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15 January 2006 Kelvin transform for Grushin operators and critical semilinear equations
Roberto Monti, Daniele Morbidelli
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Duke Math. J. 131(1): 167-202 (15 January 2006). DOI: 10.1215/S0012-7094-05-13115-5

Abstract

We study positive entire solutions u=u(x,y) of the critical equation Δxu+(α+1)2|x|2αΔyu=-u(Q+2)/(Q-2)inn=m×k, where (x,y)m×k, α>0, and Q=m+k(α+1). 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 u of (1), after a suitable scaling and a translation in the variable y, the function v(x)=u(x,0) satisfies the equation divx(pxv)-qv=-pv(Q+2)/(Q-2),|x|<1, with a mixed boundary condition. Here, p and q are appropriate radial functions. In the last part, we prove that if m=k=1, the solution of (2) is unique and that for m3 and k=1, problem (2) has a unique solution in the class of x-radial functions

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Roberto Monti. Daniele Morbidelli. "Kelvin transform for Grushin operators and critical semilinear equations." Duke Math. J. 131 (1) 167 - 202, 15 January 2006. https://doi.org/10.1215/S0012-7094-05-13115-5

Information

Published: 15 January 2006
First available in Project Euclid: 15 December 2005

zbMATH: 1094.35036
MathSciNet: MR2219239
Digital Object Identifier: 10.1215/S0012-7094-05-13115-5

Subjects:
Primary: 35H20
Secondary: 34B15

Rights: Copyright © 2006 Duke University Press

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Vol.131 • No. 1 • 15 January 2006
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