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

We consider an equation L α , β , γ ( u ) ≡ u x x + u y y + u z z + 2 α x u x + 2 β y u y + 2 γ z u z = 0 in a domain R 3 + ≡ { ( x , y , z ) : x > 0 , y > 0 , z > 0 } . Here α , β , γ are constants, moreover 0 < 2 α , 2 β , 2 γ < 1 . The main result of this paper is a construction of eight fundamental solutions for the above-given equation in an explicit form. They are expressed by Lauricella’s hypergeometric functions of three variables. Using the 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 → 0 . Furthermore, some properties of these solutions, which will be used for solving boundary-value problems for the aforementioned equation are shown.