Acta Mathematica Sinica

, Volume 22, Issue 6, pp 1695–1704 | Cite as

Some Results on Subelliptic Equations

ORIGINAL ARTICLES
First Online:
Received:
Revised:
Accepted:

Abstract

In this paper, we consider the principal eigenvalue problem for Hormander’s Laplacian on R n , and we find a comparison principle for such principal eigenvalues. We also study a related semi-linear sub-elliptic equation in the whole R n and prove that, under a suitable condition, we have infinitely many positive solutions of the problem.

Keywords

principal eigenvalue Hormander’s Laplacian positive solution 

MR (2000) Subject Classification

35J65 35H20 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Citti, G., Garofalo, N., Lanconelli, E.: Harnack inequality for sum of square of vector .elds plus a potential. Amer. J. Math., 115, 699–734 (1993)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Gromov, M.: Carnot–Caratheodory spaces seen from within, 79–339, in Sub-Riemannian Geometry, Eds. A. Bellaiche, J. J. Risler, Birkhauser–Verlag, Basel, 1996Google Scholar
  3. 3.
    Xu, C. J.: Regularity for quasilinear second order subelliptic equations. Comm. Pure Appl. Math., 45, 77–96 (1992)MATHMathSciNetGoogle Scholar
  4. 4.
    Jerison, D., Lee, J. M.: The Yamabe problem on CR manifolds. J. Diff. Geom., 25, 167–97 (1987)MATHMathSciNetGoogle Scholar
  5. 5.
    Jost, J., Xu, C. J.: Subelliptic harmonic maps. Trans. AMS, 350, 4633–49 (1998)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Birindelli, I., Prajapat, J.: Nonlinear Liouville theorems in the Heisenberg group via the moving plane method. Comm. Partial Differential Equations, 24, 1875–1890 (1999) MATHMathSciNetGoogle Scholar
  7. 7.
    Garofalo, N., Vassilev, D.: Symmetry properties of positive entire solutions of Yamabe-type equations on the groups of Heisenberg type. Duke Math. J., 106, 411–448 (2001)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Lu, G., Wei, J.: On positive entire solutions to the Yamabe-type problem on the Heisenberg and strati.ed groups. Electron. Res. Announc. Amer. Math. Soc., 3, 83–89 (1997)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Nagel, A., Stein, E. M., Wainger, S.: Balls and metrics de.ned by vector .elds I: basic properties. Acta Math., 155, 103–147 (1985)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Hajlasz, P., Koskela, P.: Sobolev met Poincare. Mem. Amer. Math. Soc., 145, 688 (2000)MathSciNetGoogle Scholar
  11. 11.
    Hormander, L.: Hypoelliptic second order di.erential equations. Acta Math., 119, 141–171 (1967)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bony, J. M.: Principe du maximum, inegalite de Harnack et unicite du probleme de Chauchy pour les operateurs elliptiques degeneres. Ann. Inst. Fourier, 19, 277–304 (1969)MATHMathSciNetGoogle Scholar
  13. 13.
    Garofalo, N., Nhieu, D.: Isoperimetric and Sobolev inequalities for Carnot–Caratheodory spaces and existences of minimal surfaces. Comm. Pure Appl. Math., 49, 1084–1144 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Garofalo, N.: Unique continuation for the square of vector fields plus a potential. J. Diff. Equations, 104, 117–146 (1993)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math., 33, 199–211 (1980)MATHMathSciNetGoogle Scholar
  16. 16.
    Liu, W. A., Lu, G.: Viscosity solutions of monotonous functional parabolic PDE. Acta Mathematica Sinica, English Series, 20(4), 739–748 (2004)MATHMathSciNetGoogle Scholar
  17. 17.
    Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 1977Google Scholar
  18. 18.
    Berysticki, H., Nirenberg, L., Varadhan, S. R. S.: Principal eigenvalue and maximum principle for second order ellptic operators in general domains. Comm. Pure Appl. Math., 47, 47–94 (1994)MathSciNetGoogle Scholar
  19. 19.
    Jin, Z. R.: Principal eigenvalues with inde.nite weight functions. Trans. AMS, 349, 1945–1959 (1997)MATHCrossRefGoogle Scholar
  20. 20.
    Birindelli, I., Capuzzo Dolcetta, I., Curtri, A.: Infinite semi-linear equations on the Heisenberg Group: a priori bounds and existence. Comm. Partial Differential Equations, 23, 1123–1157 (1998)MATHMathSciNetGoogle Scholar
  21. 21.
    Lin, F. H.: On the elliptic equation –∂(a ij (x)∂j u)+k(x)u – K(x)u p = 0. Proc. AMS, 95, 219–226 (1985)MATHCrossRefGoogle Scholar
  22. 22.
    Ma, L.: Mountain pass on a closed convex set. J. Math. Anal. and Applications, 205, 531–536 (1997)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Du, Y., Ma, L.: Logistic type equations on R N by a squeezing method involving boundary blow-up solutions. Journal LMS, 64, 107–124 (2001)MATHGoogle Scholar
  24. 24.
    Ni, W. M.: On the elliptic equation Δu + K(x)u (N+2)/(n-2) = 0, its generalizations, and applications in Geometry. Indiana University Math. J., 31, 493–529 (1982)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijing 100084P. R. China

Personalised recommendations