Kelvin transform for Grushin operators and semilinear critical equations

We study positive entire solutions u = u(x, y) of the critical equation xu + (α + 1)2|x|2αyu = −u(Q+2)/(Q−2) in Rn = Rm × Rk, (1) where (x, y) ∈ Rm ×Rk,α > 0, and Q = m+k(α+1). In the first part of the article, exploiting the invariance of the equationwith 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 (p∇xv) − qv = −pv(Q+2)/(Q−2), |x| < 1, (2) 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 ≥ 3 and k = 1, problem (2) has a unique solution in the class of x-radial functions.