Kelvin transform for Grushin operators and critical semilinear equations

Abstract

We study positive entire solutions $u=u(x,y)$ of the critical equation ${\displaystyle {\Delta }_{x}u+(\alpha +1){}^2\left|x\right|{}^{2\alpha }{\Delta }_{y}u=-u^{(Q+2)/(Q-2)}\hspace{1em}\text{in}{ℝ}^n={ℝ}^m\times {ℝ}^k,}$ where $(x,y)\in {ℝ}^m\times {ℝ}^k$, $\alpha >0$, 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 ${\displaystyle \mathrm{div}{}_x(p{\nabla }_{x}v)-qv=-p{v}^{(Q+2)/(Q-2)},\hspace{1em}\left|x\right|<1,}$ 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 $m\ge 3$ and $k=1$, problem (2) has a unique solution in the class of $x$-radial functions