Fundamental solutions for a class of three-dimensional elliptic equations with singular coefficients

Abstract

We consider an equation $$ L_{\alpha ,\beta ,\gamma} (u) \equiv u_{xx} + u_{yy} + u_{zz} + \displaystyle \frac{{2\alpha}}{x}u_x + \displaystyle \frac{{2\beta}}{y}u_y + \displaystyle \frac{{2\gamma}}{z}u_z = 0 $$ in a domain ${\bf R}_3^ + \equiv {{({x,y,z}): x > 0, y > 0, z > 0}}$. Here $\alpha ,\beta ,\gamma$ are constants, moreover $0 < 2\alpha, 2\beta, 2\gamma < 1$. Main result of this paper is a construction of eight fundamental solutions for above-given equation in an explicit form. They are expressed by Lauricella's hypergeometric functions with three variables. Using expansion of Lauricella's hypergeometric function by products of Gauss's hypergeometric functions, it is proved that the found solutions have a singularity of the order $1/r$ at $r \to 0$.