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Asymptotic estimates and nonexistence results for critical problems with Hardy term involving Grushin-type operators

  • Annunziata LoiudiceEmail author
Article
  • 28 Downloads

Abstract

We provide the asymptotic behavior of solutions, at the singularity and at infinity, for a class of subelliptic Dirichlet problems with Hardy perturbation and critical nonlinearity of the type \(-\,{\mathcal {L}}_\alpha u -\mu \dfrac{\psi ^2}{d^2} u =K(z)|u|^{2^*-2}u\) in \(\varOmega \), where \({\mathcal {L}}_\alpha = \varDelta _x + |x|^{2\alpha }\varDelta _y\), \(\alpha >0\) is the so-called Grushin operator, \(\varOmega \) is an open subset of \({\mathbb {R}}^N\), \(0\in \varOmega \), d is the gauge norm naturally associated with \({\mathcal {L}}_\alpha \), \(\psi :=|\nabla _\alpha d|\), where \(\nabla _\alpha \) is the Grushin gradient, \(K\in L^\infty \) and \(0\le \mu < {\overline{\mu }}\), where \({\overline{\mu }}\) is the best Hardy constant for \({\mathcal {L}}_\alpha \). Furthermore, we establish some Pohozaev-type nonexistence results.

Keywords

Subelliptic critical problem Grushin operator Hardy potential Asymptotic behavior Pohozaev-type identity Nonexistence results 

Mathematics Subject Classification

35J70 35J75 35B40 

Notes

Acknowledgements

The author is a member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), and she is partially supported by the GNAMPA research Project 2018 “Metodi di analisi armonica e teoria spettrale per le equazioni dispersive.”

References

  1. 1.
    Beckner, W.: On the Grushin operator and hyperbolic symmetry. Proc. Am. Math. Soc. 129(4), 1233–1246 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bieske, T., Gong, J.: The p-Laplace equation on a class of Grushin-type spaces. Proc. Am. Math. Soc. 134(12), 3585–3594 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cao, D., Han, P.: Solutions to critical elliptic equations with multi-singular inverse square potentials. J. Differ. Equ. 224, 332–372 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D’Ambrosio, L.: Hardy inequalities related to Grushin-type operators. Proc. Am. Math. Soc. 132(3), 725–734 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    D’Ambrosio, L., Lucente, S.: Nonlinear Liouville theorems for Grushin and Tricomi operators. J. Differ. Equ. 193(2), 511–541 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dou, J., Niu, P.: Hardy–Sobolev type inequalities for generalized Baouendi–Grushin operators. Miskolc Math. Notes 8(1), 73–77 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Federer, H.: Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969)Google Scholar
  8. 8.
    Felli, V., Ferrero, A., Terracini, S.: Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential. J. Eur. Math. Soc. 13, 119–174 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Felli, V., Schneider, M.: A note on regularity of solutions to degenerate elliptic equations of Caffarelli–Kohn–Nirenberg type. Adv. Nonlinear Stud. 3(4), 431–443 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Felli, V., Schneider, M.: Compactness and existence results for degenerate critical elliptic equations. Commun. Contemp. Math. 7, 37–73 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Franchi, B., Lanconelli, E.: Une metrique associée à une classe d’opérateurs elliptiques dégénérés. In: Proceedings of the Meeting “Linear Partial and Pseudodifferential Operators”, Rend. Sem. Mat. Univ. e Politec. Torino (1982), pp. 105–114 (1982)Google Scholar
  12. 12.
    Franchi, B., Lanconelli, E.: Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients. Ann. Sc. Norm. Sup Pisa Cl. Sci. 10(4), 523–541 (1983)zbMATHGoogle Scholar
  13. 13.
    Franchi, B., Lanconelli, E.: An embedding theorem for Sobolev spaces related to non-smooth vector fields and Harnack inequality. Commun. Partial Differ. Equ. 9, 1237–1264 (1984)CrossRefzbMATHGoogle Scholar
  14. 14.
    Garofalo, N.: Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension. J. Differ. Equ. 104, 117–146 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Garofalo, N., Lanconelli, E.: Existence and nonexistence results for semilinear equations on the Heisenberg group. Indiana Univ. Math. J. 41, 71–98 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Garofalo, N., Vassilev, D.: Strong unique continuation properties of generalized Baouendi–Grushin operators. Comm. Partial Differ. Equ. 32(4–6), 643–663 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Han, P.: Asymptotic behavior of solutions to semilinear elliptic equations with Hardy potential. Proc. Am. Math Soc. 135, 365–372 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jannelli, E.: The role played by space dimension in elliptic critical problems. J. Differ. Equ. 156, 407–426 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kogoj, A.E., Lanconelli, E.: X-elliptic operators and X-control distances. Contributions in honor of the memory of Ennio De Giorgi. Ric. Mat. 49(Suppl.), 223–243 (2000)zbMATHGoogle Scholar
  20. 20.
    Kogoj, A.E., Lanconelli, E.: On semilinear \(\Delta _\lambda \)-Laplace equation. Nonlinear Anal. 75, 4637–4649 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kogoj, A.E., Sonner, S.: Hardy type inequalities for \(\Delta _\lambda \)-Laplacians. Complex Var. Elliptic Equ. 61(3), 422–442 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kombe, I.: On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi–Grushin operators. Discrete Contin. Dyn. Syst. 33(11–12), 5167–5176 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lascialfari, F., Pardo, D.: Compact embedding of a degenerate Sobolev space and existence of entire solutions to a semilinear equation for a Grushin-type operator. Rend. Sem. Mat. Univ. Padova 107, 139–152 (2002)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Loiudice, A.: Sobolev inequalities with remainder terms for sublaplacians and other subelliptic operators. NoDEA Nonlinear Differ. Equ. Appl. 13, 119–136 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Loiudice, A.: Asymptotic behaviour of solutions for a class of degenerate elliptic critical problems. Nonlinear Anal. 70(8), 2986–2991 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Loiudice, A.: \(L^p\)-weak regularity and asymptotic behavior of solutions for critical equations with singular potentials on Carnot groups. NoDEA Nonlinear Differ. Equ. Appl. 17, 575–589 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Loiudice, A.: Critical growth problems with singular nonlinearities on Carnot groups. Nonlinear Anal. 126, 415–436 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Loiudice, A.: Local behavior of solutions to subelliptic problems with Hardy potential on Carnot groups. Mediterr. J. Math. 15(3), 20 (2018) (Art. 81)Google Scholar
  29. 29.
    Loiudice, A.: Optimal decay of \(p\)-Sobolev extremals on Carnot groups. J. Math. Anal. Appl. 470(1), 619–631 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Luyen, D.T., Tri, N.M.: Existence of infinitely many solutions for semilinear degenerate Schrödinger equations. J. Math. Anal. Appl. 461(2), 1271–1286 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mokrani, H.: Semilinear subelliptic equations on the Heisenberg group with a singular potential. Commun. Pure Appl. Math. 8, 1619–1636 (2009)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Monti, R.: Sobolev inequalities for weighted gradients. Commun. Partial Differ. Equ. 31, 1479–1504 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Monti, R., Morbidelli, D.: Kelvin transform for Grushin operators and critical semilinear equations. Duke Math. J. 131, 167–202 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Monticelli, D.: Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators. J. Eur. Math. Soc. 12, 611–654 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Pohozaev, S.I.: Eigenfunctions of the equation \(\Delta u +\lambda f(u)=0\). Dokl. Akad. Nauk. SSSR 165(1), 33–36 (1965)MathSciNetGoogle Scholar
  36. 36.
    Smets, D.: Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities. Trans. Am. Math. Soc. 375, 2909–2938 (2005)CrossRefzbMATHGoogle Scholar
  37. 37.
    Terracini, S.: On positive entire solutions to a class of equations with singular coefficient and critical exponent. Adv. Differ. Equ. 1, 241–264 (1996)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Thuy, P.T., Tri, N.M.: Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations. NoDEA Nonlinear Differ. Equ. Appl. 19, 279–298 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Tri, N.M.: On the Grushin equation. Mat. Zamet. 63(1), 95–105 (1998)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Wang, C., Wang, Q., Yang, J.: On the Grushin critical problem with a cylindrical symmetry. Adv. Differ. Equ. 20(1–2), 77–116 (2015)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Yang, Q., Su, D., Kong, Y.: Improved Hardy inequalities for Grushin operators. J. Math. Anal. Appl. 424, 321–343 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di BariBariItaly

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