pp 1–24 | Cite as

Singular Neumann (pq)-equations

  • Nikolaos S. Papageorgiou
  • Calogero Vetro
  • Francesca VetroEmail author


We consider a nonlinear parametric Neumann problem driven by the sum of a p-Laplacian and of a q-Laplacian and exhibiting in the reaction the competing effects of a singular term and of a resonant term. Using variational methods together with suitable truncation and comparison techniques, we show that for small values of the parameter the problem has at least two positive smooth solutions.


Singular term Resonant nonlinearity Nonlinear regularity Truncation and comparison Nonlinear strong maximum principle (p, q)-equation 

Mathematics Subject Classification

35J92 35P30 



The authors wish to thank a knowledgeable referee for his/her corrections and remarks.


  1. 1.
    Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints. Mem. Am. Math. Soc. 196(915), 1–70 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cherfils, L., Il’yasov, Y.: On the stationary solutions of generalized reaction diffusion equations with \(p\)&\(q\)-Laplacian. Commun. Pure Appl. Anal. 4(1), 9–22 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Gasiński, L., Papageorgiou, N.S.: Nonlinear Analysis, Series Mathematical Analysis and Applications, vol. 9. CRC Press, Boca Raton (2006)zbMATHGoogle Scholar
  5. 5.
    Gasiński, L., Papageorgiou, N.S.: Nonlinear elliptic equations with singular terms and combined nonlinearities. Ann. Henri Poincaré 13(3), 481–512 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Giacomoni, J., Schindler, I., Takáč, P.: Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6(1), 117–158 (2007)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hirano, N., Saccon, C., Shioji, N.: Brezis–Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem. J. Differ. Equ. 245(8), 1997–2037 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kyritsi, S., Papageorgiou, N.S.: Pairs of positive solutions for singular \(p\)-Laplacian equations with a \(p\)-superlinear potential. Nonlinear Anal. 73(5), 1136–1142 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lair, A.V., Shaker, A.W.: Entire solution of a singular semilinear elliptic problem. J. Math. Anal. Appl. 200(2), 498–505 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lieberman, G.M.: The natural generalization of the natural conditions of Ladyzhenskaya and Ural\(^\prime \)tseva for elliptic equations. Commun. Partial Differ. Equ. 16(2–3), 311–361 (1991)CrossRefGoogle Scholar
  11. 11.
    Marano, S.A., Mosconi, S.J.N.: Some recent results on the Dirichlet problem for (\(p, q\))-Laplace equations. Discrete Contin. Dyn. Syst. Ser. S 11(2), 279–291 (2018)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Marano, S.A., Papageorgiou, N.S.: Positive solutions to a Dirichlet problem with \(p\)-Laplacian and concave-convex nonlinearity depending on a parameter. Commun. Pure Appl. Anal. 12(2), 815–829 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)CrossRefGoogle Scholar
  14. 14.
    Papageorgiou, N.S., Rădulescu, V.D.: Combined effects of singular and sublinear nonlinearities in some elliptic problems. Nonlinear Anal. 109, 236–244 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Papageorgiou, N.S., Rădulescu, V.D.: Multiple solutions with precise sign for nonlinear parametric Robin problems. J. Differ. Equ. 256(7), 2449–2479 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Papageorgiou, N.S., Rădulescu, V.D.: Nonlinear nonhomogeneous Robin problems with superlinear reaction term. Adv. Nonlinear Stud. 16(4), 737–764 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Papageorgiou, N.S., Rădulescu, V.D., Repovs̆, D.D.: Pairs of positive solutions for resonant singular equations with the \(p\)-Laplacian. Electron. J. Differ. Equ. 2017(249), 1–22 (2017)MathSciNetGoogle Scholar
  18. 18.
    Papageorgiou, N.S., Smyrlis, G.: A bifurcation-type theorem for singular nonlinear elliptic equations. Methods Appl. Anal. 22(2), 147–170 (2015)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Papageorgiou, N.S., Vetro, C.: Superlinear \((p(z), q(z))\)-equations. Complex Var. Elliptic Equ. 64(1), 8–25 (2019)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Perera, K., Zhang, Z.: Multiple positive solutions of singular \(p\)-Laplacian problems by variational methods. Bound. Value Probl. 2005(3), 377–382 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser, Basel (2007)CrossRefGoogle Scholar
  22. 22.
    Sun, Y., Wu, S., Long, Y.: Combined effects of singular and superlinear nonlinearities in some singular boundary value problems. J. Differ. Equ. 176(2), 511–531 (2001)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Technical UniversityAthensGreece
  2. 2.Department of Mathematics and Computer ScienceUniversity of PalermoPalermoItaly
  3. 3.Nonlinear Analysis Research GroupTon Duc Thang UniversityHo Chi Minh CityVietnam
  4. 4.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam

Personalised recommendations