Through this article we will use a notation \begin{ equation}\label{ alfaLap} T_{ \alpha}u(x)=(1-|x|^2)\Delta u(x)+2 \alpha \langle x,\nabla u(x)\rangle + (n-2-\alpha) \alpha u(x). \end{ equation} Here, $|x|<1$ and $\alpha>-1$. Also, for $x=x(x_1,x_2,\ldots,x_n)\in \mathbb{ R}^n$ we use $|x|=\sqrt{ x_1^2+x_2^2+\ldots+x_n^2}, \nabla =(\frac{ \partial}{ \partial x_1},\frac{ \partial}{ \partial x_2},\ldots,\frac{ \partial}{ \partial x_n}),\Delta=\frac{ \partial^2}{ \partial x_1^2}+\frac{ \partial^2}{ \partial x_2^2}+\ldots+\frac{ \partial^2}{ \partial x_n^2}.$ The purpose of this paper is to investigate a Dirichlet problem, corresponding to above mentioned PDE. We will specificaly consider non-homogenous boundary value problem. In that purpose the explicit formula for Green function assosiated to the operator (\ref{ alfaLap}) will be calculated, and also, we will present the corresponding representation theorem.
Categories: math.CV math.AP