Abstract
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.
Article PDF
Similar content being viewed by others
References
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory, Grundlehren Math. Wiss., vol. 314. Springer, Berlin (1996)
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)
Baouendi M.S.: Sur une Classe d’Operateurs Elliptiques Degeneres. Bull. Soc. Math. France 95, 45–87 (1967)
Barros-Neto J., Cardoso F.: Bessel integrals and fundamental solutions for a generalized Tricomi operator. J. Funct. Anal. 183, 472–497 (2001)
Bocea M., Mihăilescu M.: A Caffarelli–Kohn–Nirenberg inequality in Orlicz–Sobolev spaces and applications. Appl. Anal. 91, 1649–1659 (2012)
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)
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)
D’Ambrosio L.: Hardy inequalities related to Grushin type operators. Proc. Am. Math. Soc. 132, 725–734 (2004)
D’Ambrosio L.: Hardy-type inequalities related to degenerate elliptic differential operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. 4, 451–486 (2005)
D’Ambrosio L., Lucente S.: Nonlinear Liouville theorems for Grushin and Tricomi operators. J. Differ. Equ. 193, 511–541 (2003)
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)
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)
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)
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)
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)
Gossez J.P.: Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients. Trans. Am. Math. Soc. 190, 163–205 (1974)
Grushin V.V.: On a class of hypoelliptic operators. Math. USSR-Sb 12, 458–476 (1970)
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)
Kogoj A.E., Lanconelli E.: On semilinear \({\Delta_{\lambda}}\) -Laplace equations. Nonlinear Anal. Theory Methods Appl. 75, 4637–4649 (2012)
Lamperti J.W.: On the isometries of certain function-spaces. Pac. J. Math. 8, 459–466 (1958)
Monti R., Morbidelli D.: Kelvin transform for Grushin operator and critical semilinear equations. Duke Math. J. 131, 167–202 (2006)
Musielak, J.: Orlicz Spaces and Modular Speces. Academic Press, New York (1975)
Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker Inc., New York (1991)
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)
Struwe, M.: Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer, Heidelberg (1996)
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)
Tri N.M.: On Grushin’s equation. Math. Notes 63, 84–93 (1998)
Tricomi F.G.: Sulle Equazioni Lineari alle Derivate Parziali di 2o Ordine di Tipo Misto. Mem Lincei 14, 133–247 (1923)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Gheorghe Moroşanu on the occasion of his sixty-fifth anniversary
Rights and permissions
About this article
Cite this article
Mihăilescu, M., Stancu-Dumitru, D. & Varga, C. On the spectrum of a Baouendi–Grushin type operator: an Orlicz–Sobolev space setting approach. Nonlinear Differ. Equ. Appl. 22, 1067–1087 (2015). https://doi.org/10.1007/s00030-015-0314-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00030-015-0314-5