Green function for $T_{\alpha }$-Laplacian in higher dimensions

Abstract

Through this article we will use a notation \begin{equation}\label{alfaLap} T_αu(x)=(1-|x|^2)Δu(x)+2 α\langle x,\nabla u(x)\rangle + (n-2-α) αu(x). \end{equation} Here, $|x|<1$ and $\alpha >-1$. Also, for $x=x(x_1,x_2,\dots ,x_n)\in {\mathbb{R}}^n$ we use $|x|=\sqrt{x_1^2+x_2^2+\dots +x_n^2},\nabla =(\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_2},\dots ,\frac{\partial }{\partial x_n}),\Delta =\frac{{\partial }^2}{\partial x_1^2}+\frac{{\partial }^2}{\partial x_2^2}+\dots +\frac{{\partial }^2}{\partial x_n^2}.$ 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.