On the spectrum of a Baouendi–Grushin type operator: an Orlicz–Sobolev space setting approach

  • Mihai MihăilescuEmail author
  • Denisa Stancu-Dumitru
  • Csaba Varga


We show that the spectrum of a nonhomogeneous Baouendi–Grushin type operator subject with a homogeneous Dirichlet boundary condition is exactly the interval \({(0,\infty)}\) . This is in sharp contrast with the situation when we deal with the “classical” Baouendi–Grushin operator (i.e., an operator of type \({-\Delta_x-|x|^{\xi}\Delta_y}\)) when the spectrum is an increasing and unbounded sequence of positive real numbers. Our proofs rely on a symmetric mountain-pass argument due to Kajikiya. In addition, we can show that for each eigenvalue there exists a sequence of eigenfunctions converging to zero.


Baouendi–Grushin type operator Orlicz–Sobolev space Hypoelliptic equation Nonlinear eigenvalue problem Variational methods 

Mathematics Subject Classification

47J10 47J30 49R05 35H10 46E30 


  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)Google Scholar
  2. 2.
    Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory, Grundlehren Math. Wiss., vol. 314. Springer, Berlin (1996)Google Scholar
  3. 3.
    Anh C.T., Hung P.Q., Ke T.D., Phong T.T.: Global attractor for a semilinear parabolic equation involving Grushin operatot. Electron. J. Differ. Equ. 32, 1–11 (2008)MathSciNetGoogle Scholar
  4. 4.
    Baouendi M.S.: Sur une Classe d’Operateurs Elliptiques Degeneres. Bull. Soc. Math. France 95, 45–87 (1967)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Barros-Neto J., Cardoso F.: Bessel integrals and fundamental solutions for a generalized Tricomi operator. J. Funct. Anal. 183, 472–497 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bocea M., Mihăilescu M.: A Caffarelli–Kohn–Nirenberg inequality in Orlicz–Sobolev spaces and applications. Appl. Anal. 91, 1649–1659 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Clément P.H, Garía-Huidobro M., Manáservich R., Schmitt K.: Mountain pass type solutons for quasilinear elliptic equations. Calc. Var. PDEs 11, 33–62 (2000)zbMATHCrossRefGoogle Scholar
  8. 8.
    Clément P.H, de Pagter B., Sweers G., de Thélin F.: Existence of solutions to a semilinear elliptic system through Orlicz–Sobolev spaces. Mediterr. J. Math. 1, 241–267 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    D’Ambrosio L.: Hardy inequalities related to Grushin type operators. Proc. Am. Math. Soc. 132, 725–734 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    D’Ambrosio L.: Hardy-type inequalities related to degenerate elliptic differential operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. 4, 451–486 (2005)zbMATHMathSciNetGoogle Scholar
  11. 11.
    D’Ambrosio L., Lucente S.: Nonlinear Liouville theorems for Grushin and Tricomi operators. J. Differ. Equ. 193, 511–541 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Franchi, B., Lanconelli, E.: Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10(4), 523–541 (1983)Google Scholar
  13. 13.
    Franchi B., Lanconelli E.: An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality. Commun. Partial Differ. Equ. 9(13), 1237–1264 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Franchi, B., Lanconelli, E.: Une métrique associée à une classe d’opérateurs elliptiques dégénérés. (French) [A metric associated with a class of degenerate elliptic operators]. In: Conference on Linear Partial and Pseudodifferential Operators (Torino, 1982). Rend. Sem. Mat. Univ. Politec. Torino 1983, pp. 105–114 (1984) (special issue)Google Scholar
  15. 15.
    Fukagai N., Ito M., Narukawa K.: Positive solutions of quasilinear elliptic equations with critical Orlicz–Sobolev nonlinearity on \({{\mathbb{R}}^N}\) . Funkcial. Ekvac. 49, 235–267 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Garcia-Huidobro M., Le V.K., Manásevich R., Schmitt K.: On principal eigennvalues for quasilinear elliptic differential operators: an Orlicz–Sobolev setting. NoDEA Nonlinear Differ. Equ. Appl. 6, 207–225 (1999)zbMATHCrossRefGoogle Scholar
  17. 17.
    Gossez J.P.: Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients. Trans. Am. Math. Soc. 190, 163–205 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Grushin V.V.: On a class of hypoelliptic operators. Math. USSR-Sb 12, 458–476 (1970)CrossRefGoogle Scholar
  19. 19.
    Kajikiya R.: A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. J. Funct. Anal. 225, 352–370 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Kogoj A.E., Lanconelli E.: On semilinear \({\Delta_{\lambda}}\) -Laplace equations. Nonlinear Anal. Theory Methods Appl. 75, 4637–4649 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Lamperti J.W.: On the isometries of certain function-spaces. Pac. J. Math. 8, 459–466 (1958)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Monti R., Morbidelli D.: Kelvin transform for Grushin operator and critical semilinear equations. Duke Math. J. 131, 167–202 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Musielak, J.: Orlicz Spaces and Modular Speces. Academic Press, New York (1975)Google Scholar
  24. 24.
    Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker Inc., New York (1991)Google Scholar
  25. 25.
    Sawyer, E.T., Wheeden, R.L.: Hölder continuity of weak solutions to subelliptic equations with rough coefficients. Mem. Am. Math. Soc. 180(847), x+157 (2006)Google Scholar
  26. 26.
    Struwe, M.: Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer, Heidelberg (1996)Google Scholar
  27. 27.
    Thuy P.T., Tri N.M.: Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations. Nonlinear Differ. Equ. Appl. 19, 279–298 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Tri N.M.: On Grushin’s equation. Math. Notes 63, 84–93 (1998)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tricomi F.G.: Sulle Equazioni Lineari alle Derivate Parziali di 2o Ordine di Tipo Misto. Mem Lincei 14, 133–247 (1923)Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Mihai Mihăilescu
    • 1
    • 2
    Email author
  • Denisa Stancu-Dumitru
    • 3
  • Csaba Varga
    • 3
  1. 1.Department of MathematicsUniversity of CraiovaCraiovaRomania
  2. 2.Research Group of the Project PN-II-ID-PCE-2012-4-0021“Simion Stoilow” Institute of Mathematics of the Romanian AcademyBucharestRomania
  3. 3.Department of MathematicsBabeş-Bolyai UniversityCluj-NapocaRomania

Personalised recommendations