Monatshefte für Mathematik

, Volume 177, Issue 2, pp 203–233 | Cite as

Parametric nonlinear nonhomogeneous Neumann equations involving a nonhomogeneous differential operator

  • Said El Manouni
  • Nikolaos S. Papageorgiou
  • Patrick Winkert


This work is concerned with the existence of solutions to parametric elliptic equations driven by a nonhomogeneous differential operator with a nonhomogeneous Neumann boundary condition. The assumptions on the operator involve the \(p\)-Laplacian, the \((p,q)\)-Laplacian, and the generalized \(p\)-mean curvature differential operator. Based on variational tools combined with truncation and comparison techniques we prove the existence of at least three nontrivial solutions provided the parameter is sufficiently large whereby the first solution is positive, the second one is negative and the third one has changing sign (nodal).


Constant sign and nodal solutions Nonlinear regularity Critical point of mountain pass type Extremal solutions Local minimizers Maximum principle 

Mathematics Subject Classification (2010)

35J20 35J60 35J92 58E05 


  1. 1.
    Abreu, E.A.M., Marcos do Ó, J., Medeiros, E.S.: Multiplicity of positive solutions for a class of quasilinear nonhomogeneous Neumann problems. Nonlinear Anal. 60(8), 1443–1471 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Brezis, H., Nirenberg, L.: \(H^1\) versus \(C^1\) local minimizers. C. R. Acad. Sci. Paris Sér. I Math. 317(5), 465–472 (1993)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Čaklović, L., Li, S.J., Willem, M.: A note on Palais-Smale condition and coercivity. Diffe. Integral Equ. 3(4), 799–800 (1990)zbMATHGoogle Scholar
  4. 4.
    Díaz, J.I., Saá, J.E.: Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. C. R. Acad. Sci. Paris Sér. I Math. 305(12), 521–524 (1987)zbMATHGoogle Scholar
  5. 5.
    Dunford, N., Schwartz, J.T.: Linear Operators. I. General Theory. Interscience Publishers, Inc., New York (1958)zbMATHGoogle Scholar
  6. 6.
    Fernández Bonder, J., Rossi, J.D.: Existence results for the \(p\)-Laplacian with nonlinear boundary conditions. J. Math. Anal. Appl. 263(1), 195–223 (2001)Google Scholar
  7. 7.
    Fernández Bonder, J.: Multiple positive solutions for quasilinear elliptic problems with sign-changing nonlinearities. Abstr. Appl. Anal. 2004(12), 1047–1055 (2004)Google Scholar
  8. 8.
    Fernández Bonder, J.: Multiple solutions for the \(p\)-Laplace equation with nonlinear boundary conditions. Electron. J. Differ. Equ. 37, 1–7 (2006)Google Scholar
  9. 9.
    García Azorero, J.P., Peral Alonso, I., Manfredi, J.J.: Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 2(3), 385–404 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Gasiński, L., Papageorgiou, N.S.: Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems. Chapman & Hall/CRC, Boca Raton, FL (2005)zbMATHGoogle Scholar
  11. 11.
    Gasiński, L., Papageorgiou, N.S.: Nonlinear Analysis. Chapman & Hall/CRC, Boca Raton, FL (2006)zbMATHGoogle Scholar
  12. 12.
    Khan, A.A., Motreanu, D.: Local minimizers versus \(X\)-local minimizers. Optim. Lett. 7(5), 1027–1033 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Lê, A.: Eigenvalue problems for the \(p\)-Laplacian. Nonlinear Anal. 64(5), 1057–1099 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Li, C., Li, S.: Multiple solutions and sign-changing solutions of a class of nonlinear elliptic equations with Neumann boundary condition. J. Math. Anal. Appl. 298(1), 14–32 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12(11), 1203–1219 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Liu, C., Zheng, Y.: Linking solutions for \(p\)-Laplace equations with nonlinear boundary conditions and indefinite weight. Calc. Var. Partial Differ. Equ. 41(1–2), 261–284 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Martínez, S.R., Rossi, J.D.: Isolation and simplicity for the first eigenvalue of the \(p\)-Laplacian with a nonlinear boundary condition. Abstr. Appl. Anal. 7(5), 287–293 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Martínez, S.R., Rossi, J.D.: Weak solutions for the \(p\)-Laplacian with a nonlinear boundary condition at resonance. Electron. J. Differ. Equ. 27, 1–14 (2003)Google Scholar
  19. 19.
    Martínez, S.R., Rossi, J.D.: On the Fučik spectrum and a resonance problem for the \(p\)-Laplacian with a nonlinear boundary condition. Nonlinear Anal. 59(6), 813–848 (2004)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Motreanu, D., Papageorgiou, N.S.: Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator. Proc. Amer. Math. Soc. 139(10), 3527–3535 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser Verlag, Basel (2007)zbMATHGoogle Scholar
  22. 22.
    Winkert, P.: Comparison principles and multiple solutions for nonlinear elliptic equations. Ph.D. thesis, Institute of Mathematics, University of Halle, Halle, Germany (2009)Google Scholar
  23. 23.
    Winkert, P.: Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values. Adv. Differ. Equ. 15(5–6), 561–599 (2010)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Winkert, P.: Local \(C^1(\overline{\Omega })\)-minimizers versus local \(W^{1, p}(\Omega )\)-minimizers of nonsmooth functionals. Nonlinear Anal. 72(11), 4298–4303 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Winkert, P.: Multiplicity results for a class of elliptic problems with nonlinear boundary condition. Commun. Pure Appl. Anal. 12(2), 785–802 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Winkert, P., Zacher, R.: A priori bounds for weak solutions to elliptic equations with nonstandard growth. Discrete Contin. Dyn. Syst. Ser. S 5(4), 865–878 (2012)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Zhao, J.-H., Zhao, P.-H.: Existence of infinitely many weak solutions for the \(p\)-Laplacian with nonlinear boundary conditions. Nonlinear Anal. 69(4), 1343–1355 (2008)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 2014

Authors and Affiliations

  • Said El Manouni
    • 1
  • Nikolaos S. Papageorgiou
    • 2
  • Patrick Winkert
    • 1
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Department of MathematicsNational Technical UniversityAthensGreece

Personalised recommendations