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  4. Nonlinear elliptic equations with variable exponents involving singular nonlinearity

Nonlinear elliptic equations with variable exponents involving singular nonlinearity

MMC.
2021;
Volume 8, Number 4
: pp. 705–715
https://doi.org/10.23939/mmc2021.04.705
Received: May 23, 2021
Accepted: June 07, 2021

Mathematical Modeling and Computing, Vol. 8, No. 4, pp. 705–715 (2021)

Authors:
  1. H. Khelifi
  2. Y. El Hadfi
1
Department of Mathematics and Informatics, University of Algiers, Algiers
2
Laboratory LIPIM, National School of Applied Sciences Khouribga, Sultan Moulay Slimane University, Morocco

In this paper, we prove the existence and regularity of weak positive solutions for a class of nonlinear elliptic equations with a singular nonlinearity, lower order terms and L1 datum in the setting of Sobolev spaces with variable exponents.  We will prove that the lower order term has some regularizing effects on the solutions.  This work generalizes some results given in [1–3].

Sobolev spaces with variable exponents
singular nonlinearity
elliptic equation
  1. Boccardo L., Croce G.  The impact of a lower order term in a Dirichlet problem with a singular nonlinearity.  Portugaliae Mathematica. 76 (3–4), 4075 (2019).
  2. Oliva F.  Regularizing effect of absorption terms in singular problems.  Journal of Mathematical Analysis and Applications. 472 (1), (2019).
  3. Sbai A., El Hadfi Y.  Regularizing effect of absorption terms in singular and degenerate elliptic problems.  arXiv preprint arXiv:2008.03597 (2020).
  4. Callegari A., Nashman A.  A nonlinear singular boundary-value problem in the theory of pseudoplastic fluids.  SIAM J. Appl. Math. 38, 275–281 (1980).
  5. Keller H. B., Chohen D. S.  Some positive problems suggested by nonlinear heat generators.  Indiana Univ. Math. J. 16 (12), 1361–1376 (1967).
  6. Ambrosetti A., Brézis H., Cerami G.  Combined effects of concave and convex nonlinearities in some elliptic problems.  J. Funct. Anal. 122 (2), 519–543 (1994).
  7. El Ouardy M., El Hadfi Y., Ifzarne A.  Existence and regularity results for a singular parabolic equations with degenerate coercivity.  Discrete & Continuous Dynamical Systems – S. 1–25 (2021).
  8. Sbai A., El Hadfi Y.  Degenerate elliptic problem with a singular nonlinearity.  arXiv e-prints arXiv:2005.08383 (2020).
  9. Canino A., Sciunzi B., Trombetta A.  Existence and uniqueness for p-Laplace equations involving singular nonlinearities.  Nonlinear Differential Equations and Applications. 23, Article number: 8  (2016).
  10. De Cave L M., Oliva F.  On the regularizing effect of some absorption and singular lower order terms in classical Dirichlet problem with L1 data.   Journal of Elliptic and Parabolic Equations. 2, 73–85 (2016).
  11. Chu Y., Gao R., Sun Y.  Existence and regularity of solutions to a quasilinear elliptic problem involving variable sources.  Boundary Value Problems. 2017, Article number: 1 (2017).
  12. Fan X., Zhao D.  On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$.  Journal of Mathematical Analysis and Applications. 263 (2), 424–446 (2001).
  13. Kováčik O., Rákosník J.  On spaces $L^{p(x)}(\Omega)$ and $W^{k,p(x)}(\Omega)$.  Czechoslovak Mathematical Journal. 41 (4), 592–618 (1991).
  14. Diening L., Harjulehto P., Hästö P., Ruzicka M.  Lebesgue and Sobolev spaces with variable exponents.  Lecture notes in mathematics. Heidelberg, Springer (2017).
  15. Růžička M.  Electrorheological fluids: modeling and mathematical theory.  Lecture notes in mathematics. Berlin, Springer-Verlag (2000).
  16. Carmona J., Martínez-Aparicio Pedro J., Rossi Julio D.  A singular elliptic equation with natural growth in the gradient and a variable exponent.  Nonlinear Differential Equations and Applications.  22 (6), 1935–1948 (2015).
  17. Fan X.  Positive solution to p(x)-Laplacian Dirichlet problems with sign-changing non-linearities. Glasgow Mathematical Journal. 52 (3), 505–516 (2010).
  18. Papageorgiou N. S., Winkert P.  Applied Nonlinear Functional Analysis. De Gruyter, Berlin (2018).
  19. Papageorgiou N. S., Rădulescu V. D., Repovš D. D.  Nonlinear Analysis – Theory and Methods. Springer, Cham (2019).
  20. Brezis H.  Analyse fonctionnelle theorie et applications.  New York, Springer (2010).
  21. Stampacchia G.  Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus.  Annales de l'Institut Fourier. 15 (1), 189–257 (1965).
  22. Zhang Q.  A strong maximum principle for differential equations with nonstandard p(x)-growth conditions.  Journal of Mathematical Analysis and Applications. 312 (1), 24–32 (2005).