Kelvin transform for Grushin operators and critical semilinear equations
The authors consider the positive entire solutions \(u(x,y)\) of the critical semilinear Grushin type semilinear problem \[ \Delta_x u + (\alpha + 1)^2 \delta_y u = -u^{\frac{Q+2}{Q-2}} \] \((x,y) \in \mathbb R^n = \mathbb R^m \times \mathbb R^k\), \(\alpha >0\) and \(Q = m+k(\alpha +1)\). The paper is divided into different parts. In the first part the ``spherical symmetry'' of the solutions is proved using the invariance of the equation under a suitable conformal inversion. In the second part a reduction of the dimension of the problem is performed by a hyperbolic symmetry argument. The authors prove that, after a suitable scaling and translation in the variable \(y\), the function \(v(x) = u(x,0)\) satisfies the equation \[ \text{div}_x (p \nabla_x v) - qv = - pv^{\frac{Q+2}{Q-2}} \] \(| x| < 1\), with a mixed boundary condition, where \(p,q\) are suitable radial functions which degenerate for \(| x| =1\). In the third and last part the uniqueness of the solution of the last problem is proved for \(m=k=1\) and, in the class of radial functions, for \(m \geq 3\) and \(k=1\).
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