Positive solutions of anisotropic Yamabe-type equations in $\mathbb {R}^n$
HTML articles powered by AMS MathViewer
- by Roberto Monti and Daniele Morbidelli PDF
- Proc. Amer. Math. Soc. 136 (2008), 4295-4304 Request permission
Abstract:
We study entire positive solutions to the partial differential equation in $\mathbb {R}^{n}$, \[ \Delta _{x}u + (\alpha + 1)^2 |x|^{2\alpha } \Delta _{y}u = - |x|^{2\alpha } u^{\frac {n+2}{n-2}}, \] where $x\in \mathbb {R}^2$, $y\in \mathbb {R}^{n-2}$, $n\geq 3$ and $\alpha > 0$. We classify positive solutions with second order derivatives satisfying a suitable growth near the set $x=0$.References
- Richard Beals, Peter Greiner, and Bernard Gaveau, Green’s functions for some highly degenerate elliptic operators, J. Funct. Anal. 165 (1999), no. 2, 407–429. MR 1698952, DOI 10.1006/jfan.1999.3421
- William Beckner, On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc. 129 (2001), no. 4, 1233–1246. MR 1709740, DOI 10.1090/S0002-9939-00-05630-6
- Bruno Franchi and Ermanno Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 4, 523–541. MR 753153
- Nicola Garofalo and Dimiter Vassilev, Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type, Duke Math. J. 106 (2001), no. 3, 411–448. MR 1813232, DOI 10.1215/S0012-7094-01-10631-5
- Nicola Garofalo and Dimiter Vassilev, Strong unique continuation properties of generalized Baouendi-Grushin operators, Comm. Partial Differential Equations 32 (2007), no. 4-6, 643–663. MR 2334826, DOI 10.1080/03605300500532905
- V. V. Grushin, On a class of hypoelliptic operators, Math. USSR Sbornik 12 (1970), 458–476.
- YanYan Li and Lei Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math. 90 (2003), 27–87. MR 2001065, DOI 10.1007/BF02786551
- Daniela Lupo and Kevin R. Payne, Conservation laws for equations of mixed elliptic-hyperbolic and degenerate types, Duke Math. J. 127 (2005), no. 2, 251–290. MR 2130413, DOI 10.1215/S0012-7094-04-12722-8
- Roberto Monti, Sobolev inequalities for weighted gradients, Comm. Partial Differential Equations 31 (2006), no. 10-12, 1479–1504. MR 2273962, DOI 10.1080/03605300500361594
- Roberto Monti and Daniele Morbidelli, Kelvin transform for Grushin operators and critical semilinear equations, Duke Math. J. 131 (2006), no. 1, 167–202. MR 2219239, DOI 10.1215/S0012-7094-05-13115-5
- D. Morbidelli, Liouville theorem, conformally invariant cones and umbilical surfaces for Grushin-type metrics, Israel J. Math. (to appear).
- Morio Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry 6 (1971/72), 247–258. MR 303464
- J. Prajapat and G. Tarantello, On a class of elliptic problems in ${\Bbb R}^2$: symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 4, 967–985. MR 1855007, DOI 10.1017/S0308210500001219
Additional Information
- Roberto Monti
- Affiliation: Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste, 63, 35121 Padova, Italy
- Email: monti@math.unipd.it
- Daniele Morbidelli
- Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato, 5, 40127 Bologna, Italy
- Email: morbidel@dm.unibo.it
- Received by editor(s): October 19, 2007
- Published electronically: June 11, 2008
- Communicated by: Matthew J. Gursky
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 4295-4304
- MSC (2000): Primary 35J60; Secondary 35J70
- DOI: https://doi.org/10.1090/S0002-9939-08-09579-8
- MathSciNet review: 2431043