Best constants in the Harnack inequality for the Weinstein equation
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.
scientific article; zbMATH DE number 4065389