Rendiconti del Circolo Matematico di Palermo

, Volume 50, Issue 1, pp 47–66 | Cite as

Nonlinear elliptic problems of Neumann-type

  • Shouchuan Hu
  • N. S. Papageorgiou


In this paper we study a nonlinear elliptic differential equation driven by thep-Laplacian with a multivalued boundary condition of the Neumann type. Using techniques from the theory of maximal monotone operators and a theorem of the range of the sum of monotone operators, we prove the existence of a (strong) solution.

1991 Mathematica Subject Classification

35J60 35J25 

Key words and phrases

Neumann problems maximal monotone operators subdifferentials boundedly inversely compact equality in the sense of distributions trace maps and spaces 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ash R.,Real Analysis and Probability, Academic Press, New York, 1972.Google Scholar
  2. [2]
    Boccardo L., Drabek P., Giachetti D., Kucera M.,Generalization of Fredholm alternative for nonlinear operators, Nonl. Anal.,10 (1986), 1083–1103.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Brezis H., Haraux A.,Image d'une somme d'opérateurs monotones et applications, Israel J. Math.,23 (1976), 165–186.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Calvert B., Gupta C.,Nonlinear elliptic boundary value problems in L p -spaces and sums of ranges of accretive operators, Nonl. Anal.,2 (1978), 1–26.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Costa D., Magalhaes C.,Existence results for perturbations of the p-Laplacian, Nonl. Anal.,24 (1995), 409–418.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Dal Maso G.,An Introduction to Γ-Convergence, Birkhäuser, Boston, 1993.Google Scholar
  7. [7]
    Del Pino M., Elgueta M., Manasevich R.,A homotopic deformation along p of a Leray-Schauder degree result and existence for (|u′| p−2 u′)′+f(t,u)=0, u(0)=u(T)=0, p>1, J. Diff. Eqns.,80 (1983), 1–13.Google Scholar
  8. [8]
    Diaz J. I.,Nonlinear Partial Differential Equations and Free Boundaries I: Elliptic Equations, Research Notes in Math., no. 106, Pitman, London, 1985.MATHGoogle Scholar
  9. [9]
    Drabek P.,Solvability of boundary value problems with homogeneous ordinary differential operator, Rendiconti dell'Istituto Matematico, Trieste,6 (1985), 105–124.Google Scholar
  10. [10]
    Hachimi A. El., Gossez J. P.,A note on a nonresonant condition for a quasilinear elliptic problem, Nonl. Anal.,22 (1994), 229–236.MATHCrossRefGoogle Scholar
  11. [11]
    Guo Z.,Boundary value problems of a class of quasilinear ordinary differential equations, Diff. Integ. Eqns.,6 (1993), 705–719.MATHGoogle Scholar
  12. [12]
    Gupta C., Hess P.,Existence theorems for nonlinear noncoercive operators and nonlinear elliptic boundary value problems, J. Diff. Eqns.,22 (1976), 305–313.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Hess P.,On semicoercive nonlinear problems, Indiana Univ. Math. J.,23 (1974), 645–654.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Ioffe A., Tichominov V.,Theory of Extremal Problems, North-Holland, Amsterdam, 1979.MATHGoogle Scholar
  15. [15]
    Kenmochi N.,Pseudomonotone operators and nonlinear elliptic boundary value problems, J. Math. Soc. Japan,27 (1975), 121–149.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    Kesavan S.,Functional Analysis and Applications, Wiley, New York, 1989.MATHGoogle Scholar
  17. [17]
    Landesman E., Lazer A.,Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech.,19 (1969), 609–623.MathSciNetGoogle Scholar
  18. [18]
    Menou S.,Famille mesurable d'operateurs maximaux monotones, CRAS Paris,290 (1980), 741–744.MATHMathSciNetGoogle Scholar
  19. [9]
    Rockafellar R. T.Conjugate Duality and Optimization, Reg. Conf. Series in Appl. Math., vol. 16, SIAM, Philadelphia, 1973.Google Scholar
  20. [20]
    Schatzman M.,Problèmes aux limites non-lineaire non-coercifs, Ann. Scuola Norm. Sup. Pisa,27 (1973), 641–686.MathSciNetGoogle Scholar
  21. [21]
    Zeidler E.,Nonlinear Functional Analysis and Its Applications II, Springer-Verlag, New York, 1990.Google Scholar

Copyright information

© Springer 2001

Authors and Affiliations

  • Shouchuan Hu
    • 1
  • N. S. Papageorgiou
    • 2
  1. 1.Department of MathematicsSouthwest Missouri State UniversitySpringfiled
  2. 2.Department of MathematicsNational Technical UniversityAthensGreece

Personalised recommendations