L -Estimates for nonlinear elliptic Neumann boundary value problems

  • Patrick Winkert


In this paper we prove the L -boundedness of solutions of the quasilinear elliptic equation
$$ \begin{array}{ll} Au \, = f(x,u,\nabla u) &\quad \rm{in }\, \Omega, \\ \dfrac{\partial u}{ \partial \nu} \, = g(x,u) &\quad \rm{on }\, \partial \Omega, \end{array} $$
where A is a second order quasilinear differential operator and \({f:\Omega \times \mathbb{R} \times \mathbb{R}^N \rightarrow \mathbb{R}}\) as well as \({g: \partial \Omega \times \mathbb{R} \rightarrow \mathbb{R}}\) are Carathéodory functions satisfying natural growth conditions. Our main result is given in Theorem 4.1 and is based on the Moser iteration technique along with real interpolation theory. Furthermore, the solutions of the elliptic equation above belong to \({C^{1,\alpha}(\overline{\Omega})}\), if g is Hölder continuous.

Mathematics Subject Classification (2000)

35B45 35J25 35J60 


A priori estimates Neumann boundary values Nonlinear elliptic equations 


  1. 1.
    Amann H.: Linear and Quasilinear Parabolic Problems. Birkhäuser Boston Inc., Boston (1995)zbMATHGoogle Scholar
  2. 2.
    Carl, S., Le, V.K., Motreanu, D.: Nonsmooth variational problems and their inequalities. Springer Monographs in Mathematics. Springer, New York (2007)Google Scholar
  3. 3.
    Dobrowolski M.: Applied functional analysis. Functional Analysis, Sobolev Spaces and Elliptic Differential Equations. Springer, Berlin (2006)zbMATHGoogle Scholar
  4. 4.
    Dobrowolski, M.: private communication, 2008Google Scholar
  5. 5.
    Drábek P.: The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems. Math. Bohem. 120, 169–195 (1995)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Drábek P.: Nonlinear eigenvalue problem for p-Laplacian in R N. Math. Nachr. 173, 131–139 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Drábek P., Hernández J.: Existence and uniqueness of positive solutions for some quasilinear elliptic problems. Nonlinear Anal. 44, 189–204 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Drábek P., Kufner A., Nicolosi F.: Quasilinear elliptic equations with degenerations and singularities. Walter de Gruyter & Co., Berlin (1997)zbMATHGoogle Scholar
  9. 9.
    Lê A.: Eigenvalue problems for the p-Laplacian. Nonlinear Anal. 64, 1057–1099 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lieberman G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Motreanu D., Motreanu V.V., Papageorgiou N.S.: Nonlinear neumann problems near resonance. Indiana Univ. Math. J. 58(3), 1257–1279 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Runst T., Sickel W.: Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations. Walter de Gruyter & Co., Berlin (1996)zbMATHGoogle Scholar
  13. 13.
    Sickel, W.: private communication, 2008Google Scholar
  14. 14.
    Triebel H.: Interpolation theory, function spaces, differential operators. VEB Deutscher Verlag der Wissenschaften, Berlin (1978)Google Scholar
  15. 15.
    Triebel H.: Theory of Function Spaces. Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig (1983)Google Scholar
  16. 16.
    Triebel, H.: Theory of function spaces II. Monographs in Mathematics, vol. 84. Birkhäuser Verlag, Basel (1992)Google Scholar
  17. 17.
    Wei J., Xu X.: Uniqueness and a priori estimates for some nonlinear elliptic Neumann equations in \({\mathbb{R}^3}\). Pacific J. Math. 221, 159–165 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Zeidler E.: Nonlinear Functional Analysis and its Applications. II/B. Springer, New York (1990)zbMATHGoogle Scholar
  19. 19.
    Zhu M.: Uniqueness results through a priori estimates. I. A three-dimensional Neumann problem. J. Differ. Equ. 154, 284–317 (1999)zbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  1. 1.Department of MathematicsMartin-Luther-University Halle-WittenbergHalleGermany

Personalised recommendations