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Existence Results for a Perturbed Dirichlet Problem Without Sign Condition in Orlicz Spaces

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Ukrainian Mathematical Journal Aims and scope

We deal with the existence result for nonlinear elliptic equations of the form Au + g(x, u,∇u) = f, where the term –div (a(x, u, ∇u)) is a Leray–Lions operator from a subset of \( {W}_0^1{L}_M\left(\Omega \right) \) into its dual. The growth and coercivity conditions on the monotone vector field a are prescribed by an N-function M, which not necessarily satisfies a Δ2-condition. Therefore, we use Orlicz–Sobolev spaces that are not necessarily reflexive and assume that the nonlinearity g(x, u,∇u) is a Carathéodory function satisfying solely a growth condition without any sign condition. The right-hand side f belongs to \( {W}^{-1}{L}_{\overline{M}}\left(\Omega \right). \)

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References

  1. L. Aharouch, E. Azroul, and M. Rhoudaf, “Existence of solutions for unilateral problems in L1 involving lower order terms in divergence form in Orlicz spaces,” J. Appl. Anal., 13, 151–181 (2007).

    Article  MathSciNet  Google Scholar 

  2. L. Aharoucha, A. Benkirane, and M. Rhoudaf, “Existence results for some unilateral problems without sign condition with obstacle free in Orlicz spaces,” Nonlinear Anal., 68, 2362–2380 (2008).

    Article  MathSciNet  Google Scholar 

  3. L. Aharouch and M. Rhoudaf, “Existence of solution for unilateral problems with L1 data in Orlicz spaces,” Proyecciones, 23, No. 3, 293–317 (2004).

    MathSciNet  Google Scholar 

  4. D. Apushkinskaya, M. Bildhauer, and M. Fuchs, “Steady states of anisotropic generalized Newtonian fluids,” J. Math. Fluid Mech., 7, 261–297 (2005).

    Article  MathSciNet  Google Scholar 

  5. A. Benkirane and A. Elmahi, “Almost everywhere convergence of the gradients of solutions to elliptic equations in Orlicz spaces and application,” Nonlin. Anal., 11, No. 28, 1769–1784 (1997).

    Article  MathSciNet  Google Scholar 

  6. A. Benkirane and A. Elmahi, “A strongly nonlinear elliptic equation having natural growth terms and L1 data,” Nonlin. Anal., 39, 403–411 (2000).

    Article  Google Scholar 

  7. A. Benkirane, J. Benouna, and M. Rhoudaf, “Some remarks on a sign condition for perturbations of nonlinear problems,” in: Recent Developments in Nonlinear Analysis, World Scientific Publishing Co., Inc., Hackensack, NJ (2010), pp. 30–42.

  8. A. Bensoussan, L. Boccardo, and F. Murat, “On a nonlinear partial differential equation having natural growth terms and unbounded solution,” Ann. Inst. H. Poincaré Anal. Non Linéaire, 5, No. 4, 149–169 (1988).

    Article  MathSciNet  Google Scholar 

  9. M. Bildhauer, M. Fuchs, and X. Zhong, “On strong solutions of the differential equations modeling the steady flow of certain incompressible generalized Newtonian fluids,” St. Petersburg Math. J., 18, 183–199 (2007).

    Article  MathSciNet  Google Scholar 

  10. L. Boccardo, F. Murat, and J. P. Puel, “L∞ estimate for some nonlinear elliptic partial differential equations and application to an existence result,” SIAM J. Math. Anal., 2, 326–333 (1992).

    Article  MathSciNet  Google Scholar 

  11. L. Boccardo, F. Murat, and J. P. Puel, “Existence of bounded solutions for nonlinear elliptic unilateral problems,” Ann. Mat. Pura Appl. (4), 183–196 (1988).

  12. H. Brézis and F. Browder, “A property of Sobolev spaces,” Comm. Partial Differential Equations, 4, 1077–1083 (1979).

    Article  MathSciNet  Google Scholar 

  13. Y. Chen, S. Levine, and M. Rao, “Variable exponent, linear growth functionals in image restoration,” SIAM J. Appl. Math., 66, 1383–1406 (2006).

    Article  MathSciNet  Google Scholar 

  14. A. Elmahi and D. Meskine, “Existence of solutions for elliptic equations having natural growth terms in Orlicz spaces,” Abstr. Appl. Anal., 12, 1031–1045 (2004).

    Article  MathSciNet  Google Scholar 

  15. A. Elmahi and D. Meskine, “Nonlinear elliptic problems having natural growth and L1 data in Orlicz spaces,” Ann. Mat. Pura Appl. (4)., 184, 161–184 (2005).

    Article  Google Scholar 

  16. A. Fqayeh, A. Benkirane, M. El Moumni et al., “Existence of renormalized solutions for some strongly nonlinear elliptic equations in Orlicz spaces,” Georgian Math. J., 22, No. 3, 305–321 (2015).

    Article  MathSciNet  Google Scholar 

  17. J.-P. Gossez, “Nonlinear elliptic boundary-value problems for equation with rapidly or slowly increasing coefficients,” Trans. Amer. Math. Soc., 190, 217–237 (1974).

    Article  MathSciNet  Google Scholar 

  18. J.-P. Gossez, “Nonlinear elliptic boundary-value problems for equations with rapidly or slowly increasing coefficients,” Trans. Amer. Math. Soc., 190, 163–205 (1974).

    Article  MathSciNet  Google Scholar 

  19. J.-P. Gossez and V. Mustonen, “Variational inequalities in Orlicz spaces,” Nonlin. Anal., 11, 379–492 (1987).

    Article  MathSciNet  Google Scholar 

  20. J.-P. Gossez, “Some approximation properties in Orlicz–Sobolev,” Studia Math., 74, 17–24 (1982).

    Article  MathSciNet  Google Scholar 

  21. P. Gwiazda, P. Wittbold, A. Wróblewska-Kamińska, and A. Zimmermann, “Renormalized solutions to nonlinear parabolic problems in generalized Musielak–Orlicz spaces,” Nonlin. Anal., 129, 1–36 (2015).

    Article  MathSciNet  Google Scholar 

  22. P. Gwiazda, I. Skrzypczak, and A. Zatorska-Goldstein, “Existence of renormalized solutions to elliptic equation in Musielak–Orlicz space,” J. Different. Equat., 264, No. 1, 341–377 (2018).

    Article  MathSciNet  Google Scholar 

  23. P. Gwiazda, M. Bulí\( \tilde{\mathrm{c}} \)ek, J. Málek, and A. Świerczewska-Gwiazda, “On unsteady flows of implicitly constituted incompressible fluids,” SIAM J. Math. Anal., 44, No. 4, 2756–2801 (2012).

  24. S. Hadj Nassar, H. Moussa, and M. Rhoudaf, “Renormalized Solution for a nonlinear parabolic problems with noncoercivity in divergence form in Orlicz spaces,” Appl. Math. Comput., 249, 253–264 (2014).

    MathSciNet  MATH  Google Scholar 

  25. O. Kováčik and J. Rákosník, “On spaces Lp(x) and Wk,p(x),” Czechoslovak Math. J., 41, 592–618 (1991).

    Article  MathSciNet  Google Scholar 

  26. J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris (1969).

    MATH  Google Scholar 

  27. H. Moussa, F. O. Gallego, and M. Rhoudaf, “Capacity solution to a coupled system of parabolic-elliptic equations in Orlicz–Sobolev spaces,” NoDEA Nonlinear Differential Equations Appl. 25, Article number 14 (2018).

  28. J. Musielak, “Orlicz spaces and modular spaces,” Lect. Notes Math., 1034, Springer, Berlin (1983).

  29. H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co., Ltd., Tokyo (1950).

    MATH  Google Scholar 

  30. P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell., 12, 629–639 (1990).

    Article  Google Scholar 

  31. A. Porretta, “Nonlinear equations with natural growth terms and measure data,” Electron. J. Different. Equat., Conf. 09, 181–202 (2002).

    MATH  Google Scholar 

  32. K. R. Rajagopal and M. Ružička, “Mathematical modeling of electrorheological materials,” Contin. Mech. Thermodyn., 13, 59–78 (2001).

    Article  Google Scholar 

  33. M. Ružička, “Electrorheological fluids: modeling and mathematical theory,” Lect. Notes Math., Springer, Berlin (2000).

  34. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory,” Math. USSR, Izv., 29, No. 1, 33–66 (1987).

    Article  Google Scholar 

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Authors and Affiliations

  1. University of Sultan Moulay Slimane, Beni-Mellal, Morocco

    H. Moussa

  2. University of Moulay Ismail, Meknes, Morocco

    M. Rhoudaf

  3. University of Ibn Tofail, Kénitra, Morocco

    H. Sabiki

Authors
  1. H. Moussa
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  2. M. Rhoudaf
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  3. H. Sabiki
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Correspondence to H. Moussa.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 4, pp. 509–526, April, 2020.

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Moussa, H., Rhoudaf, M. & Sabiki, H. Existence Results for a Perturbed Dirichlet Problem Without Sign Condition in Orlicz Spaces. Ukr Math J 72, 585–606 (2020). https://doi.org/10.1007/s11253-020-01802-0

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  • DOI: https://doi.org/10.1007/s11253-020-01802-0

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