Nonlinear elliptic problems in weighted variable exponent Sobolev spaces by topological degree

  • M. Ait University Sidi Mohamed Ben Abdellah,
  • E. Azroul University Sidi Mohamed Ben Abdellah.

Resumen

In this paper, we prove the existence of solutions for the nonlinear p(·)-degenerate problems involving nonlinear operators of the form − div a(x, ∇u) = f(x, u, ∇u) where a and f are Carathéodory functions satisfying some nonstandard growth conditions.

Biografía del autor/a

M. Ait, University Sidi Mohamed Ben Abdellah,
FSDM.
E. Azroul, University Sidi Mohamed Ben Abdellah.
FSDM.

Citas

I. Aydin, “Weighted variable Sobolev spaces and capacity”, Journal of function spaces and applications, vol. 2012, pp. Article ID 132690, 2012, doi: 10.1155/2012/132690.

E. Azroul, M. Benboubker and A. Barbara, “Quasilinear elliptic problems with nonstandard growth”, Electronic journal of differential equations, vol. 2011, no. 62, pp. 1-16, 2011. [On line]. Available: https://bit.ly/2BewUWI

J. Berkovits, On the degree theory for nonlinear mappings of monotone type, vol. 58. Helsinki: Suomalainen Tiedeakatemia, 1986.

J. Berkovits, “Extension of the Leray–Schauder degree for abstract Hammerstein type mappings”, Journal of differential equations, vol. 234, no. 1, pp. 289–310, Mar. 2007, doi: 10.1016/j.jde.2006.11.012

J. Berkovits, and V. Mustonen, “On topological degree for mappings of monotone type”, Nonlinear analysis, vol. 10, no. 12, pp. 1373-1383, Dec. 1986, doi: 10.1016/0362-546X(86)90108-2

J. Berkovits and V. Mustonen, Nonlinear mappings of monotone type I. Classification and degree theory, Preprint no 2/88, Mathematics, University of Oulu, 1988.

M. Boureanu, A. Matei and M. Sofonea, “Nonlinear problems with p(.)-growth conditions and applications to antiplane contact models”, Advanced nonlinear studies, vol. 14, no. 2, pp. 295-313, May 2014, doi: 10.1515/ans-2014-0203.

L. Brouwer, “Uber abbildung von mannigfaltigkeiten”, Mathematische annalen, vol. 71, no. 1, pp. 97-115, Mar. 1912, doi: 10.1007/BF01456931.

L. Diening, P. Harjulehto, P. Hästö, and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, vol. 2017. Berlin: Springer, 2011, doi: 10.1007/978-3-642-18363-8.

X. Fan and Q. Zhang, “Existence for p(x)-Laplacien Dirichlet problem”, Non linear analysis, vol. 52, no. 8, pp. 1843-1852, Mar. 2003, doi: 10.1016/S0362-546X(02)00150-5.

X. Fan, D. Zhao, “On the spaces Lp(x) (Ω) and Wm,p(x) (Ω)”, Journal of mathematical analysis and applications, vol. 263, no. 2, pp. 424-446, Nov. 2001, doi: 10.1006/jmaa.2000.7617.

O. Kováčik and J. Rákosník, “On spaces Lp(x) and Wk,p(x)”, Czechoslovak mathematical journal, vol. 41, no. 4. pp. 592-618, 1991 [On line]. Available: http://bit.ly/2VQzJqg

B. Lahmi, E. Azroul and K. El Haiti, “Nonlinear degenerated elliptic problems with dual data and nonstandard growth”, Mathematical reports, vol. 20, no. 1, pp. 81-91, 2018. [On line]. Available: http://bit.ly/31u0pi8

J. Leray and J. Schauder, “Topologie et équations fonctionnelles”, Annales scientifiques de lÉcole normale supérieure. 3e séri., vol. 51, pp. 45–78, 1934, doi: 10.24033/asens.836

K. Rajagopal and M. Ružička, “Mathematical modeling of electrorheological materials”, Continuum mechanics and thermodynamics, vol. 13, no. 1, pp. 59-78, Jan. 2001, doi: 10.1007/s001610100034.

M. Ružička, Electrorheological fluids: modeling and mathematical theory, vol. 1748. Berlin: Springer, 2000, doi: 10.1007/BFB0104029.

S. Samko, “Density of C₀∞0(Rn) in the generalized Sobolev spaces Wm,p(x) (Rn)”, Doklady akademii nauk, vol. 369, no, pp. 451-454, 1999. [On line]. Available: http://bit.ly/32oJHSj

M. Sanchon, J. Urbano, “Entropy solutions for the p(x)-Laplace equation”, Transactions of the american mathematical society, vol. 361, no. 12, pp. 6387-6405, 2009, doi: 10.1090/S0002-9947-09-04399-2.

C. Unal and I. Aydin, “Weighted variable exponent Sobolev spaces with zero boundary values and capacity estimates”, Sigma journal of engineering and natural sciences, vol. 36, no. 2, pp. 373-388, 2018. [On line]. Available: http://bit.ly/2VTUHVA

C. Unal and I. Aydin, “Compact embedding on a subspace of weighted variable exponent Sobolev spaces”, Advances in operator theory, vol. 4, no. 2, pp. 388-405, 2019, doi: 10.15352/AOT.1803-1335

E. Zeidler, Nonlinear monotone operators, vol. 2-B. New York, NY: Springer, 1990, doi: 10.1007/978-1-4612-0981-2.

V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory”, Mathematics of the USSR-Izvestiya, vol. 29, no. 1, pp. 33–66, 1987, doi: 10.1070/IM1987v029n01ABEH000958

Publicado
2019-10-21
Cómo citar
[1]
M. Ait Hammou y E. H. Azroul, «Nonlinear elliptic problems in weighted variable exponent Sobolev spaces by topological degree», Proyecciones (Antofagasta, En línea), vol. 38, n.º 4, pp. 733-751, oct. 2019.
Sección
Artículos