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Characterization of the Critical Value for a Quasilinear Elliptic Equation with Arbitrary Growth with Respect to the Gradient

  • Fatima AqelEmail author
  • Nour Eddine Alaa
Article
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Abstract

The aim of this work is to study a quasilinear elliptic equation where we are particularly interested in the characterization of the critical value, which appears as the Lagrange multiplier in the functional minimization associated with the dual problem, over one close convex subset of \(L^\infty \times (W^{1,\infty })^N\). We are going to give a generalization of Alaa (Etude d’équations elliptiques non linéaires à dépendance convexe en le gradient et à données mesures. Ph.D. Thesis, University of Nancy I, 1989) method in the case of an upper space dimension \((N\ge 2)\).

Keywords

Critical value quasilinear equation functional minimization lower semicontinuity data measures 

Mathematics Subject Classification

26A15 26A51 35A01 35B09 58J10 35J62 

Notes

Acknowledgements

The authors are very thankful to the anonymous referee for his/her careful reading of the manuscript and valuable comments on this paper.

References

  1. 1.
    Alaa, N.: Etude d’équations elliptiques non linéaires à dépendance convexe en le gradient et à données mesures. Ph.D. Thesis, University of Nancy I (1989)Google Scholar
  2. 2.
    Alaa, N.: Quasilinear elliptic equation with arbitrary growth nonlinearity and data measures. Extr. Math. 11, 405–411 (1996)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Alaa, N., Pierre, M.: Weak solutions of some quasilinear elliptic equations with data measures. SIAM J. Math. Anal. 24(1), 23–35 (1993)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Alaa, N., Iguernane, M.: Weak periodic solutions of some quasilinear parabolic equations with data measures J. Inequal. Pure Appl. Math. 3(3), (Article 46) (2002)Google Scholar
  5. 5.
    Amann, H., Quittner, P.: Elliptic Boundary problems involving measures: existence, regularity and multiplicity. J. Adv. Differ. Equ. 3(6), 753–813 (1998)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary values problems. J. Funct. Sp. Differ. Oper. Nonlinear Anal. (H.J. Schmeisser, H. Triebel, eds.), B. G. Teubner Verlagsgesellschaft, Teubner-texte Math. 133 Stuttgart 133, 9–126 (1993)Google Scholar
  7. 7.
    Boccardo, L., Gallouët, T.: Non-linear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149–169 (1989)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chouquet-Bruhat, Y., Leray, J.: Sur le probléme de Dirichlet quasilinéaire d’ordre deux. Annales de la Faculté des sciences de Toulouse Mathématique Série 5 Tome 1(1), 9–25 (1979)Google Scholar
  9. 9.
    Lions, P.: Résolution de problèmes elliptiques quasilinéaires. Arch. Ration. Mech. Anal. 74(4), 335–353 (1980)CrossRefGoogle Scholar
  10. 10.
    Oskolkov, A.: On the solvability of the Dirichlet problem for quasilinear elliptic equations in unbaunded domains J. Boun. Value Probl. Math. Phys. Relat. Probl. Funct. Theory, Part 4, Zap. Nauchn. Sem. LOMI, 14, “Nauka”, Leningrad. Otdel., Leningrad 88, 173–190 (1969)Google Scholar
  11. 11.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of Cadi Ayyad Faculty of Sciences and Techniques Laboratory of LAMAIMarrakechMorocco

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