Duke Mathematical Journal

Kelvin transform for Grushin operators and critical semilinear equations

Roberto Monti and Daniele Morbidelli

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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

Article information

Source
Duke Math. J., Volume 131, Number 1 (2006), 167-202.

Dates
First available in Project Euclid: 15 December 2005

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1134666124

Digital Object Identifier
doi:10.1215/S0012-7094-05-13115-5

Mathematical Reviews number (MathSciNet)
MR2219239

Zentralblatt MATH identifier
1094.35036

Subjects
Primary: 35H20: Subelliptic equations
Secondary: 34B15: Nonlinear boundary value problems

Citation

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


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