Multiple Solutions for \({\varvec{(p,2)}}\)-Equations with Resonance and Concave Terms

  • Leszek GasińskiEmail author
  • Nikolaos S. Papageorgiou
Open Access


We consider parametric Dirichlet problems driven by the sum of a p-Laplacian (\(p>2\)) and a Laplacian ((p, 2)-equation) and with a reaction term which exhibits competing nonlinearities. We prove two multiplicity theorems. In the first the competing terms are not decoupled, the dependence on the parameter is not necessarily linear and the reaction term has a general polynomial growth, possibly supercritical. We produce three nontrivial solutions for small values of the parameter. We provide sign information for all solutions (two of constant sign and the third nodal). Then we decouple the competing nonlinearities and allow for resonance to occur at \(\pm \,\infty \). We produce six nontrivial smooth solutions for small values of the parameter. We provide sign information for five of these solutions.


Competing nonlinearities concave term constant sign and nodal solutions multiplicity theorems critical groups resonance 

Mathematics Subject Classification

35J20 35J60 35J92 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), 70 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Nodal solutions for \((p,2)\)-equations. Trans. Am. Math. Soc. 367, 7343–7372 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Benci, V., D’Avenia, P., Fortunato, D., Pisani, L.: Solitons in several space dimensions: Derrick’s problem and infinitely many solutions. Arch. Ration. Mech. Anal. 154, 297–324 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chang, K.-C.: Methods in Nonlinear Analysis. Springer, Berlin (2005)zbMATHGoogle Scholar
  6. 6.
    Cherfils, L., Il’yasov, Y.: On the stationary solutions of generalized reaction diffusion equations with \(p\)&\(q\)-Laplacian. Commun. Pure Appl. Anal. 4, 9–22 (2005)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cingolani, S., Degiovanni, M.: Nontrivial solutions for \(p\)-Laplace equations with right hand side having \(p\)-linear growth at infinity. Commun. Partial Differ. Equ. 30, 1191–1203 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Diaz, J.I., Saa, J.E.: Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. C. R. Acad. Sci. Paris Sér. I Math. 305, 521–524 (1987)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fadell, E.R., Rabinowitz, P.H.: Generalized cohomological index theories for Lie group actions with an applications to bifurcation questions for Hamiltonian systems. Invent. Math. 45, 139–174 (1978)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Filippakis, M.E., Papageorgiou, N.S.: Multiple constant sign and nodal solutions for nonlinear elliptic equations with the \(p\)-Laplacian. J. Differ. Equ. 245(7), 1883–1922 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gasiński, L., Klimczak, L., Papageorgiou, N.S.: Nonlinear Dirichlet problems with no growth restriction on the reaction. Z. Anal. Anwend. 36(2), 209–238 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gasiński, L., Papageorgiou, N.S.: Nonlinear Analysis. Mathematical Analysis and Applications, 9. Chapman & Hall, Boca Raton, FL (2006)zbMATHGoogle Scholar
  13. 13.
    Gasiński, L., Papageorgiou, N.S.: Nodal and multiple constant sign solutions for resonant \(p\)-Laplacian equations with a nonsmooth potential. Nonlinear Anal. 71(11), 5747–5772 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gasiński, L., Papageorgiou, N.S.: Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential. Set-Valued Var. Anal. 20, 417–443 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gasiński, L., Papageorgiou, N.S.: Multiplicity of positive solutions for eigenvalue problems of \((p,2)\)-equations. Bound. Value Probl. 2012(152), 1–17 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gasiński, L., Papageorgiou, N.S.: On generalized logistic equations with a non-homogeneous differential operator. Dyn. Syst. Int. J. 29, 190–207 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gasiński, L., Papageorgiou, N.S.: A pair of positive solutions for \((p, q)\)-equations with combined nonlinearities. Commun. Pure Appl. Anal. 13(1), 203–215 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gasiński, L., Papageorgiou, N.S.: Dirichlet \((p,q)\)-equations at resonance. Discrete Contin. Dyn. Syst. Ser. A 34(5), 2037–2060 (2014)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gasiński, L., Papageorgiou, N.S.: Nonlinear elliptic equations with a jumping reaction. J. Math. Anal. Appl. 443(2), 1033–1070 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Gasiński, L., Papageorgiou, N.S.: Multiplicity theorems for \((p,2)\)-equations. J. Nonlinear Convex Anal. 18(7), 1297–1323 (2017)MathSciNetGoogle Scholar
  21. 21.
    Guedda, M., Véron, L.: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 13, 879–902 (1989)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Vol. I. Theory, Mathematics and its Applications, Vol. 419. Kluwer Academic Publishers, Dordrecht (1997)Google Scholar
  23. 23.
    Ladyzhenskaya, O.A., Uraltseva, N.: Linear and Quasilinear Elliptic Equations. Mathematics in Science and Engineering, vol. 46. Academic Press, New York (1968)Google Scholar
  24. 24.
    Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Liang, Z., Su, J.: Multiple solutions for semilinear elliptic boundary value problems with double resonance. J. Math. Anal. Appl. 354(1), 147–158 (2009)MathSciNetCrossRefGoogle Scholar
  26. 26.
    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
  27. 27.
    Mugnai, D., Papageorgiou, N.S.: Wang’s multiplicity result for superlinear \((p, q)\)-equations without the Ambrosetti–Rabinowitz condition. Trans. Am. Math. Soc. 366, 4919–4937 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Papageorgiou, N.S., Rădulescu, V.D.: Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance. Appl. Math. Optim. 69, 393–430 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Papageorgiou, N.S., Rădulescu, V.D.: Resonant \((p,2)\)-equations with asymmetric reaction. Anal. Appl. 13(5), 481–506 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Papageorgiou, N.S., Rădulescu, V.D.: Noncoercive resonant \((p,2)\)-equations. Appl. Math. Optim. 76(3), 621–639 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Papageorgiou, N.S., Smyrlis, G.: On nonlinear nonhomogeneous resonant Dirichlet equations. Pac. J. Math. 264, 421–453 (2013)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Papageorgiou, N.S., Winkert, P.: Resonant \((p,2)\)-equations with concave terms. Appl. Anal. 94(2), 342–360 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Pucci, P., Serrin, J.: The Maximum Principle. Progress in Nonlinear Differential Equations and their Applications, 73. Birkhäuser Verlag, Basel (2007)zbMATHGoogle Scholar
  34. 34.
    Sun, M.: Multiplicity of solutions for a class of the quasilinear elliptic equations at resonance. J. Math. Anal. Appl. 386, 661–668 (2012)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Sun, M., Zhang, M., Su, J.: Critical groups at zero and multiple solutions for a quasilinear elliptic equation. J. Math. Anal. Appl. 428, 696–712 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Troianiello, G.M.: Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York (1987)CrossRefGoogle Scholar
  37. 37.
    Yang, D., Bai, C.: Nonlinear elliptic problem of \(2\)-\(q\)-Laplacian type with asymmetric nonlinearities. Electron. J. Differ. Equ. 170, 1–13 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsPedagogical University of CracowKrakówPoland
  2. 2.Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland
  3. 3.Department of MathematicsNational Technical UniversityAthensGreece

Personalised recommendations