Duke Mathematical Journal
- Duke Math. J.
- Volume 131, Number 1 (2006), 167-202.
Kelvin transform for Grushin operators and critical semilinear equations
We study positive entire solutions of the critical equation where , , and . In the first part of the article, exploiting the invariance of the equation with respect to a suitable conformal inversion, we prove a “spherical symmetry result for solutions”. In the second part, we show how to reduce the dimension of the problem using a hyperbolic symmetry argument. Given any positive solution of (1), after a suitable scaling and a translation in the variable , the function satisfies the equation with a mixed boundary condition. Here, and are appropriate radial functions. In the last part, we prove that if , the solution of (2) is unique and that for and , problem (2) has a unique solution in the class of -radial functions
Duke Math. J., Volume 131, Number 1 (2006), 167-202.
First available in Project Euclid: 15 December 2005
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Monti, Roberto; Morbidelli, Daniele. Kelvin transform for Grushin operators and critical semilinear equations. Duke Math. J. 131 (2006), no. 1, 167--202. doi:10.1215/S0012-7094-05-13115-5. https://projecteuclid.org/euclid.dmj/1134666124