysis and its application, New York: Springer, 1986.