Robin Boundary Value Problems With Natural Growth Term in Variable Exponent Space

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-23-2025-36
Published: Feb 14, 2025
Cite as:
Boulaye Yira, Ibrahime Konaté, Mamadou Diop, Robin Boundary Value Problems With Natural Growth Term in Variable Exponent Space, Int. J. Anal. Appl., 23 (2025), 36.

Main Article Content

Boulaye Yira, Ibrahime Konaté, Mamadou Diop

Abstract

The main purpose of this paper is to investigate a nonlinear elliptic problem with a natural growth term under Robin boundary conditions. Using approximation techniques and surjectivity criteria of an operator mapping from a Banach space into its dual, we prove the existence of a sequence of weakly approximated solutions and take its limit to establish the existence of a renormalized or entropy solution for the initial problem.

Article Details

References

  1. Y. Akdim, C. Allalou, N. El Gorch, Existence of Renormalized Solutions for Nonlinear Elliptic Problems in Weighted Variable-Exponent Space with L 1 -Data, Gulf J. Math. 6 (2018), 151–165. https://doi.org/10.56947/gjom.v6i4.254.
  2. Y. Akdim, C. Allalou, Existence of Renormalized Solutions of Nonlinear Elliptic Problems in Weighted VariableExponent Space, J. Math. Stud. 48 (2015), 375–397. https://doi.org/10.4208/jms.v48n4.15.05.
  3. Y.A. Youssef Akdim, M.O. Morad Ouboufettal, Existence of Solution for a General Class of Strongly Nonlinear Elliptic Problems Having Natural Growth Terms and L 1 -Data, Anal. Theory Appl. 39 (2023), 53–68. https://doi.org/10.4208/ata.OA-2020-0049.
  4. F. Andreu, N. Igbida, J.M. Mazón, J. Toledo, L 1 Existence and Uniqueness Results for Quasi-Linear Elliptic Equations with Nonlinear Boundary Conditions, Ann. Inst. Henri Poincaré C Anal. Non Linéaire 24 (2007), 61–89. https://doi.org/10.1016/j.anihpc.2005.09.009.
  5. F. Andreu, J.M. Mazon, S.S. de Léon, J. Toledo, Quasi-Linear Elliptic and Parabolic Equations in L 1 With Nonlinear Boundary Conditions, Adv. Math. Sci. Appl. 7 (1997), 183-213.
  6. S.N. Antontsev, J.F. Rodrigues, On Stationary Thermo-Rheological Viscous Flows, Ann. Univ. Ferrara 52 (2006), 19–36. https://doi.org/10.1007/s11565-006-0002-9.
  7. E. Azroul, H. Hjiaj, A. Touzani, Existence and Regularity of Entropy Solutions for Strongly Nonlinear p(x)-Elliptic Equations, Electron. J. Differ. Equ. 2013 (2013), 68.
  8. E. Azroul, M.B. Benboubker, R. Bouzyani, H. Chrayteh, Renormalized Solutions for Some Nonlinear Nonhomogeneous Elliptic Problems with Neumann Boundary Conditions and Right Hand Side Measure, Bol. Soc. Paran. Mat. 39 (2021), 81–103. https://doi.org/10.5269/bspm.41896.
  9. M.B. Benboubker, E. Azroul, A. Barbara, Quasilinear Elliptic Problems with Nonstandard Growth, Electron. J. Differ. Equ. 2011 (2011), 62.
  10. P. Bénilan, L. Boccardo, T. Gallouët, et al. An L1-Theory of Existence and Uniqueness of Solutions of Nonlinear Elliptic Equations, Ann. Scuola Norm. Super. Pisa – Cl. Sci., Ser. 4, 22 (1995), 241-272.
  11. F. Murat, A. Bensoussan, L. Boccardo, On a Non Linear Partial Differential Equation Having Natural Growth Terms and Unbounded Solution, Ann. Inst. Henri Poincaré C, Anal. Non Lin. 5 (1988), 347–364. https://doi.org/10.1016/s0294-1449(16)30342-0.
  12. B.K. Bonzi, I. Nyanquini, S. Ouar, Existence and Uniqueness of Weak Solution and Entropy Solutions for Homogeneous Neumann Boundary-Value Problems Involving Variable Exponents, Electron. J. Differ. Equ. 2012 (2012), 12.
  13. L. Boccardo, F. Murat, J.P. Puel, Existence of Bounded Solutions for Non Linear Elliptic Unilateral Problems, Ann. Mat. Pura Appl. 152 (1988), 183–196. https://doi.org/10.1007/BF01766148.
  14. L. Boccardo, T. Gallouët, F. Murat, A Unified Presentation of Two Existence Results for Problems with Natural Growth, in: Progress in Partial Differential Equations, The Metz Surveys, 2 (1992), Pitman Research Notes in Mathematics Series, vol. 296, pp. 127–137, Longman Scientific and Technical, Harlow (1993).
  15. L. Boccardo, T. Gallouet, Strongly Nonlinear Elliptic Equations Having Natural Growth Terms and L 1 Data, Nonlin. Anal.: Theory Methods Appl. 19 (1992), 573–579. https://doi.org/10.1016/0362-546X(92)90022-7.
  16. I. Boccardo, T. Gallouet, Nonlinear Elliptic Equations with Right Hand Side Measures, Commun. Partial Differ. Equ. 17 (1992), 189–258. https://doi.org/10.1080/03605309208820857.
  17. H. Brezis, Opérateurs Maximaux Monotones et Semigroupes de Contraction dans les Espaces de Hilbert, North Holland, Amsterdam, 1973.
  18. Y. Chen, S. Levine, M. Rao, Variable Exponent, Linear Growth Functionals in Image Restoration, SIAM J. Appl. Math. 66 (2006), 1383–1406. https://doi.org/10.1137/050624522.
  19. M.C.H. Moulay Cherif Hassib, Y.A. Youssef Akdim, E.A. Elhoussine Azroul, A.B. Abdelkrim Barbara, Existence and Regularity of Solution for Strongly Nonlinear p(x)-Elliptic Equation with Measure Data, J. Partial Differ. Equ. 30 (2017), 31–46. https://doi.org/10.4208/jpde.v30.n1.3.
  20. L. Diening, Theoretical and Numerical Results for Electrorheological Fluids, PhD Thesis, University of Frieburg, Germany, 2002.
  21. R.J. DiPerna, P.L. Lions, On the Cauchy Problem for Boltzmann Equations: Global Existence and Weak Stability, Ann. Math. 130 (1989), 321–366. https://doi.org/10.2307/1971423.
  22. X. Fan, Anisotropic Variable Exponent Sobolev Spaces and p(x)-Laplacian Equations, Complex Var. Elliptic Equ. 56 (2011), 623–642. https://doi.org/10.1080/17476931003728412.
  23. X. Fan, D. Zhao, On the Spaces L p(x) (Ω) and W1,p(x) (Ω), J. Math. Anal. Appl. 263 (2001), 424–446. https://doi.org/10.1006/jmaa.2000.7617.
  24. X. Fan, D. Zhao, On the Generalized Orlicz-Sobolev Space Wk,p(x) (Ω), J. Gansu Educ. Coll. 12 (1998), 1–6.
  25. P. Gwiazda, A. Swierczewska-Gwiazda, A. Wróblewska, Monotonicity Methods in Generalized Orlicz Spaces for ´ a Class of Non-Newtonian Fluids, Math. Methods Appl. Sci. 33 (2010), 125–137. https://doi.org/10.1002/mma.1155.
  26. P. Gwiazda, P. Wittbold, A. Wróblewska, A. Zimmermann, Renormalized Solutions of Nonlinear Elliptic Problems in Generalized Orlicz Spaces, J. Differ. Equ. 253 (2012), 635–666. https://doi.org/10.1016/j.jde.2012.03.025.
  27. P. Harjulehto, P. Hästö, Sobolev Inequalities for Variable Exponents Attaining the Values 1 and n, Publ. Mat. 52 (2008), 347–363. https://doi.org/10.5565/PUBLMAT_52208_05.
  28. I. Konate, I. Idrissa, S. Ouaro, Nonlinear Problem Having Natural Growth Term and Measure Data, An. Univ. Oradea Fasc. Mat. 31 (2024), 85–113.
  29. O. Kovácik, J. Rákosník, On Spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41 (1991), 592–618. https://doi.org/10.21136/CMJ.1991.102493.
  30. J. Leray, J.L. Lions, Quelques Résultats de Višik Sur Les Problèmes Elliptiques Non Linéaires Par Les Méthodes de Minty-Browder, Bull. Soc. Math. France 79 (1965), 97–107. https://doi.org/10.24033/bsmf.1617.
  31. D. Motreanu, A. Sciammetta, E. Tornatore, A Sub-Supersolution Approach for Neumann Boundary Value Problems with Gradient Dependence, Nonlinear Anal.: Real World Appl. 54 (2020), 103096. https://doi.org/10.1016/j.nonrwa.2020.103096.
  32. I. Nyanquini, S. Ouaro, S. Safimba, Entropy Solution to Nonlinear Multivalued Elliptic Problem With Variable Exponents and Measure Data, Ann. Univ. Craiova - Math. Comput. Sci. Ser. 40 (2013), 174–198.
  33. I. Nyanquini, S. Ouaro, Entropy Solution for Nonlinear Elliptic Problem Involving Variable Exponent and Fourier Type Boundary Condition, Afr. Mat. 23 (2012), 205–228. https://doi.org/10.1007/s13370-011-0030-1.
  34. S. Ouaro, A. Ouédraogo, S. Soma, Multivalued Homogeneous Neumann Problem Involving Diffuse Measure Data and Variable Exponent, Nonlinear Dyn. Syst. Theory, 16 (2016), 102–114.
  35. O. Stanislas, L 1 Existence and Uniqueness of Entropy Solutions to Nonlinear Multivalued Elliptic Equations with Homogeneous Neumann Boundary Condition and Variable Exponent, J. Partial Differ. Equ. 27 (2014), 1–27. https://doi.org/10.4208/jpde.v27.n1.1.
  36. S. Ouaro, A. Ouedraogo, S. Soma, Multivalued Problem with Robin Boundary Condition Involving Diffuse Measure Data and Variable Exponent, Adv. Nonlinear Anal. 3 (2014), 209–235. https://doi.org/10.1515/anona-2014-0010.
  37. S. Ouaro, N. Sawadogo, Structural Stability of Nonlinear Elliptic p(u)-Laplacian Problem with Robin Type Boundary Condition, in: G.M. N’Guérékata, B. Toni (Eds.), Studies in Evolution Equations and Related Topics, Springer, Cham, 2021: pp. 69–111. https://doi.org/10.1007/978-3-030-77704-3_5.
  38. S. Ouaro, A. Tchousso, Well-Posedness Result for a Nonlinear Elliptic Problem Involving Variable Exponent and Robin Type Boundary Condition, Afr. Diaspora J. Math. (N.S.) 11 (2011), 36-64.
  39. A. Prignet, Conditions aux Limites Non Homogènes pour des Problèmes Elliptiques avec Second Membre Mesure, Ann. Fac. Sci. Toulouse 6 (1997), 297–318. http://www.numdam.org/item?id=AFST_1997_6_6_2_297_0.
  40. K.R. Rajagopal, M. Ruzicka, Mathematical Modeling of Electrorheological Materials, Continuum Mech. Thermodyn. 13 (2001), 59–78. https://doi.org/10.1007/s001610100034.
  41. M. Ružiˇcka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer, Berlin, Heidelberg, 2000. https://doi.org/10.1007/BFb0104029.
  42. S. Ouaro, N. Sawadogo, Nonlinear Multivalued Homogeneous Robin Boundary p(u)-Laplacian Problem, Ann. Math. Comput. Sci. 24 (2024), 57–84. https://doi.org/10.56947/amcs.v24.308.
  43. L.-L. Wang, Y.-H. Fan, W.-G. Ge, Existence and Multiplicity of Solutions for a Neumann Problem Involving the p(x)-Laplace Operator, Nonlinear Anal.: Theory Methods Appl. 71 (2009), 4259–4270. https://doi.org/10.1016/j.na.2009.02.116.
  44. J. Yao, Solutions for Neumann Boundary Value Problems Involving p(x)-Laplace Operators, Nonlinear Anal.: Theory Methods Appl. 68 (2008), 1271–1283. https://doi.org/10.1016/j.na.2006.12.020.
  45. C. Yazough, E. Azroul, H. Redwane, Existence of Solutions for Some Nonlinear Elliptic Unilateral Problems with Measure Data, Electron. J. Qual. Theory Differ. Equ. 2013 (2013), 43.