Existence of Continuous Eigenvalues for a Class of Parametric Problems Involving the \((p,2)\)-Laplacian Operator

  • Tilak BhattacharyaEmail author
  • Behrouz Emamizadeh
  • Amin Farjudian


We discuss a parametric eigenvalue problem, where the differential operator is of \((p,2)\)-Laplacian type. We show that, when \(p\neq 2\), the spectrum of the operator is a half line, with the end point formulated in terms of the parameter and the principal eigenvalue of the Laplacian with zero Dirichlet boundary conditions. Two cases are considered corresponding to \(p>2\) and \(p<2\), and the methods that are applied are variational. In the former case, the direct method is applied, whereas in the latter case, the fibering method of Pohozaev is used. We will also discuss a priori bounds and regularity of the eigenfunctions. In particular, we will show that, when the eigenvalue tends towards the end point of the half line, the supremum norm of the corresponding eigenfunction tends to zero in the case of \(p>2\), and to infinity in the case of \(p < 2\).


Fibering method Continuous eigenvalues \(p\)-Laplacian 

Mathematics Subject Classification (2010)

35J60 35P30 



The authors wish to thank the anonymous referee for their corrections and constructive suggestions.


  1. 1.
    Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Nodal solutions for \((p,2)\)-equations. Trans. Am. Math. Soc. 367(10), 7343–7372 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aris, R.: Mathematical Modelling Techniques. Research Notes in Mathematics, vol. 24. Pitman, London (1978) zbMATHGoogle Scholar
  3. 3.
    Bartolo, R., Candela, A., Salvatore, A.: On a class of superlinear \((p,q)\)-Laplacian type equations on \(\mathbb{R}^{N}\). J. Math. Anal. Appl. 438(1), 29–41 (2016) MathSciNetCrossRefzbMATHGoogle 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(4), 297–324 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chaves, M.F., Ercole, G., Miyagaki, O.H.: Existence of a nontrivial solution for a \((p,q)\)-Laplacian equation with \(p\)-critical exponent in \(\mathbb{R}^{N}\). Bound. Value Probl. 2014(1), 236 (2014). CrossRefzbMATHGoogle Scholar
  6. 6.
    Chaves, M.F., Ercole, G., Miyagaki, O.H.: Existence of a nontrivial solution for the \((p,q)\)-Laplacian in \(\mathbb{R}^{N}\) without the Ambrosetti-Rabinowitz condition. Nonlinear Anal. 114, 133–141 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, C., Bao, J., Song, H.: Multiple solutions for a class of fractional \((p,q)\)-Laplacian system in \(\mathbb{R}^{N}\). J. Math. Phys. 59(3), 031505 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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
  9. 9.
    DiBenedetto, E.: \(C^{1+\alpha}\) local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7(8), 827–850 (1983) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fãrcãşeanu, M., Mihãilescu, M., Stancu-Dumitru, D.: On the set of eigenvalues of some PDEs with homogeneous Neumann boundary condition. Nonlinear Anal. 116(0), 19–25 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fife, P.C.: Mathematical Aspects of Reacting and Diffusing Systems. Lecture Notes in Biomathematics. Springer, Berlin (1979) CrossRefzbMATHGoogle Scholar
  12. 12.
    Figueiredo, G.M.: Existence of positive solutions for a class of \(p\&q\) elliptic problems with critical growth on \(\mathbb{R}^{N}\). J. Math. Anal. Appl. 378(2), 507–518 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Figueiredo, G.M.: Existence and multiplicity of solutions for a class of \(p \& q\) elliptic problems with critical exponent. Math. Nachr. 286(11–12), 1129–1141 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gasiński, L., Papageorgiou, N.S.: Asymmetric \((p,2)\)-equations with double resonance. Calc. Var. Partial Differ. Equ. 56(3), 88 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001) zbMATHGoogle Scholar
  16. 16.
    Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968) Google Scholar
  17. 17.
    Lv, Y., Ou, Z.Q.: Existence of weak solutions for a class of \((p,q)\)-Laplacian systems. Bound. Value Probl. 2017(1), 1 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mercuri, C., Squassina, M.: Global compactness for a class of quasi-linear elliptic problems. Manuscr. Math. 140(1), 119–144 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mihailescu, M.: An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue. Commun. Pure Appl. Anal. 10, 701–708 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Papageorgiou, N.S., Rădulescu, V.D.: Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance. Appl. Math. Optim. 69(3), 393–430 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Papageorgiou, N.S., Rădulescu, V.D.: Noncoercive resonant \((p,2)\)-equations. Appl. Math. Optim. 76(3), 621–639 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Papageorgiou, N.S., Rădulescu, V.D., Repovs̆, D.D.: On a class of parametric \((p,2)\)-equations. Appl. Math. Optim. 75(2), 193–228 (2017). MathSciNetCrossRefGoogle Scholar
  23. 23.
    Papageorgiou, N.S., Rădulescu, V.D., Repovs̆, D.D.: Existence and multiplicity of solutions for resonant \((p,2)\)-equations. Adv. Nonlinear Stud. 18(1), 105–129 (2017) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Pohozaev, S.I.: The fibering method and its applications to nonlinear boundary value problem. Rend. Ist. Mat. Univ. Trieste 31(1–2), 235–305 (1999) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Radulescu, V., Chorfi, N.: Continuous spectrum for some classes of \((p,2)\)-equations with linear or sublinear growth. Miskolc Math. Notes 17(2), 817–826 (2016) MathSciNetzbMATHGoogle Scholar
  26. 26.
    Serrin, J.: On the strong maximum principle for quasilinear second order differential inequalities. J. Funct. Anal. 5(2), 184–193 (1970). MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Struwe, M.: Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4th edn. Springer, Berlin (2008) zbMATHGoogle Scholar
  28. 28.
    Vélin, J.: On an existence result for a class of \((p,q)\)-gradient elliptic systems via a fibering method. Nonlinear Anal. 75(16), 6009–6033 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Vélin, J.: Multiple solutions for a class of \((p, q)\)-gradient elliptic systems via a fibering method. Proc. R. Soc. Edinb., Sect. A 144, 363–393 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wu, M., Yang, Z.: Positive solutions for a class of \(p\)-\(q\)-Laplacian type equations with potentials in \(\mathbb{R}^{N}\). J. Abstr. Differ. Equ. Appl. 2(2), 93–109 (2012) MathSciNetzbMATHGoogle Scholar
  31. 31.
    Yin, H., Yang, Z.: A class of \(p\)-\(q\)-Laplacian type equation with concave-convex nonlinearities in bounded domain. J. Math. Anal. Appl. 382(2), 843–855 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Yin, H., Yang, Z.: Multiplicity of positive solutions to a \(p\)-\(q\)-Laplacian equation involving critical nonlinearity. Nonlinear Anal. 75(6), 3021–3035 (2012) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Tilak Bhattacharya
    • 1
    Email author
  • Behrouz Emamizadeh
    • 2
  • Amin Farjudian
    • 3
  1. 1.Department of MathematicsWestern Kentucky UniversityBowling GreenUSA
  2. 2.School of Mathematical SciencesUniversity of Nottingham Ningbo ChinaNingboP.R. China
  3. 3.School of Computer ScienceUniversity of Nottingham Ningbo ChinaNingboP.R. China

Personalised recommendations