(p, 2)-Equations with a Crossing Nonlinearity and Concave Terms

  • Nikolaos S. Papageorgiou
  • Calogero Vetro
  • Francesca Vetro


We consider a parametric Dirichlet problem driven by the sum of a p-Laplacian (\(p>2\)) and a Laplacian (a (p, 2)-equation). The reaction consists of an asymmetric \((p-1)\)-linear term which is resonant as \(x \rightarrow - \infty \), plus a concave term. However, in this case the concave term enters with a negative sign. Using variational tools together with suitable truncation techniques and Morse theory (critical groups), we show that when the parameter is small the problem has at least three nontrivial smooth solutions.


p-Laplacian Concave term Crossing nonlinearity Nonlinear regularity Nonlinear maximum principle Critical groups Multiple smooth solutions 

Mathematics Subject Classification

35J20 35J60 58E05 


  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)MathSciNetMATHGoogle Scholar
  2. 2.
    Allegretto, W., Huang, Y.X.: A Picone’s identity for the $p$-Laplacian and applications. Nonlinear Anal. 32(7), 819–830 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14(4), 349–381 (1973)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ambrosetti, A., Brezis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122(2), 519–543 (1994)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chang, K.-C.: Methods in Nonlinear Analysis. Springer, Berlin (2005)MATHGoogle Scholar
  6. 6.
    D’Aguì, G., Marano, S.A., Papageorgiou, N.S.: Multiple solutions to a Robin problem with indefinite weight and asymmetric reaction. J. Math. Anal. Appl. 433(2), 1821–1845 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    de Paiva, F.O., Massa, E.: Multiple solutions for some elliptic equations with a nonlinearity concave at the origin. Nonlinear Anal. 66(12), 2940–2946 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    de Paiva, F.O., Presoto, A.E.: Semilinear elliptic problems with asymmetric nonlinearities. J. Math. Anal. Appl. 409(1), 254–262 (2014)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gasiński, L., Papageorgiou, N.S.: Nonlinear Analysis. Series in Mathematical Analysis and Applications, vol. 9. Chapman and Hall/CRC Press, Boca Raton (2006)MATHGoogle Scholar
  11. 11.
    Gasiński, L., Papageorgiou, N.S.: Resonant equations with the Neumann $p$-Laplacian plus an indefinite potential. J. Math. Anal. Appl. 422(2), 1146–1179 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gasiński, L., Papageorgiou, N.S.: Exercises in Analysis, Part 2. Nonlinear Analysis. Springer, Cham (2016)CrossRefMATHGoogle Scholar
  13. 13.
    Guo, Z., Zhang, Z.: $W^{1, p}$ versus $C^1$ local minimizers and multiplicity results for quasilinear elliptic equations. J. Math. Anal. Appl. 286(1), 32–50 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)MATHGoogle Scholar
  15. 15.
    Liang, Z., Su, J.: Multiple solutions for semilinear elliptic boundary value problems with double resonance. J. Math. Anal. Appl. 354(1), 147–158 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12(11), 1203–1219 (1988)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Marano, S.A., Papageorgiou, N.S.: Multiple solutions to a Dirichlet problem with $p$-Laplacian and nonlinearity depending on a parameter. Adv. Nonlinear Anal. 1(3), 257–275 (2012)MathSciNetMATHGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Motreanu, D., Motreanu, V.V., Papageorgiou, N.S.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)CrossRefMATHGoogle Scholar
  20. 20.
    Palais, R.S.: Homotopy theory of infinite dimensional manifolds. Topology 5(1), 1–16 (1966)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Papageorgiou, N.S., Rǎdulescu, V.D.: Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance. Appl. Math. Optim. 69(3), 393–430 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Papageorgiou, N.S., Rǎdulescu, V.D.: Neumann problems with indefinite and unbounded potential and concave terms. Proc. Am. Math. Soc. 143(11), 4803–4816 (2015)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Papageorgiou, N.S., Rǎdulescu, V.D.: Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities. Discret. Contin. Dyn. Syst. 35(10), 5003–5036 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Papageorgiou, N.S., Rǎdulescu, V.D.: Nonlinear nonhomogeneous Robin problems with superlinear reaction term. Adv. Nonlinear. Stud. 16(4), 737–764 (2016)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Papageorgiou, N.S., Winkert, P.: Resonant ($p$,2)-equations with concave terms. Appl. Anal. 94(2), 342–360 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Perera, K.: Multiplicity results for some elliptic problems with concave nonlinearities. J. Differ. Equ. 140(1), 133–141 (1997)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser, Basel (2007)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Nikolaos S. Papageorgiou
    • 1
  • Calogero Vetro
    • 2
  • Francesca Vetro
    • 3
  1. 1.Department of MathematicsNational Technical UniversityAthensGreece
  2. 2.Department of Mathematics and Computer ScienceUniversity of PalermoPalermoItaly
  3. 3.Department of Energy, Information Engineering and Mathematical ModelsUniversity of PalermoPalermoItaly

Personalised recommendations