Existence result for the CR-Yamabe equation

Vittorio Martino

Abstract


In this note we will prove that the CR-Yamabe equation has infinitely many changing-sign solutions. The problem is variational but the associated functional does not satisfy the Palais-Smale compactness condition; by mean of a suitable group action we will define a subspace on which we can apply the minimax argument of Ambrosetti-Rabinowitz. The result solves a question left open from the classification results of positive solutions by Jerison-Lee in the '80s.


Keywords


Reeb vector field; mountain-pass with symmetry

Full Text:

PDF (English)

References


A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349-381.

A. Bahri, S. Chanillo, The difference of topology at infinity in changing-sign Yamabe problems on S3 (the case of two masses), Comm. Pure Appl. Math. 54 (2001), no. 4, 450-478.

A. Bahri, Y. Xu, Recent progress in conformal geometry, ICP Advanced Texts in Mathematics, 1. Imperial College Press, London, 2007.

D.E. Blair, Riemannian geometry of contact and symplectic manifolds. Second edition, Progress in Mathematics, 203. Birkhauser Boston, Inc., Boston, MA, 2010.

L.A. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271-297.

M. del Pino, M. Musso, F. Pacard, A. Pistoia, Torus action on Sn and sign changing solutions for conformally invariant equations, preprint.

M. del Pino, M. Musso, F. Pacard, A. Pistoia, Large energy entire solutions for the Yamabe equation, to appear on Journal of Dierential Equations.

W.Y. Ding, On a conformally invariant elliptic equation on Rn, Comm. Math. Phys. 107 (1986), no. 2, 331-335.

G.B. Folland, E.M. Stein, Estimates for the @b complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522.

D. Jerison, J.M. Lee, The Yamabe problem on CR manifolds, J. Dierential Geom. 25 (1987), no. 2, 167-197.

D. Jerison, J.M. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), no. 1, 1-13.

A. Maalaoui, V. Martino, Changing-sign solutions for the CR-Yamabe equation, Dierential and Integral Equations, Volume 25, Numbers 7-8, (2012), 601-609

R.S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), no. 1, 19-30.

N. Saintier, Changing sign solutions of a conformally invariant fourth-order equation in the Euclidean space, Comm. Anal. Geom. 14 (2006), no. 4, 613624.

Y.B. Zhang, The contact Yamabe flow on K-contact manifolds, Sci. China Ser. A 52 (2009), no. 8, 1723-1732.




DOI: 10.6092/issn.2240-2829/4017

Refbacks

  • There are currently no refbacks.


Copyright (c) 2013 Vittorio Martino

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 Unported License.

Creative Commons License 

ISSN 2240-2829
Registration at Bologna Law Court no. 8138 on 24th November, 2010

The journal is hosted and mantained by ABIS-AlmaDL [privacy]