## Duke Mathematical Journal

### Kelvin transform for Grushin operators and critical semilinear equations

#### Abstract

We study positive entire solutions $u = u(x,y)$ of the critical equation $\begin{equation} \label{sceriffo} \Delta_x u+{(\alpha+1)^2}|x|^{2\alpha} \Delta_y u=-u^{({Q+2})/({Q-2})} \textrm{in}\mathbb{R}^{n}= \mathbb{R}^m\times\mathbb{R}^k, \end{equation}$ where $(x,y)\in \mathbb{R}^m\times \mathbb{R}^k$, $\alpha>0$, and $Q=m+k(\alpha+1)$. 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 $\begin{equation} \label{pistolero} {\mathrm{div}\!_x}(p\nabla_xv )-q v=-p v^{({Q+2})/({Q-2})},|x|\lt 1,\end{equation}$ 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 $m\ge 3$ 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

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