Best constants in the Harnack inequality for the Weinstein equation (Q1107665)

Let U be the class of positive solutions of the equation \[ \sum^{n}_{1}\partial ^ 2u/\partial x^ 2_ i+(k/x_ n)\partial u/\partial x_ n+(\ell /x\quad 2_ n)u=0\quad in\quad {\mathbb{R}}\quad n_+=\{x_ n>0\} \] where \(k\in {\mathbb{R}}\) and \(-\infty <\ell \leq ((k- 1)/2)^ 2\). The author gives a representation theorem for u (modifying some known results when \(\ell =0).\) Then using a Möbius transformation \(T: B^ n\to {\mathbb{R}}\) \(n_+\), the above representation is considered in the unit ball \(B^ n\) to determine the best constants in the Harnack inequality for U. Also found here are representation theorems for positive axially symmetric solutions of certain (Grušin's and Heisenberg's) partial differential equations.